Market Equilibrium Model Management Ordinary Share

439

A MARKET EQUILIBRIUM MODEL FOR

THE MANAGEMENT OF ORDINARY

SHARE PORTFOLIOS

by

R. S. CLARKSON, B.Sc., F.F.A., F.I.M.A.

[Submitted to the Faculty on 16th March 1981. A synopsis of the paper will be found on page 571.]

“ For the investigation of the laws of history, we must completely change the subject of observations, must let kings and ministers and generals alone, and study the homogeneous, infinitessimal elements by which masses are led. No one can say how far it has been given to man to advance in that direction in understanding the laws of history. But it is obvious that only in that direction lies any possibility of discovering historical laws ; and that the human intellect has hitherto not devoted to that method of research one millionth part of the energy that historians have put into the description of the doings of various kings, ministers, and generals, and the exposition of their own views on those doings.”

TOLSTOY

1. Introduction

1.1. The price of a particular ordinary share represents an equilibrium position between the views of those participants in the market who wish to buy and those who wish to sell. Most participants, and certainly all institutional investors, have access to a vast amount of background information, and share prices adjust continuously as these participants revise their buying or selling prices in the light of new information or in the light of a changed interpretation of existing information. If an explicit price model can be developed solely from the principle that prices are in equilibrium once all participants in the market have acted on their interpretation of the information available to them, this model will be of considerable assistance in the management of ordinary share portfolios.

1.2. This paper describes the construction and application of such a price model and discusses the optimal extent to which mathematical

Richard Kwan

TFA 37 (1979-1981) 439-607

Richard Kwan

440 A Market Equilibrium Model for the

and statistical methods can be employed in portfolio management. A recent paper by Clarkson (1980) describes the stage of development that had been reached at the end of 1978 ; the present paper expands on the underlying concept of market equilibrium and on the practical implications for the management of institutional portfolios of ordinary shares.

1.3. The theoretical derivation of the model is described in Part I. Particular attention is paid to statistical stability and to the balance between accurate representation of important market features and generality of approach.

1.4. Part II describes the detailed steps in the construction of a model for the U.K. ordinary share market and shows how the early results were tested to check the validity of the assumptions used in the theoretical development.

1.5. Practical applications, such as the formulation of price projections and the analysis of other investment assessments, are discussed in Part III. In the last section, the practical experience is used to obtain a detailed description of the price formation process in an ordinary share market.

1.6. The technical methods involved in following the market equilibrium concept through to a disciplined approach to investment management are often complex in the extreme. Since the principles underlying the practical application of the model are much more important than the intermediate complexities, a discussion of various aspects of these principles constitutes Part IV of the paper.

1.7. There are numerous references in the actuarial literature to the limitations of any formalised approach to ordinary share investment. Accordingly, the last section of Part IV discusses the role of mathematics in the construction and application of the market equilibrium model and shows how the model can be regarded as a framework which allows the judgment and experience of the investment manager to be utilised in a systematic and efficient manner.

1.8. Stockmarket theorists such as Cootner and Fama have challenged the validity of traditional methods of stock selection and have put forward instead the proposition that is now generally known as the Efficient Market Hypothesis. In the absence of any academically acceptable demonstration that the Efficient Market Hypothesis

Management of Ordinary Share Portfolios 441

is not a satisfactory description of stockmarket behaviour, the theorists have tended to move away from the question of stock selection and have concentrated instead on the development of various portfolio optimisation models under the generic name of Modern Portfolio Theory. Part V of the paper attempts to reconcile the apparent contradictions between these theoretical researches and the theory and results of the market equilibrium model as set out in Parts I to IV.

PART I: THEORETICAL DERIVATION OF THE MODEL

2. Preliminary considerations

2.1. Before attempting the detailed construction of the model, it is necessary to define the problem being tackled and to describe in outline the essential properties that the model must possess if it is to be of practical use in the management of ordinary share portfolios.

2.2. The central aim is to express the price of any particular share in such a way that the complex problem of forecasting the future relative price performance of the share is reduced to a series of less complex judgments. As stated in the opening paragraph of the paper, the underlying principle is to be that prices are in equilibrium once all participants in the market have acted on their interpretation of the information available to them. This implies that some form of statistical best fit criterion will have to be employed to relate prices to the available information and also that the available information must be expressed in quantitative rather than qualitative form.

2.3. The use of a statistical best fit criterion means that a “ model ” price, or expected price, can be calculated for each share and then used as a reference point for the projection of future prices. It is essential that these projections, which will use the model as a frame of reference, can be compared and contrasted with investment conclusions that are available from other sources. If this were not the case, the practical application of the model would tend to replace rather than complement the judgment of the investment manager.

2.4. Since the principal application of the model will be the assessment of future share price movements relative to the general market level, it is essential that all the individual components of the model are statistically stable. Also, the investment manager, before


attempting to use the model for future assessments, will wish to satisfy himself that the model reflects all the salient features of the present price structure. Accordingly, the goodness of fit achieved by the model must be satisfactory. However, with any mathematical model where a statistical best fit criterion is employed, there is a direct conflict between goodness of fit and statistical stability. If too few variable parameters are allowed for, the goodness of fit will be poor and the general validity of the model will be in doubt. If, on the other hand, the number of variables is increased, the goodness of fit improves but the stability of the fitted parameters deteriorates. Considerable care will therefore be required in the construction of the model to achieve a satisfactory balance between statistical stability and goodness of fit.

2.5. The principal benefit resulting from the ownership of ordinary shares in a company is the right to receive a proportionate share of the distributions made by the company by way of dividends out of the net profits that result, after the payment of all charges and taxes, from the utilisation by the company of the assets it owns. There are certain types of company, such as investment trusts, where the principal function of the company is to provide a marketable investment vehicle for the collective ownership of a certain class of asset. In these cases the net asset value rather than the net profits or dividends may be the predominant factor in determining the share price, and no attempt will be made to incorporate such companies within the model. For the purposes of the model, the entitlement to dividends is therefore assumed to be the sole benefit arising from ownership of ordinary shares, and the expected price of a particular share will be expressed as a function of current net profits and dividends and of forecast future net profits and dividends.

2.6. Any attempt to approach ordinary share investment in a formalised and systematic manner is seriously impeded by the practical complications that result from the subjectiveness of much of the relevant background information. With fixed interest securities, on the other hand, there are relatively few practical complications, and it is accordingly very much easier to formulate, and then test, statistical models. The present author has also applied the market equilibrium concept to the U.K. Government Securities market, which, in terms of marketability, trading activity and variety of stocks, is almost certainly the best example anywhere in the world of what the theorists would call a perfect market. There are many similarities, both in the construction and in the applications,


between the present model for ordinary shares and the model devised for gilt-edged stocks. These similarities suggest that the price formation processes in the two markets are closely related and that the theory of the market equilibrium model is of general validity in any capital market.

3. The market equilibrium concept

3.1. The fundamental idea developed in this paper is that the prices of individual ordinary shares follow a regular and determinable pattern. As stated in the opening sentence of the paper, the price of a particular ordinary share represents an equilibrium position between the views of those participants in the market who wish to buy and those who wish to sell. In coming to a decision on whether to buy or sell a particular share, any one investor will assess various attributes of the share in accordance with the criteria that he, and he alone, considers appropriate. Since it is reasonable to expect that when assessing different shares he will be consistent in applying these criteria, each investor will tend, to a greater or lesser degree, to impose an ordering on the prices of different shares that reflects his assessments of the various individual attributes.

3.2. Although different investors may use differing assessment criteria, the overall effect of all the buying and selling that takes place in the market should still be that prices are ordered in a regular manner that reflects the aggregate assessment of each individual attribute.

3.3. Many who have considerable investment experience will be reluctant to accept that the price structure within an ordinary share market follows an orderly pattern as suggested above. Indeed, very considerable efforts have been made by economists, statisticians and others to investigate the price formation process with a view to determining successful trading strategies, yet very little of this work has any practical application in the day-to-day management of ordinary share portfolios.

3.4. The suggested regularity of structure, if it exists, can easily remain undetected, since nearly all methods of investment appraisal deal with disconnected units selected arbitrarily from the total structure. It is impossible, from an examination of isolated parts of the market, to determine the structure of the whole market.


3.5. In the assessment of a particular company, for example, various abstract concepts, such as the quality of its management or the strength of its product range, are often discussed at great length even although other factors, such as general economic conditions, have a much greater influence on future profits and dividends, both in absolute terms and relative to other companies.

3.6. There is also a tendency to concentrate on one specific attribute, particularly one that is easy to quantify, and to formulate investment conclusions in such a way that other important attributes, such as earnings per share and dividends, are virtually disregarded. It is therefore not unusual to find recommendations such as “ buy because of the high proportion of profits earned overseas ” or “ sell because of the high level of borrowings ”, where the implication is that all other attributes are of little consequence.

3.7. Sector research is yet another important example of how attention is focused on isolated parts of the market rather than on the overall structure. Companies in a particular industry sector are analysed in considerable detail in an attempt to identify those which offer the best investment value within that sector. However, particularly in stockbroking firms, each sector is often researched by a different analyst, with the result that standardisation of recommendations between sectors is difficult, if not impossible, to achieve.

3.8. Finally, where the assessment criteria of a particular participant are formulated in sufficient detail (as, for example, is the case with price models based on a discounted value of future dividends), the “ correct ” prices of different shares as implied by these criteria will form a regular price pattern. It is often assumed that this type of price pattern, which has been deduced from a particular theory or from the actions of one particular investor, will be reflected in the price structure of the market as a whole. However, the price formation process in the market arises out of the interaction of all the buying and selling that takes place, and it cannot be assumed that the resulting equilibrium position can be explained in terms of the actions and theories of specific participants.

3.9. For all these reasons, nearly all investment research is carried out in such a way that only a small part of the total market is studied at a time, and it is unlikely that any generalisations made from these studies of disconnected units will give a satisfactory description of the entire market. It is obvious that the postulated


regularity of price structure can only be discovered by identifying the various individual attributes of all shares in the market and by studying the equilibrium position in terms of these attributes.

4. Mathematical and statistical formulation

4.1. Before the market equilibrium concept can be used to identify the postulated regularity of prices, it is necessary to describe the preferences of investors for particular attributes in such a form that the equilibrium position can be expressed in mathematical terms.

4.2. Suppose that the different attributes which affect prices are A, B, C, ... and the values of these attributes for all shares at a particular time have been quantified as A1, B1, C1, ..., A2, B2, C2, ...,

A3, B3, C3, .... If for two shares r1 and r2 the values of all corresponding attributes are equal (i.e. Ar1 = Ar2, Br1 = Br2, Cr1 = Cr2, ...). then the prices of the two shares must be identical. If this were not the case, the consistency of investors’ assessments, which is part of the market equilibrium hypothesis, would mean that every holder of the share with the higher price would regard it as being overpriced compared with the other and would switch accordingly until the prices were equal. Such a switching opportunity would contradict the hypothesis that prices are in equilibrium, and hence the two prices must be identical. We may therefore express the price of each share in terms of the function

P = P(A, B, C, ...)

so that the price of share r is

Pr = P(Ar, Br, Cr, ...).

4.3. A simplified example is the case of a market where there is only one relevant attribute A, namely the historic earnings per share, and where the price-earnings ratios of all shares are identical at any time. Assuming that A>0, this gives

P = P(A) = kA,

where k is a positive number which may vary over time.

4.4. Returning now to the general case, we build up through a series of steps a mathematical statement of the market equilibrium concept. It is convenient to make the following simplifying assumptions to ensure that the important points of principle involved are not obscured by practical complications:


(i) All participants in the market use identical values of individual attributes.

(ii) There are no transaction costs. (iii) Negative holdings of shares (i.e. short positions) cannot occur. (iv) A participant who identifies that a share he holds is overpriced

compared with certain other shares on the basis of his own assessment criteria will sell that share and reinvest in one or more of those other shares.

(v) A switch from share r1 into share r2 causes the ratio to

decrease. (vi) Attribute A is such that all participants in the market prefer

to hold share r2 rather than share r1 if all three of the following conditions apply :

(a) Pr1 ≥ Pr2 (b) Ar1 < Ar2 (c) Br1 = Br2, Cr1 = Cr2, ....

4.5. Consider first the case where there are only two participants, X and Y, and where there are only two shares, so that X and Y must always hold between them the total amounts in issue of shares 1 and 2. Suppose that A1 < A2, and that

B1 = B2, C1, = C2, ....

Suppose also that the assessment criteria used by Y are such that he attaches much more importance than X to attribute A. To highlight the mechanics of the price formation process, let us assume that, for the particular value of A involved, X is indifferent as to which share he holds provided

and that the corresponding indifference ratio for Y is 1.3. The equilibrium position is illustrated in Figure 1, where price is shown as a function of attribute A.

The distance QS represents P1, and the distance RV represents P2. Also, TU = 0.1QS, and TW = 0.3QS. If the prices P1 and P2 are in equilibrium, then

since otherwise both X and Y would consider that share 2 was overpriced relative to share 1, and either or both would switch from


FIGURE 1

share 2 into share 1. Since one or other of X and Y must at all times have a holding in share 2, equilibrium cannot occur if

Similarly it follows from consideration of a switch from share 1 into share 2 that

Hence

so that V must lie between U and W or be coincident with either U or W.

4.6. Consider now the extension of this result to the case where there is a larger number N of participants. The indifference ratios of these participants corresponding to the values of 1.1 and 1.3 above are represented in Figure 2 by the ratios

Some of the points Xi and Xj can be coincident. The nearer V is to XN the larger will be the number of participants

who consider share 2 overpriced relative to share 1 and who therefore would switch into share 1 if they held share 2. Conversely, the nearer V is to X1 the larger will be the number of participants who would switch into share 2 if they held share 1. The equilibrium


FIGURE 2

position SV is therefore such that the downward pressure on the ratio

caused by switching action from share 2 into share 1 is exactly

balanced by the upward pressure on this ratio caused by switching action from share 1 into share 2.

4.7. The next case is where there are three shares in the market, with A1 < A2 < A3, and all other factors are as in 4.6. The indifference

ratios for are represented in Figure 3 by

FIGURE 3


The equilibrium position for the ratio is represented by the ratio

again this must be such that the downward pressure on the

ratio caused by switching from share 3 into share 2 is exactly balanced by the upward pressure caused by switching in the opposite direction.

Similarly, the ratios and must simultaneously be in equilibrium

under switching action.

4.8. Consider now a large number of shares with differing values of A but the same values for all other attributes. In view of the assumed consistency of application of assessment criteria we deduce that the equilibrium prices will follow a regular pattern as shown in Figure 4.

FIGURE 4

In 4.7 it is implied that the same participants could all be involved in the equilibrium process for the three shares. However, it is quite possible that, for different ranges R,(A) and R,(A) of A, participants who have holdings in R,(A) might not have holdings in R,(A), and vice versa.

4.9. We deduce from the above simplified cases that, for equal values of all other attributes of each share, the market equilibrium position is such that the price of a particular share can be represented by a smooth increasing function of A, as shown in Figure 5.

and


FIGURE 5

4.10. If the smooth curve in Figure 5 relates to fixed values

B0, C0, ... of the other attributes, consider now the case where C0, D0, ... remain fixed but a different value B'0 is used for attribute B. Figure 6 shows the case corresponding to Figure 2 where there are four shares with the following values of attributes:

share 1 A1, B0, C0, ... share 2 A2, B0, C0, ... share 3 A1, B'0, C0, ... share 4 A2, B'0, C0, ...

FIGURE 6


In a similar notation to that used in Figure 2, the indifference

ratios for are

From the consistency of application of assessment criteria, these ratios should be similar to the ratios

since corresponding pairs relate to the proportionate increase in price of two shares at which the participant is indifferent to the same change in attribute A. The equilibrium position, which can be interpreted as a weighted average of indifference ratios, should also be similar. By interpreting A2 – A1 as a small quantity AA, we have

and

Then

and

From the similarity of equilibrium positions we deduce that these

two limits will be similar, and hence that will be relatively

insensitive to changes in B. This is represented in Figure 7 by the

similarities in shape and location of the three curves of for the

three values B0, B1, and B2, with B0< B1 < B2.

4.11. The discussion so far relates to prices at a particular point in time. If the preferences of individual participants change over time (i.e. if, in the simplified case shown in Figure 2, the indifference

ratios vary with time), the equilibrium position

as measured by will also vary with time. Using once more the

consistency of application of the assessment criteria, we expect that

4.52 A Market Equilibrium Model for the

FIGURE 7

this variation over time will be regular. This is represented in

Figure 8 by the similarity of the curves representing at three

different points in time.

FIGURE 8

4.12. Condition (vi) in 4.4 was introduced purely to simplify the

illustrative sketches and can now be removed. This allows

to assume negative values. In general, will be either mono-

tonic increasing or monotonic decreasing.


4.13. Although the above discussion of the price formation process is by no means rigorous, it should be clear that the market equilibrium concept has been developed in such a way that mathematical and statistical analysis can now be applied. In particular, the postulated regularity of price structure can now be expressed as follows :

“ As a result of the collective action of all participants in the market, each of whom uses his own criteria when assessing the various attributes of individual shares, share prices at any par-

ticular time vary with these attributes in such a way that

(where P is the share price and A is any attribute) is a smooth function of A and is relatively insensitive to changes in the other attributes.”

4.14. The above statement relates solely to the existence of the underlying regularity within the price structure. The next stage in the development of the model is to describe in general terms the statistical methods that are required to identify the equilibrium position in practice.

4.15. As outlined in 2.4, considerable care will be required in the construction of the model to achieve a satisfactory balance between statistical stability and goodness of fit. It is obvious from the variety and quantity of the relevant background information that the possible number of attributes that could be included in the model is very large. However, some attributes can be expected to play a key role in the price formation process, whereas others will have so little effect that their existence can, for most practical purposes, be ignored. To identify which attributes must be taken into account explicitly in the model, the relative importance of each attribute has to be considered.

4.16. A suitable measure of this relative importance is the price sensitivity with respect to attribute A defined by

where AUQ, AM and ALQ are the upper quartile, median, and lower quartile values of A respectively. In Figure 9, S(A) is represented

by the ratio

B


FIGURE 9

4.17. This definition of price sensitivity relates to a particular point in time. As discussed in 4.11, it is possible that the equilibrium position may change over time, in which case the price sensitivity may also change. We define the variation in price sensitivity over time as :

4.18. For the goodness of fit to be satisfactory, the model must be formulated in such a way that those attributes with the largest price sensitivity are allowed for explicitly. Also, where the variation in price sensitivity of attribute A is significant, the formulation of the

model must allow to vary over time. In general this will

require one or more variable parameters in the function chosen to

represent with the values of these parameters being obtained

by a best fit criterion.

4.19. In view of the tendency for investment research to be carried out in such a way that only a small part of the total market is studied at a time, it is highly unlikely that share prices are always “ logical ” in the sense that they can be represented exactly in terms of a smooth price function P(A, B, C, . . .). We must therefore expect a large amount of random noise, and the process of statistical fit will thereby be made very much more difficult. If, for example, a variable parameter is used in the representation of an attribute


with a relatively low price sensitivity, this random noise rather than any underlying change in the equilibrium position could cause a variation in the fitted value of the parameter.

4.20. As discussed in 4.10, we expect that expressions of the type

will be small in magnitude. By assuming that such

expressions are identically zero, we can improve the statistical stability significantly without affecting the goodness of fit materially. We then have

where P1, P2, P3, . . . are functions of a single variable. The problem of determining the general price function has now been reduced to the determination of a series of subsidiary functions, each of which represents a smooth curve of the type shown in Figure 5.

4.21. An example of a subsidiary function P,(A) is shown in Figure 10. For an attribute A which can assume values from 0 to 1, we have

so that P1(0) = a, P1(l) = 1, and P,(A) is a linear function of A.

FIGURE 10

If the variation V[S(A), t1, t2] is relatively small, a fixed value of a can be used to specify the equilibrium position in relation to attribute A. If, however, the variation is relatively large, it may be necessary to allow a to be a variable parameter whose value is determined using


a criterion of best fit. In this case, if E is the error function minimised in the best fit process, E can be regarded as a function of the parameter a and of any other variable parameters. For a stable value of a to result from the best fit process, E must have a well- defined minimum with respect to a, as shown in Figure 11.

FIGURE 11

This is equivalent to the existence of a unique value where is

zero and where is positive and sufficiently large to ensure

that the position of the minimum is unlikely to be materially affected by random noise.

4.22. There may be some attributes where experience of investors’ attitudes suggests that variation in the price sensitivity will occur over time but where flexibility in the equilibrium position through the use of a variable parameter cannot be allowed because of these considerations of statistical stability. This inability to measure certain specific factors affecting the equilibrium position may appear unsatisfactory, but it is an inescapable feature of the general statistical process that an attempt to measure a relatively weak element in the system will reduce the statistical stability of the entire measurement process. It is interesting to note that there is an exact parallel in the physical world : the Heisenberg Uncertainty Principle states that, when measuring the characteristics of electrons, we have to accept an accurate knowledge of wavelength and ignorance of position, or the converse, or inaccurate knowledge of both.


5. Ordering relationships

5.1. The next stage in the construction of the market equilibrium model is to specify, in as efficient a manner as possible, the most important attributes involved in the price formation process.

5.2. The specification of these attributes must take into account the very difficult conflict between accurate representation of important market features and generality of approach. If an essential factor in the price formation process is left out of account, it is likely that the resulting model will give too poor a representation of the price structure to be of any practical use ; if, on the other hand, the model involves any relationship between price and certain attributes which is not of general validity, it cannot then be claimed that the resulting price structure has been developed solely from the market equilibrium hypothesis and this lack of generality would again seriously restrict the usefulness of the model.

5.3. There are in particular three major areas of difficulty that stand in the way of an explicit statement of the price formation process :

(i) Time horizon

Ordinary shares can be bought or sold on very short term considerations, as short in some cases as a week or two. Again, the expected holding period can be very long, such as when an institution or individual regards a purchase as a “ core ” portfolio holding.

(ii) Investment objective Some investors are primarily concerned with dividend income

while others are relatively insensitive to immediate income and regard their holdings more as a store of capital value than as a source of income.

(iii) Variability of forecasts Forecasts of future profits and dividends for individual companies

involve large margins of error ; economic and political uncertainties make it very difficult to forecast even aggregate company profits a year or so ahead.

5.4. As a starting point, we assume for the moment that all participants can forecast future earnings per share and dividends exactly for all companies in the market. We then derive from a


series of simplified examples a set of ordering relationships to describe the mechanics of the market equilibrium process. The logic underlying these examples is sufficiently general to ensure that differing investor attitudes arising out of items (i) and (ii) of 5.3 do not affect the conclusions.

5.5. To simplify the analysis, we assume that, in respect of a general share whose price is P, annual results have just been declared showing a dividend per share of D and earnings per share (i.e. net profits per share available for distribution as dividends) of E, with

and

The corresponding earnings per share that will be declared in 1, 2, 3, . . . years’ time are denoted by E1, E2, E3, . . . .

It is also assumed that the dividend payout ratio in each future

year is the payout ratio for the year for which results have just

been announced. Where different shares are being compared, the corresponding

notation adopted for share x is

5.6. Suppose that for two shares x and y

and for n = 1, 2, . . .

where

The dividends and earnings, present and future, which represent the benefits of a holding of M shares in company 2 are identical to the dividends and earnings attributable to a holding of k.M shares in company y, where M is an arbitrary large positive integer.

If any investor with a holding in share x could switch into share y and increase both the dividend and earnings attributable to his holding in each year. Accordingly, every participant would regard share x as overpriced compared with share y. For prices to be in equilibrium, we therefore require


Similarly, from consideration of a switch in the opposite direction, price equilibrium requires

These results together give

5.7. By putting k = 1 above we deduce that where two shares have identical dividends and earnings, both present and future, then the two prices must be identical. In view of this uniqueness property, the price P of a particular share can be expressed as

where the expression on the right hand side represents a function P, as yet unknown, of the independent variables E, D, E1, E2, . . . .

5.8. For a general value of k, the result in 5.6 can be stated in the form

5.9. Suppose that for two shares z and y

for n = 1, 2, . . . and

This situation is represented in Figure 12 with k = l.25.

If any investor holding share y could switch into share x and increase both the dividend and the earnings per share each year.

FIGURE 12


As in 5.6, every participant would regard share y as overpriced compared with share z and the two share prices could therefore not be in equilibrium. Accordingly, the equilibrium position must be such that

If then a switch from share z into share y increases the earnings per share each year while keeping the dividend each year unchanged. An investor who assesses shares on the basis of dividend payment alone would not switch from share z into share y since the switch does not increase the amount of dividend received in any year. However, it is recognised in 5.3 that the investment objectives of investors vary greatly and in particular that some investors are relatively insensitive to immediate income. The other limiting case is where an investor assesses shares on the basis of earnings per share alone, and such an investor would be indifferent as to which share he held provided We therefore deduce that there must

exist an indifference ratio for which characterises the ratio of the

two prices at which a particular investor is indifferent as to which share he holds and that this ratio will vary between 1 and k, the former value corresponding to an investor who is concerned only with dividend and the latter to an investor concerned only with earnings. This is precisely the type of situation represented in Figure 2, and we deduce that the equilibrium ratio is r1 where

Accordingly, the equilibrium position is such that

5.10. Consider now the case where

and

This situation is represented in Figure 13 with k = 1.25.

Using similar arguments to those of 5.9, we deduce the existence

of an indifference ratio for which varies from k for an investor

concerned only with dividend to 1 for an investor concerned only with earnings. When prices are in equilibrium, the ratio is r2 where

and hence


FIGURE 13

5.11. The results of 5.8, 5.9 and 5.10 are inter-related in that

5.12. Consider two shares x and y where

for n = 1, 2, . . . and

This situation is illustrated in Figure 14 with k = 1.25.

FIGURE 14


If an investor holding share y could switch into share x and increase both the dividend and earnings each year. For the two prices to be in equilibrium, we therefore require

If an investor holding share y could switch into share 2 and increase the dividend and earnings each future year ; the historic dividends and earnings are identical.

If the future earnings and dividends attributable to a holding in share z are exactly equal to those attributable to a holding of equal value in share y ; the historic earnings and dividends are lower in the case of the holding of share z.

We therefore deduce that the limiting cases for the indifference

ratio for are 1 (for an investor who is concerned only with historic

earnings and dividends) and k (for an investor who completely disregards the historic earnings and dividends). At the equilibrium position, the indifference ratio will be s1, where

Since it is unrealistic to have share prices determined solely by historic earnings and dividends, we discard the case s1 = 1 so that

5.13. Consider now a generalisation of the example in 5.12 where earnings and dividends are identical each year up to year N-1 (N > 1) and where earnings and dividends are higher in the case of share 5 by a factor of k (k > 1) for year N and all subsequent years.

If the indifference ratio for in this case is sN, then by arguments

similar to those above the equilibrium position is such that

5.14. The examples so far have involved a regular variation of earnings or dividends by a factor of k as between share x and share y. The same arguments can be applied to a more general case to show that provided

for n = 1, 2, . . .

and strict inequality holds for at least one value of n.


5.15. Suppose that for a certain subset of shares it is possible to define the function G such that

and strict inequality holds for at least one value of n. This function G can be interpreted as a measure of earnings

growth. For shares in this subset, the price P can now be expressed in terms of the three independent variables E, D, and G as

Again in respect only of the shares in this subset in which G is defined, the principal results contained in 5.8, 5.9, 5.10 and 5.14 respectively can now be restated as follows :

5.16. By defining

where R = is the dividend payout ratio, the general price function

can be expressed as

Since F is a function of only two independent variables, this represents a significant simplification of the structure of the price function. Also, the number of attributes involved has been reduced to three, namely :

E the historic earnings per share R the dividend payout ratio G a measure of earnings growth.

This representation of the general price function is now in a sufficiently explicit form to be used as the functional core of the market equilibrium model.


6. The general model

6.1. We now proceed to the detailed construction of the market equilibrium model. Since the results of Section 5 are based on the simplifying assumption that investors can forecast future earnings and dividends exactly, the first step is to relax this assumption and then reformulate the results accordingly. A further restriction in Section 5 is that the earnings growth function G is defined only on certain subsets of shares. For the general model, the definition of this function must be such that it applies to all shares in the market.

6.2. In effecting these generalisations, we take into account certain important features of the U.K. ordinary share market, since the principal applications discussed in Part II and Part III relate to this market.

6.3. At various stages in the construction of the model, considerations of goodness of fit and statistical stability arise. Since a large amount of random noise is anticipated, the statistical development must proceed along the lines discussed in Section 4 to ensure that reliable measurements of the equilibrium position can be obtained.

6.4. We now drop the assumption that future earnings and dividends can be forecast, and build up, through a series of examples, price relationships which are based on a set of less restrictive conditions.

6.5. Consider first the case where two investors, U and V, are projecting the likely future earnings per share of three particular companies, z, y and z. U bases his estimates on a higher rate of economic growth than that used by V, but in all other respects their assessments of the companies and their projection bases are identical. The general patterns of their earnings projections (based on unit historic earnings per share) are shown in Figure 15.

Companies x, y and z can be considered as representing high, average, and low earnings growth respectively. Assume that the dividend payout ratios and all other attributes of the three companies are identical. The arguments of 5.12 still apply in the context of U’s forecasts of future earnings. We therefore deduce that U would switch from share z into share y if and would switch from share y into share z if Despite the differing earnings forecasts, the same results can be deduced in the case of V. If all participants in the market agree with the earnings growth rankings


FIGURE 15

as forecast by both U and V, the equilibrium position will be such that

6.6. Since earnings are a residual item affected by a wide variety of essentially unpredictable economic factors, it is unrealistic to assume that all investors make explicit forecasts of each future year’s earnings for all companies in the market. It is usually possible to make a reasonable estimate of earnings a year ahead, but at two years the estimate is very unreliable, and at durations beyond that no meaningful figure can be obtained in view of the various economic and political uncertainties. Accordingly, most investors can only classify growth prospects in a qualitative manner, e.g. “ very good ”, “ slightly above average ”, “ fairly poor ”, etc. These qualitative assessments generally relate to long term prospects over, say, a complete economic cycle.

6.7. We can formalise this qualitative approach by saying that each investor regards the future earnings of a particular company as following one of a family of N non-intersecting curves of the type illustrated in Figure 16.

As shown in 6.5, it is the ranking order rather than the forecast values at any particular duration that determines the equilibrium position. Accordingly, we define, for share x, the function G(z) as the number of the curve in the family which corresponds to the estimated future earnings of share 5. Thus G(z) is always an integer, with minimum value 1 and maximum value N. A particular


FIGURE 16

investor’s indifference ratio for is determined by his growth

rankings G(z) and G(y) as follows :

6.8. If we assume for the moment that all investors agree exactly in their rankings, this function G is an attribute of the type analysed in Section 4. For fixed values of all other attributes, the price P will therefore be an increasing function of G.

6.9. Beyond, say, two years into the future, any detailed earnings projections are tenuous in the extreme. For any forecast of earnings at durations t1 and t2, where it can therefore be assumed for all practical purposes that

For durations up to two years, however, more accurate forecasts are possible, and the values of E1 and E2 must be taken into account. Accordingly, we must replace the earnings base E in the expression E.F(R, G) by a more general function which involves E1 and E2.

6.10. We now drop the assumption that all companies have just announced annual results. As the date of the next announcement


of annual results approaches, the historic value of earnings per share becomes progressively less important and the forecast value of E, becomes increasingly reliable. If we now let E0 denote the historic earnings per share and let t denote the fraction of a year which has elapsed since the last announcement of annual results, the required earnings base function can be defined as E(E0, E1, E2, t). The precise formulation must depend on various practical circumstances.

6.11. It is instructive to review at this stage the way in which the three areas of difficulty referred to in 5.3 have been overcome in the construction of the model. In Section 5, ordering relationships are derived on the assumption that future earnings and dividends can be forecast exactly. Since differing emphasis as between capital and income is allowed for explicitly, varying investment objectives are successfully accommodated within the model. Possible complications arising from different time horizons are avoided by deriving ordering relationships which are always satisfied regardless of the detailed assessment criteria applied by individual investors. By being invariant with respect to time horizon or any other particular element of an investor’s assessment criteria, these ordering relationships provide a functional framework for the model which meets the generality requirement discussed in 5.2. The restriction in the results of Section 5 imposed by the function G being defined only on certain subsets of all shares in the market arises because the price relationship between two shares is indeterminate if the earnings profiles cross over at any duration. However, once the assumption of exact forecasting is dropped, the projected earnings profiles to be used in practice do not cross over beyond three years. Since the earnings base function, as described in 6.10, takes account of forecast values of earnings at shorter durations, the restriction in the definition of G is removed. Accordingly, the general validity of the price model is not impaired by any of the possible difficulties discussed in 5.3.

6.12. The price model is now of the form

where the various functions and variables are as defined in 5.16 and 6.10. The remainder of this section deals with the practical problems encountered in applying this form of price model to the U.K. ordinary share market. In particular, the implementation of the model must be such that the statistical stability of the resulting price structure is not in doubt.


6.13. As a first step towards ensuring adequate statistical stability, we employ the technique discussed in 4.20 and assume that

and

This gives

where F1 and F2 are functions of the single independent variables R and G respectively.

We now investigate the general properties of the functions E, F1 and F2 to identify how many variable parameters are required.

6.14. As discussed in 4.18, the number of variable parameters required depends on the variability of the functions S(Ar), where S(Ar) is the price sensitivity of attribute A,. as defined in 4.16. It is clear from the method of derivation of E as an earnings base function that it can be formulated in terms of E0, E1, E2 and t without any variable parameter being required. Accordingly, we need only investigate the general properties of F,(R) and F2(G) and hence of S(R) and S(G).

6.15. In 5.9 some of the properties of F1(R) are derived in terms of indifference ratios of prices. In particular, an increase in dividend from D to k. D results in an increase in price from P to r1.P, where

The value of r1 cannot be determined accurately at this

stage. However, as a first estimate we assume that Consider now the general levels of dividend cover in the U.K. ordinary share market. On the customary definition of dividend cover, a value of about 1.5 is regarded as somewhat low and a value of 2.5 is regarded as fairly high. If we take these values to represent respectively the upper quartile and lower quartile for dividend payout ratio, an increase in dividend from the lower quartile payout ratio to the upper quartile corresponds to giving

6.16. If the price sensitivities with respect to both dividend payout ratio and growth rate were identical, then a change in the growth rate from the lower quartile value to the upper quartile value would correspond to a 30% increase in the price. In terms of the price- earnings ratio relative to the market (i.e. the price-earnings ratio of


a particular share divided by an average value for all shares in the market) this corresponds to relative values of, say, 88% for the lower quartile growth rate and 115% for the upper quartile growth rate. General experience of the U.K. market suggests that the variation of price-earnings relatives with respect to growth rates is somewhat higher than this, and accordingly we must construct the model on the basis that the price sensitivity of growth rate is higher than the price sensitivity of dividend payout ratio.

6.17. Since the scale employed for G is arbitrary, the interpretation of the results of the model can be simplified by defining G and hence F2(G) in such a way that equal proportionate changes in price result from equal changes in G. This is equivalent to

where g> 0 and Go is a fixed value of G. On this definition of F2(G),

has the value of log g for all values of G, as shown in Figure 17.

FIGURE 17

6.18. Suppose that all investors, having previously used the earnings projections derived by V as described in 6.5, now revise their underlying assumptions and base their share assessments on the earnings projections derived by U. If the various changes in the earnings patterns are exactly as shown in Figure 15 (i.e. the change in basis results in a relatively small increase in earnings for the low growth company and a relatively large increase in earnings for the

C


high growth company), each investor will increase his indifference

ratio for both and and the values of these ratios at the

equilibrium position will increase. Since all other attributes remain constant, this variation in the price structure must relate wholly to changed values of F2(G). Also, since G and G0 remain constant by definition, the changes in F2(G) must result from an increased value of g at the new equilibrium position. We therefore expect that V[S(G), t1, t2] (i.e. the variation in price sensitivity with respect of G over time as defined in 4.17) may be significant, and accordingly g must be regarded as a variable parameter, with its value at any particular time being found by a best fit process.

6.19. The argument employed in 6.18 should not be regarded as an attempt to derive the general properties of the function g. At this stage we are interested solely in constructing a statistical description of the overall price structure in such a way that the values of variable elements in the price formation process can be measured accurately. The above example shows that any statistical description based on a fixed value of g would be unsatisfactory in that the variation of an important element of the price formation process would be suppressed.

6.20. It is in fact possible to construct an example where a downgrading of general economic growth gives rise to a higher value of g rather than the lower value that would be expected from the above example. A number of companies produce “ added value ” statements, and the efficiency of a particular company in terms of value added often gives a satisfactory indication of its earnings growth prospects. In general, high growth companies have a high value of sales per employee and can generate most of their capital requirements from retained earnings. Consider a deterioration in economic conditions which takes the form of a sudden increase in wage costs and a consequential increase in interest rates, either through the market mechanism in response to the higher rate of inflation or as the result of Government action to reduce the rate of inflation. Typical added value profiles for a high growth company and a low growth company, both before and after this change in economic conditions, might be as shown below.

The low growth company is in a very vulnerable position, since a 4% increase in wage costs and a 20% increase in interest costs leave the company with no retained earnings. The high growth company,


on the other hand, is far less susceptible to these increased costs and

suffers only a 10% fall in the amount available for re-investment in

the business. High growth Low growth

company company

Before

Wages 50 Taxation 10 Interest 5

Dividends 5

Re-investment in the business 30

100

After Before After

52 70 73

10 10 10

6 10 12

5 5 5

27 5 Nil

100 100 100

6.21. We can use the price model in the form

and the general properties of P(E, D, G) as set out in 5.15 to derive

the following properties of F1(R):

Since these properties are based on the assumption that the dividend

payout ratio remains constant, special consideration is required

where it is the company policy to pay no dividend (or only a token

amount of dividend) in the foreseeable future so that all earnings

are re-invested in the business. In such cases, we can interpret R

as the expected average dividend payout ratio over, say, each of the

next five years. Since the relative price-earnings ratios of shares in

this type of company are rarely very low, it is obvious that F1(R)

does not tend to zero as R tends to zero. If we assume for the moment that F1(1) = 1, the function F,(R) has the general properties of the

curve shown in Figure 18, with the gradient at any point being

positive and less than the gradient of the straight line joining that

point and the origin, and with

6.22. In all other cases it is assumed that the dividend payout

ratio will remain constant, at some specified value. A satisfactory


F1(R)

FIGURE 18

definition for practical purposes is R = where D1 and E1 are the

forecast values respectively of dividend and earnings per share in

respect of the next annual results to be announced by the company.

6.23. Given that F1(R) is to be a smooth curve satisfying the

properties discussed above, the two principal characteristics that

have still to be specified are

(i) the limiting value lim F1(R), r say, and

(ii) for a fixed value of r, the height of the curve in the middle of

the range.

A suitable two-parameter family of curves which allows these two

characteristics to assume all appropriate values is

where

and

The curves of F1(R) for r = 0·5, and d = 0·5, 1 and 2 are shown in Figure 19.

6.24. If the dividend payout ratios of shares x and y are 1 and

R respectively and the values of all other attributes are identical, then


F1(R)

FIGURE 19

As discussed in 5.9, this indifference ratio describes the equilibrium

position arising out of the actions of different classes of investor

whose preferences for immediate income may vary greatly. Accord-

ingly, any changes in the relative influence of these different classes

of investor could result in a change in the equilibrium value of

E1(R). In addition, the preferences of particular investors might

change over time. Improved economic prospects, for example, might

cause certain investors to change from a defensive policy, which was

biased towards income rather than capital appreciation, to a more

aggressive policy where immediate income is accorded a much lower

priority. Since this confidence factor relating to the balance between

dividends and earnings may be a key element in the price formation

process, we clearly wish to measure its effect and hence we must

allow one at least of the two parameters in F,(R) to be determined

by a best fit process.

6.25. Since the price sensitivity with respect to growth rate is

expected to be greater than the price sensitivity with respect to

dividend payout ratio, and since F2(G) involves only one variable

parameter, it would be statistically unsound to allow two variable

parameters in F1(R). Either d or r must therefore remain constant,

with the other being determined by a best fit process. Since r can be

interpreted as the indifference ratio for the extremes of dividend

payout ratio, namely R = 0 for share y and R = 1 for share x, it is

clearly r rather than d which must be chosen as the variable

parameter.


6.26. In the simplified example of 4.3, where the price earnings

ratio of all shares is constant, the price function P involves a positive

number k which is in effect a scaling factor. Since the functions

F2(G) and E(E0, E1, E2, t) do not incorporate a scaling factor and the

last function to be discussed, namely F,(R), is defined in such a way

that F1(1) = 1, a positive number k must now be introduced into the

price model as a scaling factor. The price model therefore becomes

where the individual elements are as defined above.

6.27. In the above discussion of the growth measure G it is

assumed that all companies operate in the same economic environ-

ment. However, an important feature of the U.K. ordinary share

market is the relatively high proportion of corporate profits that

arise from overseas manufacturing capacity or from exports from

the U.K. At a time when the U.K. economy is depressed and the

pound sterling is weak against other currencies, companies with a

relatively large overseas involvement can obtain the double benefit

of operating in a more favourable economy and then achieving a

further improvement in profits in sterling terms as a result of currency

appreciation on consolidation. Conversely, if the domestic economy

is strong and sterling appreciates against other currencies, the balance

swings in favour of companies with a relatively low overseas

involvement.

6.28. On a change in economic conditions which favours companies

with overseas interests, the improvement in the projected earnings

pattern depends on the relative size of the overseas involvement. If

three companies with high, average and low overseas involvement

had the same projected earnings patterns, the revised profiles taking

into account such a change in economic conditions would be as shown

in Figure 20.

If we assume that the growth rate G for a particular company

incorporates long-term forecasts of the various currencies and

economies involved, we can accommodate the shorter term fluctua-

tions in the balance of advantage as between U.K. and overseas

involvement by replacing the function F2(G) by F2(G). F3(A) where

A represents the attribute “overseas exposure”. Although the

precise definition of A must depend on various practical considera-

tions such as the availability of suitable data, a convenient scale for

A will generally be 0 to 1, with the former value relating to a company

with no overseas exposure and the latter value relating to a company


FIGURE 20

with 100% overseas exposure. Since the effect of a relevant change

in economic conditions will in general be proportional to the overseas

exposure, we define F3(A) as follows:

F3(A) = 1 + a. A,

where a is positive when the balance of advantage is in favour of

overseas exposure and negative when the converse applies.

6.29. It is apparent from the above discussion that the value of a

is expected to change over fairly short time scales to reflect changes

in economic conditions and currency levels. Accordingly, we allow

a to be a variable parameter and determine its value by a best fit

process. Since F3(A) is by definition a function which modifies

F,(G) in accordance with certain short-term influences, we expect

that S(A), the price sensitivity with respect to overseas exposure, will

be smaller than S(G), the price sensitivity with respect to growth rate.

6.30. In the same way as the forecast values of G involve a

central long-term forecast in relation to overseas exposure, these

values of G will also involve central forecasts of interest rates and

levels of economic growth. However, one of the hazards of ordinary

share investment is that unforseen extremes of economic conditions

can materialise with little warning. If economic conditions become

so harsh that corporate profits fall significantly, companies that are

highly geared in terms of borrowings will be very seriously affected.

On the other hand, in a period of rapid economic growth a company

where borrowings are already very large in relation to shareholders’


funds may be unable to raise the additional working capital required

to finance a higher volume of turnover.

6.31. To allow for these extremes of economic conditions, we

define G as being based on a central economic forecast and regard

the level of borrowings as an attribute to be incorporated in the price

model. We therefore introduce a function F,(B), where B is an

appropriate measure of the level of borrowings and F4(B) decreases

as B increases. The future earnings profile is now represented in the

model by the product of three functions:

6.32. Since the absolute level of borrowings that is generally

accepted as prudent varies between companies in different industries,

the definition of B will involve serious problems of standardisation,

and F4(B) is therefore unlikely to vary over time in a sufficiently

regular manner to justify the incorporation of a variable parameter.

6.33. A further, and more fundamental, reason for not incorpora-

ting a variable parameter in F4(B) is that the level of borrowings is

one of the many factors that has to be taken into account in estimating

the growth rate G. If two variable parameters are affected by a

common influence, problems of multicollinearity would arise and the

statistical robustness of the whole model would be seriously impaired.

6.34. An attribute of a quite different nature from those already

discussed is the marketability of a share. Where two shares x and y

are identical in respect of all attributes except that the marketability

of share x is significantly poorer than that of share y, investors would

in general prefer to buy share y rather than share x if xP = yP. We

therefore deduce that at the equilibrium position xP < yP.

6.35. In most cases the marketability will depend on the market

capitalisation of the share. If C is the ratio of the market capitali-

sation of the share to the total market capitalisation of all shares in

the market, marketability could be allowed for if necessary in the

price model by the incorporation of some function F5(C) which

increases with C. In the applications discussed in Part II no

allowance for marketability is required and accordingly the function

F5(C) is not included in the final formulation of the general model.

6.36. The general price model is now of the form:


where the individual elements are as defined above. This expression

is the product of four items each of which is totally independent of

the others:

k a scaling factor which is constant for all

shares;

E(E0, E1, E2, t) an earnings base relating to current and forecast near term future values of earnings per share;

F1(R), a function involving the dividend payout

ratio;

F2(G).F3(A).F4(B) a compound function involving the

forecast relative rates of growth of

earnings per share.

6.37. Although nearly all of the simplifying assumptions used in

the initial stages of the construction of the model have now been

relaxed, this formulation is based on the fundamental assumption

that the share price is a function of current values, and of forecast

future values, of earnings and dividends.

Where, for instance, a take-over bid or a significant disposal of

assets is likely, this assumption is clearly not realistic, and the price

model must now be generalised accordingly.

6.38. As an example of how the market equilibrium concept can

be used to effect this generalisation, consider the case of a company

which decides to dispose of certain assets by forming a new company

to hold these assets and then issuing shares in this new company to

existing shareholders free of payment. Suppose that on the basis of

the price model, taking into account earnings and dividends only, the

share price would have been 100p, that the relevant asset value per

share is 20p, and that the new shares are expected to stand at a 25%

discount to net asset value. The additional value per share attribu-

table to the disposal of assets is then 15p, and a multiplicative factor

of 1·15 must be introduced into the price model to take the expected

price to 115p. This factor of 1·15 can be interpreted as the indif-

ference price ratio where share x and share y are identical in

respect of all attributes except that share y has entitlement to

the above disposal of assets.

6.39. Where it was likely, but not certain, that such a disposal

of assets would be made, a smaller factor, 1.1 say, might be appro-

priate to allow for the lower probability involved. In general, the


multiplicative factor introduced into the model to allow for the

special factor z is of the form 1 + f(z), where f(z) takes into account

not only a forecast value of the relevant indifference price ratio on

the basis that the event occurs but also, if appropriate, the probability

that the event occurs. Since there may be more than one special

factor involved for any particular share, the additional term required

in the price model is

where N is the number of such special factors.

6.40. This completes the construction of the general price model,

which now expresses the expected price of a share in the form:

where

k is a scaling factor which is constant for all shares

R is a measure of dividend payout ratio

G is a measure of relative earnings growth rate

A is a measure of overseas exposure

r, g and a are variable parameters

G0 is a fixed value of G

a is a positive constant

and the general properties of the functions E(E0, E1, E2, t), F4(B) and

are as discussed in 6.10, 6.31 and 6.39 respectively.

6.41. We now express xP, the actual price of share x, in terms of

the expected price as follows:

so that

For each share, is a function of the variable parameters r, g and a

until values are assigned to those parameters. In view of the use of

indifference price ratios at various stages in the development of the


market equilibrium model, a natural criterion of best fit for the

purposes of assigning values to r, g and a is to minimise

6.42. Since represents the proportionate change in price that

would occur if the price moved to become equal to the expected

price, this definition is very convenient in practical applications.

6.43. Most formalised price models are constructed in such a way

that the relative price residual can be used as the primary tool in

practical applications. The present model, however, has been

constructed in such a way that it can measure the equilibrium

position in terms of the individual elements of the price formation

process. The remainder of this section shows how the practical

application of the model must be guided by the principal mathe-

matical and statistical properties that result from the philosophy

underlying its construction.

6.44. In Section 3 reference is made to the way in which most

methods of investment appraisal deal only with disconnected units

rather than with the whole structure. In particular, given the

practical difficulties of obtaining consistent forecasts of the future

earnings profiles of different companies and the general lack of

standardisation between sectors, it is highly unlikely that the values

of the growth rates G used in the model will be in close agreement

with the “consensus” values implicit in actual prices. Since G is

in many ways the most important component of the expected price,

this will give rise to some values of the relative price error that

remain either strongly positive or strongly negative for long periods

of time and hence to a relatively poor statistical fit.

6.45. In much the same way as the perceived attractions of

overseas exposure can vary over relatively short periods as exchange

rates and economic prospects change, the growth prospects for

individual companies, even on a relative rather than an absolute

basis, will also appear to many investors to change over fairly short

time scales as the general business background relevant to these

companies improves or deteriorates. As discussed in 6.28, the values of G are central values which average out the various influences, both

favourable and unfavourable, that may be present at different points

in an economic cycle. However, general experience of investor

attitudes and of U.K. investment appraisal methods suggests that

these short-term influences play an important part in the price

formation process.


6.46. If the values of G are revised at frequent intervals to reflect

these influences, the robustness of the model as a frame of reference

would be seriously affected and little importance could then be

attached to the values of the variable parameters. Since this would

greatly reduce the usefulness of the model for formulating future

projections, we retain the definition of G as a long-term average

value and allow the various short-term influences to be reflected in

the relative price residual

6.47. Short-term variations in the perceived attractions of

overseas involvement relative to domestic operations have been

allowed for explicitly in the function F3(A). If investors in general

attach far too much importance at a particular time to the advantages

of overseas exposure and thereby raise the prices of shares with high

overseas exposure to unsustainably high levels compared with shares

with low overseas exposure, it is the value of the variable parameter

a and not the relative price residual which indicates that the prices

of shares of the former class may be vulnerable to a setback. The current value of a must therefore be compared with the past history

of values to assess the relative attractiveness of shares with either

high or low overseas exposure.

6.48. If it is found that the values of the parameters g or r vary

significantly over time, similar assessments must be made to decide

whether shares with high values of growth rate or dividend payout

ratio are likely to change in price relative to shares with low values

of the corresponding attribute.

6.49. Since there are three variable parameters, namely r, g and a,

in the general model, the term F1(R). F2(G). F3(A) can be regarded as

defining a surface in 4-dimensional space. A change in the equilibrium

position with respect to any of the three attributes where a variable

parameter is incorporated gives rise to an altered set of values for

r, g and a and hence to a different surface in 4-dimensional space.

The price model can therefore be regarded as a space time co-ordinate

system in that all the attributes which affect the price of a share are

described in terms of numerical scales and all the measurable changes

over time in the equilibrium position correspond to changes in the

position of a surface in 4-dimensional space.

6.50. Future projections of relative share price movements

involve comparisons between the present state of this space time

co-ordinate system and the expected state of the system at various


times in the future. From the above discussion it can be seen that

these comparisons are highly complex. In particular, different

elements change in value over quite different time scales: the values

of individual attributes for a particular company change only slowly

but the variable parameters and the relative price residual may

change fairly rapidly. In addition, the relative price residual may

reflect estimation errors, which are essentially long-term in nature,

as well as short-term fluctuations in investor attitudes. Finally,

since the entire system will exhibit heavy random noise, the

interpretation of the results will require a high degree of skill and

judgment.


PART II: DETAILED MODEL FOR THE U.K. MARKET

7. Conventional methods of share appraisal

7.1. The market equilibrium model derived in Part I is a direct

extension of conventional methods of share appraisal that were

employed by a Scottish life office in the early seventies for the

management of U.K. ordinary share portfolios. In this section we

explain why the market equilibrium concept was chosen as the

starting point in the construction of a more systematic approach.

7.2. The basic principle employed by the life office in its share

appraisal methods was that the price-earnings relative (i.e. the

price-earnings ratio of a particular share divided by an average for

all shares in the market) represented the most satisfactory single

statistic on which to base comparisons of different shares. In

general, the higher the rate of expected earnings growth the higher

would be the price-earnings relative. Also, if two companies were

expected to show the same rate of earnings growth, but one had, say,

a higher dividend payout ratio or a stronger balance sheet than the

other, then this advantage over the other company would be reflected

in a higher price-earnings relative. The relative rather than the

absolute price-earnings ratio was used so that the numerical values

involved in these comparisons were not affected by changes in the

general market level.

7.3. Although the concept of the price-earnings relative appears

straightforward, its application to share appraisal gives rise to serious

practical problems of standardisation and interpretation.

7.4. A simplified example of the use of price-earnings relatives in

share comparisons is set out below:

Comparative Company x Company y factor

Earnings growth rate Significantly Slightly

above below

average average 1·2

Dividend payout ratio Average High 0.9

Balance sheet Very strong Average 1·05

Price-earnings

relative % 119 100 —


For equal values of all other attributes, it is judged that the higher

earnings growth rate for share x should give rise to a price-earnings

relative of 1·2 times that of share y. The corresponding comparative factors for dividend payout ratio and balance sheet are 0·9 and

1·05 respectively. Based on the value of 100 for share y, the

expected price-earnings relative for share x would therefore be

100 x 1·2 x 0·9 x 1·05, i.e. 113·4. Since the actual value is 119, i.e. 113·4 x 1·049, we conclude that share x is overpriced by about 5%

compared with share y.

7.5. Comparisons such as these are of considerable practical value in portfolio management since they can accommodate nearly all of

the investment research material relating to individual companies

that is available to institutional investors. However, although the general principles underlying these comparisons are soundly based,

many of the detailed steps suffer from a serious lack of rigour. We discuss below no fewer than five unsatisfactory aspects of the

simplified example described in 7.4.

7.6. The first and most obvious unsatisfactory feature is that the assessments of each of the three attributes involved are qualitative

and hence highly subjective in nature. The statement that the

dividend payout ratio is “average”, for instance, implies that the

frequency distribution of all the numerical values of this attribute is

known. Estimation errors will arise even in assessing what value of

dividend payout ratio can be classed as “average”.

7.7. In classifying earnings growth rates into various qualitative

categories, the fund manager will in the first instance base his con-

clusions on the opinions of analysts who research particular groups

of companies. If company x is an engineering company and company

y is a food retailing company, it is highly unlikely that direct com-

parisons will be available. Instead, most of the research on the

earnings prospects of company x will concentrate on comparisons

between company x and other engineering companies, and a further

comparison between the earnings prospects of engineering companies

in general and those for the whole market will be necessary before the

earnings growth rate of company x can be classified. Since the same

applies to company y, any attempt to differentiate between the

earnings growth rates of the two companies involves four separate

comparisons, each of which is subject to estimation errors.


7.8. The comparative factor of 1·2 for the differential in earnings

growth rate reflects the belief that, for equal values of all other

attributes, there is a one-to-one correspondence between earnings

growth rate and price-earnings relative, and that for the particular

earnings growth rates involved the two price-earnings relatives

should be in the proportion 1·2 : 1. The choice of 1·2 as the appro-

priate value clearly involves a high degree of uncertainty. If the

fund manager carrying out the comparison was asked to be more

specific about his choice of this value, he might say that in his opinion

the lowest possible value was 1·1, that the highest possible value was

1·4, and that the most likely value was 1·2. His beliefs can be

represented by a frequency distribution of the type illustrated in

Figure 21.

FIGURE 21

Suppose that the price structure of the market is such that the

true value of this factor is 1·26 and that another portfolio manager

carries out the comparison described in 7.4 using this value. He

calculates that the price-earnings relative of share x should be

100 x 1·26 x 0·9 x 1·05, i.e. 119·1, and concludes that share x is

correctly priced compared with share y. Although the true value of

1·26 is not unreasonable in the context of the first portfolio manager’s

beliefs (he assesses the probability of the value being 1·26 or higher as

approximately 0·3), his investment conclusion is invalid as a result

of his estimation error. Estimation errors of this type can be

eliminated only if the salient features of the price structure can be

measured with sufficient accuracy that any frequency distributions

implicit in the formulation of investment conclusions have very small

variances.


7.9. The comparison between share x and share y does not

indicate whether either share is correctly priced in relation to the

market as a whole. If we accept that share x is 5% overpriced

compared with share y, share x could be 5% overpriced and share y

correctly priced relative to the market, or share x could be correctly

priced and share y about 5% underpriced.

7.10. A further inherent disadvantage of comparisons of the type

described in 7.4 is that they are essentially static in nature and do not

take into account dynamic aspects such as changes over time in the

differentials in ratings between high growth shares and low growth

shares.

7.11. Despite all the inefficiencies that might arise out of the lack

of rigour, comparisons of the above type are implicit in the relative

value assessment criteria of the vast majority of investors. To allow

for this potential inefficiency in the price formation process, we must

in the first instance use a very weak description of investor behaviour

when constructing a more systematic framework for share appraisal.

The market equilibrium concept employed in Part I is therefore

formulated in the following terms:

“In coming to a decision on whether to buy or sell a particular

share, any one investor will assess various attributes of the share

in accordance with the criteria that he, and he alone, considers

appropriate.”

Any stronger statement of the price formation process could lead to

serious loss of generality and consequently to a highly inaccurate

representation of the market structure.

7.12. In the development of this concept into a practical model, a

primary objective is the attainment of a high degree of rigour through-

out so that the final version of the model avoids, as far as possible, the

unsatisfactory features discussed in 7.6 to 7.10. The market equili-

brium model can therefore be regarded as a restatement of con-

ventional share appraisal principles in such a form that the analysis

is brought within the reach of scientific measurement.

8. General development programme

8.1. Work began in May 1975 to develop the life office’s price-

earnings relative approach into a more systematic framework for

share appraisal.

D


8.2. Since the theoretical development of the market equilibrium

model led to a completely novel measure of earnings growth, pilot

tests were first of all carried out to check whether such an approach

could be expected to be successful in practice.

8.3. Although the market equilibrium model is multiplicative in

nature, these preliminary tests were carried out on an additive basis

for simplicity. The expected long-term earnings growth rate, the

dividend payout ratio and balance sheet strength were assumed to

be the only factors that affected the price-earnings relative of a

share. After careful consideration of what importance the market

appeared to pay to each factor, it was decided that, as a first approxi-

mation, the price-earnings relative of a share could be expressed as

price-earnings relative = 75 + factor total,

where the values given to each of the three factors were based on the

following scales:

earnings growth: – 20 (very low) to + 50 (very high)

dividend payout ratio: – 15 (very low) to + 15 (very high)

balance sheet: – 20 (very weak) to + 10 (very strong).

8.4. Since it would have been unrealistic to expect these ad hoc

rules to reproduce the market structure exactly, a process of least

squares fit was then applied, giving:

price-earnings relative = A + B x (factor total),

where A and B should be approximately equal to 75 and 1

respectively.

8.5. Around 100 shares were assessed in this way in June 1975,

with three different investment analysts each covering about one-

third of the total. The factor values for dividend payout ratio and

balance sheet were calculated on prescribed quantitative bases, but

the factor value for growth rate was based on purely qualitative

assessments. To test for any bias in the way the analysts estimated

the earnings growth factor, a separate least squares fit was carried

out for each of the three groups of shares. The resulting regression

lines, together with the line corresponding to A = 75 and B = 1,

were as shown in Figure 22.

Although the factor ranges were remarkably accurate overall, very

considerable analyst bias was present. In particular, one analyst

(analyst 3 in Figure 22) was using a very much bolder interpretation


FIGURE 22

of the guidelines than the other two, giving a wide range of factor

totals and a compensating low value of B.

8.6. In view of this bias, no attempt was made to fit a single

regression line. The cheapness of each share was therefore assessed

using the ratio of its expected price-earnings relative (calculated from

the appropriate analyst’s regression line) to its actual price-earnings

relative. For each of the three groups of shares, a very strong

correlation between this assessment of cheapness and the relative

price performance was apparent within a few weeks.

8.7. These tests demonstrated that even a highly simplified

version of the market equilibrium model could detect the general

market structure and identify anomalies.

8.8. In August 1975 it was decided to develop the market equi-

librium model as the primary tool in the management of the life

office’s U.K. ordinary share portfolios. Since comprehensive records

of earnings, dividends, balance sheet strength and overseas exposure

were already in existence for a large number of companies, the

determination of earnings growth rates accounted for most of the

preparatory work. The first operational results were obtained in

October 1975 for a group of 136 industrial and commercial companies.

8.9. As discussed in Part I, theoretical considerations suggest that

the model should incorporate three variable parameters to take

account of those changes in investors’ preferences that are likely to


have a significant effect on the equilibrium position. However,

there could be no possible way of assessing the statistical stability of

such a model until operational results had been studied for a sufficient

length of time. The development of the statistical fitting process has

therefore proceeded in three stages:

stage 1 : October 1975 to December 1976

stage 2 : December 1976 to September 1978

stage 3 : from September 1978.

8.10. During stage 1 only the earnings growth parameter g was

allowed to vary. Specified values of the dividend payout parameter

r and the overseas exposure parameter a were used. Throughout this

period all the calculations, including the statistical fit, were carried

out manually on a fortnightly basis.

8.11. During stage 2 the same formulation of the model was used

but the calculations were carried out by computer on a weekly basis.

From time to time the effects of varying the other two parameters

were tested to ensure that the fixed values used were reasonably close

to the values which gave the best statistical fit.

8.12. Stage 3, which allows all three parameters to vary, was

implemented after comprehensive tests at stage 2 showed that the

model possessed exceptionally good statistical stability.

8.13. Apart from the introduction of a current cost earnings

adjustment in 1976, the detailed methods in use at present are little

changed from those used in the first applications of the model some

five years ago.

9. Earnings base

9.1. The general properties of the earnings base function

E(E0, E1, E2, t)

are described in 6.10. We consider first of all the problems of

standardisation and interpretation that arise in the calculation of

earnings per share.

9.2. Many of the serious problems caused at present by differing

accounting bases are described by the Chairman of Transport

Development Group in his report on 1979:


“Any consideration of accounts these days is materially affected

by the standards now being imposed by the accounting bodies in

the interests of uniformity. The initial effect of such standards

is to make comparison with earlier years more difficult. This is a

temporary problem but uniformity of treatment will still not

produce ready comparability between companies. Businesses are

too diverse for this to be so and it is a pity that integrity of

judgement and professional competence seem to be left such small

room. The resulting answers may be tidy but they are often

meaningless and sometimes downright misleading.

“A case in point is the treatment accorded to deferred tax under

SSAP 15. This treatment has been adopted with great reluctance

in the 1979 accounts; when applied to historic accounts it renders

them a hopeless guide as to the amount of distributable profits.

“Current cost accounts on pages 26 and 27 attempt to show

profits adjusted for inflation and the replacement worth of assets

employed. They cannot be precise, but give a broad indication.

In the case of the Group, with assets being regularly replaced,

profits remain sufficient to cover the dividend payment but leave

scarce funds for future expansion. Many enterprises these days

are not even in that position.”

9.3. The general approach adopted is to calculate

E(E0, E1, E2, t)

on historic cost principles and then to incorporate an explicit

adjustment for current cost profits.

9.4. Since the tax charge arising under SSAP 15 reflects some

aspects of current cost accounting whereas the pre-tax profit figure from

which it is deducted is calculated on historic cost principles, reported

distributable profits involve so many inconsistencies that any figures

of earnings per share derived from them would be of little practical

value in inter-company comparisons. For the purposes of the market

equilibrium model, where comparability is of crucial importance, we

therefore calculate historic cost earnings per share on a fully taxed

basis.

9.5. Some companies report profits after an additional depreciation

charge or other special charge which reflects some of the impact of

inflation. In such cases it is necessary to adjust the reported profits

by adding back the special charge.


9.6. The price-earnings relative approach developed by the life

office in the early seventies is based on “current earnings” Et given by

where E0 is the historic earnings per share

E1 is the forecast prospective earnings per share

and t is the fraction of a year since the last annual results were

announced.

9.7. We use this current earnings function Et as the starting

point in the construction of the earnings base function and then

incorporate two adjustments to reflect the general importance that

the market appears to attach to E0, E1 and E2. To reflect weights

that approximate to the following pattern:

t E0 E1 E2 0 0·5 0·5 0

0·5 0·25 0·5 0·25

1 0 0.5 0.5

we define the earnings base function as

where

The detailed structure of this earnings base function has very little

effect on the final results.

9.8. Having defined the earnings base function in terms of earnings

per share figures calculated on historic cost principles, we can regard

the proportionate change in profits as between historic cost and

current cost bases as an additional attribute to be incorporated in the

model by means of a multiplicative factor.

9.9. Current cost accounting principles attempt to give a more

realistic measure of distributable profits during inflationary con-

ditions. If two shares differ on account only of the reduction in

profits under current cost accounting, the share with the smaller

proportionate fall will merit the higher price since the dividend cover

under current cost accounting is higher. The scale of values incor-

porated in the model is based on the ratio of current cost profits to

historic cost profits as shown in Figure 23.


FIGURE 23

10. Earnings growth rate

10.1. The earnings growth rate is determined by assessing the following four factors in the light of all available information :

1. Rate of growth of turnover. 2. Profit margins. 3. Ability to finance expansion. 4. Operational flexibility.

Numerical values are assigned to each factor, and the total over all factors gives G, the ranking measure of earnings growth used in the model.

10.2. In assessing turnover, the general growth prospects for the relevant industries are first of all reviewed. Energy, aerospace and electronics clearly offer very attractive growth prospects at present, whereas activities such as food manufacturing can be expected to remain static or decline in terms of economic importance. Some areas such as engineering and retailing are so diverse that the products and marketing concepts of each company have to be studied before an assessment of the general growth prospects can be made. Within any particular industry, different companies will vary greatly in the degree to which they anticipate and exploit potentially profitable markets. An assessment of these management qualities must also be incorporated in the turnover factors for each company.

10.3. Both the general level and the stability of profit margins have to be taken into account. Price sensitivity, the importance of


the product, possible sources of competition, and price controls (formal or informal) are some of the many considerations involved.

10.4. In assessing a company’s ability to finance expansion, the main considerations are the general capital requirements of the principal activities, the level of retained earnings, the capital structure and the amount of unused borrowing powers.

10.5. The fourth factor, operational flexibility, takes into account the extent to which a company can adapt to changing economic circumstances. A specialist chemical company which is totally dependent on the level of activity in one particular user industry is a good example of a company that has very little operational flexibility. At the other extreme is a company with, say, four main divisions operating in diverse industries. Provided that the necessary management skills are present, this company can expand or contract particular operations to ensure that available resources are channelled into those areas that are most profitable in the prevailing economic circumstances.

10.6. From 1975 until July 1978 each factor was assessed on a scale of 1 (very poor) to 7 (exceptionally good), giving values of G that ranged from 9 to 25.

10.7. Practical experience of the market equilibrium model soon showed that the market appeared to attach undue importance to past earnings growth. Since this caused some values of the relative price residual to remain strongly positive or strongly negative for long periods, the statistical fit was fairly poor. When only one variable parameter was used, as was the case until September 1978, this poor statistical fit was of no practical consequence. However, there was a risk of instability when three variable parameters were used.

10.8. As a precautionary measure, earnings growth rates much closer to the past experience were used when the number of variable parameters was increased to three in September 1978. As a result of this change of basis, the root mean square error was reduced from 14% to 11%.

10.9. After running the model on this new basis for two months it became clear that the statistical stability was more than adequate to cope with the three variable parameters. It was therefore decided to revert gradually to the previous earnings growth basis.


11. Other factors

11.1. Of the various functions described in 6.40, only a, d, B and F4(B) have still to be defined.

11.2. A convenient definition of a, the measure of overseas exposure, is the proportion of profits arising from overseas operations plus the proportion of profits derived from exports from the U.K. These proportions can normally be estimated with sufficient accuracy from published accounts.

11.3. For d, the value 1.5 is used. Changes in d have little effect on the final results.

11.4. The level of borrowings, B, is defined as the ratio of total borrowings less cash deposits to equity capital plus reserves. The function used in the model to represent F4(B) is shown in Figure 24.

FIGURE 24

11.5. In calculating the level of borrowings, various problems of standardisation can arise. However, in view of the relatively minor effect of the function F4(B), these problems are of no practical significance.

12. Minimisation procedures

12.1. For each share, the expected price x and the relative price residual x involve four unknown quantitities, the scaling factor k and the three variable parameters a, r and g. The required values of the

variable parameters are those which minimise


12.2. Consider first the case where only g varies and a and r are

assigned the values a0 and r0 respectively. The expected price x can be expressed as

where Yx involves no unknown quantities and hence can be calculated directly.

12.3. The situation is represented in Figure 25.

FIGURE 25

Provided the relative price residuals are small, the values of lie

close to a smooth curve k.(1+g)G–G0 for suitable values of k and g. The value of k determines the height of the curve at G0 and the value of g determines the gradient of the curve.

12.4. The required values of k and g are those which minimise and it is obvious that these values can be found by elementary

least squares methods. These values of k and g, and the corresponding minimum values of depend on a0 and r0.

12.5. Since the error function is a smooth function of a, r and

g, the overall minimum value can be obtained by carrying out the above procedure for different values of a, and r0, interpolating to


find the required values of a and r, and finally repeating the above procedure for these values of a and r to obtain the required value of g. The minimisation results for 24th June 1980, when 171 shares were involved, are shown below to illustrate the simplicity of the method.

12.6. The previous week’s values of a and r were 0·1135 and 0·5128 respectively. Using combinations of these values and values 0·05 higher and lower, the following values of minimised with

respect to g, were obtained :

r

a 0·4628 0·5128 0·5628

0·0635 3·0331 2·9297 3·0005 0·1135 3·0092 2·9140 2·9951 0·1635 3·0237 2·9395 3·0331

Interpolation on the basis of constant second differences with respect to both a and r gives a = 0·1070 and r = 0·5152. The final application of the least squares fit for g using these values then gives g = 0·0437 and = 2·9133.

13. Results

13.1. The first operational results were obtained using prices and other data as at 28th October 1975. The early results were tested in various ways to check the validity of the assumptions used in the theoretical construction of the model.

13.2. The theory suggests that the relative price residual ε x should be unbiased. However, the use of inappropriate functions in the construction of the model or the use of inaccurate values of specified parameters could lead to bias in ε x which would seriously restrict the usefulness of the model.

13.3. The 136 shares included at 28th October 1975 were first of all ranked in descending order of relative price residual and par- titioned into five groups, Buy, Buy/hold, Hold, Hold/sell, and Sell, based on the implied short-term assessment of attractiveness relative to the market. Since the earnings growth rate is by far the most important element of the model, the frequency of high, medium and low growth rates within the five groups was examined. The results are shown in Table 1.


TABLE 1

Frequency of high, medium and low growth rates in assessment groups

at 28th October 1975

Group

Growth rate Buy Buy/hold Hold Hold/sell Sell

High: 19-25 1 3 2 3 9 Medium: 13-18 17 20 23 19 16 Low: 9-12 9 4 3 5 2

Total 27 27 28 27 21

13.4. It was apparent from the large number of low growth shares in the Buy group and the large number of high growth shares in the Sell group that serious bias existed. The function

is defined in such a way that equal proportionate changes in price result from equal changes in G. The original scale used for G was not widely enough spaced at the upper end to satisfy this condition. This was corrected by applying a transformation of scales as shown in Figure 26 to increase the spacing at the upper end of the range.

FIGURE 26

This transformation successfully eliminated any bias with respect to growth rate in subsequent applications of the model.

13.5. When similar tests were carried out for dividend payout ratio it was found that the Buy and Buy/hold groups had a modest bias in favour of low payout. Closer examination showed that the companies with a very low dividend payout ratio tended to be those that had been prevented by statutory controls from increasing their dividends in line with rapidly rising earnings. Since it appeared at the time that the market might be attaching too much importance to immediate income, it was decided that the slight bias was, on balance, a desirable feature and accordingly no attempt was made to eliminate it.


13.6. To test whether the relative price residuals could be used to predict future short-term price movements, the average market value of each of the five assessment groups was calculated at fortnightly intervals for a period of six months. The results for 28th October 1975 are shown in Table 2.

TABLE 2

Market values of initial assessment groups at fortnightly intervals based

on 28th October 1975 = 100

Group Number in group

28th October 1975 11th November 25th November 9th December 23rd December 6th January 1976 20th January 3rd February 17th February 2nd March 16th March 30th March 13th April 27th April

Buy 27

100·0 105·2 107·3 104·3 104·6 110·0 115·6 118·7 116·9 119·1 116·3 117·4 118·2 119·7

Buy/hold 27

100·0 100·4 106·4 102·8 102·4 107·8 110·8 114·4 112·1 113·5 112·6 113·3 112·7 112·7

Hold Hold/sell Sell 28 27 27

Buy + Sell

100·0 100·0 100·0 1·000 104·0 103·6 101·9 1·032 104·1 104·8 101·2 1·060 100·2 102·8 99·9 1·044 100·6 102·6 99·1 1·055 104·9 105·8 103·5 1·063 108·8 109·1 109·3 1·058 109·6 111·7 113·0 1·050 107·8 109·5 110·4 1·059 110·0 110·9 110·5 1·078 109·5 109·4 108·4 1·073 110·5 110·6 109·8 1·069 109·1 109·6 109·2 1·082 110·0 108·5 108·3 1·105

13.7. Similar highly regular patterns of differential price performance were obtained using the output of the model at later dates. The ratio of the market value of the Buy group to that of the Sell group was usually in the range 1·07 to 1·11 after six months, irrespec- tive of whether the market had been rising or falling over the period.

13.8. The wide divergence in performance and the highly regular dispersion of the five groups represent very powerful evidence that the market equilibrium model can identify the market structure and measure accurately the extent to which a particular share price deviates from that structure.


PART III : PRACTICAL APPLICATIONS

14. Practical aspects

14.1. In this section we discuss some of the important practical considerations that arise in the operation of the market equilibrium model.

14.2. The universe of shares covered by the model has first of all to be chosen. Around 170 U.K. ordinary shares are included at present, and these are mainly, but not exclusively, the largest in terms of market capitalisation. Some sectors, such as banks, consist of only a few large companies, and these are all included in the model. In other sectors, such as stores, which contain a large number of companies of widely differing market capitalisation, all the larger companies are again included, but only those smaller companies with good earnings growth prospects are included.

14.3. The earnings growth rates of all the individual companies must be determined on a standardised basis which incorporates long- term economic assumptions. It will generally be the investment manager who specifies these economic assumptions and overlooks the earnings growth rates to ensure that standardisation is achieved.

14.4. Estimates of pre-tax profits will in many cases be based on stockbrokers’ research. The estimates used in the model must reflect various short-term economic assumptions such as likely trends in interest rates, retail sales and stockbuilding.

14.5. As stated in 7.1, the market equilibrium model is a direct extension of conventional methods of share appraisal, based on price- earnings relatives, that were employed by a Scottish life office in the early seventies for the management of U.K. ordinary share portfolios. These price-earnings relatives are based on current earnings, as defined in 9.6. The market average price-earnings ratio used approxi- mates to the median price-earnings ratio (using current earnings) of the universe of shares described in 14.2. Weekly changes in this market average are calculated from the equally weighted geometric average. As an illustration of the type of graph obtained, Chart 1 shows the price-earnings relative history of Whitbread “ A ” from January 1976 to September 1980. In this case the price-earnings


relative has generally been in the range 110 to 130, but remained below 110 for six months during 1977 and has been above 130 for most of the time since the middle of 1979.

14.6. The weekly results of the model are printed out in a detailed format which shows the effect of each attribute and expresses the expected price of each share in terms of price-earnings relatives. Using this format, the results as at 24th June 1980 for Electro- components, Whitbread “ A ” and Tube Investments (representing three very dissimilar companies operating respectively in electronic components distribution, brewing and engineering) are as follows :

Electro- Whitbread Tube components “ A ” Investments

Earnings base adjustment for E1

Earnings base adjustment for E2

Dividend payout ratio Current cost earnings Earnings growth rate Overseas exposure Balance sheet Special factors

100 105 107

110 100 101

56 75 100 95 91 75

349 151 115 101 100 106 110 101 100 100 115 100

Expected price-earnings relative

Actual price-earnings relative

227 126 98

219 144 101

Relative price residual % +3·5 –12·8 –2·6

With the exception of the earnings growth rate factor, which incorporates the scaling constant k, the factors are based on the scales described in Section II. The expected prices-earnings relative is then the product of the eight factor values shown.

14.7. All the analytic data used in the model are stored in the computer and printed out in full each week in sector order.

14.8. To facilitate comparisons between the results obtained from the model and other assessments of relative attractiveness, stock-


brokers’ assessments of individual shares are stored in the computer in the following classified form:

Buy 5 Qualified buy (e.g. buy for income) 4 Hold 3 Qualified sell (e.g. fully valued) 2 Sell 1 No assessment 0

To monitor the consensus view and identify any wide divergence of opinion, the arithmetic average and standard deviation are calculated for each share.

14.9. As discussed in 6.44, the relative price residuals,, while having zero expected value over the long term, may remain either strongly positive or strongly negative over the medium term. To allow for this behaviour, x is analysed using Mean Absolute Deviation techniques of the type described by Plymen and Prevett (1972). These techniques involve an exponentially weighted moving average of the time series, the exponentially weighted moving average of the absolute value of the current deviation from the average (the mean absolute deviation), and upper and lower control limits which are placed a fixed number of mean absolute deviations above and below the average.

14.10. Mean Absolute Deviation techniques involve two parameters, the weekly exponential smoothing factor and the ratio of the distance between the average and a control limit to the mean absolute deviation. These parameters have to be chosen in such a manner that major turning points will normally be successfully indicated by penetration of the control limits. Practical experience shows that values of 0·13 and 1·6 respectively are highly satisfactory.

14.11. For each company, four separate graphs are produced :

1. Price-earnings relative. 2. Relative price residual (including moving average and control

limits). 3. Price relative to the F.T.-Actuaries All-Share Index. 4. Average value of stockbrokers’ assessments.

15. Analysis of the relative price residual

15.1. Chart 2 shows the graphs for Whitbread “ A ” of the relative price residual (together with moving average and control limits) and


the price relative to the F.T.-Actuaries All-Share Index for the period January 1979 to September 1980. The vertical scale of the relative price residual is reversed, so that penetration of the upper control limit indicates that the share may be dear and penetration of the lower limit indicates that the share may be cheap.

15.2. Towards the end of March 1979 the share price rose strongly against the market on the news that the company had been granted a 1p a pint interim rise on beer prices by the Price Commission and was to seek a further increase of 2p a pint, and the relative price residual rose above the upper control limit. However, in view of the likelihood of a Conservative victory in the imminent general election, this penetration of the control limit was not interpreted as a “ sell ” signal.

15.3. When the promised abolition of the Price Commission was taken into account in profit projections after the general election, the relative price residual returned to the middle of the range. Following revisions to earnings and dividend projections after the 1978-79 preliminary results were announced in May 1979, the relative price residual penetrated the lower control limit. Since the company had indicated its confidence in the future by announcing with the results a £230 million investment programme, this penetration of the control limit was interpreted as a “ buy ” signal and holdings were increased early in June at the point on the graphs shown by the vertical broken line.

15.4. During the next 8 weeks the share price rose by 25% relative to the market and early in August the relative price residual penetrated the upper control limit. At this point, when it looked as though the shares had become overvalued against the market, the position was reviewed carefully to determine whether in fact holdings in the company should be reduced. The fairly optimistic current year earnings and dividend prospects had been reflected in the input data, the benefits to accrue from the Chiswell Street property development had been taken into account by means of a special factor of 1·08, and the price-earnings relative was standing at 151 as against highs for 1976, 1977 and 1978 of 135, 127 and 130 respectively. It was judged that the shares had in fact risen to an unsustainably high rating, and accordingly holdings were reduced at the point shown by the second vertical broken line.

15.5. Despite the announcement of highly satisfactory interim results in November 1979 and the upgrading of estimates of the value

E


to Whitbread of the Chiswell Street development, the share price declined sharply against the market in January 1980, causing the relative price residual to penetrate the lower control limit. The background news was again reviewed in detail. Although it appeared that the weakness in the share price had been caused by fears of consumer resistance to increased beer prices, the earnings and dividend estimates incorporated in the model reflected some deterioration in the trading background and it was judged that the price weakness had been overdone. Holdings were therefore increased at the point shown by the third vertical broken line.

15.6. In the following 3 months the shares rose by 25% relative to the market, taking the relative price residual very close to the upper control limit. When the trading background was once again reviewed ahead of the 1979-80 preliminary results due in May, it appeared that the general squeeze on consumer expenditure and the very sharp increases in beer prices might soon have a serious effect on sales volume. Since the 1979-80 results tended to confirm this view, holdings were reduced immediately after the announcement of these results at the point shown by the fourth vertical broken line.

15.7. Apart from a short-lived rally towards the end of June, the share price steadily underperformed the market after these results. At the annual general meeting in July, the chairman warned shareholders that “ if the group suffered a similar fall away in business as had been seen in June and July, some elements of the group’s investment plan would have to be postponed ”.

15.8. This example highlights the sensitivity of share prices, even in well researched sectors like breweries, to short-term changes in market sentiment. During the relatively short period under review, two buying opportunities were identified and in both cases selling opportunities after 25% outperformance were identified a few months later. On all four occasions, holdings of Whitbread “ A ” shares were increased or decreased accordingly.

16. Earnings growth rates

16.1. One of the most important features of the market equilibrium model is that the values of the earnings growth rate G do not require to be revised at frequent intervals. The overseas exposure function F3(A) takes account of medium term changes in general company profitability within the U.K., and the relative price residual then takes into account short-term changes in the profitability of an


individual company. The Whitbread example described in Section 15 illustrates these properties.

16.2. Apart from one change during August 1979, when a small increase was made to reflect the company’s very large investment plan, the earnings growth rate remained constant throughout the period.

16.3. The general increase in expected U.K. economic growth (and hence in profits arising from U.K. operations) as perceived by the market around the time of the 1979 general election, and the subsequent downgrading of these expectations during 1980, are taken into account by the overseas exposure function F3(A). Since Whit-

bread has negligible overseas exposure, the behaviour of the overseas exposure market parameter, as described in 18.15, gave rise to an increase in F3(A) during the early part of 1979 and a decrease in F3(A) during 1980.

16.4. Since Whitbread’s profits are very sensitive to price controls and sales volume, the short-term profitability increased more rapidly during the first half of 1979 than the average for a company with the same level of overseas exposure, and subsequently declined more rapidly than the average during the middle of 1980. Relative price

movements arising out of these short-term fluctuations in perceived profitability are reflected in the relative price residual, and the Mean Absolute Deviation techniques help identify major turning points in the relative price residual and hence in the performance of the shares relative to the market.

16.5. As can be seen from the examples in 14.6, the earnings growth rate function F2(G) is by far the most important term in the expected value. However, estimation errors in G do not affect short- term performance assessments of the type described in Section 15, since Mean Absolute Deviation techniques are used in the analysis of the relative price residual.

16.6. The earnings growth rate G relates to the average experience over a complete economic cycle of, say, 5 years. Since conventional share appraisal methods usually involve a much shorter time horizon, it is important to check that the earnings growth rates used in the model can discriminate with sufficient reliability between high growth and low growth shares as measured over a 5-year period.


16.7. A very direct test is possible using the earnings growth rates calculated in 1975 when the model was being set up. In assessing the component of earnings growth that relates to the financing of expansion, particular regard was paid to the dangerously high rate of inflation in the U.K. at that time and to the likelihood that high rates of inflation would persist for several years. It appeared that certain capital-intensive companies, including many engineering companies, would be unable to generate sufficient finance internally to support the rate of profits growth previously expected; During August and September 1975, holdings in some of these companies were sold and re-invested in companies expected to show higher earnings growth.

16.8. Although numerous other smaller holdings were also involved, the order of magnitude of the profitability of this switching operation can be judged by considering the performance of the two shares which accounted for more than 50% of the sales proceeds, namely Guest Keen and Tube Investments, and the performance of the five shares in which the largest amounts were re-invested, namely Arthur Bell, General Electric, Great Universal Stores “ A ”, Im- perial Group and Shell. The average performance relative to the F.T.-Actuaries All-Share Index of Guest Keen and Tube Investments, ignoring expenses and dividends, and the corresponding performance for the five shares representing the re-investment, are shown in Figure 27 at quarterly intervals to September 1980.

FIGURE 27


Guest Keen and Tube Investments outperformed the market very strongly until the second quarter of 1976. Thereafter the performance showed little change for more than a year and then began a steady decline. The purchases, on the other hand, having moved only slightly ahead of the market over the first year, outperformed significantly for the next 2 years and then levelled off. After 5 years, the value of the purchases was three and a half times the value of the sales.

16.9. The important point about this example is that the earnings growth rates obtained in 1975 were calculated on the basis of a complete economic cycle, whereas most other assessments of companies like Guest Keen and Tube Investments at that time concentrated on the prospects for profits during 1976, when a cyclical recovery was expected.

16.10. Practical experience over the 5 years that the model has been in existence suggests that the four component method of determining the earnings growth rate is highly satisfactory. It is systematic without being too rigid, and it facilitates standardisation between different companies.

16.11. Although the relativities between sectors may change if the underlying economic assumptions are altered significantly, there is a very high degree of stability in the rankings of earnings growth rates within sectors.

17. Special factors 17.1. As explained in 6.37, it is sometimes necessary to incorporate

within the model a special factor to take account of non-standard features such as the likelihood of a take-over bid or a significant disposal of assets.

17.2. In the case of Whitbread, the Chiswell Street development was allowed for by means of a special factor of 1·08 throughout most of 1979. Towards the end of 1979, when there was extensive press comment suggesting that the company would shortly be able to realise its profit on the development, the factor was increased to 1·15.

17.3. In the case of possible take-over bids, factors in the range 1·10 to 1·30 are normally appropriate. Since these values incorporate a probability element and are generally kept constant, the relative price residual may become strongly negative if the share


price rises sharply on take-over rumours. Conversely, if little attention is focussed on the take-over possibilities for a period, the relative price residual may become positive.

17.4. Asset value is taken into account in the model only in special cases. Examples might be a construction company with a property portfolio and a stores company which owns a significant amount of freehold property and which is earning a poor rate of return on its assets compared with other companies operating in similar fields.

18. Market parameters 18.1. Theoretical considerations of statistical stability and possible

variations in investor preferences show that the market equilibrium model should incorporate three variable market parameters, namely r, g and a, which relate to dividend payout ratio, earnings growth rate, and overseas exposure respectively. If the model is to be used as a frame of reference for future projections, it is essential to verify that the overall structure of the model is statistically stable and that any major variations in the market parameters are consistent with changes in investor preferences in the aggregate.

18.2. After the process of statistical fit had been altered in Sep- tember 1978 to include all three market parameters, various adjustments were made to the earnings growth rates to improve the standardisation between sectors. This fine-tuning of the model was completed in December 1978.

18.3. The values of the three market parameters, together with the root mean square of the relative price residual and the F.T.- Actuaries All-Share Index, are shown in the appendix at weekly intervals from 2nd January 1979 to 30th September 1980.

18.4. It is clear that all three market parameters exhibit a very high degree of statistical stability. The significant variations over time in the values of the parameters confirm the theoretical arguments set out in Part I.

18.5. Although the value of the root mean square error is well above the level that would be regarded as acceptable in most statistical models, the theoretical construction of the market equilibrium model takes explicit account of the expected heavy random noise within ordinary share markets. The Whitbread example in Section


15 shows how volatile the share price can be, even in the case of a well researched major company with a highly stable profits record.

18.6. The behaviour of the root mean square error at a time when share prices are moving very rapidly is further evidence that the high values reflect random noise rather than distortion in the model. In

June and July 1980, when share prices generally were rising very rapidly, the behaviour of the root mean square error was as follows :

Date 3:6:80

10:6:80 17:6:80 24:6:80

1:7:80 8:7:80

15:7:80 22:7:80

F.T.-Actuaries Root mean All-Share Index square error

245·28 0·1236 255·70 0·1224 268·95 0·1259 267·86 0·1305 267·01 0·1436 282·14 0·1457 282·71 0·1326 284·43 0·1323

Similar behaviour occurred during the second quarter of 1979.

18.7. The most convenient method of analysing the practical significance of the value of a particular market parameter is to calculate the associated price sensitivity function as defined in 4.16. Chart 3 shows the price sensitivity functions S(R), S(G) and S(A) for dividend payout ratio, earnings growth rate, and overseas exposure respectively, together with the F.T.-Actuaries All-Share Index, at weekly intervals from 2nd January 1979 to 30th September 1980. The vertical scales of the dividend payout ratio and overseas exposure functions are reversed to facilitate interpretation.

18.8. The variable market parameters are incorporated in the model to take account of changes in investor preferences. The function S(R) is thus a measure of investor preference, in the aggregate, for earnings as opposed to dividends, with a higher preference for earnings reflected in a lower value of S(R). Similarly, the function S(G) measures investor preference, in the aggregate, for high earnings growth companies as opposed to low earnings growth companies, with a higher preference for high earnings growth reflected in a higher value of S(G). Finally, the function S(A) measures investor preference, in the aggregate, for U.K. (i.e. non-overseas) profits as opposed to overseas profits, with a higher preference for U.K. profits reflected in a lower value of S(A).


18.9. If the U.K. ordinary share market is rising strongly, supported by increasing investor confidence and expectations of increased U.K. economic growth, as was the case during March and April 1979, we would expect S(G) to increase and S(R) and S(A) to decrease. Since the vertical scales for S(R) and S(A) are reversed in Chart 3, the strong rises in each of the three graphs from early March 1979 until the market top in May 1979 demonstrate this behaviour very clearly. This provides strong evidence that the variable parameters in the model do in fact reflect changes in the equilibrium position that arises out of changes in investor preferences.

18.10. After the market top in May 1979, the dividend sensitivity graph declines very sharply for two months, recovers briefly, and then trends downwards once again. After reaching its lowest value in May 1980, the graph then rises (as does the F.T.-Actuaries All-Share Index) for the remainder of the period but remains below the levels of the first quarter of 1979.

18.11. The earnings growth sensitivity graph, on the other hand, continues to rise during May, June and July 1979 against a falling market, remains fairly steady until May 1980, and then rises again during June, July and August 1980.

18.12. The continuing rise in this graph during May, June and July 1979, when there was a strong downward trend in the market, was investigated very closely at the time since it appeared to be inconsistent with past experience at similar market peaks. As wage costs were rising rapidly and the new Conservative government were using high interest rates to regain control of the economy, all the evidence pointed to the explanation outlined in 6.20, namely that the profits and cash flow of low growth companies had come under severe pressure in the new economic environment whereas the profits of high growth companies were only marginally affected.

18.13. Confirmation of this divergence in performance between high growth and low growth companies from the middle of 1979 onwards is provided by two external sources. The “ Investors Chron- icle ” has for many years set up a new high yield portfolio every six months using a screening process to isolate those high yielding shares which involve a relatively low risk. An article in the “ Investors Chronicle ” of 20th June 1980 describes the unusually poor performance of the two high yield portfolios introduced in 1979 and suggests the following explanation :


“ There is no doubt that High Yielders are generally second-rank stocks, at least when they go into a portfolio. They are selected on the theory that the market is undervaluing them in comparison with other shares in the stock market whose yields are close to, or below average.

“ The effect of high interest rates and correspondingly the high cost of borrowing may be affecting the companies’ performances. There are cases throughout the lists where interim dividends have been cut or omitted altogether, though there are no situations yet where the total dividend has been passed or cut to such an extent to warrant sales.”

18.14. Further confirmation of the existence of significant differentials in performance between high growth shares and low growth (and hence high yield) shares appears in the “ Investors Chronicle ” of 29th August 1980. In a review of the performance of unit trusts managed by stockbrokers, based on statistics taken from “ Money Management “, August 1980, the individual trusts are classified into

general funds, growth funds and income funds. The summary table is as follows :

Average results of £1000 investment held for

General funds Growth funds Income funds

7 yrs 5 yrs 3 yrs 2 yrs 1 yr £ £ £ £ £

1733 2169 1550 1324 1125 1679 2141 1598 1377 1153 1706 2182 1377 1129 999

Over 5 years and 7 years, the results differ only marginally. This suggests that, if the comparison period is long enough to cover a complete market cycle, the different investment policies can lead to very similar results. Over each of 1, 2 and 3 years, however, the growth funds have the best performance, followed closely by the general funds, with the income funds a long way behind.

18.15. The overseas sensitivity graph reaches a peak 2 weeks after the market top in May 1979 and then declines fairly steadily, with the only major upwards move against this trend occurring during July and August 1980. This decline from May 1979 onwards indicates a change in investor preference from U.K. profits to overseas profits, which is consistent with the squeeze on U.K. profits described in 18.12.


18.16. Since the dividend sensitivity function can be regarded as a measure of investor confidence and changes in the growth rate and overseas sensitivity functions reflect the impact of economic forces on company profits, major variations in the market parameters are in fact consistent with changes in investor preferences in the aggregate. The predictability of future values of the market parameters is discussed in Section 19.

18.17. The magnitudes of the market parameters and of the associated sensitivity functions are of considerable interest since they highlight the extent to which the assumptions made in the theoretical development of the model are verified by direct measurement of the market structure using the model as a frame of reference.

18.18. Throughout stage 1 in the development of the model a constant value of 0·6 was used for the dividend parameter r. During 1979 and 1980 the value of r has varied from 0·48 to 0·62.

18.19. The values of the growth rate G incorporate a central long- term forecast of the effect of overseas exposure, and this should lead to a long-term average value of zero for the overseas exposure parameter a. The value of a is negative from January 1979 to March 1980 and positive thereafter, which suggests that the long-term average value is close to zero.

18.20. The three market parameters behave very differently over the period and hence can be regarded as statistically independent.

18.21. In the theoretical development it is assumed that the price sensitivity with respect to growth rate is higher than the price sensitivity with respect to dividend. The ratio of the growth sensitivity to the dividend sensitivity varies from a minimum of 1·03 on 23rd January 1979 (the market low for 1979) to a maximum of 1·72 on 8th May 1979 (the market high for 1979).

18.22. Variable parameters are incorporated in the model for attributes where it is expected that the variation function V defined in 4.17 will be large. For each of the three market parameters, the maximum and minimum price sensitivity values and the corresponding variations are shown below :


Price sensitivity Maximum Minimum Variation

Growth parameter g 0.280 0.161 0.119 Dividend parameter r 0.213 0.120 0.093 Overseas parameter a 0.085 –0.130 0.215

These figures confirm that significant changes in the general structure of the market can occur within relatively short periods of time.

18.23. Since there are various indications that the behaviour of the growth rate parameter g after the market top in May 1979 was highly untypical, it is likely that the value of g will, over a sufficiently long period, show a reasonably strong correlation with the market index. This correlation will be such that high growth shares will tend to outperform in a rising market and underperform in a falling market. Accordingly, if the monthly price changes of a share are plotted as a scatter diagram against the monthly price changes in the market index, the gradient of the straight line fitted to these points will be greater than one in the case of a high growth share and less than one in the case of a low growth share.

18.24. In Modern Portfolio Theory, the gradient of this straight line is defined as the beta co-efficient, or systematic risk, of the share, and projections of future performance are based on the assumption that “ high beta ” shares will outperform in a rising market and underperform in a falling market. It is, however, the behaviour of the market parameter g, not the behaviour of the market index, that determines whether “ high beta ” shares perform better over any given period. On the basis of Modern Portfolio Theory, high growth shares should have underperformed the market from May 1979 to November 1979, when the F.T.-Actuaries All-Share Index fell more than 20%. As discussed above, high growth shares outperformed the market significantly during this period.

18.25. The conceptual foundations of Modern Portfolio Theory are discussed in detail in Part V of the paper.

19. Primary trends 19.1. It is clear that the behaviour of the market parameters is

closely linked to changes in general investor confidence and to changes in company profitability. Since these changes are largely responsible for determining whether the trend in the market level is upwards or downwards, forecasts of the future values of the market parameters must be consistent with forecasts of market movements.


19.2. In practice, stockmarket indices do not move smoothly from one terminal value to another, and movements against the general trend are often wrongly interpreted at the time as a change in the trend.

19.3. It is apparent from Chart 3 that the price sensitivity functions derived from the market parameters behave in a much more regular manner than the F.T.-Actuaries All-Share Index. This suggests that the behaviour of these market parameters may be useful in identifying trends in market movements.

19.4. If we define the primary trend of the market as being upwards when the behaviour of all three market parameters is consistent with improving investor confidence and expectations of improving company profitability, it is clear that from January 1979 until the market top in May 1979 the primary trend was upwards.

19.5. Conversely, we define the primary trend as being downwards when the behaviour of all three market parameters is consistent with deteriorating investor confidence and expectations of deteriorating company profitability. The graphs of the dividend and overseas price sensitivity functions turn downwards immediately after the market top in May 1979 and the continuing rise in the graph of the growth sensitivity function is the result of a deterioration in company profitability. The primary trend is therefore downwards from May 1979 until May 1980.

19.6. The behaviour of the dividend and overseas parameters in the three months after May 1980 is shown below :

Date

27:5:80 3:6:80

10:6:80 17:6:80 24:6:80

1:7:80 8:7:80

15:7:80 22:7:80 29:7:80 5:8:80

12:8:80

F.T.-Actuaries All-Share Index

246.16 245.28 255.70 268.95 267.86 267.01 282.14 282.71 284.43 281.76 279.01 280.36

r

0.5075 0.5117 0.5082 0.5128 0.5152 0.5232 0.5070 0.5019 0.5170 0.5076 0.5081 0.5190

a 0.0465 0.0677 0.1077 0.1135 0.1070 0.1642 0.1744 0.1448 0.1320 0.1327 0.1259 0.1098


Despite the strong upward trend in the market, the dividend parameter remains virtually unchanged. The overseas parameter increases sharply from 0.0465 to 0.1744 when the market is rising and then falls back to 0.1098 while the market stabilises at the new level. This suggests that most of the initial buying power was directed towards shares with high overseas exposure and that the strength of the market rise subsequently attracted buying interest, mainly from other investors who missed the initial rise, in shares with low overseas exposure. The rise can therefore be classified as technically weak, in that neither the dividend parameter nor the overseas parameter behaves in a manner which is consistent with an upward primary trend.

19.7. From the middle of August until the end of September, when the market again rises strongly, the overseas parameter increases sharply to 0.1953. This is strong evidence that the primary trend is still downwards. We therefore conclude that the primary trend has been downwards throughout the period from May 1979 to September 1980.

19.8. Although it may appear inconsistent to classify the primary trend as downwards at a time when the F.T.-Actuaries All-Share Index was achieving new all time highs, the market parameters and the market index relate to quite different types of measurement. In the market equilibrium model, shares can be regarded as points in 4-dimensional space. The relative price residual measures the vertical distance between a particular share and the surface that represents the equilibrium position, and the market parameters determine the shape of this surface. The F.T.-Actuaries All-Share Index is a scalar quantity which represents price changes weighted by market capitalisation. In August and September 1980, the very high market capitalisation of oils and electricals caused the relatively small number of upward price changes in these and other companies to outweigh the larger number of price falls in less highly capitalised companies where profits were under severe pressure. As would be expected, the F.T. 30 Industrial Ordinary Share Index, which contains much lower weightings of oils and electricals, remained well below its all time high achieved in May 1979.

20. Fundamental analysis

20.1. All but the largest institutional investors will rely on stockbrokers’ research for much of the background information that has to be taken into account when assessing the future profitability of


individual companies. This research generally takes the form of fundamental analysis which first of all attempts to identify the “ intrinsic value ” of a share and then compares this with the current price to obtain an investment recommendation.

20.2. As a preliminary step towards describing the various ways in which the market equilibrium model can assist in the formulation of investment recommendations, we examine the practical problems that arise in reviewing fundamental analysis of the type contained in stockbrokers’ research.

20.3. In the period between the announcement of Whitbread’s preliminary results for 1979-80 and the annual general meeting in July 1980, there was a very wide divergence of investment view. The 16 stockbrokers’ recommendations held on file at 24th June 1980 consisted of 4 Buys, 3 Buy/holds, 5 Holds, 2 Hold/sells and 2 Sells. This distribution, on the classification scale described in 14.8, gives a standard deviation of 1.35, which is very high.

20.4. Since the four Buy recommendations are very similar in nature, the causes of the differing investment conclusions can be found by studying the following three recommendations, one of the Buys and the two Sells, where the underlying lines of reasoning are set out in detail :

A. Buy recommendation “ Although Whitbread shares have had a strong performance

recently, in anticipation of these good results, the outlook is still favourable, and they are attractive as the leading major in a sector which has growth prospects in the coming year, despite the difficult conditions.

“ Whitbread should continue to make progress in the current year, with a good portfolio of ales and lagers, strong regional identity, and a comparatively efficient distribution network. While interest costs will continue to be a problem in the first half, there should be some compensation in the comparison with a period when the Price Commission’s interference cost over £4m in profits. Thereafter, the disposal of the Chiswell Street development (on which some announcement could be made soon) will build cash reserves and aid cash flow. As Magor comes on stream, later in the year, it should also cease being a drain on cash.

“ The current year will be more difficult than the last, with conditions and competition more stringent. But we would expect


Whitbread to show a reasonable advance in trading profits. The outcome at the pre-tax level will depend on the details of the Chiswell Street arrangements, at present impossible to forecast.”

B. Sell recommendation “ Whitbread’s pre-tax profits of £61.8m (£54.4m in 1978-79), an

increase of 18% after adjustment for an extra week’s trading in the comparable period, confirm its image as an efficient and expan- ding company. Whilst the trading/financial strength of the company means that it will weather the recession better than industry in general and the brewing industry in particular, a prospective fully taxed price-earnings ratio at a 48% premium to the market has more than discounted these positive factors. Our estimate of £68m pre-tax for 1980-81 (£61.8m in 1979-80) is based on assumptions of a dull trading performance relieved by the interest saving on Chiswell Street funds received later in the year. In our view the risk posed by a sharper than expected decline in consumer spending in the latter part of 1980 makes the rating vulnerable.”

C. Sell recommendation “ Pre-tax profits for the year to end-February of £61.8m were

closely in line with our own and most expectations. The figure represented an adjusted year-on-year increase of 18%, and the second-half performance, whilst representing a slow-down in the underlying growth rate, compares favourably with the few other results for the winter half so far to hand.

“ Beer volume so far in 1980-81 is, we gather, at best unchanged on a year earlier, and the group is not optimistic on the industry outlook for the rest of the year. It is also taking a cautious view on margins, given the signs of consumer resistance to present prices, and the need—with inflation running at 20%—for an increase within the next few months.

“ We continue to believe that the industry’s allegedly defensive qualities have been exaggerated, and that by this time next year, moreover, investors’ attention could be turning to those areas likely to benefit from a recovery in world trade and in manufacturing industry. Within the sector, signs of recovery amongst the recent laggards could make life more difficult for the leaders, such as Whitbread.

“ The shares have outperformed the market by 28% over the past 12 months, and now command a massive premium in both price-earnings relative and yield terms, which more than takes


account of the group’s good performance under inflation accounting and the possibility of slightly above-average profits growth this year. We recommend that profits now be realised.”

20.5. All three recommendations identify the same principal positive aspects (i.e. the financial strength of the company, the absence of price controls, increasing profits at a time when many other companies are suffering sharply reduced profitability, and the benefits to accrue from the Chiswell Street development) and the same principal negative aspects (i.e. the expected deterioration in trading conditions and the high rating). However, although each recommendation in isolation appears to use a valid chain of logic, dia- metrically opposed investment conclusions can be obtained.

20.6. Despite the differing investment conclusions, the three assessments are identical in general structure. Assumptions are first of all made regarding the trading background of the company, and numerical values are then obtained for future earnings per share and dividends and other attributes referred to explicitly in the assessment. The third element is the chain of logic employed in the determination of the intrinsic value, and the fourth element is the application of this chain of logic, making use of input material that may be either quantitative or qualitative in nature, to determine the intrinsic value. The final element is the investment conclusion obtained by assessing the extent to which the actual price and the intrinsic value will move into line.

20.7. Since all investment assessments based on fundamental analysis may be resolved into these five elements, we can analyse any assessment on a standardised basis by examining each element in turn. It is clear that the most important of these elements is the chain of logic employed in the determination of the intrinsic value.

20.8. As explained in 6.50, future projections of relative price movements using the market equilibrium model involve comparisons between the present state of a space time co-ordinate system and the expected state of this system at various times in the future. Theo- retical considerations suggest that these comparisons will be highly complex in nature, since different elements can be expected to change in value over quite different time scales.

20.9. In the following section we examine how individual steps in the chain of logic underlying investment assessments relate to the


time scales over which elements of the market equilibrium model change.

21. Time scales 21.1. The behaviour of the relative price residual in relation to

its moving average, and the earnings growth rate, which represent the short-term and long-term components of the market equilibrium model respectively, are discussed in detail in Sections 15 and 16.

21.2. In the medium term, which may involve a time scale of up to two years, variations in the values of the market parameters play a very important part in relative price movements. Changes in the moving average of the relative price residual (which, as discussed in 6.46, reflect changes in the profitability of a company over an economic cycle) have also to be taken into account in medium-term projections.

21.3. Investment assessments such as those in 20.4 generally relate to the expected performance over a period of 6 to 12 months, and accordingly the longer term growth prospects are not discussed in detail. However, many stockbrokers carry out major reviews of companies from time to time where an assessment of the long-term earnings growth forms the main part of the investment conclusion. When earnings growth rates are being calculated or checked for consistency, this type of review is particularly useful.

21.4. Since most investment assessments involve qualitative judgments, they cannot identify short-term turning points in relative performance as accurately as the market equilibrium model, where every element is quantified and the relative price residual is analysed statistically. However, these assessments may contain background information which, while not directly affecting the attribute values used in the model, might cause certain investors to either buy or sell the shares in question and thereby affect the share price. These assessments, together with any press comment, must be monitored closely to supplement the information obtained from the analysis of the relative price residual.

21.5. The three investment assessments in 20.4 illustrate the difficulties involved in analysing assessments of relative performance over the medium term. In all three cases it is agreed that the trading background will deteriorate over the following year. Assessment A argues that the average company will show a far more severe fall in

F


profitability over the same period and that Whitbread’s superior profits performance will merit a continuation of the high rating and hence will cause the share price to outperform the market. Assess- ments B and C argue that the deterioration in Whitbread’s profitability will cause the price-earnings relative to decline to such an extent that this will outweigh the better than average profits performance. Assessment C argues further that the rating differentials between relatively recession-proof companies and manufacturing companies will be reduced significantly once investors anticipate a recovery in the profitability of the latter class of company.

21.6. In the context of the market equilibrium model, this last argument is equivalent to the expectation that the growth rate parameter g will decline significantly during 1981 as a result of changing investor preferences.

21.7. By analysing the logic of the three assessments in this way, it can be seen that the major area of disagreement between assessment A and assessment B is the effect that the reduced (but still above average) profitability will have on the price-earnings relative, and that assessment C is very similar to assessment B but also contains an argument based on a general narrowing of rating differentials throughout the market.

22. Random events

22.1. It is sometimes argued that random events can invalidate many of the conclusions reached by means of fundamental analysis. In the short term, occurrences such as extremes of weather, strikes, plant failures, government controls and tax changes can seriously affect the profitability of individual companies or sectors. In the longer term, changes in general economic, political and social conditions, all essentially unpredictable in nature, can materially affect even the viability of certain companies as well as the differentials in profitability between various companies. We discuss briefly how these random events are allowed for in the market equilibrium model.

22.2. Although the timing of any short-term random event is by definition unpredictable, the general level of exposure to the various risks or opportunities is taken into account in the earnings growth rate. For a reasonably large sample of companies where similar earnings growth rates are used, variations in profitability should be unbiased and hence of no practical significance.


22.3. The method of calculating the earnings growth rate takes explicit account of long-term random events. The turnover and margin components allow for management skills in adapting to new trading environments, while the operational flexibility component assesses the extent to which the profile of the company can change to reflect changes in the profitability of different areas of its operations.

22.4. The “ random walk ” model of stockmarket prices is based on the hypothesis that price changes are the result of new information and that there is no reason to expect that information to be non- random in its appearance. Accordingly, it is argued, period-to-period price changes of a share should be random movements, statistically independent of one another. In the market equilibrium model, the earnings growth rate acts as a filter, with random alterations in the business and economic environment tending to have a positive effect on the earnings and hence the price of a high growth rate share and a corresponding negative effect on the earnings and hence the price of a low growth rate share.

23. Price projections

23.1. Earlier sections show how the market equilibrium model is used to analyse the short-term, medium-term and long-term components of the price formation process. We summarise here the final steps involved in formulating price projections.

23.2. The financial press, economic surveys, company reports and stockbrokers’ research form the main sources of information regarding the trading background of individual companies, economic trends, and possible primary trends in the market. Detailed analysis of all this material leads to the short-term economic background required in the standardisation of estimates of attribute values and also the pattern of primary trends that will be used in estimating future values of the market parameters.

23.3. The methods used to analyse short-term movements are described in detail in Section 15. Since no meaningful estimate of short-term performance is possible if the relative price residual is near its moving average, the only shares that require careful analysis at a particular time are those where the relative price residual is near a control limit. Shares are classified into one of five groups, namely Buy, Buy/hold, Hold, Hold/sell and Sell, on the basis of the expected short-term price performance relative to the market.


23.4. Medium-term projections are generally the most difficult to formulate. Before a final judgment can be made, all relevant investment assessments must be reviewed carefully, with each of the five elements described in 20.6 being analysed separately. In the case of the three assessments in 20.4, this analysis highlights two areas of disagreement, namely the effect that the reduced profitability will have on the price-earnings relative, and the likelihood of a narrowing of rating differentials. The first can be examined using earnings estimates, the behaviour of the price-earnings relative in previous business cycles, and the moving average of the relative price residual, while the second involves a judgment on when an upward primary trend in the market will begin. The medium-term projections relative to the market are again classified into five groups.

23.5. The earnings growth rate provides a direct measure of expected long-term relative performance. These values generally involve a time horizon very much longer than that implicit in fundamental analysis and hence cannot be used in isolation for portfolio selection.

24. Price formation. process

24.1. In the light of the practical experience described above, we can obtain a more detailed description of the price formation process in an ordinary share market.

24.2. We note first of all that the practical experience is consistent in every respect with the hypothesis used in the construction of the model, namely :

“ Each participant, in deciding whether to buy or sell a particular share, will assess various attributes of that share in accordance with the criteria that he, and he alone, considers appropriate.”

24.3. It is clear that the main attributes involved are those that relate to future earnings and dividends and that the criteria that any one participant uses in his assessment procedures are dependent on his expectations of future financial and economic conditions.

24.4. The discussion on fundamental analysis shows that an investment assessment can be regarded as a logical proposition involving five distinct elements and that differing attribute values, differing preferences for individual attributes, or differing assessment periods can lead to differing investment conclusions being derived from the same background information.


24.5. The significant variations in the values of the market parameters show that the preferences underlying the assessment procedures of any participant can change over time.

24.6. We therefore deduce that the price formation process in an ordinary share market is governed by the following axioms :

1. All participants in the market are aware that the future prices of individual shares depend on future earnings and dividends of these shares and on future financial and economic conditions. 2. Each participant, in deciding whether to buy or sell a particular share, will assess various attributes of that share in accordance with the criteria that he, and he alone, considers appropriate. 3. Different participants may hold mutually inconsistent beliefs about the attribute values they use in their assessment procedures. 4. Different participants may have differing preferences for individual attributes. 5. Different participants may base their assessment procedures on differing time scales. 6. The preferences underlying the assessment criteria of any participant may change over time.

24.7. The market equilibrium model is a frame of reference which identifies the importance of the various elements in this price formation process. In addition, it highlights what forecasts and judgments are required for future price projections and supplies the past history as a guide to the formulation of these forecasts and judgments.

24.8. An illuminating description of the importance of different elements of the price formation process in a capital market is given by Keynes (1936) :

“ It might have been supposed that competition between expert professionals, possessing judgment and knowledge beyond that of the average private investor, would correct the vagaries of the ignorant individual left to himself. It happens, however, that the energies and skill of the professional investor and speculator are mainly occupied otherwise. For most of these persons are, in fact, largely concerned, not with making superior long term forecasts of the probable yield of an investment over its whole life, but with fore- seeing changes in the conventional basis of valuation a short time ahead of the general public. They are concerned, not with what an investment is really worth to a man who buys it “ for keeps ”, but with what the market will value it at, under the influence of mass psychology, three months or a year hence. Moreover, this behaviour


is not the outcome of a wrongheaded propensity. It is an inevitable result of an investment market organized along the lines described. For it is not sensible to pay 25 for an investment of which you believe the prospective yield to justify a value of 30, if you also believe that the market will value it at 20, three months hence.

“ Thus the professional investor is forced to concern himself with the anticipation of impending changes, in the news or in the atmos- phere, of the kind by which experience shows that mass psychology of the market is most influenced.”

24.9. In the case of the U.K. ordinary share market, the vast majority of investment assessments relate to what Keynes calls the expected value “ under the influence of mass psychology, three months or a year hence ”. Since the underlying assessment criteria relate mainly to expected short-term trends, the price formation process arising out of this type of investment research is inherently unstable. This instability in both sentiment and price is illustrated very vividly in the recent Whitbread experience described in Section 15.

24.10. In the market equilibrium model, the prospective yield referred to by Keynes is measured by the earnings growth rate, the “ influence of mass psychology ” is reflected in the behaviour of the relative price residual in relation to the control limits, and the distance between the control limits gives a direct measure of the degree of instability in the price formation process.

24.11. If, as argued above, competition between expert professionals causes a higher degree of instability, the distance between the control limits will tend to increase with the research coverage. Also, for companies that attract similar research coverage, a wide divergence of investment recommendation should result in a high degree of instability and hence lead to a larger distance between the control limits.

24.12. This behaviour is in fact confirmed very clearly by the practical results. As an illustration of the order of magnitude of the differentials, the ratios of the difference between the control limit values to their mean value, together with the standard deviation of the classified recommendations (on the scale described in 14.8), are shown below for three chemical companies and four pharmaceutical companies using data as at 24th June 1980.


Control limit ratio

Chemical companies Fisons 0.145 Imperial Chemical Industries 0.112 Rentokil 0.085

Standard deviation

of recommendations

1.11 0.83 0.47

Pharmaceutical companies Beecham Glaxo Reckitt & Colman Smith & Nephew

0.143 0.90 0.165 0.77 0.099 0.84 0.112 0.83

Rentokil attracts relatively little research coverage, while Fisons and Imperial Chemical Industries are comprehensively researched. In the case of the pharmaceutical companies, where the standard deviations are very similar, Beecham and Glaxo attract considerably more research coverage and press comment than the other two companies.

24.13. This behaviour is, of course, the exact opposite of what would occur if the well known “ stockmarket efficiency ” arguments were valid. The Efficient Market Hypothesis states that, in a well- developed capital market, the price of a security will be a good estimate of its intrinsic value as estimated by a large number of professional investors, a high proportion of whom are assumed to be well informed and intelligent. Any substantial disparity between price and intrinsic value would reflect inefficiency, or incorrect pricing. In a well-developed capital market, such inefficiencies, it is argued, will be rare, since alert analysts and fund managers will quickly identify these disparities and, by seeking to exploit them, will cause the temporary inefficiencies to disappear.

24.14. The fallacy in this line of argument is the assumption that the “ intrinsic value ” of a share exists and can be determined accurately by all professional investors. Both the theory and the practical experience described in this paper provide numerous counter- examples which show that the price formation process is much more complex and that the individual elements can be identified and measured only by using the market equilibrium model as a frame of reference.


PART IV: GENERAL PROPERTIES OF THE MODEL

25. Appraisal of the model

25.1. Before describing the general features of the market equilibrium model and in particular the implications for practical investment management, we investigate whether the model provides a satisfactory description of the price structure in the U.K. ordinary share market. We must first be able to define precisely what we mean by “ the model ”.

25.2. Jewell (1980) has suggested the following operational definition of a model :

“ A model is a set of verifiable mathematical relationships or logical procedures which is used to represent observed, measurable real-world phenomena, to communicate alternative hypotheses about the causes of the phenomena and to predict future behavior of the phenomena for the purposes of decision making.”

25.3. This definition is particularly illuminating since it embraces no fewer than five important elements of model-building :

1. The basic aim is to increase our understanding of “ observed, measurable real-world phenomena ”.

2. A working hypothesis is required before model-building can commence.

3. The appropriateness of the mathematical relationships that constitute the model can be verified by experimentation.

4. The expression “ is used to represent ” signifies that the mathematical relationships will not take into account all the complexities of the real-world situation. The model is therefore a simplified logical framework which is substituted for the complex real-world situation.

5. Prediction of the future behaviour of the phenomena is an important area of application.

25.4. Parts II and III of the paper contain a large body of evidence which suggests that the market equilibrium model constructed in Part I provides a satisfactory practical system for the management of portfolios of U.K. ordinary shares. For each of the five headings


listed in 25.3 we now give some of the more important elements of this evidence.

25.5. The model expresses relative price movements in terms of well-defined functions which can be analysed separately. The model thereby significantly increases our understanding of price movements.

25.6. The hypothesis underlying the model is neat and aesthetic in nature and yet can be developed very easily into the mathematical framework for the model. Also, the hypothesis is consistent, virtually by definition, with all traditional methods of ordinary share analysis.

25.7. The stability, absolute magnitude and general behaviour over time of each of the three variable parameters provide very strong evidence that the mathematical relationships used in the model are satisfactory.

25.8. Although the model does not, and could not, incorporate all the complexities of the real-life situation, the explicit allowance for possible non-standard attributes minimises the risk of any material factor being overlooked.

25.9. Since the model can identify and measure features of the price structure that had not been observed before, its quantitative precision and statistical stability represent a significant improvement over all previous methods of analysis. Also, the model highlights what forecasts and judgments are necessary and supplies the past history as a guide to the formulation of these forecasts and judgments. Finally, using the model as a frame of reference, all share assessments from other sources can be expressed as logical propositions which can then be analysed in detail.

25.10. In terms of Jewell’s definition, therefore, the market equilibrium model offers a satisfactory theoretical framework for the management of U.K. ordinary share portfolios.

25.11. This appraisal of the model deals of course with the more obvious tests that it must be expected to pass rather than with the very much more difficult question of whether or not a better model, based on a quite different hypothesis, exists. However, in view of the simplicity of the market equilibrium hypothesis and the discussion in Section 24 of the practicalities of an ordinary share market, it seems


highly unlikely that any model based on an alternative hypothesis could match, let alone improve upon, the quantitative precision of the market equilibrium model.

26. Generalised statements

26.1. In this section we summarise the salient features of the price formation process within an ordinary share market as ex- emplified by the practical experience described in Part III and then state the principal characteristics that the market equilibrium model must possess in order to be able to detect and measure the resulting price structure.

26.2. The price formation process in an ordinary share market is governed by the following axioms :

1. All participants in the market are aware that the future prices of individual shares depend on future earnings and dividends of these shares and on future financial and economic conditions.

2. Each participant, in deciding whether to buy or sell a particular share, will assess various attributes of that share in accordance with the criteria that he, and he alone, considers appropriate.

3. Different participants may hold mutually inconsistent beliefs about the attribute values they use in their assessment procedures.

4. Different participants may have differing preferences for individual attributes.

5. Different participants may base their assessment procedures on differing time scales.

6. The preferences underlying the assessment criteria of any participant may change over time.

26.3. Under this type of price formation process, the price performance of a particular share relative to a market index can be regarded as consisting of three components :

1. A long-term time scale component relating to the earnings and dividends of that share that emerge over time.

2. A medium-term time scale component arising out of changes in the aggregate preferences of participants for particular attributes.

3. A short-term time scale component which can be regarded as random noise arising from four main sources :

(i) price changes are discrete rather than continuous in nature,


(ii) different participants may act on the basis of mutually inconsistent views,

(iii) standardisation of attribute values is very difficult, and (iv) identification and measurement of the market structure is

very difficult.

26.4. To identify the importance of the various elements in the price formation process and to be able to make future projections for the three components described in 26.3 it is necessary to use a market equilibrium model of the following type :

1. The price of P of an ordinary share at a particular time is expressed as

P = P(A1, A2, . . . ; B1, B2, . . . ; C1, C2, . . . ; a1, a2, . . . ; k ; ε )

where A1, A2, . . ., B1, B2, . . . are attributes of the share that relate to current and future earnings and dividends

C1, C2, ... are other attributes of the share

a1, a2, . . . are variable market parameters associated with

A1, A2, . . . k is a scaling factor

and ε is a relative price residual which has zero expected value and is independent of all the other variables.

2. The function P, when expressed in the form P = P (E, D, G) using the definitions of 5.15, satisfies the conditions :

(i) P(kE, kD, G) = k.P(E, D, G)

(ii) >0

(iii) >0

(iv) >0.

3. The detailed construction, including the choice of the variable parameters, is carried out in such a way that the model possesses a very high degree of statistical stability.

4. The values of the variable parameters are determined by minimising the equally weighted sum of the squares of the relative price residuals.

5. Mean Absolute Deviation methods are used to analyse the relative price residuals.

82ERUGIF


27. The dynamics of price movements

27.1. Before discussing the investment management function, we show how price movements relative to the market can be regarded as consisting of five independent components which involve three quite separate time scales. Each component can be studied in isolation using the market equilibrium model.

27.2. Consider first three groups of shares with high, average and low growth rates respectively. It is assumed that each group is homogeneous with respect to other attributes such as overseas exposure, dividend payout ratio and relative price residual. The relative price performance of each of the three groups will be as shown in Figure 28.

This divergence of performance relates to the long-term component associated with the earnings and dividends that emerge over time.

27.3. Consider now three groups of shares which are homogeneous with respect to all attributes except overseas exposure. Depending on whether the initial value of the corresponding market parameter is at a minimum, average or maximum value, the relative price performances will follow one of the patterns shown in Figure 29.

This component is medium-term in nature, and the components relating to changes in the market parameters for dividend payout ratio and earnings growth will behave in a similar manner.


FIGURE 30


27.4. Consider finally three groups that are homogeneous with respect to all attributes except relative price residual. The corresponding relative price performances will be as shown in Figure 30.

This component is short-term in nature, in that most of the divergence in performance occurs within the first three months or so.

27.5. When projections of future price performance are being made, the relative importance of each of these components will vary with the time scale over which the projection is being made. For example, if projections over a very short period such as three months are required, the short-term component corresponding to the relative price residual will clearly be the most important, whereas over five years it will be the long-term earnings growth component that is dominant. Over one year, the short-term component and the three medium-term components together might be of roughly equal importance. This is illustrated in Figure 31.

27.6. In practice there will of course be various complications. Values of attributes for individual companies will involve estimation errors, and future values of market parameters can never be forecast accurately. However, this method of resolving the price performance into independent components ensures that each element of the price formation process is analysed in as systematic a manner as possible.

28. The investment management function

28.1. In this section we describe the main stages involved in the management of ordinary share portfolios using the market equilibrium model.


FIGURE 31

28.2. The first stage is the assembly of all the input information. This will include various financial newspapers, annual reports and accounts of companies, stockbrokers’ research, and also a considerable range of background economic material.

28.3. The next stage involves the calculation, on standardised bases, of the values of earnings per share, earnings growth rates and other attributes included in the model. Once the system is operational, this analysis tends to fall into two main types. There is first of all the routine company analysis required after the announcement of results or the publication of accounts. The second type of analysis involves the periodic review of particular sectors to ensure that all estimates used in the model reflect current trading news from companies in the sector and take account of any expected changes in economic conditions.

28.4. The third stage is the actual running of the model. If changes in attribute values are fed into the system on a daily basis, the main work involved in the weekly computer run is the entering of current share prices. This stage also includes the updating of the various graphs for each company.

28.5. The next stage involves the regular monitoring of each company to relate its recent price behaviour to current news affecting the company. Section 15, which uses Whitbread as an example, shows what types of analysis and judgment are required. The general


aim is to assess whether the share price has deviated significantly from its expected value as calculated by the model, but all relevant factors, and not just those used explicitly in the model, have to be taken into account.

28.6. The projection of the various market parameters represents the fifth stage in the implementation of the model. It should normally be possible to forecast the direction, if not the terminal value, of the next major move in each parameter.

28.7. The sixth stage is to bring together the assessments at stages four and five to arrive at projections of price performance for each share over the short term (three months, say) and the medium term (six to twelve months, say). In practice, it is convenient to classify each of these projections into one of five categories. A measure of the long-term attractiveness is of course already available in the form of the earnings growth rate.

28.8. The seventh and final stage is the selection of the shares to buy or sell for the various portfolios under management. Most of the detailed work has already been carried out by the end of the sixth stage. For each portfolio the time scales over which price projections are required will depend on the general investment objectives, the amount of new money, and the degree of switching activity. The attractiveness of each share over any particular time scale can then be assessed, in accordance with the principles described in 27.5, from the assessments of attractiveness over the short term, medium term and long term.

29. Dividend valuation models

29.1. In many areas of financial analysis it is regarded as axiomatic that the value of an asset is the present value of the stream of payments it is expected to generate. The application of this concept to ordinary share analysis gives rise to dividend valuation models which relate the share price to the present value of future dividends.

29.2. There are two types of dividend valuation model, those which use the same discount rate throughout to calculate an expected value for each share, and those where the discount rate is variable and where the return on a share is calculated as the discount rate at which the value of future dividends is equal to the share price. In the former case the relative attractiveness of a share is found by comparing the actual and expected prices, whereas in the latter case the return is used as the measure of attractiveness.


29.3. To avoid the difficulties that might arise from a high rate of dividend growth in perpetuity, dividends are forecast for, say, 3, 4 or 5 years and are assumed thereafter to increase in line with corporate profits generally.

29.4. The main input of a dividend valuation model is the set of forecast dividend growth rates. Since the ranking of these dividend growth rates could be expected to be similar to the ranking of earnings growth rates used in the market equilibrium model, it might be thought that the two classes of model have similar properties. This, however, is not the case.

29.5. The most important difference is that absolute values of growth rates are required in the dividend valuation model, whereas only the ranking is required in the market equilibrium model. To take account of changing economic and financial conditions, frequent revisions of the dividend growth rates would be required, but these revisions would then seriously weaken the statistical stability of the dividend valuation model and thereby reduce its usefulness as a frame of reference for future projections.

29.6. An obvious disadvantage of the dividend valuation model is that other important attributes such as dividend payout ratio, overseas exposure and balance sheet strength cannot be allowed for explicitly.

29.7. The most serious weakness of the dividend valuation approach is that the market structure is assumed to be logical in compound interest terms. Given the highly complex nature of the price formation process, the use of arbitrary compound interest functions is likely to lead to serious bias in the resulting measure of attractiveness for individual shares. As described in Section 13, any such bias in the market equilibrium model can be identified and, if necessary, corrected. In the case of the dividend valuation model, however, fine- tuning of this nature is impossible in view of the inflexibility of the functional core of the model.

30. The Weaver and Hall model

30.1. There are numerous similarities between the market equilibrium model and the linear regression model described by Weaver and Hall (1967).

G


30.2. The Weaver and Hall model can be stated as follows:

where P is the share price D is the dividend per share E is the current value of earnings per share X2 is the forecast short-term dividend growth rate X3 is the forecast long-term earnings growth rate X4 is the historical earnings variability

and X5 is the historical earnings growth rate. Multiple regression techniques are used to determine a0, a1, ..., a5.

30.3. If, for simplicity, we put X4 = 0 and X2 = X3 = X5 = 1 + g, the model becomes:

This is very nearly identical to the general formulation of the market equilibrium model at the stage of development described in 6.26:

30.4. The only important difference is in the function used to represent the dividend payout ratio. In the market equilibrium model this function tends to r as the payout ratio tends to zero, whereas in the Weaver and Hall model the corresponding function tends to zero. Practical experience of the market equilibrium model shows that r is generally in the range 0·5 to 0·6 and that the relative price residuals are independent of the dividend payout ratio. It would therefore appear that the dividend payout ratio function in the Weaver and Hall model is unsatisfactory in that it contains a bias in favour of shares with a high dividend payout ratio. Comments along similar lines were made by several speakers when the Weaver and Hall paper was discussed at the Institute of Actuaries.

30.5. When the simplifications in 30.3 are discarded, the Weaver and Hall formulation of the growth rate function is

whereas the market equilibrium model formulation is simply


In the market equilibrium model, the historical earnings variability X4 is taken into account in the operational flexibility component of the earnings growth rate G, the historical earnings growth rate X5 does not appear explicitly but will have been taken into account in the assessment of the earnings growth rate G, and the short-term dividend growth rate X2 can be regarded as being taken into account in the earnings base function. The two earnings growth rates, X3 and G, correspond exactly in terms of concept.

30.6. This comparison highlights the principal difference between the two models. Although very similar analytic inputs are used for the growth rate functions, the market equilibrium model has only one associated variable parameter whereas the Weaver and Hall model has four. The statistical stability of the market equilibrium model is accordingly very much better than that of the Weaver and Hall model.

30.7. Although the Weaver and Hall model does not contain an explicit adjustment for overseas exposure or balance sheet strength, these factors can to some extent be taken into account in the values of the various inputs.

30.8. In the application of the Weaver and Hall model, the unadjusted price residuals are used to assess the attractiveness of individual shares over a twelve month period. With the market equilibrium model, on the other hand, the various short-term, medium-term and long-term components of relative price performance are in the first instance projected separately, and, in particular, Mean Absolute Deviation techniques are used to analyse the relative price residuals. By combining these components as discussed in 27.5 the attractiveness over any particular time scale can then be assessed.

31. Comparisons with gilt-edged

31.1. The market equilibrium concept was first used by Clarkson (1978) in connection with the gilt-edged market. We describe below the striking similarities, both in the construction and in the applications, between the gilt-edged and ordinary share models.

31.2. In the gilt-edged market, we assume in the first instance that stocks differ on account of only two attributes, the outstanding term to maturity n and the coupon g. The price of a general stock can then be defined as :

P = P(n, g).


The corresponding definition in the case of ordinary shares is:

P = P(E, D, G).

31.3. The market equilibrium concept is applied to the gilt-edged market by assuming that prices are in equilibrium under the switching action of all participants in the market. This leads to the following definition :

Prices are in equilibrium under switching action if and only if no switch exists from any one stock into any combination of stocks of the same term as that stock which results in :

(i) a higher capital amount at maturity and maintained income, or

(ii) higher income and a maintained capital amount at maturity, or

(iii) higher income and a higher capital amount at maturity.

31.4. From this definition we can deduce the following results : A smooth positive function P of the two independent variables, term n and coupon g, represents a price structure which is in equilibrium under switching action if and only if :

and

where

31.5. The corresponding results for ordinary shares are:

and

31.6. As with the ordinary share model, the detailed construction of the final gilt-edged model involves a delicate balance between goodness of fit and statistical stability. Too flexible a fit may result in an unstable model which has no predictive properties as regards


the relative cheapness or dearness of individual stocks, whereas the use of arbitrary mathematical functions in any part of the model may result in certain important features of the market not being reflected in the price model.

31.7. When the auxiliary functions have been specified, the variable parameters are estimated in exactly the same manner as in the ordinary share model, namely by minimising the equally weighted sum of squares of the relative price residuals.

31.8. Once the model has been fitted to actual prices on a daily basis, the relative price residuals are analysed using the Mean Absolute Deviation techniques described in 14.9. As would be expected, the spread between the upper and lower control limits is very much smaller than for ordinary shares. A typical value for long-dated stocks is 0·5%, compared with an average of around 12% for ordinary shares.

31.9. In the ordinary share market, investors’ preferences as between high and low values of a particular attribute can change over time as financial and economic conditions change. Similarly, in the gilt-edged market, investors’ preferences regarding maturity date and coupon can change over time, and the resulting change in the equilibrium position within the market can be measured. As an example, the ratio of the expected prices of two 20-year stocks, with coupons of 6% and 15% respectively, is shown in Figure 32 for the period March to September 1980.

FIGURE 32


Since the spread between the maximum and minimum values is about 3·5%, it is clear that these medium-term changes in the equilibrium position offer attractive switching opportunities.

31.10. Future prices in the gilt-edged market are determined by future interest rates. Accordingly, the long-term component of price movements relates to future interest rates, whereas in the case of ordinary shares it relates to future earnings and dividends.

31.11. Consider now the generalised statements set out in Section 26. If, under item 1 of the price formation process, we replace “ future earnings and dividends of these shares ” by “ future interest rates ”, the six axioms can be taken as a statement of the price formation process in the gilt-edged market. If we make a similar change in item 1 of the statement of relative price performance and delete items 3 (iii) and 3 (iv) (since the values of term and coupon are known exactly), the statement relates to the relative price performance within the gilt-edged market. Finally, with the substitution of P(n, g) for P(E, D, G), the general description of the market equilibrium model applies equally well to the model used for the gilt-edged market.

32. Systematic relative value analysis

32.1. In view of the strong similarities between the price formation processes in the gilt-edged market and the U.K. ordinary share market, it would appear that the general description of the price formation process given in 26.2 is valid in all capital markets.

32.2. To be of completely general validity, the first axiom can be restated in the form:

1. All participants in the market are aware that the future prices of individual shares depend on the future values of primary attributes of these shares and on future financial and economic conditions.

32.3. Each participant in a capital market acts in what he considers to be a rational manner. However, as discussed in Section 24, much of the trading activity reflects expected price changes over relatively short periods of time rather than expected long-term rates of return. In addition, the relative importance of individual factors in the assessment criteria will tend to change over time in response to general changes in financial and economic conditions.


32.4. As a consequence of this type of behaviour, the price per-

formance of a particular share relative to a market index can be

regarded as consisting of the three general components described

earlier, namely:

1. A long-term time scale component relating to the primary

attributes of that share that emerge over time.

2. A medium-term time scale component arising out of changes in

the aggregate preferences of participants for particular attri-

butes.

3. A short-term time scale component which can be regarded as

random noise arising out of instability in the price formation

process and structural inefficiencies in the market.

32.5. The scientific study of these three components using a

market equilibrium model involves a form of financial analysis which

is quite different from all traditional forms of security analysis.

Since the essential feature of this new approach is that all elements

of the price formation process are studied in as systematic a manner

as possible, an appropriate description of the analytic methods em-

ployed in the application of a market equilibrium model in any

capital market might be “ systematic relative value analysis ”.

33. The role of mathematics

33.1. The main aims of this paper are, firstly, to create, at any

instant of time, order out of apparent disorder within an ordinary

share market and, secondly, to investigate the “ laws of motion ”

exhibited over time by the price system. In this section we review

the mathematical methods that are required at various stages in the

pursuit of these aims.

33.2. In the construction of the model, the first stage is to develop

the market equilibrium concept into such a form that further

mathematical and statistical analysis can be applied. The postulated

regularity of price structure is thereby expressed as follows:

“ As a result of the collective action of all participants in the

market, each of whom uses his own criteria when assessing the

various attributes of individual shares, share prices at any particular

time vary with these attributes in such a way that (where P is

the share price and A is any attribute) is a smooth function of A and

is relatively insensitive to changes in the other attributes.”

33.3. The next stage is the derivation of the main properties of

the price function P. Using a first principles approach which


expresses the general consequences of various aspects of investor behaviour in mathematical terms, a series of ordering relationships is built up. These ordering relationships are then expressed in terms of partial derivatives to obtain the following concise description of the main properties of the price function P :

33.4. The third and most complex stage is the detailed construction of the model. Various steps, such as the choice of attributes and the number of variable parameters, involve a combination of general investment experience and an awareness of the mathematical and statistical consequences of different approaches. Every step has to be completed in such a way as to ensure that the statistical stability of the entire system will be satisfactory.

33.5. At each of these three stages, mathematics is employed in a conceptual form to construct a measurement system capable of detecting the detailed price structure within an ordinary share market.

33.6. The minimisation procedure, which constitutes the first stage in the application of the model, is a computational form of mathematics. In view of the complexity of the price structure and the heavy random noise, the human brain alone cannot determine the equilibrium position from the crude data.

33.7. The second application stage is the analysis of the relative price residual. The Mean Absolute Deviation time series approach involves a higher order of logic than that employed in conventional investment assessments. In particular, the value of the mean absolute deviation for an individual share can be regarded as the “ memory ” of the level of random noise that characterises its price formation process.

33.8. The third stage in the application of the model is the analysis of the market parameters. Again the human brain alone is unable to detect changes in market structure that correspond to changes in the values of these parameters.


33.9. A more general area of application is the formulation of investment policy guidelines. Unless the short-term volatilities of individual shares and the effects of changes in the general market structure can be quantified, the investment manager cannot with any certainty decide whether a particular type of switching activity is likely to be profitable. It is only from the past results of the market equilibrium model that the magnitudes of the various short-, medium- and long-term components of the price formation process can be determined.

33.10. In each of the areas referred to above, mathematics is used in such a way that it complements, summarises, or builds upon the judgment and experience of the investment manager. At no stage does the formalised approach over-ride that judgment and experience.

33.11. The market equilibrium model can therefore be regarded as a formalised description of traditional methods for managing ordinary share portfolios, with mathematics being used as the unifying discipline to bring the system within the bounds of scientific measurement.


PART V: COMPARISONS WITH MODERN PORTFOLIO THEORY

34. General background

34.1. Parts I to IV of the paper can be regarded as a formalised description of traditional methods for managing ordinary share portfolios, with mathematics being used as the unifying discipline. However, stockmarket theorists such as Cootner and Fama have challenged the validity of these traditional methods and have put forward instead the proposition that is now generally known as the Efficient Market Hypothesis. In the absence of any academically acceptable demonstration that the Efficient Market Hypothesis is not a satisfactory description of stockmarket behaviour, the theorists have tended to move away from the question of stock selection and have concentrated instead on the development of various portfolio optimisation methods under the generic name of Modern Portfolio Theory.

34.2. Although Modern Portfolio Theory (MPT) is based on theoretical work which has been carried out almost exclusively in the United States whereas the detailed construction of the market equilibrium model is based on practical experience of the U.K. ordinary share market, it is highly unlikely that the price formation process differs markedly between the two countries. However, the principles underlying the practical application of the market equilibrium model have virtually nothing in common with the MPT methods that are currently being used in the United States.

34.3. Since the two approaches lead to radically different formats for the investment management function, this paper would not be complete without a detailed comparison of the respective merits of the two incompatible systems.

34.4. It is no exaggeration to say that capital market theorists in the U.S. and investment managers of institutional portfolios in the U.K. operate in different scientific worlds. There is no overlap in training, experience or professional literature, and the absence of any such overlap makes it very difficult to carry out an unbiased appraisal of whether MPT or the market equilibrium model offers the better scientific framework for the management of ordinary share portfolios.


Someone trained in MPT will reject the concept that prices reflect fundamental attributes such as earnings and dividends, because the textbooks from which he has learned the theory of his trade expound the view that there is no empirical evidence to support this concept. On the other hand, the investment manager of a U.K. life office or pension fund will regard it as obvious from practical experience that share prices depend on future profits and dividends and will dismiss without detailed investigation the MPT approach which states that statistical measures constructed from past prices can be used to predict future prices.

34.5. Because of the differing conceptual foundations, the conflict between the two approaches cannot be resolved by evaluative procedures. Instead, it is necessary to return to first principles and to examine the whole body of theory on which each is based. In terms of Jewell’s operational definition of a model, we wish to investigate the “ alternative hypotheses about the causes of the phenomena ” to see which approach offers the more satisfactory explanation of the price formation process.

34.6. Parts I to IV of this paper set out the theory of the market equilibrium model and the empirical evidence, based on U.K. experience, that it provides a highly satisfactory framework for the management of ordinary share portfolios. The theory on which MPT is based, however, has been evolving for more than a quarter of a century. This makes it difficult, when making reference to only a very limited number of background papers, to do justice to the high degree of coherence that has been achieved in MPT theory. Fortun- ately there are two excellent books of readings by Cootner (1964) and Lorie and Brealey (1978), each containing extensive commentaries by the editors. The former deals with “ random walk ” investigations, the latter with a very wide range of topics, and between them they contain a selection of MPT literature that is sufficiently comprehensive for the purposes of the present discussion.

34.7. The preface to Lorie and Brealey contains a very clear statement of the three most important concepts of MPT:

“ The essays that we selected were distinguished for their lucid explanation of three pivotal ideas. These ideas are : (1) capital markets in the U.S.A. are highly efficient-meaning that current prices reflect in an unbiased way what is knowable about the companies whose securities are traded ; (2) portfolio management is a different subject from security analysis, (security analysis is


designed to indicate the likely or possible returns from investing in particular securities, whereas portfolio management has to do with the selection and surveillance of a bundle of securities that match the aspirations, fortitude and tax status of the beneficiary) ; and (3) the relative prices of securities are determined by the expected return to the investor and also by the uncertainty about the return. Modern theory has taught us something about the way in which the market determines the premium for enduring uncertainty or risk.”

34.8. These three concepts, namely efficiency, portfolio theory and capital asset pricing, are discussed in turn in the following three sections and contrasted with the experience of the market equilibrium model.

35. The Efficient Market Hypothesis

35.1. In his introduction, Cootner explains the background to the “ random walk ” investigations and highlights the importance of being able to understand the price formation process :

“ Wherever there are valuable commodities to be traded, there are incentives to develop markets to organize that trade more efficiently. In modern complex societies the securities markets are usually among the best organized and virtually always the largest in terms of value of sales. The prices of such securities are typically very sensitive, responsive to all events, both real and imagined, that cast light into the murky future. The subject of this book is the attempts by skilled statisticians and economic theorists to probe into this process of price formation.”

35.2. Most of these investigations represented attempts to answer the following question in the context of U.S. markets :

“ Is there any evidence that any historical data about the price of a stock will enable us to improve our forecasts of the future profit from holding this stock?”

35.3. When nearly all of these statistical investigations concluded that there was no such evidence, the idea of stockmarket efficiency was advanced as a possible explanation for the negative results. Cootner gives a very clear description of a “ perfect market ” that would behave in accordance with this empirical evidence :

“ While individual buyers or sellers may act in ignorance, taken as a whole, the prices set in the marketplace thoroughly reflect the


best evaluation of currently available knowledge. If any substantial group of buyers thought prices were too low, their buying would force up the prices. The reverse would be true for sellers. Except for appreciation due to earnings retention, the conditional expectation of tomorrow’s price, given today’s price, is today’s price.

“ In such a world, the only price changes that would occur are those which result from new information. Since there is no reason to expect that information to be non-random in its appearance, the period-to-period price changes of a stock should be random movements, statistically independent of one another. The level of stock prices will, under these conditions, describe what statisticians call a random walk, and physicists call Brownian motion.”

35.4. Despite these negative results, Cootner does not believe that a random walk model is appropriate and suggests instead a simplified model of the stockmarket that would explain the departures from randomness. He describes two classes of investor, non-professionals, who do not possess a high degree of knowledgeability, and professionals, who are very knowledgeable and can accurately assess the expected price of a stock. Professionals, Cootner argues, can make profits from observing the random walk of the stockmarket prices produced by the non-professionals until the price wanders sufficiently far from the expected price to warrant the prospect of an adequate return. When the price has deviated enough from the expected price, future surprises should force prices towards their mean more often than not. Using statistical tests based on transition matrices, Cootner concludes that there is in fact a mild tendency for price changes to move towards the mean.

35.5. Other researchers also believed that the random walk model was not wholly appropriate, but none of the statistical tests could be adapted to identify undervalued stocks. This led to a reformulation of the efficiency hypothesis to take into account the information set against which the hypothesis was tested.

35.6. The “ Efficient Market Hypothesis ” is now stated in three forms, corresponding to different information sets. A convenient statement is provided by Fama (1970) in the introduction to a comprehensive review paper on the subject :

“ This paper reviews the theoretical and empirical literature on the efficient markets model. After a discussion of the theory, empirical


work concerned with the adjustment of security prices to three relevant information subsets is considered. First, weak form tests, in which the information set is just historical prices, are discussed. Then semi-strong form tests, in which the concern is whether prices efficiently adjust to other information that is obviously publicly available (e.g. announcements of annual earnings, stock splits, etc.) are considered. Finally strong form tests concerned with whether given investors or groups have monopolistic access to any information relevant for price formation are reviewed. We shall conclude that, with but a few exceptions, the efficient markets model stands up well.”

35.7. One of the most thorough tests of the semi-strong form is the investigation by Fama, Fisher, Jensen and Roll (1969). The introduction explains the reasons for the investigation and describes the nature of the test :

“ There is an impressive body of empirical evidence which indicates that successive price changes in individual common stocks are very nearly independent. Recent papers by Mandelbrot and Samuelson show rigorously that independence of successive price changes is consistent with an ‘ efficient ’ market, i.e. a market that adjusts rapidly to new information.

“ It is important to note, however, that in the empirical work to date the usual procedure has been to infer market efficiency from the observed independence of successive price changes. There has been very little actual testing of the speed of adjustment of prices to specific kinds of new information. The prime concern of this paper is to examine the process by which common stock prices adjust to the information (if any) that is implicit in a stock split.”

35.8. The authors conclude that there is no evidence of inefficiency: “ In sum, in the past stock splits have very often been associated with substantial dividend increases. The evidence indicates that the market realizes this and uses the announcement of a split to re-evaluate the stream of expected income from the shares. Moreover, the evidence indicates that on the average the market’s judgments concerning the information implications of a split are fully reflected in the price of a share at least by the end of the split month but most probably almost immediately after the announcement date. Thus the results of the study lend considerable support to the conclusion that the stock market is ‘ efficient ’ in the sense that stock prices adjust very rapidly to new information.”


35.9. The MPT literature contains very few comprehensive tests of the strong form of the Efficient Market Hypothesis. Reference is usually made, however, to the paper by Jensen (1968) in which he derives the following conclusions :

“ The evidence on mutual fund performance discussed above indicates not only that these 115 mutual funds were on average not able to predict security prices well enough to outperform a buy- the-market-and-hold policy, but also that there is very little evidence that any individual fund was able to do significantly better than that which we expected from mere random chance. It is also important to note that these conclusions hold even when we measure the fund returns gross of management expenses (that is assume their book-keeping, research, and other expenses except brokerage commissions were obtained free). Thus on average the funds apparently were not quite successful enough in their trading activities to recoup even their brokerage expenses.”

35.10. It is important to realise that the measure of portfolio performance used in this investigation is based on theoretical results derived from capital asset pricing models. In order to test one part of MPT, Jensen constructs a measurement system that is based on another part of MPT and uses it to assess the investment performance of practitioners whose beliefs regarding the price formation process are incompatible with MPT.

35.11. As stated in 34.5, the conflict between MPT and traditional methods of security analysis cannot be resolved by evaluative procedures that depend on the conceptual foundations of one or other approach. Jensen’s conclusions depend wholly on his use of one part of MPT and cannot therefore be regarded as independent evidence of the validity of the strong form of the Efficient Market Hypothesis.

35.12. Despite a considerable amount of further research in recent years, the conclusion reached more than ten years ago by Fama (1970) is still generally accepted amongst the proponents of MPT :

“ In short, the evidence in support of the efficient markets model is extensive, and (somewhat uniquely in economics) contradictory evidence is sparse. Nevertheless, we certainly do not want to leave the impression that all issues are closed. The old saw, ' much remains to be done ', is certainly relevant here. Indeed, as is often the case in successful scientific research, now that we


know where we have been in the past, we are able to pose and, hopefully, to answer an even more interesting set of questions for the future. In this case the most pressing field of future endeavor is the development and testing of models of market equilibrium under uncertainty. When the processes generating equilibrium expected returns are better understood, we will have a better framework for more sophisticated tests of market efficiency.”

35.13. Although most tests of market efficiency have been carried out in the U.S., it has been suggested, mainly in academic circles, that the U.K. ordinary share market is so efficient that MPT techniques should be used in the management of institutional portfolios.

35.14. The theory and results of Parts I to IV of this paper are of course totally incompatible with the Efficient Market Hypothesis.

35.15. Since the market equilibrium model has been constructed with the requirements of institutional investors in mind, the information set for the purposes of testing for efficiency can be defined as “ information readily available to an institutional investor at reasonable cost ”. This obviously includes :

1. Financial newspapers. 2. Annual reports and accounts of companies. 3. Stockbrokers’ research on companies and sectors.

35.16. The question to be answered now becomes :

“ Is there any evidence that we can construct a set of decision rules which can be applied by investment analysts to information readily available to an institutional investor at reasonable cost to predict the future relative price performance of a large number of shares ? ”

35.17. In Section 27 many of the earlier results of the paper are summarised to show how the market equilibrium model can be used to select pairs of groups of shares which will exhibit divergent price performance over long-, medium- and short-term time scales respectively. Since the information sets employed are of the type described in 35.15 and the detailed analytic methods used are an extension of traditional methods of security analysis, Parts I to IV of this paper should be acceptable to most people involved in investment work as evidence that the Efficient Market Hypothesis


is totally inappropriate in the context of the U.K. market and that none of the investment strategies based on the Efficient Market Hypothesis should have any place in the management of portfolios of U.K. ordinary shares.

35.18. Some theorists, however, may attempt to dismiss this evidence for a variety of reasons and to claim that the general validity of the Efficient Market Hypothesis is not impaired. It might be argued, for instance, that the results obtained are heavily dependent on the investment experience of the user of the model.

35.19. It is obvious that the long-term projections and the medium- term projections require considerable investment experience in addition to the theoretical framework of the market equilibrium model before measurable success can be expected. With the short- term projections, however, the effects of any estimation errors are much less serious, and accordingly any attempts to carry out independent tests of the Efficient Market Hypothesis using the methods of this paper should concentrate initially on the relative price residuals.

35.20. Two “ catch-all ” arguments have been used by theorists to defend the Efficient Market Hypothesis when it has come under attack, namely :

1. Although it may be possible to find decision rules which have been successful in the past, there is no guarantee that they will continue to be successful in future. 2. Even if there is any validity in a particular method, once it becomes generally known and widely used it would destroy the kind of anomalies it claims to identify and it would thereafter cease to be of any use.

35.21. It should be clear from Part III of the paper that simplistic arguments such as these can be ignored since they display a lack of understanding of the realities of capital markets and of the day-today decisions with which professional investors are confronted. There will always be disagreement between investors about long- term economic prospects, about medium-term factors such as general stockmarket levels, and about short-term factors which affect individual companies. In such an environment, a model which highlights what forecasts and judgments have to be made, and which supplies the past history as a guide to the formulation of these forecasts and judgments, will always be a valuable practical tool.

H


36. Portfolio theory

36.1. The portfolio analysis techniques that form an important part of MPT are based on Markowitz’ definition of an efficient portfolio and on the concept that an investor can select the particular efficient portfolio that is best suited to his preferences.

36.2. Markowitz (1952) first of all assumes that it is possible to obtain the expected value of the return on each security in the market and the associated covariances and then defines a portfolio of securities as efficient if no other one gives either :

(i) a higher expected return and the same variance of return, or (ii) a lower variance of return and the same expected return.

36.3. This definition is illustrated in Figure 33.

FIGURE 33

If E and V denote respectively the expected return and the variance of return, and the attainable E-V combinations are represented by the convex region, then the efficient sets are those lying along the boundary ABC.

36.4. The statistical formulation is as follows :

Let Ri be the return from security i E(Ri) be the expected value of Ri Cij be the covariance between Ri and Rj

and Xi be the proportion of the portfolio in security i.


Then

and

where

36.5. In portfolio theory it is axiomatic that risk, as perceived by

an investor, can be regarded as being equivalent to the variance of

the return on the portfolio. Since the associated covariance function

Cij bears no direct resemblance to any of the functions used in

Parts I to IV of the paper, we examine some of the general properties

of Cij by means of simplified examples.

36.6. Consider the case where the fund manager of a pension

scheme is considering a switch from a low earnings growth share A

into a high earnings growth share B because he believes that high

growth shares, as a class, will outperform low growth shares in the

near future. Based on current prices of 100 for both shares, he

describes the expected prices PA and PB at some future date in terms

of the following probabilities:

Probability 0·05 0·05 0·15 0·55 0·2

PA 110 105 100 95 90

PB 90 95 100 105 110

The expected values of PA and PB are 96 and 104 respectively, so that

E(RA) = – 0·04 and E(RB) = 0·04. The covariance CAB is calcu-

lated as follows:

Probability (RA + 0·04) (RB – 0·04) product

0·05 x (0·1 + 0·04) x (– 0·1 – 0·04) = – 0·00098

0·05 x (0·05 + 0·04) x (– 0·05 – 0·04) = – 0·000405

0·15 x (0 + 0·04) x (0 – 0·04) = – 0·00024

0·55 x (– 0·05 + 0·04) x (0·05 – 0·04) = – 0·000055

0·2 x (– 0·1 + 0·04) x (0·1 – 0·04) = – 0·00072

CAB = – 0·0024

36.7. If a second fund manager holds similar beliefs but expresses

his expectations in the following terms:

Probability 0·3 0·6 0·1

PA 100 95 90

PB 100 105 110

he obtains the same expected returns but calculates the covariance

as follows:


Probability (RA + 0·04) (RB – 0·04) product 0·3 x (0 + 0·04) x (0 – 0·04) = – 0·00048

0·6 x (– 0·05 + 0·04) x (0·05 – 0·04) = – 0·00006 0·1 x (– 0·1 + 0·04) x (0·1 – 0·04) = – 0·00036

CAB = – 0·0009

36.8. Suppose finally that a trustee of the pension scheme which

employs the first fund manager holds a contrary view and expresses

the relevant probabilities as follows:

Probability 0·2 0·55 0·15 0·05 0·05

PA 110 105 100 95 90

PB 90 95 100 105 110

36.9. Although the two fund managers’ beliefs are for all practical

purposes identical, the slight differences in the probabilities used to

describe these beliefs give rise to very significantly different values

for the covariance CAB. The efficient portfolios constructed by the

two fund managers may therefore have markedly different pro-

portions in shares A and B purely as a result of slight differences in

probabilistic inputs that have in themselves no practical significance.

36.10. When more than one attribute is involved, the associated

probability distributions are far too complex to be specified explicitly,

even for a limited number of shares. The pure Markowitz approach,

as well as being sensitive to minor variations in the input probabilities,

is therefore totally impractical for institutional portfolios.

36.11. An even more serious problem, however, arises from the

concept that risk can be regarded as being equivalent to the variance

of the portfolio return. In the above example, the first fund manager

may report to his trustee that the portfolio has a holding in share B

but no holding in share A, and that the portfolio is efficient in the

Markowitz sense and therefore takes appropriate account of risk.

But the trustee, being a prudent man and believing that share A is

more attractive than share B, asks the fund manager if there is not

a risk of poorer performance than would otherwise be the case as a

result of the decision to hold share B rather than share A.

36.12. The trustee’s perception of risk relates to the likelihood of

inferior performance; the fund manager derives his measures of risk

from the probability distributions underlying his investment strategy,


without regard to whether, in all the circumstances, this investment

strategy is appropriate.

36.13. The fund manager’s conception of risk is internal in nature,

being dependent only on the probability distributions that describe

his investment strategy. The trustee’s perception of risk, on the

other hand, is external in nature since it relates to the appropriateness

or otherwise of the investment strategy.

36.14. A fund manager will, in general, base his investment

actions on those economic and stockmarket scenarios that he thinks

are most likely to occur. The better he is at identifying the most

likely scenarios, the more successful will be his investment per-

formance. It is therefore reasonable to regard risk as being related

to the fund manager’s understanding of the price formation process,

and this is in accordance with the trustee’s perception of risk. The

Markowitz definition of risk, however, takes no account of the fund

manager’s understanding of the price formation process and therefore

cannot be accepted as a suitable yardstick for the purposes of a

trustee or other person whose responsibilities involve the review of

the fund manager’s activities.

36.15. Most practical applications of the Markowitz approach are

based on the diagonal model suggested by Sharpe (1963). In this

model, the return R on a security is expressed as

R = A + BI + C

where A and B are parameters, C is a random variable with an

expected value of zero, and I is the level of some index.

36.16. If we consider small changes in earnings per share, in the

three market parameters and in the relative price residual, the

corresponding return using the market equilibrium model can be

expressed, in an obvious notation, as

If second-order small quantities are ignored, this gives

There is a very strong similarity between this expression and the

expression used in the Sharpe diagonal model.


36.17. Sharpe, however, tackles the implementation of the diagonal

model by dividing the risk of a security into two components, market

risk and non-market risk, thereby weakening the link between the

Sharpe model and any diagonal model based on traditional methods

of security analysis. This link is then broken completely when

Sharpe considers how the required risk measures can be obtained:

“The problem does not differ in any significant way from that of

estimating, say, future earnings per share. Careful analysis of past

data, expert knowledge about the firm and the industry, inter-

views with management, study of investor psychology-all these

ingredients can be brought to bear on the problem. But this may

provide accuracy that costs more than it is worth. It is important

to remember that the values obtained for individual securities are

to be combined to calculate comparable values for a portfolio.

And the accuracy of a portfolio estimate is far more important than

the accuracy of the individual security estimates.

“A number of rather inaccurate estimates for securities may

combine to form an exceptionally accurate estimate for a portfolio,

thanks to the law of large numbers. The estimate for one security

may be too high, and another too low, with the result that the

average is ‘just right’. To borrow the statistician’s jargon: if

predictions about securities are subject to error but unbiased,

predictions about fairly well-diversified portfolios may be quite

accurate.

“This suggests the possibility of using past data exclusively to

obtain estimates of the relevant attributes of securities.”

36.18. The practical method suggested by Sharpe is to plot the

monthly price changes of a security against the monthly price

changes in the market index; the gradient of the straight line fitted

to this diagram gives B, the market risk of the security, and the

goodness of fit of the straight line gives C, the non-market risk.

36.19. Starting from the Markowitz definition of risk, which, as

discussed above, is unsatisfactory in concept, Sharpe assumes—

without proof-that the stability of some statistical measure for an

entire portfolio is more important than an accurate understanding

of the price formation process for individual securities.

36.20. In the market equilibrium model formulation of the

diagonal model, the term E relates to earnings growth, and the

term relates to short-term price movements relative to an expected

price. The examples of the switches carried out in 1975, described in


16.7, and the recent price history of Whitbread, described in Section

15, provide practical examples of how these two aspects of investment

management can be analysed in the context of the market equilibrium

model. These types of analysis cannot be carried out within the

framework of MPT.

36.21. The term and the corresponding term

BI in the Sharpe diagonal model relate to changes in the general

market structure. As described in Section 18, the three market

structure parameters do not in general move in phase with a market

index. If only one such component is used, as is the case with the

Sharpe diagonal model, many important features of the market

structure will remain undetected.

36.22. Applications of the Sharpe model depend on the assumption

that the values of B are stable from one period to the next. The recent experience described in Section 18 shows that even this

assumption is incorrect and can lead to unsatisfactory investment

performance.

36.23. It is difficult to avoid the conclusion that portfolio theory,

far from being a useful investment tool, tends to distract attention

from many of the difficult forecasts and judgments that should

constitute an important part of the investment manager’s responsi-

bilities.

37. The capital asset pricing model

37.1. The capital asset pricing model, which states that prices are

determined by the expected return to the investor and by the

uncertainty of the return, is perhaps the most important single part

of MPT.

37.2. The model is generally stated in the following form:

where E(Rj) is the expected one-period return (capital gain plus

dividend divided by initial price) for security j

RF is the one-period risk free interest rate

Bj is a measure of the systematic risk of security j

and E(RM) is the expected one-period return on the market

portfolio.


37.3. This is equivalent to saying that the expected return on any

asset is equal to the risk free rate plus a risk premium given by the

product of the systematic risk of the asset and the difference between

the expected return on the market portfolio and the risk free rate.

37.4. This relationship can be represented in diagrammatic form

as shown in Figure 34.

FIGURE 34

According to the theory, an investor can choose any point along

the capital market line in accordance with the degree of risk that he

is prepared to accept.

37.5. It is important to realise that the capital asset pricing model

is not based on practical experience of investment management or

even on empirical measurements carried out in any particular market.

Instead, it is a purely theoretical attempt to describe the manner in

which risk (as defined by the theorists) and investor preferences

interact to determine the observed prices of capital assets.

37.6. The model can be regarded as a theoretical justification of

Sharpe’s diagonal model. This, as described in 36.15, asserts that

the return on a security is

R = A + BI + C

where A and B are parameters, C is a random variable with an expected

value of zero and I is the level of some index.


37.7. Sharpe defines C as “non-market risk”, and assumes that,

by diversification, the non-market risk of a portfolio can be reduced

by a considerable extent. Accordingly, C is ignored for the purposes

of the capital asset pricing model, so that the return on a security

becomes a linear function of B, its market risk.

37.8. A convenient definition of the assumptions behind the

capital asset pricing model is given by Jensen (1968):

1. All investors are averse to risk, and are single period expected

utility of wealth maximisers.

2. All investors have identical decision horizons and homogeneous

expectations regarding investment opportunities.

3. All investors are able to choose among portfolios solely on the

basis of expected returns and variances of returns.

37.9. In deriving the model from these assumptions, it is taken as

axiomatic that there exist riskless assets which, by definition, have

a return equal to the risk free interest rate. This concept of the risk

free interest rate is a special case of one of the fundamental beliefs of

financial theory, namely that prices in capital markets will adjust

until an equilibrium expected rate of return exists. In 35.12, for

example, Fama refers to “the processes generating equilibrium

expected returns ” rather than to the price formation process.

37.10. The concept of an equilibrium expected rate of return is discussed by Miller and Modigliani (1961) in a paper which investi-

gates the price formation process for ordinary shares. In describing

first of all the price formation process under conditions of certainty,

Miller and Modigliani make the following assumptions:

1. In “ perfect capital markets”, no buyer or seller (or issuer) of

securities is large enough for his transactions to have an

appreciable impact on the then ruling price. All traders have equal and costless access to information about the ruling price

and about all other relevant characteristics of shares. No

transaction costs are incurred when securities are bought, sold

or issued, and there are no tax differentials either between

distributed and undistributed profits or between dividends and

capital gains.

2. “Rational behaviour” means that investors always prefer more

wealth to less and are indifferent as to whether a given increment

to their wealth takes the form of cash payments or an increase

in the market value of their holdings of shares.


3. “Perfect certainty” implies complete assurance on the part of

every investor as to the future investment programme and the

future profits of every corporation.

37.11. The authors then deduce that:

“Under these assumptions the valuation of all shares would be

governed by the following fundamental principle: the price of

each share must be such that the rate of return on every share will

be the same throughout the market over any given interval of

time.”

37.12. The authors show that, on their assumptions, the dividend

payout ratio of a particular company has no effect on its share price;

this result is known as the dividend irrelevance proposition.

37.13. These results relate to conditions of perfect certainty. The

authors then relax these conditions and deduce that, even under

uncertainty, dividend payout ratio is not a determinant of market

value. After discussing tax considerations, they conclude:

“Finally, we may note that since the tax differential in favor of

capital gains is undoubtedly the major systematic imperfection in

the market, one clearly cannot invoke ‘imperfections’ to account

for the difference between our irrelevance proposition and the

standard view as to the role of dividend policy found in the litera-

ture of finance. For the standard view is not that low-payout

companies command a premium; but that, in general, they will

sell at a discount! If such indeed were the case-and we, at least,

are not prepared to concede that this has been established-then

the analysis presented in this paper suggests there would be only

one way to account for it; namely, as a result of systematic

irrationality on the part of the investing public.”

37.14. Returning now to the market equilibrium model, we

describe the implications of dividend preference in 5.3 as follows:

“Some investors are primarily concerned with dividend income

while others are relatively insensitive to immediate income and

regard their holdings more as a store of capital value than as a

source of income.”

37.15. These differing objectives are allowed for in the construction

of the model by introducing an indifference ratio which can assume

any value between two limiting values, the first relating to investors


who are concerned only with dividends and the second relating to

investors who are concerned only with earnings. In the final model,

this indifference ratio is measured by the dividend parameter r, and

the values corresponding to these two limiting cases are 0 and 1

respectively. As was expected before even the pilot tests were

carried out, values of around 0·6 are obtained for the U.K. ordinary

share market. Also, the dividend parameter changes over time in a

manner that is consistent with changes in investor preferences in the

aggregate.

37.16. Miller and Modigliani, on the other hand, assume that

investors are concerned only with earnings and define “investor

rationality” accordingly. Given the U.K. experience as measured

directly by the market equilibrium model and what is referred to as

the “standard view” in the U.S., it is not unreasonable to conclude

that Miller and Modigliani fail to allow correctly for investor pre-

ferences, that their dividend irrelevance proposition is not in

accordance with observed market structure, and that there is no

theoretical justification for the concept of the equilibrium rate of

return.

37.17. A similar interaction between investor preferences and

rates of return can be demonstrated in the gilt-edged market.

Before the application of the market equilibrium model to the gilt-

edged market, all models, including those based on an expectations

hypothesis, satisfied the condition

However, once it is recognised that different investors need not have

a uniform effect throughout all regions of the market, the market

equilibrium model approach shows that the strongest general

condition that can be deduced is

and that, in practice, strict inequality holds. For a fixed value of

term, there is in general no income tax rate at which the net re-

demption yields of stocks with differing coupon are equal.

37.18. The capital asset pricing model is based on the concept of

an equilibrium interest rate and on the three assumptions listed in

37.8. The above discussion suggests that neither the concept of an


equilibrium interest rate nor the first assumption (which involves a

definition of investor behaviour that is similar to the one used by

Miller and Modigliani) has a sound theoretical foundation. Using

the price model in the form P = P(E, D, G), the Miller and Modigliani

assumptions give = 0 whereas the U.K. experience shows that

Also, the second assumption is unrealistic in the light of

practical experience and is not required in the case of the market

equilibrium model. Finally, there is no empirical evidence to suggest

that investors base their actions on variances of portfolio returns.

37.19. In conclusion, the capital asset pricing model has a

somewhat insecure theoretical foundation and does not appear to

add anything to our understanding of the price formation process

within a capital market.

38. MPT applications

38.1. The general approach in the application of MPT is to control

portfolio “risk” (in the MPT sense) in such a way that proper

account is taken of the preferences of the investor and of his stock

selection ability.

38.2. Black (1976), for example, introduces a paper on investment

policy in the following manner:

“Investment policy, as I use the term, means such things as the

choice of a risk level for an investor’s portfolio, the division of the

portfolio among broad classes of investments such as common

stocks and bonds, the policy of taking capital losses whenever

possible and letting gains go unrealized, or the reverse policy of

taking capital gains and letting losses go unrealized. It means

such things as the decision on how widely diversified a portfolio

is to be or, conversely, how much it is to be concentrated in particu-

lar investments considered especially attractive.

“I will assume, in this article, that the stock and bond markets

are extremely efficient, so that the expected gains from trading

stocks and bonds or from attempting to outguess movements in

the market as a whole are exceeded by the total costs of doing the

required analysis and executing the resulting transactions. This

assumption is consistent with most of the empirical evidence on

the performance of professionally managed portfolios. Only a

very few portfolios give evidence of consistently superior per-

formance.”


38.3. Risk analysis based on MPT can be regarded as the imple-

mentation of the following principles:

1. Risk is equivalent to the variance of return.

2. The risk of a security can be split into two components,

systematic (or market-related) risk and residual (or non-market)

risk.

3. Systematic risk is measured as the gradient of the straight line

fitted to a scatter diagram of price change against market

change.

4. Residual risk is measured by the goodness of fit of the straight

line described in 3.

5. Systematic risk is utilised by increasing its value ahead of a rising market and vice versa.

6. Residual risk is to be avoided and is reduced in practice by

diversification.

38.4. The following comments can be made regarding these

principles:

1. As discussed in 36.14, the MPT definition of risk is unsatis-

factory.

2. This is merely a preliminary step before implementation of the

risk concept, but it breaks the link with earnings and dividends.

3. A statistical measure derived from past prices is easy to

calculate, but current and future profits and dividends are

likely to be of much greater importance in the price formation

process; the measure is also medium-term, rather than long-

term, in nature.

4. The likely direction of price variation, although far more

important than the statistical variability, cannot be measured.

5. This application depends on weak evidence of continuity of

experience; recent U.K. experience provides a strong counter-

example.

6. No attempt can be made to identify over-priced or under-priced

securities.

38.5. As with the concepts of portfolio theory, it is difficult to

avoid the conclusion that risk analysis is likely to distract the investor from those areas of analysis that will assist him in his

pursuit of superior performance. In terms of the three components

of relative price movement described in 26.3, the long-term com-

ponent relating to future earnings and dividends is not allowed for

explicitly under MPT, the medium-term changes in market structure


cannot be identified let alone measured, and finally no attempt is

made to profit from short-term random noise that arises out of

structural inefficiencies in the market.

38.6. As stated in 34.3, MPT and the market equilibrium model

approach lead to radically different formats for the investment

management function. In view of the unsatisfactory nature of the MPT definition of risk and the serious deficiencies of the MPT

approach as regards identifying and measuring salient features of the

price structure, we conclude that none of the concepts of MPT need

be incorporated in applications of the market equilibrium model.

39. Divergence of development

39.1. To conclude the discussion on the relative merits of MPT

and the market equilibrium model, we investigate the very interesting

question of why the two approaches, both of which claim to be

scientific and systematic in nature, can have so little in common.

39.2. In any new area of scientific work, there are many possible

starting points as regards the collection of empirical data. The state of knowledge in a relatively new field will therefore be strongly

influenced by those experiments that happen, either by chance or as

a result of prior experience on the part of the investigators, to be

carried out first.

39.3. In the case of MPT, investigations of “random walk”

phenomena dominated the early work and led directly to the formu-

lation of the Efficient Market Hypothesis. The Markowitz definition of an efficient portfolio, which lay dormant for some years until the

diagonal model was proposed by Sharpe, also played an important

part in much of the early work.

39.4. Then came one single development which irrevocably changed

the direction of all future empirical work. Sharpe, in the passage quoted in 36.17, suggested that practical measures of risk should be calculated from past prices rather than from future projections

of fundamental attributes such as earnings and dividends. Influenced

presumably by the increasing body of empirical work that supported

the concept of stockmarket efficiency, Sharpe believed that the

stability of statistical measures for an entire portfolio was more

important than an accurate understanding of the price formation

process for individual securities.


39.5. The next important stage in the development of MPT was

the formulation of the capital asset pricing model as a theoretical

justification of the diagonal model. As discussed in Section 37, two

of the assumptions used in the construction of the capital asset

pricing model are that the market structure is consistent with the

Miller and Modigliani dividend irrelevance proposition and that

investors choose between portfolios on the basis of the variance of

the portfolio return.

39.6. The Efficient Market Hypothesis, the Markowitz concept of

risk, the diagonal model, and the capital asset pricing model, having

first made their appearance in various academic journals, gradually

became the central theories in the textbooks from which later

entrants to the field learned their trade.

39.7. By this stage in its development, MPT had diverged

irrevocably from traditional methods of security analysis. Students

and practitioners of MPT rarely had any experience of fundamental

analysis and in fact became known as quantitative analysts, since

the tools of their trade were quantitative measures derived solely

from past prices.

39.8. In many areas of science, progress within the existing

framework of accepted theory can result from the identification of

apparent disorder, the exploration of this disorder using refined

methods of measurement, and ultimately the construction of a new

and more general theory which encompasses the features that were

previously regarded as displaying disorder.

39.9. The development of the market equilibrium model for the

gilt-edged market occurred in this manner. All previous models

satisfied the condition

but empirical tests carried out in 1971 showed that

A more general model satisfying this condition was then constructed.

39.10. This type of progress cannot occur within MPT, where

prices are represented by statistical measures rather than by an


explicit model. It is impossible, for instance, to detect apparent

disorder using the capital asset pricing model, since this model

represents a particularly inefficient measurement system.

39.11. In the absence of sufficiently powerful observational

techniques, there can be no awareness of apparent disorder, no

exploration of possible anomalies, and no impetus from within MPT

to re-examine the theoretical foundations. Any such impetus must

come from a re-appraisal of MPT in the light of some competing

theory.

39.12. The theory and application of the market equilibrium model suggest that the theoretical foundations of MPT are unsatisfactory

in the following respects:

1. The Efficient Market Hypothesis is invalid; differentials in

price performance can be forecast over three quite distinct time

scales.

2. The Markowitz definition of risk is internal in nature and

therefore unsuitable for review purposes.

3. Risk assessments based on past prices alone are unrealistic.

4. U.K. experience shows that the Miller and Modigliani dividend

irrelevance proposition does not give an accurate description of

market structure.

5. There is no empirical evidence to show that investors select

portfolios on the basis of the variance of the portfolio return.

39.13. These arguments are essentially conceptual in nature and

may of course be disputed by proponents of MPT. At a more

practical level, the main argument in favour of the market equilib-

rium model is that the quantitative precision is markedly superior to

that attainable using MPT techniques.

39.14. It is explained in 34.5 that, because of the differing

conceptual foundations, the conflict between the two approaches

should be resolved by seeing which approach offers the more satis-

factory explanation of the price formation process. On this basis, it seems reasonable to conclude that the market equilibrium model,

rather than MPT, offers the better scientific framework for the

management of ordinary share portfolios.

39.15. Eventually, however, it will be free competition in

commercial life rather than conceptual arguments that decides

which approach will achieve dominance. Investment management


is a highly competitive activity, and any new approach which offers

the possibility of superior performance will either flourish or be

ignored, depending on the investment community’s appraisal of its

merit as compared with existing approaches.

40. Acknowledgements

40.1. In conclusion, I wish to express my appreciation of the

encouragement and support that I have received from my colleagues

at all stages in the development of the methods described in the paper.

40.2. My thanks are due in particular to G. L. Henshilwood,

B.Sc., F.F.A., who designed the computer systems, and to G. Bennett

and S. R. Boshell, M.A., who carried out most of the detailed analytic

work.

REFERENCES

*BLACK, F. (1976). The Investment Policy Spectrum: Individuals, Endow- ment Funds and Pension Funds. The Financial Analysts Journal, XXXII (January/February 1976), p. 23.

CLARKSON, R. S. (1978). A Mathematical Model for the Gilt-edged Market. T.F.A., vol. 36, p. 85.

CLARKSON, R. S. (1980). A Market Equilibrium Model for the Selection of Ordinary Shares for Insurance Funds. Transactions of the 21st Inter- national Congress of Actuaries, Zurich and Lausanne 1980.

COOTNER, P. H. (ed.). (1964). The Random Character of Stock Market Prices (Cambridge, Mass.: M.I.T. Press 1964).

*FAMA, E. F. (1970). Efficient Capital Markets: A Review of Theory and Empirical Work. The Journal of Finance, XXV, No. 2 (May 1970), p. 383.

*FAMA, E. F., FISHER, L., JENSEN, M. and ROLL, R. (1969). The Adjustment of Stock Prices to New Information. International Economic Review, X, No. 1 (February 1969), p. 1.

*JENSEN, M. (1968). The Performance of Mutual Funds in the Period 1945-64. The Journal of Finance, XXIII, No. 2 (May 1968), p. 389.

JEWELL, W. S. (1980). Generalized Models of the Insurance Business: Report of Introduction. Transactions of the 21st International Congress of Actuaries, Zurich and Lausanne 1980.

KEYNES, J. M. (1936). The General Theory of Employment, Interest & Money (Macmillan, 1936).

LORIE, J. and BREALEY, R. (ed.). (1978). Modern Developments in Investment Management: A Book of Readings. Second Edition. (Dryden Press, 1978).

*MARKOWITZ, H. M. (1952). Portfolio Selection. The Journal of Finance, VII, No. 1 (March 1952), p. 77.

*MILLER, M. H. and MODIGLIANI, F. (1961). Dividend Policy, Growth, and the Valuation of Shares. The Journal of Business, XXXIV, No. 4 (October 1961), p. 411.

PLYMEN, J. and PREVETT, R. M. (1972). The Computer for Investment Research. T.F.A., vol. 33, p. 143.

*SHARPE, W. F. (1972). Risk, Market Sensitivity and Diversification. The Financial Analysts Journal, XXVIII (January/February 1972), p. 74.

WEAVER, D. and HALL, M. G. (1967). The Evaluation of Ordinary Shares using a Computer. J.I.A., vol. 93, p. 165.

I

* Reprinted in Lorie and Brealey (1978)


APPENDIX

F.T.-Actuaries Date All-Share Index

2: 1:79 220·60 9: 1:79 226·10

16: 1:79 226·77 23: 1:79 221·72 30: 1:79 222·60

6: 2:79 222·22 13: 2:79 221·91 20: 2:79 225·76 27: 2:79 235·76

6: 3:79 241·15 13: 3:79 255·27 20: 3:79 253·72 27: 3:79 266·71

3: 4:79 264·62 10: 4:79 267·35 17: 4:79 269·37 24: 4:79 275·46

1: 5:79 273·59 8: 5:79 283·05

15: 5:79 269·43 22: 5:79 261·56 29: 5:79 262·10 5: 6:79 261·62

12: 6:79 259·19 19: 6:79 250·58 26: 6:79 248·46

3: 7:79 251·35 10: 7:79 248·31 17: 7:79 248·21 24: 7:79 243·19 31: 7:79 237·13

7: 8:79 244·80 14: 8:79 249·36 21: 8:79 245·86 28: 8:79 245·21 4: 9:79 248·65

11: 9:79 253·41 18: 9:79 250·30 25: 9:79 251·21 2:10:79 255·08 9:10:79 262·70

16:10:79 253·03 23:10:79 249·37 30:10:79 238·23

6:11:79 227·10 13:11:79 225·93 20:11:79 223·81 27:11:79 227·22 4:12:79 232·35

11:12:79 230·30 18:12:79 232·10

Dividend Growth Overseas parameter parameter parameter

r g a 0·5586 0·0294 –0·0392 0·5601 0·0289 –0·0435 0·5692 0·0275 –0·0464 0·5436 0·0291 –0·0200 0·5515 0·0286 –0·0260 0·5595 0·0296 –0·0399 0·5574 0·0297 –0·0571 0·5582 0·0296 –0·0597 0·5966 0·0289 –0·0363 0·5915 0·0271 –0·0842 0·5864 0·0297 –0·0820 0·5851 0·0307 –0·1003 0·5899 0·0318 –0·1058 0·6204 0·0336 –0·1472 0·6101 0·0328 –0·1587 0·6092 0·0333 –0·1595 0·6156 0·0338 –0·1677 0·6121 0·0343 –0·1656 0·6278 0·0349 –0·1797 0·6150 0·0354 –0·1656 0·5738 0·0365 –0·2019 0·5501 0·0369 –0·1939 0·5511 0·0375 –0·1816 0·5547 0·0363 –0·1902 0·5430 0·0373 –0·1770 0·5386 0·0384 –0·1720 0·5367 0·0387 –0·1772 0·5432 0·0389 –0·1821 0·5445 0·0388 –0·1738 0·5455 0·0397 –0·1679 0·5192 0·0411 –0·1380 0·5109 0·0407 –0·1303 0·5046 0·0421 –0·1408 0·5046 0·0425 –0·1473 0·5068 0·0423 –0·1397 0·5114 0·0414 –0·1427 0·5234 0·0422 –0·1474 0·5443 0·0398 –0·1306 0·5440 0·0411 –0·1359 0·5275 0·0418 –0·1188 0·5276 0·0423 –0·1177 0·5368 0·0418 –0·1031 0·5253 0·0426 –0·1050 0·5243 0·0423 –0·0946 0·5280 0·0415 –0·0796 0·5260 0·0413 –0·0696 0·5527 0·0420 –0·0636 0·5289 0·0435 –0·0466 0·5189 0·0426 –0·0490 0·5106 0·0427 –0·0261 0·5155 0·0424 –0·0254

Root mean square relative price residual

0·1165 0·1170 0·1173 0·1220 0·1218 0·1215 0·1215 0·1235 0·1243 0·1247 0·1234 0·1252 0·1250 0·1225 0·1335 0·1361 0·1344 0·1366 0·1491 0·1460 0·1397 0·1368 0·1315 0·1249 0·1218 0·1268 0·1258 0·1336 0·1276 0·1279 0·1324 0·1375 0·1264 0·1221 0·1238 0·1222 0·1294 0·1421 0·1363 0·1246 0·1200 0·1250 0·1282 0·1271 0·1179 0·1163 0·1253 0·1128 0·1122 0·1218 0·1274


F.T.-Actuaries Date All-Share Index

25:12:79 230·46 1: 1:80 229·79 8: 1:80 228·22

15: 1:80 239·93 22: 1:80 245·64 29: 1:80 252·82 5:2:80 247·91

12: 2:80 261·45 19: 2:80 260·63 26: 2:80 257·97 4: 3:80 258·82

11: 3:80 253·99 18: 3:80 241·85 25: 3:80 240·74 1: 4:80 243·70 8: 4:80 241·36

15: 4:80 246·47 22: 4:80 247·57 29: 4:80 247·59 6: 5:80 252·76

13: 5:80 251·85 20: 5:80 249·37 27: 5:80 246·16 3: 6:80 245·28

10: 6:80 255·70 17: 6:80 268·95 24: 6:80 267·86

1: 7:80 267·01 8: 7:80 282·14

15: 7:80 282·71 22: 7:80 284·43 29: 7:80 281·76 5: 8:80 279·01

12: 8:80 280·36 19: 8:80 284·43 26: 8:80 289·52 2: 9:80 282·57 9: 9:80 293·54

18: 9:80 294·65 23: 9:80 291·69 30: 9:80 290·26

Dividend Growth Overseas parameter parameter parameter

r g a 0·5257 0·0406 –0·0200 0·5274 0·0407 –0·0201 0·5269 0·0402 –0·0331 0·5051 0·0418 –0·0279 0·5116 0·0420 –0·0353 0·5277 0·0422 –0·0223 0·5241 0·0412 –0·0250 0·5160 0·0406 +0·0010 0·5234 0·0401 –0·0045 0·5247 0·0412 –0·0056 0·5253 0·0410 +0·0017 0·5140 0·0419 –0·0014 0·5036 0·0410 +0·0011 0·4905 0·0412 –0·0001 0·4987 0·0413 +0·0205 0·5008 0·0411 +0·0370 0·4993 0·0410 +0·0293 0·5022 0·0424 +0·0421 0·4830 0·0418 +0·0529 0·4849 0·0419 +0·0475 0·4806 0·0429 +0·0574 0·5051 0·0427 +0·0729 0·5075 0·0431 +0·0465 0·5117 0·0432 +0·0677 0·5082 0·0439 +0·1071 0·5128 0·0432 +0·1135 0·5152 0·0437 +0·1070 0·5232 0·0453 +0·1642 0·5070 0·0442 +0·1744 0·5019 0·0457 +0·1448 0·5170 0·0463 +0·1320 0·5076 0·0473 +0·1327 0·5081 0·0468 +0·1259 0·5190 0·0466 +0·1098 0·5347 0·0466 +0·1259 0·5246 0·0470 +0·1122 0·5294 0·0469 +0·0913 0·5138 0·0477 +0·1344 0·5215 0·0476 +0·1620 0·5422 0·0476 +0·1735 0·5259 0·0470 +0·1953

Root mean square relative price residual

0·1172 0·1184 0·1224 0·1187 0·1106 0·1247 0·1172 0·1259 0·1290 0·1218 0·1257 0·1276 0·1234 0·1261 0·1220 0·1220 0·1254 0·1280 0·1311 0·1321 0·1249 0·1235 0·1227 0·1236 0·1224 0·1259 0·1305 0·1436 0·1457 0·1326 0·1323 0·1338 0·1344 0·1346 0·1432 0·1457 0·1419 0·1385 0·1419 0·1399 0·1442



CHART 2


CHART 3


SYNOPSIS

The paper describes the construction and application of a general price model based on the hypothesis that prices within an ordinary share market are in equilibrium after all participants have acted on their interpretation of the information available to them. The model can be regarded as a space time co-ordinate system in that all the attributes which affect the price of a share are described in terms of numerical scales and all the measurable changes over time in the equilibrium position correspond to changes in the position of a surface in 4-dimensional space.

The application of the model to the U.K. ordinary share market is described. In particular, it is shown how the relative performance of a share can be resolved into various short-, medium- and long-term components, each of which can be studied in isolation using the model as a frame of reference. In the light of this practical experience, a detailed description of the price formation process within an ordinary share market is obtained.

Since the principles underlying the practical application of the model have virtually nothing in common with the Modern Portfolio Theory methods currently in use in the United States, an attempt is made to reconcile the differing conceptual approaches. The empirical results of the market equilibrium model indicate that the theoretical foundations of Modern Portfolio Theory are somewhat insecure, and it is therefore concluded that the market equilibrium model offers the better scientific framework for the management of ordinary share portfolios.


DISCUSSION

Mr. R. S. Clarkson, introducing the paper, said:—I should first of all like to comment briefly on the general structure of the paper. Part I can be regarded as my detailed answer to the question: “ What is the correct conceptual level at which to begin the analysis of price movements within an ordinary share market? ” The price model that gradually takes shape in response to this question contains various unconventional features, and it is therefore essential to test whether its general properties are consistent with the theory underlying its construction. These empirical tests are described in Part II. Part III then answers the question: “ What types of application are possible? ” while Part IV summarises the principles involved in these applications.

Since Section 33, which discusses the role of mathematics in the construction and application of the model, completes the development of the main subject matter, Part V is virtually a separate paper in its own right and, it might be argued, could have been presented on its own at a later date. However, I strongly believe that the perennial skirmishing between the fundamentalists and the Efficient Market theorists is wasteful of professional effort, particularly in terms of educational and research resources, and should be brought to a halt as speedily as possible. Having made a careful study of the MPT literature to see if any of the methods could be of practical use in the management of institutional portfolios, I am of the opinion that MPT contains too much statistical mumbo-jumbo and not nearly enough emphasis on sound investment principles. Although some proponents of MPT may react violently to my presentation of their subject, Part V of the paper should at least ensure that both the fundamentalists and the Efficient Market theorists can come to a better understanding of the other side’s conceptual foundations.

Although extensive use is made of a computer in applying the model to the UK ordinary share market, the paper-makes very little reference to the role of the computer. This is deliberate. There is a widely held belief that computers are now making important decisions of the kind that were previously made by human beings. If the computers are being used sensibly, what in fact is happening is that they store data and do complex arithmetic and thereby provide information in a highly concentrated form which allows the human beings to make better decisions than would be the case if they had to rely on the raw data. In the applications described in the paper, the computer is being used as a mere electronic slave in precisely this way, namely storing data, doing arithmetic, printing out data and drawing graphs.

Finally, it may be of interest to update some of the practical examples in the paper. With regard to the switching operation described in Section 16.8. both Guest Keen and Tube Investments have now cut their dividend and the share prices have fallen further against the market. Last Sep- tember the value of the purchases was three and a half times the value of the sales. This ratio is now four and a half. Also, I have the suspicion that history is repeating itself, in that the recovery potential frequently alluded to in these two companies at present might be overstated in precisely the same manner as occurred in 1975.

Three Whitbread recommendations are described in Section 20.4. The sell recommendations are winning at present, as the share price has fallen by around 20% against the All-Share Index since last June.


With regard to the functions shown in Chart 3, two notable changes in trend occurred in early January this year. Throughout January and February S(G) has declined sharply and S(A) has risen sharply, suggesting that the market rise during this period has been the result of fairly substantial buying of “ recovery ” stocks, which generally have a low growth rate and high UK exposure. Since I expect these short-term trends to reverse before long, the primary trend, as defined in Section 19.4, is still— in my opinion-downwards. However, I shall be surprised, even dis- appointed, if this opinion is not challenged in the discussion.

Mr. A. T. Grant, opening the discussion, said:-The paper we have before us is an important paper, representing a good deal of hard work and experience, and I feel privileged to be asked to open the discussions tonight. I congratulate the author warmly on its production.

The problem in commenting on this work is that there are so many ideas, so many arguments and so many opinions ventilated in it that you would not thank me for working through the paper commenting on an item by item basis. I therefore propose to start by setting out shortly some general personal viewpoints which colour my acceptance of any consistent numerical system of share analysis.

Within the UK, between 1950 and the mid-sixties the background to share evaluation was on the whole stable, though it did not seem so at the time. Broadly speaking, from 1949 to 1967 the sterling exchange rate was fixed under the Bretton Woods system, from 1950 to 1965 there was a consistent corporate tax system, and until 1964 we were governed by a Conservative administration with a relatively laissez-faire economic policy.

In 1964 the political choice presented to the electorate was between a Wilson administration and a Heath administration. Whichever side won, the chosen economic doctrine was one of interventionism. Therefore, within the UK I would argue that since the mid-sixties there has been one basic rule in the investment game, namely that the rules keep changing. They are still changing, and there now exists a whole generation of investment managers that have been brought up on the basis that what you have to do is to observe changes and react to them, rather as a rat in a maze has to keep checking what doors have been opened and which closed. It follows from this that I believe in an approach which recognises the importance of exogenous variables.

Mr. Clarkson’s model gains good although not full marks on this score, since it makes explicit allowance for a number of factors including, for example, the overseas factor. In the same way I do not accept within the UK the Modern Portfolio Theory propositions that risk is a function of variability of share price and that share price variability based on past history, is an acceptable proxy for risk in present conditions. I am not prepared to lean heavily on the persistence of beta factors derived from past UK history. Within the US stock market environment the beta factor approach seems likely to be much more applicable, given that the USA has had a much more stable investment environment, with less susceptibility being demonstrated over the years to changes in taxation systems or exchange rates. etc. I therefore tend to agree with Mr. Clarkson on some of his criticisms of the beta approach, within the UK. I do not regard variability of share prices as the most important risk nowadays and am much more interested in the vulnerability of a portfolio to external factors. Mr. Clarkson’s line of approach seems to me better for UK share selection.

As a second general viewpoint, I believe the investment industry


contains many half-truths or part-truths. The fact that they are only part-truths does not mean that they should be ignored. It is quite wrong to ignore them, and equally quite wrong to claim that any have exclusive importance or relevance. They tend to have phases of importance. They also tend to be believed in with different degrees of belief by different investors. Then I am again much in sympathy with the equilibrium proposition that different groups of investors are working on entirely different bases. In everyday terms one gets accustomed to different investors pointing out, as general guiding principles for stock selection, that high yielding portfolios have tended to beat the market averages, or that recovery funds have performed well, and so on. Such guiding principles tend to contain statistical truth historically, but they are not always true. The important thing is to know in what conditions these different half- truths or part-truths have most relevance.

I cannot resist teasing the author in these terms regarding paragraph 18.14 of his paper where he claims that the table based on Money Manage- ment’s analysis of unit Trusts shows that the run of figures has been much better for growth funds than for income funds. The table demonstrates this for the year 1979/80 and for various longer periods ending in 1979/80, and from the author’s comments this appears to be used as support for the importance given to the growth factor g in the model. The form chosen for the table includes the bad performance of the income portfolio in 1979/80 in every period shown, and therefore biassed the results. I could not resist the temptation to use the author’8 own figures to extract the performance figures for successive intervals of time. The results are quite different, and do not indicate superior figures for growth funds as suggested. For the two year interval between seven years and five years earlier, income funds did fully as well as growth funds with a return of 31.6%. Between five years and three years earlier income funds rose by 38.4%, better than the 34% of growth funds. Between three and two years earlier income funds rose by 21.9%, better than the 16.0% for growth funds. The evidence is not that growth funds performed better, as suggested.

Clearly, one must not make too much of this. My point is that every dog tends to have a day (or two) and the problem is to determine when it is his day. One must draw only limited conclusions from the experience of particular periods of time and not generalise too freely. It is dangerous to assume that the answers relevant to one time period will be perpetuated.

Thus while I have a great deal of respect and admiration for the work the author has done I have to sound a warning note by revealing that he is mortal and has not been in existence to operate his model over the past hundred years. It would be unreasonable to ask him to backtrack over such a period of time. More positively, it can be said that the immediate past has been a difficult period and that one has to give good marks to the model approach since it has coped with a variety of changes in background such as changes in the dollar premium and dividend limitation.

I part company from him in places where he makes comments on the superiority of this approach over all other methods. In my view it is worth gathering flowers wherever we can, whether the ground is well cultivated or not. Thus, although I do not hold that mainstream MPT methods are well suited to UK conditions, there are some useful ideas to be gleaned there too.

From my comments you will understand that I greatly like the feature that the author’s method allows explicitly for up to date expectations of earnings growth and on the overseas factors, and that the weight and


sensitivity of these factors can be explicitly tracked. I also like the ability to adjust explicitly for such items as takeover bids, property deals and so on, and for current cost earnings.

In the course of development of the model the author has gained great familiarity with the data as expressed in this way, and his observations therefore must command respect as being based on experience. However I have some anxieties and must present them. The crucial one is that the detailed design of this model rests heavily on the proposition that the partial differential co-efficients are “ relatively insensitive to changes in the other attributes ”. This proposition includes the assumption that earnings growth and payout policy are independent. I find this hard to accept and would prefer to see some statistics setting out the degree to which these partial differential co-efficients are actually independent. It may well be that they are sufficiently independent or that they are independent locally, but the question should be asked nonetheless. If interdependence does exist, this does not mean that the model cannot work for practical purposes. If interdependence does exist and the model does not allow for it, then it will contribute an additional element of noise to the residual function.

Interdependence also affects the size and even the sign of the co-efficients in the model. This point has arisen in the past on the construction of regression models, for example in the discussions of the Weaver & Hall paper cited by the author. In general, consider the situation when you are trying to build up a model to explain share prices by increasing the number of variables gradually. As you move from a first factor such as an earnings factor by adding a dividend factor you may account for more of the residual variance, but despite the obvious truth that both dividends and earnings will be positively correlated with share price, in the resulting regression formula the earnings factor (for example) is quite likely to be lower than in the single factor formula and may even be negative. This is a result of correlation between the earnings and dividends.

The author is well aware of this type of problem as he has shown, for example, in paragraph 6.33, but it is worth making the point that if there is interdependence between the different constituents in the model the level of the parameters will depend on this. If cross-correlation exists the parameters do not exist in a vacuum, but only in concert with each other. However the author appears to claim total mutual independence for his parameters, for example in paragraph 6.36. One would expect that the dividend payout ratio for example would not be independent of earnings and growth, and I would prefer to see some substantiation of this claim. Certainly the assumption of total independence allows the use of the much neater multiplication formula, and that is well worth stretching for even if the lack of interdependence of parameters is only local. However this question of statistical proof keeps obtruding.

There are other points which may be less susceptible to statistical proof but where one wonders whether the author’s design solution is optimal. For example, in paragraph 6.17, the author states that for g (growth), the most important parameter in the model, he has chosen an arbitrary scale by defining g so that equal proportional changes in price result from equal changes in g. Although this is modified a little bit later on, in the light of the data one has to ask whether this is a reasonable design solution. Are we saying that as growth in earnings goes from 21% to 23% the proportionate effect on price is the same as a change from 0% to 2%? Is this the way fund managers actually build in growth rates with prices? One would like to see this discussed more fully. At the same time one has to sort out the extent to which g (growth) is a function of inflation, widely applicable, and


to what extent it is a function of better management. In recent years many poorer companies have been able to produce good historical cost profits through being carried along on the tide of inflation, so that the extra growth produced by good management has not attracted the premium it deserved.

In paragraphs 6.21-6.25 regarding the function F(R), a minor point which puzzled me was why the author had chosen the curve shape shown in Figures 18 and 19. I accept that the ratio should be rising, I accept that it should start above zero, but why not in linear progression? A straight- line shape would have simplified the later formula (i.e. let r vary directly, with no need for d).

In paragraph 6.28 the author constructs an allowance for overseas exposure as a modification of the growth rate 9. I am not happy with this construction, although I do feel the need to recognise the relevance of overseas economies and exchange rates. As exchange rates vary there is a great deal of difference between the experience of companies with overseas subsidiaries and those with substantial export business, and even between different exporting companies the effect of exchange rate variations can be markedly different according to the production costs and profit margins of the respective companies. The test suggested of how much profit arises overseas is simple, but one would like to go further on this point.

In paragraphs 13.6 to 13.8 we have the first demonstration of results of the use of the method, tracing selections made in October 1975. It can be seen that at the end of 6 months the five groups of selections have performed in line with the aim of the model, with the purchase recommendations doing best and the sales recommendations worst, with the intermediate groups ranking in the selected sequence. The overall difference of 10.5% between the purchase and sale recommendations would be well worthwhile.

There will obviously be considerable variability in such results. For example by 3rd February 1976 the table shows that the “ hold ” recommendations were the worst performing group, with the “ sale ” recommendations doing materially better. So it is not absolutely cast-iron that this approach is going to work. Presumably the author could have chosen other dates which would have produced different results, and there will obviously be variations in results. We therefore have to rely quite heavily on the author’s observation in paragraph 13.7 that the ratio between the Buy group and the Sell group was usually in the range 1.07 to 1.11 after six months.

A practical point is that in the UK equity market the sale of one share and the purchase of another tends to involve switch expenses of the order of 7%, including stamp duty, jobber’s spread and brokerage. The size of the profit brought out by the author’s system over a six months period on moving from the Sell group to the Buy group covers these expenses with a modest amount to spare. The question which I believe is not fully covered here is whether, having covered the dealing expenses within six months, the switching is justified in the longer term. I should appreciate comments from the author on the longer term performance achieved by the system. If the system operates only over a short term time horizon to cover switching expenses then this constitutes a valuable contribution to investment success, but its overall value must depend on how it performs over the longer term.

Part III, on “ Practical Applications ” is the section of the paper with which I am happiest. The author has set out a good deal of knowledge and experience in the commentary, with a good deal of meaty chunks and also


many provocative assertions. In view of the pressure of time, I shall limit myself to commenting on individual items.

In paragraph 14.2 the universe of UK ordinary shares used is not actually defined. Specifically, are any property shares, banks or investment trusts included? These are groups of shares where assets tend to be an important part of the practical share assessment process. The paper does not dwell on assets as a basis for valuation, and it may be that his universe excludes some of the types of shares most affected.

In the following paragraphs the author refers to stockbrokers’ research. As a stockbroker I must express my gratitude for any reliance on stockbrokers’ research services and look forward to a subsequent paper comparing their assessments with random choices. As a practical point, the monitoring of 20 reports from stockbrokers, as listed in paragraph 14.8, leaves the problem of whether one can equate one review weighing seven pounds three ounces with another of half a page. Despite difficulties of this nature, the attempt to formalise records of currently available advice is laudable.

The record of current recommendations is one of the four charts referred to in paragraph 14.11. In practical terms I like charts of this nature to add to one’s feel for timing and variance, and I would expect that prolonged use of such charts would add to one’s experience and understanding of share movements. I like the chart of the relative price residual as shown for Whitbread in Chart 2 in the appendix, with the high and low lines built in as control guides as described in paragraph 14.10. One could alternatively have built in dynamic upper and lower significance limits into the chart using computers, provided the distribution around the mean was in a recognisable distribution pattern such as the normal distribution.

In paragraph 16.6 the author makes the point that the average growth rate g should relate to the average growth over a complete economic cycle, while shorter-term earnings estimates are not normally made on this basis. In his model he has built in more specific estimates of earnings for the next two years. One could reasonably argue for extending a similar treatment for dividends before establishing the payout ratio factor.

This difference in the estimation of short term earnings or dividends as offered to longer term estimates is evidenced by the US bank which operates and offers a UK stock selection model. That bank requires that analysts input specific earnings and dividend estimates for the next 5 years, allowing inter alia for the actual economic cycle. However they are also asked to give an estimate five years hence for earnings on the formal assumption that one was at that time at the mid point of an economic cycle. That estimate is used as the base point for subsequent estimates, and the discontinuity is regarded as acceptable. This model has also appeared to have some positive results in the short time that it has been on offer.

In paragraphs 16.7 and 16.8, and in Chart 27, the author gives an example illustrating long term persistence in the benefits of a switch over some 5 years. One clearly cannot argue much from a single example, but this switch was obviously very profitable over that period. It is interesting to note that in this case it actually took two years for the lines to cross so that the switch began to be justified. One wonders whether the author would have decided on different timing tactics in the light of the full fine-tuning form of the model he refers to later, completed at the end of 1978. Many investment managers are all too familiar with the pressure from short term portfolio performance measurements to emphasise short term performance.


In paragraphs 18.5 and 18.6 the author notes that the value of the root mean square error is well above the level that would be regarded as acceptable in most statistical models. I think it is right to point this out, but I also am prepared to take it as reasonably acceptable. One has to recognise that there are other partial truths operating in the market, and their effects will add to the variability of the residual figures in a model such as this.

This last comment may appear too easy an abrogation of statistical discipline. However the stock market is an area of interplay between many different factors. One has to recognise that any model or any single approach will have to leave out of account many factors which will add to residual error.

It is quite possible in the investment field for results to be impressive and yet not statistically significant. One of my own colleagues recently ran a test on work he has carried out over three years grading stocks by current cost attributes. The gradings were into five groups of equal size. He found that over the three years the group representing his optimum selection of stocks was worth twice as much as the group representing his selection of the most unattractive stocks. The first group had risen in value from 100 to 130, and the second group had fallen from 100 to 65. Moreover the intermediate groups were in line.

Regarding this. two points immediately come up. First, this result did not prove anything except that over that particular period current cost considerations had some value in the market place. Secondly, when we subjected this good result to simple statistical testing, we found that the randomness or variation within each of these groups was so large that the result was not statistically significant at the 5% level. Despite the good results on a group basis, it would have been possible for an individual investor wishing to make a smaller selection of stocks to do badly by choosing the wrong stocks in the most attractive sector, and conversely by selling the wrong stocks in the least attractive sector.

The same question must be put on the author’s groups of selections. I should welcome comments from the author about the variability of the individual stocks within the five groups shown in the paper.

Paragraph 18.7 and the associated Chart 3 in the Appendix sets out a historical record of the overseas exposure factor S(A). If I understand this correctly, the overseas variable A is included on the basis that the weighting function F,(A) equals 1+ aA, where A lies between 0 and 1 and a is a fitted variable. The lowest value shown in the Appendix for the variable a appears to be – 0.2019 in May 1979. The highest value was +0.174 in July 1980. Thus the valuation factor applied to A, the degree of overseas exposure, varied over the period from l–0.2019, i.e. approximately 0.8, to 1+0.174. i.e. 1.17. The increase from 0.8 to 1.17 is about 46%. If I understand this correctly, one of the nice things about this model is that over this period one can compare a stock with full overseas exposure (A = 1) and one with none (A = 0) and say that the overseas element could be expected to add 46% on average to the value for the first stock and would make no contribution to that of the second stock. The con- venience of this form of construction of the model is very appealing.

This was interestingly a time when sterling appreciated, so one might have expected the value of foreign subsidiaries to decline in sterling terms. The fact that the numbers came out as favouring overseas stocks for this period shows how difficult it was to earn profits in the U.K. during this time.

I do feel that the author has made unduly sweeping assertions on alterna-


tive valuation approaches. I feel he presses the unique merit of this method too strongly, end since I believe in the validity of many partial truths in the investment field I believe other approaches can also work.

The author has concentrated more on the detailed measuring of individual stocks and has not spent much time on portfolio management as such, an area in which some of the MPT concepts can be of use.

I should like to highlight paragraph 28.6, where the author expresses hope that it should be possible to forecast the direction at least of the major moves in each parameter. Perhaps this will be the subject of a future paper.

Finally, may I comment that this method does not remove the need for investment judgement. It is not a wholly automatic system. My belief is that automatic investment systems producing single-answer solutions are intrinsically doomed to failure. If they work for a period, other people imitate them so that their actions destroy the original market equilibrium and they lose their value. If they do not work, they are of no value anyway. What the author has done here is to describe an approach to codify- ing and quantifying stock valuations using skill and judgement and I am grateful that he has done so.

Mr A. D. Wilkie:—I intend to be critical, severely critical. I have a great respect for Robert Clarkson’s ability and enthusiasm, and I hope he will not take what I have to say as rude or denigrating. My object is to be constructively critical. But I have to show what are the faulty parts that need demolishing before we can rebuild on sound foundations.

Part I purports to give a rigorous development of what the author calls the market equilibrium model. Section 4 develops the mathematical and so-called statistical formulation. Paragraph 4.2 introduces us to various attributes of shares, A, B, C, etc. Now in paragraph 3.1 we are told that “ any one investor will assess various attributes of the share in accordance with the criteria that he, and he alone, considers appropriate ”. In 4.4 we are asked to assume that “ all participants in the market use identical values of individual attributes ”. For attributes then the author has in mind features of a share that are agreed facts; opinions, assessments, judgments must be excluded. Remember this.

Attributes. according to 4.2 are quantifiable. This does not tell US whether the variables quantifying them are to be continuous or discrete values or both. Much of the mathematical rigour collapses because of this lack of specification.

Paragraph 4.3 gives us a simplified market model, where all price/ earnings ratios are constant. An alternative would be that all dividend/ price ratios, i.e. historic dividend yields, are constant. I should like you to remember these simplified models.

In 4.4 we are asked to make a great many simplifying assumptions. These are very similar to the simplifying assumptions made by those who have contributed to Modern Portfolio Theory, such as those listed in 37.10. But they are obviously not wholly valid. It is worth reminding ourselves of the realities that have been assumed away.

First, not all holders of shares are regular active participants in the market. To that extent, some holders of shares just don’t use any attributes at all. I have already mentioned that attributes must be restricted to those items about which all active participants agree.

Secondly, there are transaction costs, which are far from negligible in the UK market.

Thirdly, while most holders cannot hold short positions, companies


themselves can create and sell their own shares, either as rights issues or as the consideration for takeovers.

Fourthly, a participant who identifies that a share he holds is overpriced will only sell and reinvest if he can cover his transaction costs and possibly his conversion of a potential capital gains tax liability into a real one. Some participants are also restricted in their actions by a limit on the proportion of their portfolio they can invest in a particular share, or a limit on the proportion of the outstanding capital of the share they can hold.

Fifthly, while a large switch from one share into another will probably cause the ratio of their prices to alter, a small enough switch may not cause the prices to change. One can claim only that the ratio will not increase.

Sixthly, different participants may, perhaps perversely, treat certain attributes in different ways. In particular, the tax position of a share- holder may affect his valuation of the payout ratio. Under the UK system of corporation tax a gross fund should logically favour high payout ratios, whereas a high marginal rate taxpayer should favour low payout ratios.

Apart from this last point, and the existence of transaction costs, I don’t think these realities severely affect the author’s simplifying assumptions, but they cannot be wholly ignored.

Paragraphs 4.5 to 4.9 then purport to show that the price P(A) of a share with attribute A is a smooth increasing function of A. But in fact all that is proved is that P(A) does not decrease with an increase in A. Nothing is said about what happens if A is discrete. There is no proof that P(A) is continuous, even if A is continuous. There is no proof that P(A) is smooth, nor that its derivative exists. If we are going to be mathematically rigorous, we should do the job properly.

In reality there is only a finite number of shares, so all attributes only take a finite number of values; but we can distinguish between those that could take any value, and those that can only take a small number of values. But we don’t expect to be able to differentiate with respect to these last.

The author has also not proved that there is only one equilibrium price or any at all. In his simplest example in 4.5, if X and Y are the only participants, they can trade shares between themselves at any price such that until either X owns all of share 1 or Y owns all of share 2, when trading will stop and there will be no prices.

When we come to three or more participants and three or more shares, does the model ensure that there is only one set of relative prices for the shares? Do we not rather find market segmentation? Those who value attribute A highly end up owning shares that have a high value of attribute A, and vice versa. In order to have even marginal prices established we need enough participants for their assessments to be nearly continuous. In order to have trading we need changes in the attributes or in the assessments from one time to another. If we introduce transaction costs we have wider bands of price over which no trading will take place.

Economists can show that under certain assumptions, there may be more than one equilibrium exchange rate between two currencies, or more than one rate of interest that balances the supply and demand of loanable funds. The author has made no attempt to show either the existence or the uniqueness of his price structure. I don’t deny that there may be only one equilibrium position; only that it is not proved.

In 4.10 we are asked to assume that, when we are considering two attributes, “ the equilibrium position should also be similar ”, and we are


told that we can then deduce that (which may not exist) will be

relatively insensitive to changes in B. Why should we assume this? It is quite possible that there is an interaction between different participants’ assessments of A and B. We cannot take this for granted. It is at this stage a further simplifying assumption. If we used what statisticians call a general linear model we could allow explicitly for interaction terms. We certainly may need to investigate them. And we may be confused by correlation between attributes A and B in the population of shares under consideration.

In 4.11 we are asked to expect that the variation over time of

(which still may not exist), which results from changes in the preferences of participants, will be regular. Why should it be? Why should not the attributes also change, and change discretely? Why should external discrete events not cause a discrete reassessment of their criteria by all participants? What about the relative weights of participants? If, in the simplest example, X decides to value attribute A more highly than before, and more highly now than Y does, doesn’t the whole position change abruptly?

In 4.12 we are told that condition (vi) can now be removed. What the author really means is that if A is a bad thing rather than a good thing we can use attribute A1 = – A, and condition (vi) will still hold. If we really remove the condition we have no guarantee that all participants treat A as a good thing, or even that one participant treats A as a good thing for all

values of A. In that case can go up and down in any fashion, and we

very likely would find multiple equilibrium positions. The market equilibrium model, as developed so far, tells us very little

about actual prices. It is really a market equilibrium model for holdings, and to complete it we would need to introduce the quantity of shares in each company and the relative wealths of each participant.

But this model does not tell us why any participant would want to hold more than one share. In the simplest example, X would want all his portfolio to be in share 1, Y all in share 2; why should either want to hold a portfolio containing both, except that the number of shares is restricted. We need to explain why anyone chooses to hold a spread of shares in a portfolio.

An economist might approach this by introducing indifference ratios for each participant that varied with the size of their holdings in the shares. A portfolio theorist would justify this by introducing uncertainty or risk in some form and assuming that participants wish to reduce risk; they can do so by each having a holding in each share. The author does not follow this road because in 5.4 he assumes that “ all participants can forecast future earnings per share and dividends for all companies in the market ”. For him, risk does not exist.

Next in 4.16 we are introduced to the author’s measure of “ price sensitivity ” of P with respect to A. This measure looks like the inter- quartile range of P divided by the median of but it is defined in terms of the value of P at the upper quartile, lower quartile and median value of A, keeping B, C, etc. constant. Presumably this is the expected value of P, in which case the quartiles of P and of A correspond. But S(A) is not independent of the other attributes used. S(A) ignoring B is different from S(A) at the mean of B, if attributes A and B are correlated.

This sensitivity measure bears some relation to the “ coefficient of K


variation ”, i.e. the standard deviation divided by the mean, which in turn is related to the variance of the logarithm of the variable. If, as I shall suggest, we treat log P, or log (P/E) as the variable of interest we can use, instead of this sensitivity, the reduction in the variance of log P attributable to variable A in the usual multiple regression analysis of variance.

There is an enormous advantage in using well-known statistical measures. These have usually been properly investigated, their sampling distributions are known, and one can apply conventional statistical tests to them. Ad hoc new measures about which less is known should be avoided unless they are the only satisfactory ones.

The rest of Section 4 discusses the so-called statistical formulation. I have to say that the way in which the author uses the word “ statistical ” bears no relation whatever to the conventions of mathematical statistics. “ Statistical fit ” (4.19), “ statistical stability ” (4.20 and 4.22), “ general statistical process ” (4.22). just mean “ fit ”. “ stability ”. and “ method ” respectively. Statistics has nothing to do with the case, here or elsewhere in the paper.

Throughout, the author has made very little use of conventional statistical theory. As far as I can see the terms “ random variable ”, “ probability distribution ”, “ variance ” (or “ standard deviation ”) “ confidence interval ”, “ standard error ”, “ correlation coefficient ”, hardly appear in the paper anywhere. Certainly these concepts are entirely missing. But then, the author hardly seems to recognise the existence of risk. Ad- mittedly in 5.3 (iii) he says that “ forecasts of future profits and dividends for individual companies involve large margins of error ”. But in 5.4 he assumes “ for the moment that all participants can forecast future earnings per share and dividends exactly for all companies in the market ”. This assumption is never actually relaxed; it is simply replaced in 6.5 by assuming that investors use “ likely future earnings ” in place of certain ones.

But if this were true we would have a conventional fixed interest model, for which indeed the author develops various elementary relationships in the rest of Section 5, on similar lines to what he did in his previous paper on the Gilt-edged market. Again, he seems reluctant to use the conventional results of compound interest, which include the idea that a present value might be considered as the sum of the values of future payments, each discounted at a suitable factor for its time of payment. Yet it follows immediately from his analysis in Section 5.

Imagine a set of n securities (not exactly shares), each of which provides dividends of xDt, in year t, and zero in every other year, for t = 1 to n, together with an (n + 1)th security that provides the same as share x in year (n + 1) onwards. In what way should the value of share x differ from the value of the sum of this set of (n+ 1) securities? Remember, certainty is assumed throughout. It soon follows that a certain payment in year t can be valued by a discount factor appropriate to year t. Different (fixed interest) securities have different values to different holders, of course, because of their different tax situations; and each individual may discount at his own set of discount rates (not necessarily uniform for each term).

An example of unsound logic appears in 5.12 and 5.13. Here the author is using the technique of mathematical induction. But having proved that 1 < s k for N =1, he omits to prove the next step, that if the statement is true for some N it is also true for N+ 1. “ Arguments similar to those above ” lead eventually to the statement that “ it is unrealistic to have share prices determined solely by earnings and dividends for the first hundred (or thousand, or million) years, so that s101 < s100 or s1,000,001 <

s1,000,000· Such an argument is surely fallacious. It is quite possible for


share prices to ignore entirely what might happen after some year far into the future, just as the discount factor for a reversion due at the end of a 999 year lease might be really zero, not just very small indeed.

In 5.15 we are now asked to suppose that for a certain subset of shares it is possible to define in effect an order relation between two shares. G is surely not a function or a value at all, but a relation, like “ greater than ”, and to attribute numerical values to it is unjustified. If the author wants to introduce his own expectations of earnings growth he could do it without all this spurious formality.

I can think of several such subsets, each consisting of the ordinary and the A shares of the same company, or possibly the linked shares of Unilever Ltd. and Unilever NV, Royal Dutch and Shell. I cannot imagine any two different companies about which any investor can say with absolute certainty that in every future year the earnings per share relative to current earnings of the one will be greater than the earnings per share relative to current earnings of the other.

The extension of this subset in Section 6 to include all shares is thus without foundation. What in fact an investor buys from a share is not a stream of known future earnings, or dividends, or even their likely future values, but a stream of uncertain earnings in each year, to which one should at least attempt to attribute a probability distribution. It is a common actuarial fallacy to replace distributions by their means, appropriate when a large number of fairly independent lives are considered as one portfolio, but quite invalid when considering investments.

Even going one year ahead we have in reality share X with some probability distribution of earnings next year, and share Y with some other probability distribution. The earnings of the two companies may not be independent random variables, but may be correlated. If we are restricted to holding only X or Y but not both, we have to compare two probability distributions, not two certainties. We need a technique for doing this. If we can hold part of our portfolio in X and part in Y we have a whole range of choices of probability distribution, and we can apply our new technique to choosing the most suitable portfolio. But why not look at what others have done, and study the methods of Portfolio Theory, which the author rejects. Portfolio theorists are only using proper actuarial techniques.

I could go on further with detailed criticism of the faults of Part I, such as the statement in 6.7 that G(X) is always an integer in the range 1 to N, and differentiation with respect to G in 6.13. G is a purely arbitrary ordinal scale, i.e. it represents Number 1, Number 2 etc. like house numbers in a street. Yet we find it treated in 6.17 as if it were a proper number on which one could do arithmetic.

You will remember that I mentioned earlier that the author’s first definition of attributes implied that attributes were things about which all participants were agreed. Estimates of future growth rates cannot therefore be treated as attributes, in this sense.

I cannot pass over the statement in 6.25 that it would be “ statistically unsound to allow two variable parameters in F(R) ”. This is not true—or at least, it is not in any way “ statistical ”. An independent variable may affect the dependent one in a quadratic way, so two parameters would be quite appropriate or a variable may have three discrete categories—again two parameters are necessary.

By the end of Part I we get a model for P in a multiplicative form. Now, the author could quite well have started with the assumption in 4.3 that all P/E ratios are constant; he could have looked at the distribution of


P/E or of log (P/E) which I shall call L. Since he wants a multiplicative model he could have treated L as the dependent variable, to be “ explained ” by a multiple regression model, introducing whatever independent or explanatory variables he wanted, as in 6.40. Some would have entered in a linear form, and he could quite well have used transformations of the independent variables so that the parameters are entered linearly. He could have used category (i.e. 0 or 1) variables as well as numeric ones. He could then put where is the log (P/E) of share i at time t. He could have done a cross-sectional analysis, keeping t constant (as he has done), or alternatively have included a whole range of values of t (as he has not). He could have estimated his parameters by least squares or by maximum likelihood, presuming perhaps that the residual ES are normally distributed. He could have looked at the correlation matrix of the variables, and he could have calculated standard errors and indeed a covariance matrix for his parameter estimates. He could have tested in the usual ways whether the marginal contribution to the reduction in residual variance from any one term was “ significant ” or not. He could have applied time series models to the residual deviations (the ES) for one share over time, or to his parameter estimates over time. He could have avoided the solecism in 18.5 that “ the value of the root mean square error is well above the level that would be regarded as acceptable in most statistical models “—which is as much as to say that statisticians have some value of a standard deviation for all situations that is “ acceptable ”— what nonsense!

Alas, he does none of these things. An enormous effort of data collection and computer analysis has been spoiled by poor analysis—at least wasted as far as the rest of us are concerned. Mr Clarkson and his team may just be good at estimating future growth rates, or at least quicker than the market. Or they may just have noticed that low P/E ratio shares show better performance than high P/E ratio shares, a suggestion that has been discussed and validly tested in at least three recent papers by portfolio theorists.

Before I go on to discuss Modern Portfolio Theory I do want to praise the author for a number of ideas: one is the method of dealing with zero dividends that is shown in 6.21. This sort of technique might help to deal also with shares with negative earnings, which clearly don’t usually have negative prices. Zero or negative values are a nuisance when you want to take logarithms.

Another is the technique shown in Figure 27 in paragraph 16.8, comparing the value of portfolios of purchases and sales. Transactions costs should strictly be allowed for, by starting off the purchases some 5% below the sales, and adding dividends would make the comparison complete. This sort of chart should be compiled in every investment depart- ment for every period’s purchases and sales—keeping each analyst separately if you want. There is a technical problem in following a portfolio of shares that you have sold and that may even not exist, but it is not insuperable.

Then in 26.3 the author identifies three components of the price performance of a share, which correspond neatly with three components of the model for dividends and yields developed for the market as a whole by the Maturity Guarantees Working Party (fully in the framework of Modern Portfolio Theory), and discussed here last October.

First, dividends change in a random walk fashion over time (perhaps with short term autocorrelations).

Secondly, dividend yields wander in an autocorrelated manner around a central value.


Thirdly, there is an unexplained short term random element which remains unpredictable.

This last so swamps the first two in the short term that the random walk/Efficient Market model for the whole market is not disproved by short term investigations, as previous investigators thought. I don’t think the pure random walk model is correct in the long run. But this discovery does not invalidate all of Portfolio Theory (any more than finding that mortality rates change with time wholly invalidates the life table model).

Sadly, the author seems unable to accept the concept of risk which is at the heart of Portfolio Theory. Furthermore, he appears to be unfamiliar with the breadth of literature on the subject. For example in 34.4 he states that “ someone trained in MPT will reject the concept that prices reflect fundamental attributes such as earnings and dividends, because the textbooks from which he has learned the theory of his trade expound the view that there is no empirical evidence to support this concept ”. I have looked through 13 textbooks and books of readings on this topic, mainly American, but two British. Two of them fail to discuss future earnings and dividends, but all the rest explicitly develop share valuation models that depend on expected future dividends, or earnings minus investments, as in the Miller and Modigliani model. or indeed as in J. B. Marshall’s paper to the Faculty in 1967 (TFA 30, 95). Where they go further is in stating that changes in the assessment of future earnings and dividends are (a) correlated for different shares at one time and (5) independent from one time period to the next. This accounts for the random walk model of price changes over time and the cross-correlated Markowitz model over shares, of which Sharpe’s diagonal model is only a simplification.

The author in 34.4 then refers to “ the MPT approach which states that statistical measures constructed from past prices can be used to predict future prices ”. But this is precisely what some MPT models state can not be done, and what he suggests WA be done. MPT models necessarily use past data to estimate for example the variance of share price changes, and the correlation coefficient or beta factor between changes in the price of a share and changes in the value of an index. I know of no scientific method—or even unscientific one—that attempts to make any estimates for the future totally ignoring the past. The only person who did that was T. H. White’s Merlin, who lived backwards, and knew whet was going to happen, but not what had just happened!

In 34.7 we are told, not by the author, that “ portfolio management is a different subject from security analysis ”. Indeed it is: portfolio management is what the actuary should understand, security analysis is the job of the investment manager and his assistants. The construction of mortality tables or the setting of premium rates are different jobs from life under- writing.

In 36.5 the author states that “ in portfolio theory it is axiomatic that risk can be regarded as equivalent to the variance of the return on the portfolio ”. This is not true. If the distribution of returns can be described by two parameters (as in a normal distribution) or if the utility function of an investor can be described by one of several functions (including a quadratic one) then the variance is sufficient to describe the risk. This is a result, not an axiom. Further, some work has been done on the circumstances where variance is not sufficient.

Alas, Mr Clarkson is so antipathetic to the methods of MPT that his description and analysis of it have to be treated as of little value, and it is hardly surprising that the methods of financial economists, which mainly


consist in using statistical methods to investigate economic events, have no appeal to him.

Mr. G. L. Henshilwood:—I suppose that I must preface my few remarks this evening by declaring my personal interest in tonight’s paper, for my own experience in investments, and of the UK equity market in particular, has been gained under the direction of tonight’s author in operating the model which he has so thoroughly described in this paper before us. For my sins, I am one of the unnamed analysts numbered 1-3 though modesty forbids me from saying precisely which. However, I thought that it might be of some general interest if I described to you how interpreting the results obtained from the model has, I believe, helped me in understanding the structure of the market and in reaching investment decisions. I will leave the statistical and mathematical aspects for the author to reply to later, but, after all, ultimately it is not the statistical stability but rather how well the model works in practice which will determine its reliability and use.

In the very simplest sense, operating the model as we do in practice, on a weekly basis, imposes a discipline on the analyst responsible for each sector of the market to maintain the accuracy of the database on a continuous basis. Much more than this, interpretation of the results of each weekly run imposes a further and more important discipline for really it is the extent to which changes in the relative price residual for any particular share can be attributed either to a change in the general shape of the 4-dimensional surface or, alternatively, to changes in the input data in respect of that share which is crucial in assessing likely future price movements. The value of the model will clearly depend on to what extent the results for those shares monitored can in the aggregate logically explain relative movements in price. Once this has been understood by the analyst he can then use the divergence of anticipated versus actual price movements as a basis for dealing and he can clearly identify the basis on which future out-performance or under-performance is anticipated.

If I may borrow a definition from Oscar Wilde, I think it would be fair to describe the so-called professional market operators as “ men who know the price of everything and the value of nothing ”. I would not dispute that such operators make (and lose!) large sums of money but this is surely no sound basis for the institutional investor. Thus, the principal aim of the model must be to assist in the apportionment of intrinsic value across the market to individual shares which will broadly tie in with the intuitive appreciation which investors will have.

It is, I would suggest, an inherent weakness of many stockbroking firms and institutional investment departments that individual analysts may have a keen appreciation of the shares within the particular sector for which they are responsible, but they will have rather less appreciation of what is going on in other areas of the market and economy. There seems to me to be too little effort made to collate research centrally and to ration- alise at any time the relative value of shares in different sectors. Stock- brokers may argue that they provide the research required by their clients and I would not presume to suggest that they should change their system, but one could certainly argue that the emphasis is too often concentrated on the short term trends and on today’s immediate news rather than on any attempt to look at the longer term arguments. I would certainly argue that the time horizons of most investors are too short and this must surely be the reason for extreme market volatility.

Thus, I see the value of the model as essentially threefold:


(i) It provides a quantitative framework, and the evidence suggests that these measures are meaningful, within which qualitative preferences within the market, for example investor preference for overseas earnings, may be judged.

(ii) The structure of the model, involving input data which does not change frequently over the financial year, encourages a longer term investment perspective on the part of the individual interpreting the results.

(iii) It provides a fairly ready framework within which any shares not monitored on a continuous basis may be assessed with some measure of confidence at any time.

It would be foolish to pretend that too rigid an application of the model with its emphasis on earnings, both on historic and current cost accounting bases, and on dividends will apply to every sector of the market without qualification. There are many companies for which pre-tax profits and earnings per share are not the most reliable guide to share price movements. Into this category comes, for example, the TV rental companies where accounting practice will normally defer the incidence of profits by a heavy front-end loaded depreciation charge. In those cases an analysis of cash flow is generally used as a measure of the company’s attractiveness. However, it is a measure of the robustness of this model that these companies can be safely accommodated within the model without distorting other values by assuming a growth rate in excess of that which might otherwise apply. Similarly, composite insurance companies and banks cannot be compared with industrial companies in the context of borrowings. However, by using the shape of the function F4 (B), described in Section 11.4, applied to the solvency margin and free equity ratio respectively, the net effect on the expected price may be reproduced in a similar fashion. More widely, by applying special factors which vary either in line with any of the three variable parameters, as appropriate, or by a constant adjustment to the expected price-earnings relative, any non-standard situation can be dealt with in a meaningful and satisfactory way.

My final remarks relate to Section 19 of the paper where I must confess I am rather less happy and would take some issue with the author. What he seemed to say was that the primary downward trend, as defined in this section, would ultimately lead to lower market levels and while I would not disagree at all with that conclusion, it seems to me that the behaviour of the parameters over the period described is quite consistent with the view frequently expressed in the market at the time that the recession, particularly sharp in the UK, would prove fairly short-lived and that the end of the de-stocking cycle would bring sharp recovery in profits, earnings and dividends and so the price level of the more vulnerable shares held up rather better than one might have expected. But this conclusion that the market is vulnerable must surely depend on an accurate assessment of the underlying economic trends rather than attaching too much importance to the mechanical trends of these parameters. They are, after all, unweighted by market capitalisation and I am not sure how accurately they can therefore represent the structure of the market as a whole.

By way of conclusion, lest I attract the sympathy of the audience as an old workhorse that is about to be put out to grass, I thoroughly endorse the author’s views that a high degree of analyst awareness is involved in the input of data to the model, in accurately assessing the growth rates used and in eliminating bias and subjectivity, and that considerable skill is


involved in assessing the output results from the model. I would certainly wish to congratulate the author most sincerely and I certainly hope that ]. will continue to make practical and profitable use of the results of the model for some time to come.

Mr. J. Plymen:—I congratulate Clarkson on his most valuable contribution to Faculty proceedings. In my opinion, based on nearly 40 years of investment management and investment analysis, this paper will be recognised as one of the most important Faculty or Institute contributions to the investment scene for 50 years, since the days of Douglas and Murray in 1930!

In this appreciation of the paper, I differ radically from Professor Wilkie, who treats this contribution as a Ph.D. thesis, and criticises it unmercifully because of alleged weakness in the statistical theory. Wilkie misses the whole point and purpose of the paper: it is a serious and probably successful attempt to devise practical analytical tools to solve the two major problems of investment management, selecting the shares and timing the markets. It should be judged on the validity of this analytical work, not on minor features of the introductory mathematics.

In practice, investment techniques, which have to make use of the limited statistical material available from the Stock Exchange and other sources, cannot always satisfy rigorous mathematical tests. Consider the various Portfolio Performance systems. Do the performance testers apply significance tests to their data? A further example comes in the joint Faculty paper of 1972, by Prevett and myself, when we describe the Mean Absolute Deviation technique (see paragraph 14.9 of the present paper). Prevett and I admitted that our methods might have certain weaknesses to the statistical purist, but they appear to work well in practice.

On the other hand, there are many systems, including parts of the MPT theory, often designed by academic characters with the minimum of investment experience, which are acceptable mathematically, but useless in practice. One such example is the Markowitz system. I had some correspondence with Mr. Markowitz about 15 years ago when I was pre- paring a paper for the International Analysts Society. I had written to him asking whether he could give me examples of some practical use being made of his work. He wrote back and said he didn’t have time to develop the model further, he’d retired and he was now employed as consultant to a computer company. The fact is that the system can never be used for investment purposes, as it requires inputs which no human analyst can produce.

I put the question to you—which is more important and interesting to members of the Faculty, a good working practical system like that of Clarkson, or an alternative system, useless as a practical tool, but rigorous mathematically?

Coming back to the present paper, I must admit that Clarkson is tackling a major problem, which many people have tried to solve in the past, and largely failed. It is well known that the equity models, set up by Bank of New York and later by Messrs. Weaver & Hall, are no longer in use. Although these systems had a degree of success, I think the sponsors found that the benefits obtained were not really quite enough to justify the work involved and one of the reasons for the poor results of these systems was that they used too many inputs, and they used inputs that correlated with each other. Clarkson, I believe, has avoided the bunkers that caused so much trouble to previous operators by using a smaller number of inputs, and I think probably avoiding the inter-correlation, and has developed a


much more powerful system. Also, his system benefits from having a relatively loose fit. Although the loose fit is anathema to statisticians it’s really what we want here. You want a loose fit because you’re looking for anomalies. If you have a close graduation, you wouldn’t get any anomalies. I must admit, like the opener, I have a little bit of a question mark with regard to correlation between the payout ratio and the growth rate. Now I think in practice the author is almost certainly right in saying that this can be ignored. At first sight you think it couldn’t be ignored. It seems obvious that a company with a low payout ratio will have a high growth rate but I think when it comes to practical finance it may well be that those companies with a low payout ratio have a low payout ratio because their profitability is low and it may be necessary to plough back heavily—to maintain their business. After all, the payout ratios don’t usually have a very wide range and I think if one carried out a statistical test here one would find the payout ratio varying between 1.2 and 2.5 did not correlate more than to a tiny extent with the long-term growth rate in the companies concerned.

I have one or two comments to make on the system myself. I think it’s a very good feature that it does require such highly skilled operation. It is obvious that the determination of the earnings base and the earnings growth rate is a very difficult process. I think in practice a whole Faculty paper could be written on this theme. I think that the author’s analysts determine the growth rate on a highly sophisticated basis, probably a much more sophisticated basis than that used by most stockbrokers. I imagine they do a sort of profit and loss account and balance sheet forecast for two or three years making sure that the various components are consistent with each other and with the rate of growth of turn-over, profit margins and capital needs. At any rate, there’s no doubt about the importance of this skilled determination of this growth rate because after all despite any sort of statistics or economics what really matters about an equity is the sustainable growth rate of profits and dividends. As I say it’s a very good thing that such skill is required in this system. It disposes of the criticism that the system is something of a black box that automatically tells you what to do. It’s absolutely the reverse. It’s a very sophisticated way of applying skilled professional investment analysis to get a practical answer. I must admit I have some reservations about the effect on the determination of the profit growth rate of inflation accountancy. I would not be surprised if the author wasn’t finding something of a hiatus here. I imagine that when the system started in 1975 he did not think he’d have the problem of alternative inflation adjusted accounts. Now we have two sets of accounts and nobody really knows which set of accounts is now being taken notice of. As far as one can tell the market appears to be very slow to appreciate the inflation adjusted accounts. There are lots of companies which by reference to the old-fashioned accountancy seem to be doing very well, but are really running at a loss by realistic accountancy, making up the loss by raising capital. I would have thought that for the moment this inflation adjustment problem must make it very difficult for the author. I would think this is just a temporary problem and in two or three years’ time with a history of inflation adjusted accounts this difficulty will be got over.

Finally I have tremendous sympathy and accord with the author in his attack on all this American MPT stuff. I don’t agree with Clarkson’s statement that it’s not widely known to the investors over here, the fact is there are lots of opportunities for investors and investment managers over here to learn something about MPT. For example, you can go to the


biennial conferences of the International Analysts Society where you’ll meet some leading American proponents of MPT. I remember Professor Sharpe giving an address to them some years ago on this subject. In addition the London Business School and the City University offer seminars plugging this American material, the MPT and so on. In fact the London Business School now provides a subscription service of functions for many UK companies. All this is very commendable but I’ve yet to find any- body making any sense whatever about this business. I would like to ask is there any investment manager here or any other investor who has made any money out of the consideration of the factors. The whole policy seems to me to be wrong, that by some strange computerised system you set up a marvellous portfolio which stands for the next two or three years without any alteration and the minimum of risks. The worst feature of the system is this assumption that past factors continue unaltered. Any real analyst of any use at all will know how dangerous it is to assume that past experience and trends for a company are to continue. Companies are always making changes, takeovers and that sort of thing. Imperial Tobacco takes over a brewery, then a huge US Motel chain. Do not these changes alter the experience ? Going back further to the beginning of my investment experience, the banks of Imperial China had a high credit rating and a low in the 1920s and were lampshades in the 1940s!

There are times when it’s appropriate to move one’s portfolio for tac- tical reasons. When you see a deteriorating economy it’s common practice to move out of capital goods shares into consumer shares. I remember doing that myself with a part of an insurance portfolio many years ago. I took the view in 1950 that engineering shares should be exchanged for food shares. I developed this theme after looking at the past movement in the Actuarial Index categories to see how the consumer goods shares survived without much price change in a recession, whilst the capital goods shares dropped like a stone. This was, I must admit, an early application of the

principle, years before the Americans invented it. However, we made sure that this historic theme was confirmed and supported by careful investment analysis of the companies and industrial groups concerned.

On the other hand, Professor Sharpe, one of the High Priests of the religion, would make similar moves, just from studying past history, rejecting any up-to-date analysis. It is obvious from his writings, extracted in paragraph 36.17, that Sharpe is no analyst. He dismisses careful analysis as “ providing accuracy that costs more than it is worth ”. What nonsense when you are dealing in millions of £s worth of shares!

I would advocate the use of s derived from fundamental principles. Strangely enough this is quite easy to do. I do not know whether members are aware of the “ Z ” score system designed by Mr. Teffler of City Univer- sity. This system uses advanced OR mathematical techniques of dis- criminant analysis, picking out those features of the balance sheet and profit and loss account that from past experience have discriminated between solvent companies and those that have gone into liquidation. Just by running Extel tapes through the computer, 1,200 companies can be ranked in order of their Z score in a few minutes. The ones at the bottom of the table are the highly volatile companies that will go broke smartly under adverse conditions, but at the same time will react very favourably in terms of share prices and profits if conditions improve. These are the “ high ” companies, as determined by fundamental analysis, rather than by use of obsolete and possibly out-dated statistics!

Mr. J. D. Campbell:— I read Robert’s paper on a railway journey to and


from London. I just made it as the train was half an hour late both ways. The paper cast something of a spell over me. There is a bewitching fascina- tion about the paper and the effort put into it.

The paper describes an attempt to be systematic and rigorous in the investment analysis leading to share selection with particular reference to relative values. As the saying goes “ if you cannot measure you cannot manage ” so it all fits in with modern management techniques in general.

There is, however, no attempt to introduce the model as a “ be-all and end-all ”. In no way is it claimed that the model and its application does away with investment judgement and experience—the model is married to the judgement and experience. And the process is subjective, I think, at every stage e.g. in gauging the most important attributes involved in the price formation process.

I don’t think I am a natural for the style of the model because I have always seemed to have a strongly sceptical streak in me about any single investment—thus having lost money from time to time. In earlier years, before we had teams of analysts, and pre-computers, investment decisions tended to be made by a man and a boy or perhaps just a man or a boy. Scepticism plus spread were perhaps the main requirements, letting our brain capacity do its best. However we do have teams and computers now. And so, despite scepticism—and tempting fate in saying it—one day does tend to follow another fairly symmetrically and indeed investment decisions (without any attempt to deny the cycle) do need to be based on rational expectations of continuity. It is not possible, in large scale investment, to proceed on the basis of irrational assumptions. Everything ultimately seems to be about a matter of degree, does it not, and Robert’s model seems to probe that degree, to find comprehensively the limits of peoples’ analytical competence in leading to share selection or, as the paper itself says, “ finding the optimal extent to which mathematics and statistical methods can be employed in portfolio management ”.

I was fascinated by the discussion of time horizons. In the modern era of very keen competitive measurements of performance, practitioners must be continuously conscious that an investment to be right over two years may be wrong for one year and so on. Continual activity following perfect judgement to keep the portfolio right all the time is unimaginable. Any attempt to apply too much reasoning and judgement to analysis defeats its object through sheer exhaustion—and what is judgement anyway, very difficult to define? And yet judgement is all, ultimately.

I found the paper fascinatingly bold e.g.

(i) It takes a really good swipe at the USA and MPT. Although I hope there is some reality in Mr. Sharpe’s (of MPT) predilections for the virtues of a well-diversified portfolio.

(ii) It dares to talk of investment in terms of science. The systematic ordering of thought is no doubt a scientific style but can it lead to laws for investment?

(iii) It even coolly reflects on the great Keynes himself and fits some of his concepts into the model.

Most fascinating of all is the fact that Robert uses the model and its methods in practice and in a life office. He is straightforward about this which makes me think he is being successful. That worries me! If you can’t beat them, join them! Could it come to that? Robert is quite frank about this. Right at the end of the paper he says in effect that the “ proof of the pudding will be in the performance ”. Trouble is he may have a 10 year start, although neither I nor anyone else will be admitting to that.


The paper will surely make practitioners think and come to think of it, that seems to be what the paper is trying to tell us we must do—and hard. Who could possibly deny that advice ? Perhaps, but only occasionally, a successful investor.

I much enjoyed this thoughtful and hard-worked paper and I feel quite patriotic that it was produced only a few doors away from my own place of work.

Mr. P. J. F. Taylor:—Mr. Clarkson’s objective in the research he has done and in writing the paper is clearly set out in the introduction: he is looking for a tool to assist him in the management of ordinary share portfolios. As a pensions manager rather than an investment manager, my objectives are rather different from Mr. Clarkson’s: I am looking for the best possible investment performance at the least possible cost. With this objective in mind my thoughts on reading the paper were: this is all very well, but at the end of the day does it produce results which are more cost effective than buying a pin and a copy of the F.T.? I can excuse a great deal of faulty logic if it makes money. I should have found the paper much more helpful if it had contained an independent assessment of the investment performance achieved with the help of the methods described in the paper. The model is stated to have been in use since 1975 with little change in detail, although there was some development up to September 1978. Perhaps Mr. Clarkson could say in his reply what the performance has been in 1979 and 1980? The performance in earlier years would be interesting too as possibly giving an indication of performance improving as the model increased in sophistication.

Performance is, however, to me only half the answer. It would also be interesting to know the running costs of the model including overheads and to compare these costs with the profits made by using the model instead of a pin.

In this context you may be interested, Mr. President, in some research which some colleagues and I engaged in recently. We tried to find out if some simple objective rules could be discovered for share selection which would result in performance significantly better than either selecting shares at random or investing in the Index. There was an obvious extension to the research if successful: it could be developed to provide for automatic share selection by computer.

We were fortunate to have access to a source of very high quality data covering a large number of shares representing a considerable proportion of the Index. The data were fed into a computer which was then programmed to invest £lm in the 50 shares having the highest values of a given parameter. The parameters chosen for initial study were: P/E, yield, earnings per share. Every 2 months the computer reviewed its portfolio, selling all the shares no longer in the top 50 and investing the proceeds in the shares which had displaced them. Additional runs were made where instead of choosing the top 50, a random selection was made of up to 50 shares from all shares where the parameter or the parameter relative to the market average exceeded a predetermined figure; also for the bottom 50 instead of the top 50, and for portfolios of 20 shares instead of 50.

Deals were assumed to take place at middle market prices and in half the runs dealing expenses of 7% were allowed for.

The progress of the portfolios was monitored and the results compared over a 10 year period and separately for the first 5 and second 5 years. Additional comparisons were made with the result of holding the initial portfolios unchanged over the 10 years and with random selections.


The results were that, ignoring expenses, three strategies looked promis- ing, two of which were tested to be significant at the 1% level (the third was not tested). The three strategies were:—

1st: Lowest earnings, 2nd: Highest yield, 3rd: Lowest P/E.

Two of these would probably not come as a surprise to anyone, but the lowest earnings one was a surprise certainly to us.

When dealing expenses were allowed for, no strategy performed better than buying and holding the index, although the shortfall was not significant in the best results.

We concluded that certain automatic selection rules could show a significant theoretical improvement over what I might call “ average ” performance if expenses were ignored. Before drawing any definite conclusions, we should have liked to have undertaken further work in three main avenues:

(i) to see if an inertial rule could have been devised which would have cut down the number of transactions and hence the dealing expenses without affecting the performance adversely

(ii) to eliminate possible bias as far as possible by testing the results over a different period of time using a different source of data, and

(iii) to devise a rule for practical use.

I regret to say we were unable to do any more because of a lack of resources.

Mr. D. I. W. Reynolds:—As with other speakers, I have only had a brief time in which to read this long and interesting paper. It is quite possible, therefore, that Mr. Clarkson will be able to answer my first question by saying that I have not read the paper properly! In Table 2 he shows the differences in the market values of 5 classes of stock, but I do not see where he describes what account he took of differences in income received from the stocks. The amount of the difference in income received from the stocks in the different categories could explain a substantial part of the difference between the categories “ sale ” through to “ buy/hold ”, although the difference between the “ sale ” and “ buy” groups is probably too great to be explained in this way.

A more general point is that I am happy that the author has done what I think is a satisfactory demolition job on Modern Portfolio Theory. I have had some doubts about the concept of risk (i.e. variance of portfolio values) used in Modern Portfolio Theory for some time and I think he has confirmed those doubts. My own feeling is that risk must be much more closely associated with bias—in particular bias measured as total expected return related to the index total return. In this connection I find the diagonal model intuitively more attractive than the basic Markowitz model. Previous speakers have already mentioned the short time horizon at present in the market and in particular I think this is pressed upon investors by the vogue for the measurement of investment performance. In the author’s case the short time horizon as described in paragraph 30.6 is 6 months, and yet there are not many pieces of advice given to me as an investment manager with a time horizon as long as that. I can think of one circular issue by stockbrokers that attempted to look ahead 5 years, the author of which had to make some very general assumptions about long term economic policy. Mr. Clarkson clearly concentrates his investment analysis on the long term and this is to be welcomed.


I think it would be of considerable advantage if we could extend the stock market’s time horizon, although that would involve several changes. As I have said, one reason for the short-sightedness of the market relates to portfolio performance measurement, but, it also relates to the way in which stockbrokers are currently rewarded. Their reward is based on turnover, although the majority of services they are now providing relate to economic and investment analysis. Clearly, with the way he uses investment analysis Mr. Clarkson would be prepared to pay for the analysis—and so would I— even if it resulted in no direct dealing. The third factor is the short time horizons of our political and economic masters and there is little that we can do about that.

There are, however, disadvantages to us as investors. Directly, we are criticised for the short-term nature of the markets and our actions in them but indirectly the market’s short time horizon may create difficulties for the companies in which we invest. Consider the case where a company might be prepared to go ahead with a long-term project, but because this will hit its earnings in years 1 and 2 it may not be able to proceed because the share price will fall with consequent increasing risks of takeover. The author’s model is able to incorporate the benefits of a project of that sort and I expect that he would be quite amenable to a company management who approached him saying “ we want to do this project but it will hit our earnings in the first two years ”. However, in general the market does not look beyond current and next year’s earnings and would take the contrary attitude.

I think Mr. Clarkson’s paper is the first positive development I have seen towards lengthening the time horizons of investors and for that reason above all I welcome it.

Mr. D. C. Damant:—I speak as an Efficient Market Theorist. I take a very flexible view of many of the consequences. I take a pretty hard line on the basic truth of the theory. The efficiency of markets seems to me chiefly based not so much on theoretical considerations but on the evidence of the consequences, that is on the study of the performance of portfolios. Both abroad and in this country large portfolios have shown no ability to outperform the appropriate market averages to an extent greater than could be achieved merely by chance, and often the performance is dis- tinctly inferior to the averages. For these reasons I am convinced of the strong form of the Efficient Market Theory.

One of the important things about this discussion, which is reflected in Mr. Clarkson’s paper, is the emphasis currently given in this country to betas. Betas are a definition of risk but only part of Modern Portfolio Theory; and in addition I do not think that betas are handled properly. Perhaps betas have achieved a good deal of publicity because they are something which fund managers can add to their traditional methods without feeling too uneasy, as they might about other aspects of MPT. I agree with earlier remarks that fundamental betas might be better, and with the doubts expressed about the stability of betas. I also have evidence to support Mr. Clarkson’s view (in his paragraph 24.11) that, shares in (effectively) large capitalisation companies have higher betas than shares in smaller companies—possibly for the reasons he gives.

However, my main point on the paper is to say that Mr. Clarkson’s proposition in his paragraph 35.14, that the theory and results which he puts forward are “ of course totally incompatible with the Efficient Market Hypothesis” is false. Certainly the words “ of course ” and “ totally ” should be deleted. I would say that I could find a number of


keen MPT people who would agree completely with the sort of mechanism which Mr. Clarkson sets out. They would agree, in accordance with Mr. Clarkson’s model, that some people have estimates of this order, and some have estimates of that order, and together with various other considerations the market prices are made. If the final price were not right as an estimate of the future of a company there would be other people, not in the market so far, who would come in and put the price right; (I would agree that there might be different risk preferences). And a result (says the Efficient Market Hypothesis) is that it is very difficult to find inefficiencies. Of course they exist and it is very possible that Mr. Clarkson has the competence to find them, But there are not many people like that and, if Mr. Clarkson succeeds, he will be the exception that proves the rule.

Perhaps I could summarise the position as follows. Mr. Clarkson and some of his predecessors in this type of enquiry are trying to find a model for the way in which fund managers and other investors act in deciding which shares to buy and sell. This is of course a very interesting line of enquiry. To some extent, perhaps, what is normally called Modern Portfolio Theory reflects on this type of enquiry; but on the whole MPT is dealing with a different problem—it deals with the consequences of models such as Mr. Clarkson’s, the statistical nature of stock exchange prices and the consequences for share selection and the performance of portfolios. So perhaps we are talking about two areas of enquiry. Also if Mr. Clarkson and people like him were to change their jobs the market would eventually become inefficient, although I suspect that a very large number of analysts and fund managers would have to be eliminated before significant inefficiences arose. In the meantime I am very glad to have Mr. Clarkson working to make the market efficient so that I can operate with the quantitive techniques which MPT has developed.

Mr. D. J. Kirkpatrick, closing the discussion, said:—I am grateful for the opportunity of closing this discussion. Clearly a paper of this length and interest prompts many questions and comments. There are many detailed points I would have liked to make but I will confine myself to some more general comments. The technical aspects of this paper have been dealt with adequately already in the discussion and I will choose to ignore them.

I would like to start with the end of the paper, that is Part V, and compliment the author on what is to me a convincing criticism of Modern Portfolio Theory. As quoted in paragraph 34.7, Lorie and Brealey state that:—

“ The relative prices of securities are determined by the expected return to the investor and also by the uncertainty about the return.”

This statement I find difficult to argue with. Unfortunately, the MP Theorists then part company with the Fundamentalists by misunderstanding and misusing the concept of uncertainty. They express it as being equivalent to the variance of the return on the portfolio thus sub- stituting volatility for uncertainty. An examination of historic price movements is carried out in order to determine the sizes of the risk components and it is then assumed that the results can be extrapolated into the future. The theory wrongly assumes that any changes in these components will be gradual. The sort of conclusion reached is that there is less risk in investing in G.K.N., Courtaulds or I.C.I. than there is in investing in Marks & Spencer or G.E.C. Small wonder that the practical investment manager has difficulty in accepting MPT as a useful management tool.


Apart from the dynamic (as opposed to static) quality of the risk components, MPT has the serious weakness of being based on false assumptions, in particular the Efficient Market Hypothesis. This Hypothesis is very ably refuted in Section 35 of the paper before us. I would like to add that many of the inefficiencies in the stock market arise from the uncertainty everyone feels when trying to project future returns. Lorie and Brealey are correct in saying that uncertainty about future returns is a determinant of relative price movements. A pity that MPT does not recognise the conflict here between uncertainty and the Efficient Market Hypothesis. In summary, MPT is based on unrealistic assumptions, it does not adequately recognise that its parameters change over time and it does not allow for uncertainty about future projections.

Moving now to the main body of this paper, I am not fully convinced that the Market Equilibrium Model does not suffer from similar weaknesses.

First of all the assumptions. In Section 4 the author assumes that the attributes are independent variables. What about, for example, gearing and return on capital? Whether or not gearing is deemed to be a good thing for a company depends very much on the return that the company earns on its capital. Later, in paragraph 6.13, R and G are assumed to be independent. Unlike Mr. Plymen, I would have thought it self-evident that they arc not and indeed the author in paragraph 6.20 describes the ability of high-growth companies to finance expansion from retained

earnings. Secondly, in paragraph 4.22 it is assumed that is mono-

tonic for all attributes. Again, what about gearing—a good thing if not carried to excess? In paragraph 6.31 the function used for gearing, F,(B) is defined as decreasing as B, the level of borrowings, increases. How is that realistic? Thirdly, in paragraph 5.7 it is assumed that everyone can forecast all dividends and earnings in the future and that if two shares have identical dividends and earnings, both present and future, these prices must be equal. This is probably a harmless assumption, but it does not make any allowance for marketability, for instance; one of them may be a close company or it may be subject to takeover rumours. Fourthly, in paragraph 6.9 the author says that two years into the future any detailed earnings projections are tenuous in the extreme. He can therefore assume that the earnings for two different shares will never cross. I would disagree with the premise here. Earnings tend to be volatile for a variety of reasons and it is often easier to determine the medium or long-term trendline than it is to assess results that might be due immediately. To appreciate this point, one merely has to look (for any major company) at the range of estimates being produced by stockbrokers for results due shortly in respect of 1980, a period which is now history and about which it should be possible to know a great deal. Given the short-term volatility in the trading ex- periences of companies, it is surprising to me that more work is not done on longer-term trends. On the question of the earnings of two companies crossing in the future, I would have thought that one could project a number of cases where this would probably happen. Cyclical companies obviously, but less obviously it could be argued that property companies will show very rapid growth in earnings over the next two to three years because of the coincidence of rent reviews but that thereafter the best estimate is for growth in line with inflation. These projections will cross with many projections of more moderate but more sustainable growth over a longer period. Fifthly, in paragraph 6.35 it is assumed that no allowance for marketability is required.

Some of these assumptions are probably not very harmful but the point


I am making is that in constructing a mathematical model designed to relate to the ordinary share market both MPT and the MEM illustrate a conflict between true representation of the market and the necessity for a workable model. This conflict is recognised particularly in paragraphs 6.32 and 6.33. If I may quote:

“ Since the absolute level of borrowings that is generally accepted as prudent varies between companies in different industries, the definition of B will involve serious problems of standardisation, and F4(B) is therefore unlikely to vary over time in a sufficiently regular manner to justify the incorporation of a variable parameter. A further, and more fundamental, reason for not incorporating a variable parameter in F4(B) (the function of gearing) is that the revel of borrowings is one of the many factors that has to be taken into account in estimating the growth rate G. If two variable parameters are affected by a common influence, problems of multicollinearity would arise and the statistical robustness of the whole model would be seriously impaired.”

It seems to me that reality has given way to statistical stability. Next I would like to comment on the change of parameters over time.

I understand from the paper that the relative price residuals are used for short-term price judgements and that more medium-term judgements are made from examination of the values of the variable parameters. Current values of these are compared with historic values in order to determine whether or not they are too high or too low. How is allowance made for permanent or semi-permanent changes in the importance of attributes? A pity that such a crucial part of the problem is not included in the paper. How does the model cope with, for instance, the decline in importance of the P/E ratio over recent years? The effects of current cost accounting could gain considerable prominence in investment decisions over the next few years.

How will the model accommodate this? During the late 1960’s and early 1970’s in America for fifty or so favoured stocks the only attribute which mattered was the growth rate. The valuations of this attribute in terms of the prices paid for these shares increased remorselessly over a long period until the fundamental value of these shares in terms of the rate of return one could expect from them over the future had been left well behind. I wonder how the MEM would have operated in this situation. After all, past history would have shown that investors valued growth in earnings very highly. To me this example illustrates the danger of basing investment decisions on what it appears other investors are prepared to pay. To be fair to the author, however, he does stress in Part III of the paper that a fundamental review of a company is carried out whenever a control limit is breached in the model.

There must be a temptation, though, particularly when testing the efficacy of the model to make claims for it which are not justifiable. In Section 16 a switch from G.K.N. and Tube Investments into a group of other stocks is described. The reasoning behind this switch, as described in the paper, has little to do with the model and indeed other institutions did just such a switch at around that time using similar reasoning. Similarly, in the Whitbread example, I would have thought that similar results could have been achieved by using a standard overbought/oversold indicator. Further, in Section 19, the author examines the possible correlation between the three market parameters (the sensitivity functions for dividends, growth and overseas profits), and the All-Share Index. The author defines the primary trend as upward when all these parameters indicate improving investment confidence and conversely. He concludes that up to May 1979

L


the primary trend was up and that the market effectively peaked out then and has been in a primary down trend ever since. Unfortunately, as he mentions, the index has reached new all-time highs since then. The explanation of how this fits with a primary down trend I find unsatisfactory. The very large oil sector certainly did contribute much to the rise in the All-Share Index but since the turn of the year this sector has fallen very substantially, while it is not obvious from a chart of the Index that it is not still in a primary up trend. Perhaps the parameters are wrong. It could be argued that the rise in the market since the last election has had little to do with dividends (the reverse yield gap has widened), it has had little to do with increased growth prospects (obviously) and it has had little to do with the desire to invest in specifically U.K. assets (institutions have been investing abroad at record rates). What it has had to do with are falling inflation and improved prospects for falling interest rates and a good gilt market.

The third major problem I have with the paper is that the model does not incorporate a measure of uncertainty. After all, uncertainty about the future returns is what differentiates the ordinary share market from the gilt market. It is precisely in order to reduce this uncertainty that we do all this fundamental analysis. It is the consideration of this uncertainty that leads an institution to buy shares in Marks & Spencer rather than M.F.I. even though the best estimates of the future rates of return available from these shares might suggest the opposite course. This uncertainty is what the non-Modern Portfolio Theorist calls risk and it varies considerably from share to share.

My comments up to now have been concerned with what I believe to be the weaknesses of the model and indeed what I believe would be weaknesses of any mathematical model of this nature. The Market Equi- librium Model as described could be nevertheless a useful tool and there are, in my opinion, some attractive features to it. Had the paper been a third the size I may have had time to go into them. I would not want Robert to think that because my comments have been critical in tone I do not admire the work he has done. This is a fascinating and thought-provoking paper, but I do have reservations as to its usefulness.

Before sitting down, I would like to comment on one more topic, namely Rate of Return Analysis. In Section 29 of the paper the author criticises this approach but I do find his criticisms rather weak. It is not necessary, as stated in paragraph 29.3, to assume that after five years, dividends will grow in line with corporate profits generally. If other assumptions lead to apparently unacceptable answers, then one must think again about these assumptions or about the reasons why the conclusions appear to be wrong. In paragraph 29.5, the author states that dividend growth rates must be revised frequently to take account of changing economic and financial conditions. This is not necessary if the model uses relative growth rates adjusted for inflation. These are much more stable, particularly if cyclical influences are treated separately.

I do not understand why dividend payout ratios, overseas exposure and balance sheet strength cannot be allowed for explicitly. The effect of each of these influences must be quantified when deriving the long-term growth rate. More importantly, all possible influences can be incorporated in the derivation of the growth rate including the element of doubt or uncertainty. This last is so important in investment decision-making that any model which does not incorporate it in its projections is unacceptable. The most serious weakness of Rate of Return Analysis is said to be the assumption that the market structure is logical in compound interest


terms. I am not sure that this matters much, particularly for the larger long term fund. Sooner or later the market must reflect the real returns achieved in terms of earnings and dividends. It appears to me much more satisfactory to relate share prices to their true value in this way than to relate them to what other investors have, in the past, been prepared to pay. The fact that the market is not logical in compound interest terms of course creates the anomalies from which the follower of this approach can gain. It appears to me that Rate of Return Analysis is much less subject to constraints than is the Market Equilibrium Model as any appropriate influence or attribute can be included.

Mr. R. S. Clarkson, replying to the discussion, said:—We have had a very full and interesting discussion, covering not only matters of principle but also numerous detailed aspects. The matter of principle that has attracted most comment is the extent to which judgement should overrule the numerical results obtained from the model. While few speakers specifically said that I had in fact achieved just the right balance between calculation and judgement, I seemed to detect a roughly even split between those who criticised me for using too much judgement and those who criticised me for using too little. Perhaps, then, I can deduce that the balance I chose is reasonably satisfactory.

Several speakers asked whether it is reasonable to assume that the earnings growth rate and the dividend payout ratio are independent. Since this is a complex matter, I shall deal with it in my written reply. Mr. Reynolds asked whether differences in dividend income could affect the results set out in Table 2. At the beginning of the period, the average dividend yields of the five groups were very similar. Accordingly, differences in dividend income would have had only a very minor, and random, effect on the results, and I decided that the additional work involved in adjusting for dividend income could not be justified.

Most of the discussion has focussed on certain detailed aspects of the market equilibrium model and on the considerations involved in its practical application. Mr. Wilkie alone has challenged the whole theoretical foundation of the model and has been severely critical of many of the detailed steps in its construction. I shall do my best to answer his criticisms in my written reply after I have studied the transcript of his remarks. However, I should like to make it very clear here and now that, taken in the context of the paper as a whole, I regard many of his detailed criticisms as unjustified. I am criticised, for example, for lack of rigour when defining the attributes used in the model. Since much of the paper discusses how best to quantify certain investment criteria that are normally dealt with only in qualitative form, I feel that these criticisms are particularly unwarranted.

I have considerable sympathy for Mr. Damant’s refreshingly straightforward view of market efficiency. He believes that inefficiencies certainly exist but that most investors cannot identify them consistently. However, he accepts that a few investors may indeed have the competence to outperform on a consistent basis. This is in sharp contrast to the more usual academic viewpoint that any attempt to outperform on a consistent basis is doomed to failure.

Although Mr. Grant approved of much that I had done, he was less than happy with some of the claims I made regarding the superiority of the model over all other methods. The point I am trying to stress in these comparisons is that most investment research approaches deal either with a small part of the market or with a measure of attractiveness over one


specific time scale. Mr. Grant in fact expresses a similar view when he says that there are many half-truths, or part-truths, in the market, and that every dog tends to have his day. By covering a large number of shares and providing a framework for making investment judgements over short, medium and long time scales, the market equilibrium model is sufficiently comprehensive to keep the whole market under constant review.

Mr. D. M. Simpson wrote:—It seems to me that the long term institutional investor, when determining which ordinary shares to buy, must try to determine the value to him, using his assumptions and his judgement about the future, of the future income flow that will derive from his investment. At this stage he is probably following the description of investor behaviour outlined in paragraph 17.11 of the paper although he does not need to assume, as I think is suggested in paragraph 3.8, that the aggregate of all investors in the market place results in the price pattern being consistent with his assumptions. If he believes that his assumptions are correct then in the long run he can ignore short term variations in prices. The value to him will come out in the end.

Having said that I agree with the author that it is necessary to try to have some standardised approach to the problem and if possible to turn the criteria of assessment into some numeric or ranking form. As I see it one can approach this in three stages.

Firstly, one assesses the current position. By this I do not just mean the current dividend yield. Adjustments will require to be made to allow for the effect of current cost accounting, to allow for a particularly small, or indeed a particularly large payout ratio, and also, if this seems necessary, for the fact that the most recently available figures may be cyclically depressed or cyclically inflated. Some of these adjustments, particularly the last one, involve a substantial element of judgement on the part of the investment manager—and he will have to make a lot more judgements before he has finished. At this stage we have an adjusted initial dividend yield which in terms of the items in Mr. Clarkson’s model described in paragraph 6.36 incorporates both E and F1(R).

Secondly, the investment manager must make some assumptions about future dividend growth. I agree with Mr. Clarkson that this should be done numerically and not merely, as he discusses in paragraph 6.6, in a descriptive way such as “ very good ” etc. The factors in assessing earnings and hence dividend growth are many and varied and are outlined in Section 10. However, I would also include in the assessment of earnings growth, a contribution, which might be positive or negative, from the company’s overseas interests whether direct exports or actual overseas operations. At this stage, the investment manager has another element in his valuation process—the dividend growth. Relating this to Mr. Clarkson’s model. this element would encompass F2(G) and F3(A) and at least a part of F4(B).

Now there are various ways of putting together these two elements namely the initial adjusted dividend yield and the dividend growth and Mr. Clarkson discusses some of these in Section 29 under the heading “ Dividend Valuation Models ”. I think the method I have described of standardising initial dividend yield and of taking account in the assessment of the growth rate of overseas exposure and balance sheet strength overcome the difficulties he mentioned in paragraphs 29.5 and 29.6 although I suppose it depends on what he means by “ explicitly ” in paragraph 29.6. The investment manager’s judgement has been used, certainly, but explicit figures are arrived at. As I have already indicated, I do not agree


with his objection in paragraph 29.7—it does not seem to me that it is necessary for the market to agree with the investor’s assessment at all times provided he is happy about the value of the investment to him.

I would then have a third element to allow for uncertainty. As Mr. Clarkson says in 5.3 (iii), there are large margins for errors in forecasts. However, some are larger than others. For example, I think one can make more certain forecasts for the earnings growth for a company such as Marks & Spencer than for I.C.I. Furthermore. as he indicates in paragraph 6.30, highly geared companies may not be able to survive a temporary downturn in their fortunes even although they are operating in a market where there is long term growth. I would thus have an adjustment for the “ riskiness ” of the investments which would incorporate part of the author’s item F4(B) but would also include other items.

Incorporating this third element is more difficult and is perhaps best done by classifying shares into different categories of riskiness and then making some corresponding numerical adjustments to the ranking produced from the other two elements.

So far I have been talking about taking a long term view, i.e. about an investor who, unlike Keynes as quoted in paragraph 24.8, is prepared to buy for 25, something: he thinks is worth 30. while recognising that it is possible that the market might value it at 20 in three months’ time.

However, there is clearly a need to make shorter term judgements about price movements. This is necessary for smaller funds or for larger funds at the margin leaving the bulk of their portfolio as core holdings. It does seem to me that this is what the author’s paper is really about. The graphs that he shows in Chart 3 and the description he gives in Section 15 of the movements in Whitbread appear to vindicate his system, but I was a bit worried about the number of times he appeared to have to exercise his judgement. In 15.2 he overruled his model because he thought that the likelihood of a Conservative victory in the imminent General Election would be good for Brewery share prices; in 15.4 he introduced the special factor of 1·08 for the development of Chiswell Street—I wonder if he could give us some indication as to why he chose 1·08 and indeed why he increased this to 1·15 later on. If I understand it right, he attributes the sensitivity of share prices to short term changes in market sentiment and I agree with this, but it is not clear to me that his model has produced the answers. It does seem to me rather that the author exercised good judgement.

What he is really saying here is that the factor 1 + ƒ (Zn) gives him scope to overrule or adjust the figures produced by his model and he alters the elements that make up this factor according to his judgement.

I am all in favour of trying to standardise an investment team’s approach to different sectors and different companies and I think it is vital that an institution devises some framework within which this can be done. However, at the end of the day it does seem to me that it is the judgement of the team—whether of earnings growth, the likelihood of a general election, or the weight to be attached to the development of Chiswell Street—that is important and I am not convinced that this can be wholly reflected in a model.

Mr. A. B. FitzGerald wrote:—The development of the market equilibrium model is an interesting attempt to explain the price formation process in the market. Moreover, it is plausible that the discipline which the use of the model imposes can lead to superior investment returns in the short to medium term.


The major disadvantage in using a model of this kind, however, is that there is no attempt to evaluate the long-term returns available in the market. While there is a recognition that the consideration of long-term growth potential forms some part of the price formation process, long-term values are continually masked by the consideration of short and medium- term attributes.

Furthermore, the author has misguidedly concluded that superior long- term growth prospects automatically imply superior long-term returns (23.5 “The earnings growth rate provides a direct measure of expected long-term relative (price) performance ”). It is, of course, a question of whether these growth prospects are discounted in the current price. The market equilibrium model makes no attempt at this evaluation.

The severest criticism must be reserved, however, for the author’s attack on Modern Portfolio Theory (MPT). There are misunderstandings, misinterpretations and unfounded conclusions throughout the paper and, contrary to the author’s belief, there is not a single piece of evidence presented from which one can conclude that: “ the market equilibrium model offers the better scientific framework (than MPT) for the management of ordinary share portfolios ”. Some specific examples of these distortions are shown later.

On a general note, it is difficult to understand the author’s eagerness to refute MPT. Since the market equilibrium model concentrates on individual share evaluation, one would have thought that MPT could provide valuable, additional tools. For example, if superior returns can be obtained consistently by the use of the market equilibrium model, measures of portfolio diversification—as provided by MPT—should be utilised so that diversification is minimised and these superior returns maximised.

Investors should certainly be wary in applying some of the theoretical propositions of MPT, especially in the UK market environment. How- ever, MPT can still be of immense practical value if understood and applied intelligently.

Some specific points arising from the discussion of MPT: There is a gross misunderstanding of the nature of systematic risk (beta)

and residual risk. 18.23 and 18.24 contain the implication that high growth shares will have

beta coefficients greater than one and low growth shares will have beta coefficients less than one. Consequently the author arrives at the totally erroneous conclusion that: “ On the basis of Modern Portfolio Theory, high growth shares should have underperformed the market from May 1979 to November 1979. when the F.T.-Actuaries All-Share Index fell more than 20%”.

High growth shares do not necessarily have high beta coefficients and vice versa. If anything, high-growth shares tend to have low systematic risk and high residual risk since the growth of the company is not closely correlated with general economic growth. For example, the three shares whose attributes are discussed earlier in Part III, Electrocomponents, Whitbread and Tube Investments had historic beta coefficients at May 1979 of 0·75, 0·90 and 1·02 respectively. Electrocomponents, presumably assigned the highest earnings growth attributable at that time, had the lowest beta coefficient and outperformed the market during the following bear phase.

Looking at performance at the sector level over the period May-Novem- ber 1979, sectors with beta coefficients less than one outperformed those with beta coefficients greater than one by 10% on a capitalisation weighted basis and by 4% on an unweighted basis.


This misunderstanding of some of the basic concepts of MPT and the erroneous conclusions drawn also invalidate arguments raised in Part V of the paper.

While the author’s scepticism towards naive applications of MPT is justified there is no foundation to the claim that: “ Someone trained in MPT will reject the concept that prices reflect fundamental attributes such as earnings and dividends, because the textbooks from which he has learned the theory of his trade expound the view that there is no empirical evidence to support this concept.” (34.4).

Shackled with this delusion, it is little wonder that the author has been reluctant to investigate the possibility of combining aspects of MPT with the market equilibrium model.

Advocates of MPT do not reject the concept that prices reflect fundamental attributes. Moreover, most textbooks devote considerable attention to the association between traditional management methods and MPT and to the relationship between MPT measures and fundamental attributes.

36.22 states: “ Applications of the Sharpe model depend on the assumption that the values of B are stable from one period to the next. The recent experience described in Section 18 shows that even this assumption is incorrect and can lead to unsatisfactory investment performance ”.

As pointed out in (i), the recent experience described in Section 18 (i.e. over May-November 1979) did not undermine the validity of this assumption. Even a crude application of the Sharpe model could have been successful over that period.

Applications of the Sharpe model depend on the assumption that portfolio betas are reasonably stable from one period to the next. The fact that “individual stock betas may show instability is not disputed. Moreover, the author may also like to note that it is now common practice in the United States to predict beta coefficients by examining fundamental attributed This helps to overcome the stability problem.

The market equilibrium model appears to be a useful aid in formulating short to medium-term stock selection policies. It is difficult to understand, however, how and why its development has led the author to attack the validity and usefulness of MPT The attack, as presented in the paper, is irrational and unfounded.

Mr. Clarkson subsequently wrote—Several speakers asked whether it is reasonable to assume that the earnings growth rate and the dividend payout ratio are independent. Like Mr. Grant and Mr. Plymen, I believe that a slight departure from independence is of no practical consequence, but I agree that the matter has to be investigated. As Mr. Kirkpatrick points out, high-growth companies will tend to have lower than average payout ratios. However, after studying scatter diagrams of earnings growth against payout ratio I found that the correlation was rather weaker than I had expected, and I concluded that I was in fact justified in assuming that earnings growth and payout ratio were orthogonal to one another. If there had been any serious error in this assumption, there would have been a high degree of correlation between the earnings growth parameter g and the payout ratio parameter r. I examined the weekly values of these parameters very carefully and was relieved to find that they behaved quite independently.

Mr. Grant asked how I obtained my rather complicated expression for F1(R), and suggested that a linear function might be used instead. Since investors tend to think in terms of the dividend cover (C, say) rather than


its reciprocal, the payout ratio, R, I want a 2-parameter curve F(C) that decreases from 1 when C = 1 to the limiting value r when C tends to infinity. A reasonably obvious choice is to use an exponential decay function, giving

with the parameter d determining the speed at which F(C) approaches the limiting value r. Towards the end of “ stage 2 ” in the evolution of the statistical fitting process I experimented with various values of d and found that d = 1·5 gave the most satisfactory fit. It can be seen from Figure 19 that the curve of F1(R) is concave upwards when d = 1·5, and accordingly a linear function would not be suitable.

Mr. Kirkpatrick examined my gearing function F,(B) and—-on the premise that gearing is a good thing if not carried to excess—asked if it is realistic for F4(B) to decrease as B, the level of borrowings, increases. The various effects of a change in gearing can perhaps best be illustrated by a simple example. Suppose that a company with no borrowings has an expected earnings profile that corresponds to the earnings growth function G. Suppose now that the company decides to improve its future profits by taking on a moderate level of borrowings to finance profitable new activities. The future earnings profile is thereby improved and it now merits a higher earnings growth function Since the proportionate increase in F,(G) resulting from the upward revision in the growth rate will almost certainly outweigh the proportionate decrease in F4(B) resulting from the higher borrowings, the expected share price will in fact increase, thereby confirming Mr. Kirkpatrick’s assertion that gearing “ is a good thing ”.

Mr. Kirkpatrick also wondered how the model might have coped with the so-called “ nifty fifty ” stocks in America which rose to earnings multiples well above anything that could be justified on fundamental grounds. The relentless increases in the earnings multiples would have caused the earnings growth parameter g to increase well above its previous range, but the relative price residuals of these stocks would not, in general, have breached the upper control limits. This type of behaviour is described in 6.47 and 6.48. The model provides the framework which alerts the investor—in this case through the value of g—that a significant change in market structure has occurred and that the ratings might not be justified by fundamental values; it is then up to the investor to review the situation and decide whether or not to sell. The same two principles—measurement by means of the model followed by investment judgement based on these results and a review of the entire situation—can be seen in 15.4 in the Whitbread example. Although the share price had risen sharply, further “ buy ” recommendations were being made. However, the model indicated that the price might be too high, and further investigation suggested that the rating was unsustainable and that the shares should be sold.

Mr. Wilkie was severely critical of my whole approach, and suggested that much of the mathematical formulation lacked rigour or was based on faulty logic. His comments at the meeting may have given the impression of being a sustained and penetrating attack which uncovered serious weaknesses in the model. However, on studying the transcript of his contribution I find that I can classify nearly all of his critical comments into one of three categories. Firstly, there are criticisms about specific steps in the theoretical formulation which, in the context of the paper as a whole, appear quite unwarranted. Secondly, there are lengthy comments about potential weaknesses in my simplifying assumptions; these apparently critical comments are then followed by a brief statement that in all the circumstances the assumptions are in fact reasonable. Thirdly, there are a


few important points of principle where Mr. Wilkie and I can do no more than agree to disagree. He begins by criticising my lack of rigour in not specifying whether attributes are continuous or discrete or both. It seems obvious to me that the attributes I define are in theory continuous but that in practice it is usually more convenient to use discrete values. I do not accept that the mathematical rigour collapses as a result of this lack of specification. Although the simplifying assumptions in 4.4 are, after much detailed comment, accepted as reasonable, I should like to stress two points. Firstly, although my simplifying assumptions are indeed similar to those made in MPT, they do not include the Miller and Modigliani assumption that investors are indifferent as to whether an increment in wealth is in the form of dividends or an increase in the market value of their holdings of shares. This is a point of the greatest importance and I shall return to it later. Secondly, in pointing out that rights issues and other issues of shares by a company can invalidate assumption (iii), Mr. Wilkie does not add that special factors of this type are dealt with in detail in 6.38 and 6.39. This is only one of many instances where a point raised by Mr. Wilkie is discussed fully in a later part of the paper.

The mathematical formulation in Section 4 attracts strong criticism from Mr. Wilkie. He says, for example, that I do not prove that P(A) is con-

tinuous or that even exists. I am using the function P(A) as a

theoretical concept which will assist in the development of a practical investment model in exactly the same way as lx is used in the development of life contingencies. The force of mortality µx, which is defined as

is also a purely theoretical concept, but it is a key element in the

development of important results in life contingencies. Similarly, I am

using to translate an intuitive concept of market equilibrium into an

explicit mathematical model, and I consider Mr. Wilkie’s criticisms in this area to be unjustified. In the remainder of Section 4 I introduce my measure of price sensitivity of P with respect to A and describe in general terms the statistical methods that are required to identify the equilibrium position. My definition of price sensitivity follows directly from the earlier formulation, has an immediate practical interpretation, is statistically robust, and can be regarded as a formalisation of the various comparative factors described in Section 7. I therefore do not accept Mr. Wilkie’s criticisms of my definition, and in view of the very high level of random noise I regard his suggested alternative measures as quite unsuitable. Mr. Wilkie then suggests that I should either have used a compound interest model or have examined the distribution of log (P/E) using multiple regression techniques. The disadvantages of a compound interest approach are discussed fully in Section 29: the multiple regression approach has been explored on numerous occasions, but the resulting modes—such as the Weaver and Hall system described in Section 30—have not lived up to the expectations of their designers, mainly because it is very difficult to reconcile the numerical output with the investment conclusions arising from conventional methods of share appraisal. As explained in Section 7, my model can be regarded as a restatement of these conventional share appraisal principles in a more rigorous and quantitative form.

Mr. Wilkie’s references to allegedly unsound logic in 5.12 and 6.13 are somewhat misguided; investors have more to worry about than the limiting properties of theoretical functions which involve a time horizon of a

M


thousand years. My introduction of the earnings growth function G in 5.15 is criticised by Mr. Wilkie on the grounds that it is not a function or a value at all. I maintain that G is indeed a function in the accepted mathematical sense of a mapping by some definite rule from a domain set, in this case a subset of all shares in the market, to a set of real numbers. Mr. Wilkie then suggests that my extension of this subset in Section 6 to include all shares is without foundation since I do not make explicit allowance for the probability distribution of future earnings. Here yet again I must disagree. In view of the tenuous nature of future estimates, future variability of earnings is of far lower importance in the price formation process than their general level. Accordingly, in the construction of the price model priority must be given to accommodating the general level of earnings, and this is achieved by using the key function G. Then in 6.27 and subsequent paragraphs I discuss very carefully the factors such as overseas exposure and balance sheet strength which could lead to a change in the ranking of this growth measure if economic circumstances change, and I gradually build up the detailed earnings profile function

I know of no other price model which deals so explicitly with future variations in earnings resulting from changes in specific aspects of the economic background, and I do not accept Mr. Wilkie’s criticisms in this area.

A more general criticism levelled by Mr. Wilkie is that insufficient reference is made to the concepts of statistical theory. As described in 2.4, a very delicate balance has to be struck between statistical stability and goodness of fit if the model is to be successful as a practical investment tool. Guided wholly by general investment experience and considerations of stability, I concluded that this delicate balance could best be achieved using three fitted market parameters and the detailed formulation set out in 6.40. In view of the very high level of random noise, I very much doubt whether statistical investigations could have played any useful part in the construction of so detailed a model. Once the model has been constructed, it is of course essential to assess whether it offers a satisfactory framework for investment management. But here again I am distrustful of narrow tests of statistical significance. I prefer direct tests involving the relative price residuals together with general assessments of whether, in all the circumstances, the behaviour of the model is satisfactory. A very direct and powerful test involving the relative price residuals is discussed in 13.6, and general assessments of the broader aspects involved are summarised in Section 25.

Although Mr. FitzGerald (in his written contribution) and Mr. Wilkie were very critical of my presentation of Modern Portfolio Theory, neither of them made any attempt to refute my claim that the theoretical foundations of the capital asset pricing model, the centrepiece of the whole theory, are somewhat insecure. The market equilibrium model is, in the first instance, a static model which identifies the market equilibrium position at one point in time and measures certain features of the market structure. One of these features is the dividend parameter r, which can be interpreted as an indifference ratio whose value-would always be 1 if the Miller and Modigliani assumptions regarding rational investor behaviour were valid. Not only do I obtain values for this parameter that are significantly less than 1 (which is equivalent to the expected price of a share increasing as the payout ratio increases), but the values also change over time in a manner that is consistent with changes in investor sentiment. Furthermore, the Weaver and Hall model, which uses the multiple regres-


sion approach preferred by Mr. Wilkie, also shows that the expected price increases with payout ratio. I regard these results as strong evidence that the capital asset pricing model is unsuitable as a framework for practical investment management, since it does not take account of an important feature of market structure, namely the variation of price with dividend payout ratio. Mr. FitzGerald and Mr. Wilkie, on the other hand, recommend diversification strategies using measures of risk derived from the capital asset pricing model. This is one important area where we can only agree to disagree.

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Market Equilibrium Model Management Ordinary Share