New concepts and algorithms for portfolio choice

APPLIED STOCHASTIC MODELS AND DATA ANALYSIS, VOL. 8, 159-178 (1992)

NEW CONCEPTS AND ALGORITHMS FOR PORTFOLIO CHOICE

GUNTER DUECK IBM Germany, Heidelberg Scient$c Center, Tiergartenstraje f5, W-6900 Heidelberg, Germany

AND

PETER WINKER Universitat Konstanz, SFB 178, Postfach 5560, W- 7750 Konstanz, Germany

SUMMARY

The paper studies the design of optimal (bond) portfolios taking into account various possible utility functions of an investor. The most prominent model for portfolio optimization was introduced by Markowitz. A real solution in this model can be achieved by quadratic programming routines for mean- variance analysis. Of course, there are many reasons for an investor to prefer other utility criteria than returnlvariance of return in the Markowitz model. In the last few years, many efficient multiple purpose optimization heuristics have been invented for the needs in optimizing telephone nets, chip layouts, job shop scheduling etc. Some of these heuristics have essential advantages: they are extremely flexible and very easy to implement on computers. One example of such an algorithm is the threshold-accepting algorithm (TA). TA is able to optimize portfolios under nearby arbitrary constraints and subject to nearly every utility function. In particular, the utility functions need neither to be convex, differentiable nor ‘smooth’ in any sense. We implemented TA for bond portfolio optimization with different utility criteria. The algorithms and computational results are presented. Under various utility functions, the ‘best’ portfolios look surprisingly different and have quite different qualities. Thus, for a portfolio manager it might be useful to provide himself with such a ‘multiple-taste’ optimizer in order to be able easily to readjust it according to his own personal utility considerations.

K E Y WORDS Optimization Threshold accepting Portfolio optimization

1. INTRODUCTION

In his pioneering work Markowitz presented the standard model of mean-variance analysis in portfolio choice. In this model, an investor seeks a ‘best’ compromise between (high) return of his assets and (low) risk. The risk of a portfolio is measured in terms of variance of the return. The Markowitz model is smooth enough to allow the evaluation of efficient portfolios in practice. Nowadays, many software tools are available to compute the so-called efficient frontier of specific mean-variance models. These algorithms are essentially based on quadratic programming routines.

Of course, there are many reasons for an investor to prefer utility criteria other than returnlvariance of return. Meanwhile, there exists a fully exploited theory of utility and preference. Many different useful utility functions have been suggested. The drawback of several useful utility functions is that it is not possible to practically get solutions with common standard software. Utility functions of individuals representing preferences other than suggested by the mean-variance approach may look poor from a computer programmer’s point of view. In this case, it may be very hard to compute efficient portfolios.

875 5 -0024/92/ 030 1 59-20$15.00 01992 by John Wiley & Sons, Ltd.

Received 3 1 March 1992 Revised 10 June 1992

160 G . DUECK AND P. WINKER

In order to close this gap, we have examined the behaviour of optimization heuristics for portfolio choice. In the last few years, many powerful multiple purpose optimization heuristics have been invented for the needs of optimizing telephone nets, chip layout, job shop scheduling, etc. Most of these real life problems are mathematically awkward as well as complex and do not fit into elegant mathematical models. Nevertheless, clever heuristic algorithms can overcome these difficulties. These algorithms are heuristic, i.e. they do not compute exact optima, but solutions sufficiently near to the optimal value. The advantage of heuristics is often their flexibility and their easy structure which allows a quick implementation.

A famous heuristic is the classical ‘simulated annealing’ approach.2 In References 3 , 4 and 5, an even more efficient form, the ‘threshold accepting’ (TA) algorithm was introduced. TA is able to optimize portfolios under nearly arbitrary constraints and subject to almost every utility function. The utility functions in particular need neither to be linear nor differentiable nor ‘smooth’ in any sense. Using a TA optimizer, the investor can feel free to optimize within a model which fits into reality.

In this paper, we study a bond portfolio optimization problem under various utility considerations. We report some computational results which show, especially, that under different utility functions efficient portfolios turn out to be very different from each other.

One can conclude that the risk is not some canonical measure. Efficient portfolios in one model may be poor in another. Thus, an investor should be careful to be aware of his own understanding of risk or utility. Furthermore, it would be interesting to find out whether an investor can choose a portfolio as good as that given by a heuristic like TA. If a really excellent portfolio manager designs a portfolio, how close is it to the ‘optimal’ of a predefined model? Is the machine superior to a human being? We hope that we can test people in the near future.

2. A STANDARD MODEL FOR BOND PORTFOLIO CHOICE

2.1. The basic model

The investor is to choose a portfolio from m different available bonds. A portfolio is represented by a vector:

m

j = I (XI, x2, ..., x m ) , c x j= 1 (1)

For each j c (1, ..., m), a fraction Xj of the investor’s capital is invested in bond j . The performance of a portfolio is measured after a fixed period of time (one year, half a year, one month). Of course, the performance of a portfolio depends on the interest rate at the end of the time period. As we do not know these interest rates in advance, we have to decide under uncertainty. However, we have some idea of the evolution of interest rates in the near future. We say that the true state of the world after a certain time period may be equal to one of several imaginable scenarios.

For bond portfolios, a possible scenario is fully described by the interest rates for the different bond classes (short term, long term, ...). In our investigated model the investor describes several scenarios i E (1, ..., n) which could define the state of the world at the planning horizon.

Then the investor assigns a (subjective) probability to each of the n scenarios. This assignment represents the necessary expert knowledge for the optimization routine. Of course, for a high return on a portfolio, high quality expert knowledge is essential. However, it is not

ALGORITHMS FOR PORTFOLIO CHOICE 161

Table I. A scenario

Years to maturity Interest rate

0 1 2 3 4 5 6 7 8 9

10

8.5% 8.6% 8.6% 8.7% 8.8% 8.9% 9.0% 9.1% 9.2% 9.2% 9.2%

the subject of this paper to provide this knowledge. Here, we consider only optimization stra- tegies to choose the ‘best’ portfolios under different (subjective) probability distributions on the scenarios.

For a scenario i E (1, ..., n), let prob ( i ) be the probability that scenario i occurs at the planning horizon. In our model, a scenario for bond portfolio choice consists of Table I, which indicates for any number of years (0, 1, ..., 10) the mean interest rates for fixed deposits or bonds with that time to maturity. Once we know which scenario occurs we can evaluate the bond portfolio (rather accurately) with the help of the assigned table data, because the price of a bond at the end of the time period is a function of the interest rates at that time and the remaining years to maturity. As the effect of a change of the interest rate at the end of the time period depends on the remaining years to maturity, a covariance structure between bonds is introduced by a discrete scenario distribution (cf. Table 11). This covariance structure essentially reflects the effects of the duration, the size of the coupons and the possible changes between normal, flat and inverse interest rate structures. Bond price relations change seriously if the market scenario expectations change from, for instance, ‘interest rate falling in the future’ to ‘interest rate rising in the future’.

Table 11. Scenario Distribution 1

Scenario 1 2 3 4 5

Probability 0.3 0 - 5 0.15 0-035 0.015 -~ ~

Years to maturity

0 1 2 3 4 5 6 7 8 9

10

7.0% 7.2% 7.4% 7.6% 7.8%

8.2% 8.4% 8.6% 8.8% 9.0%

8.0%

7.5% 7.7% 7.9% 8.1% 8.3% 8.5% 8.7% 8.9% 9.1% 9.3% 9.5%

8.0%

8.4% 8.6% 8.8% 9.0‘4’0 9.2% 9.4% 9*6‘% 9.8%

10*0%

8.2% 8.5% 8.7% 8.9% 9.1% 9.3% 9.5% 9.7% 9.9%

10.1% 10.3% 10.5%

9.0% 9.2% 9.4% 9.6% 9.8%

10.0% 10.2% 10.4% 10.6% 10.8% 11 *O%

162 G. DUECK AND P. WINKER

The expected return of a portfolio is the weighted average of the returns under the different scenarios. Let rij be the return of bond j under scenario i, i.e.

rij:= interest payment for bond j for the time period -price of the bond j at the start of the time period +price of the bond j at the end of the time period under scenario i

Then

;= 1

is the return of the portfolio ( x l , ..., X m ) under scenario i . The actual return r of a portfolio is a random variable rather than a deterministic value, because it depends on the scenario which occurs at the planning horizon. The expected return of a portfolio is the expected value of the random variable r:

n n m ?:= E r := c prob(i)ri = c prob(i) ri;xj (3)

i = 1 i = l j = 1

2.2. Variance as a risk measure

An obvious goal for a portfolio manager is the maximization of the expected return. On the other hand, a high return is mostly possible only under high risk. In the classical mean-variance model of Markowitz,' the risk of a portfolio is defined as the variance of the return:

riv := Var(r) = E (r - E r)'

n

i = l (4)

If riv is small, the actual return r tends to be close to its expected value. If riv is large, r may be sometimes large, sometimes small.

If a portfolio has a small variance, the investor can be quite sure to obtain a steady cash flow. Thus, if a portfolio manager has to choose between several portfolios with the same expected return, he will prefer the one with the smallest risk. This is the basic assumption of the Markowitz model.

2.3. The mathematical problem

Mathematically, an optimizer for this model is to solve: 2

minimize 2 prob(i) (2 ri;Xj - f ) i = 1

m

j= 1 s.t. C x j = 1, O < X j < l for j = 1, ..., m and f 2 r d

Here, rd is the demanded minimum return. Traditionally, a solution can be obtained by a quadratic programming routine. With such tools it is also possible to incorporate numerous linear side conditions on the portfolio vector (XI, ..., X m ) . In general, one can solve the above


task under the k additional constraints:

a l l x l + - * . + a 1 m x m < bl

a k l X 1 + - * - + akmXm < b k

Given a set of constraints, a portfolio ( X I , ..., xm) is called efficient if it is the solution of the minimization problem. Varying the demanded return f d , one obtains a set of efficient portfolios. Graphically, this set of efficient portfolios constitutes one frontier of the set of feasible portfolios in a return/variance of return plane and therefore is called efficient frontier. We will use this term for the sets of efficient portfolios for different utility functions.

In Reference 6 , Markowitz describes conditions allowing us to find the efficient frontier by means of standard linear or quadratic programming. We will not discuss in this paper whether these conditions are or are not satisfied in practice. However, the ‘beauty’ of these algorithms demands a certain ‘smoothness’ from the data.

2.4. Deficits of the concept

It seems easier to compute an efficient portfolio for a mean-variance approach rather than one for an ‘ugly’ utility function. There are already software tools applicable to this kind of problem, whereas for many possible utility functions no efficient algorithm is known. For example, it has not been possible to include integer restrictions or to optimize utility functions that are not differentiable or not convex.

However, this advantage becomes questionable with the development of new heuristic optimization algorithms such as Simulated Annealing’ or Threshold Accepting. 3,4,5 These algorithms allow us to optimize almost every utility function object to nearly every set of constraints with a reasonable consumption of resources. Although they do not obtain an optimum exactly, the approximations obtained by these algorithms are sufficiently near to the optimal value for real life applications, as many implementations have shown. Furthermore, this observation is strengthened by the fact that these algorithms are provably convergent.

Hence, it is possible to compute a near-to-optimum solution in a utility framework which any investor can define specifically for his own use. In the next section we will discuss utility functions as alternatives to the mean-variances approach.

3. ALTERNATIVE MODELS

3.1. Variance versus generalized semi-variance

As we argued above, the variance is probably not a universal risk indicator. It might be better to consider the danger to have a poor performance as risk rather than to minimize the deviation from the mean. In Reference 9, Fishburn quotes some empirical results on decision makers’ risk attitude leading to the same conclusion that a suitable measure of risk should not depend on the mean of the portfolio. A more natural model should measure the risk that the performance will fall below some unfavourable point or line, which we will call the reference point or line. We formulate such a model as an example. The objective function measures the ‘generalized semi-variance’. The reference point or line could be represented by an alternative investment (fixed deposits, index portfolio). This measure of risk represents a special case of Bernell Stone’s Generalized risk measure (c.f. Reference lo), but it is far from being the only computable one with our TA algorithm.

1 64 G. DUECK AND P. WINKER

Let us consider the same model as above. For any scenario i, let Ri for the risk measurement. Consequently, the new problem is (rir = risk follows. Solve:

minimize rir := 2 prob(i)(ri - Ri)’ i = l

r, Q R, m

j = 1 S.t. C xj= 1, O < X j < l for j = 1, ..., rn

under the additional constraints:

be the reference point with reference line) as

allxl -I- -.. + almxnr < b~

a k i X i + * - * + UkmXm < bk

Suppose that Ri = R for all ic (1, ..., m). In this case, rir measures the riskiness of the performance being worse than R (which may represent the performance of an asset without risk). A portfolio manager is often confronted with the postulate of a customer ‘to perform better than index**’. It is easy to achieve the performance of a bond market index: One can just buy those bonds which are the base for the computation of that index. A challenging task is to beat the index. If Ri is in the above model the performance of an index portfolio under scenario i and rd is greater than the expected return of the index portfolio, then rir measures the risk not to beat that index.

Note that the defining formula of rir is not a polynomial in the Xj. Formally, rir is a sum of quadratic functions. However, we sum up only those terms for which ri ,< Ri. Hence, the subset of indices, which are relevant for the sum, depends on all the variables determining ri.

Thus, the subset of indices depend on the portfolio (XI, ..., Xm). In particular, it is not a formula which can be computed with a linear or a quadratic program. For our heuristics, this does not matter. It would be possible also to impose some ‘integer’ conditions on the xi (if one wants to invest only a multiple of a base unit in every bond j ) .

3.2. Different attitudes versus risk

We have seen in the preceding that it might be favourable to minimize semi-variance with a fixed reference line instead of variance. However, this utility function is far from being the only rational one. For example, there is no reason for an investor to judge the risk as the sum- of-squares of differences.

The utility function of an investor with low risk aversion may be of the form:

prob(i)(ri - Ri) i = l

r, 4 R,

(7)

We could even think of an exponent of the term (Ti - Ri)’ less than 1 leading to a not convex function. And a very risk averse investor might value the risk exponentially, i.e. for some constant c > 0:

. . r, < R,

If Ri represents the performance of a bond index under scenario i , the investor could also

I65 ALGORITHMS FOR PORTFOLIO CHOICE

intend to minimize the probability to perform worse than this index. Then he has to minimize:

2 prob(i) i = I

r, Q R ,

(9)

Finally a very pessimistic investor perhaps tries to maximize the return in the ‘worst case’. Then this investor is to maximize:

min ri I

The TA algorithm implemented for bond portfolio choice allows us to optimize portfolios for nearly every kind of utility function expressing the real risk attitude of an investor. We will see some computational results for some different utility functions in the last section of this paper.

Cobb-Douglas function

A practical problem in the use of mean-variance analysis as well as in some of the alternatives presented above is their two-dimensionality. Two portfolios are in general not directly comparable in their qualities. In particular, for two solutions lying on the efficient frontier the portfolio manager has to decide according to his own attitude as to the riskiness of the two solutions.

In References 11 and 12, another measure for portfolio choice is discussed. The geometric mean (also known as the Cobb-Douglas function) is a one-dimensional measure for the quality of a portfolio, i.e. there is only-one quantity representing the quality of a portfolio instead of two in the standard model. Consequently, every two portfolios are comparable under this measurement. There is a linear order of all feasible portfolios which allows us to decide which portfolio is the ‘best’.

We give a derivation of this function. Let us consider the choice of a good portfolio in the beginning of many consecutive time periods.

Assumption A

i E 11, . . ., n) the probability prob(i) that scenario i occurs at the end of the time period. The investor chooses at the beginning of each time period a portfolio; he knows for any

Assumption B

The state of the world at the end of each period is chosen to be i E (1, ..., n) with prob (i), i.e. the scenarios are distributed according to the probabilities prob (i), i = 1, ..., n. (This means that the vector of the probabilities really represents the true distribution at the end of the period.)

Assumption C

The investor reinvests his whole capital after any time period.

We look at this model as an iterated random process. The investor chooses a portfolio. Through a random generator a number i E [ 1 , . . . , n) is drawn according to prob (i), i = 1, . . . , n.


If i occurs, the capital of the investor changes from C to C * (1 + r;) , where Ti is the return of the portfolio under scenario i. Suppose the investor chooses in every round the same portfolio. Suppose furthermore, after every step k = 1, ..., 1 the scenario is ik. Then, the final capital after 1 steps is

C . (1 + ri,)(l + Ti2 ) (1 + ri,)

Now let us argue with the law of large numbers. If i is large then with high probability any scenario i E [ 1, . . . , n) occurs = 1 - prob(i) times. Hence, with high probability, the final capital is about

- -c. (1 +r,)(/*prob(l))..-(l +,.n)(/.prob(n))

One can prove the following in this model with the help of the law of large numbers. The best portfolio is that which maximizes

GM := fi (1 + ri)prob(i) i = l

where GM is the weighted geometric mean of the returns ri.

situation described by the three assumptions above. One can show the following not only for the above random process, but also for the

‘In the long run it is best to choose at the beginning of each time period a portfolio which maximizes the Cobb-Douglas function

f l k

GM := (1 + rj)probk(j) i = l

where [ l , ..., n k ] is the set of possible scenarios after period k and where probk is the assigned probability distribution.’

Since any monotone transformation of this function represents identical preferences, we may maximize the logarithm of the above function alternatively:

n k

log GM = probk(i) - log(1 -t r;) . I = 1

The main advantage of the Cobb-Douglas function GM is its one-dimensionality. We can compare portfolios directly.

There are two difficulties in using this measure for finding optimal portfolios. First of all there are no standard software tools to optimize the geometric mean under various side conditions. Furthermore it is questionable whether an investor is usually completely unrestricted to other limitations to be able to maximize utility in the long run.

In order to solve the first problem, Young and TrentI3 undertook the task of approximating GM by mean and variance. They concluded that ‘empirical evidence indicates that . . . the approximation involving only the mean and variance produces quite accurate estimates of the geometric mean. ’

Although obtaining good approximations for some examples, the authors admit that there might be cases when this method fails to give results of high quality.’


Our TA heuristic is also flexible enough to optimize portfolios directly according to the geometric mean GM.

Of course, as mentioned above, not every investor is to act ‘best possible in the long run’. It is even very probable that most investors consider their portfolio choice from a quite different point of view, especially concerning the performance in the short run.

4. FURTHER APPLICATIONS

The flexibility of the TA algorithm implemented for the portfolio choice allows us to optimize quite different utility functions as seen in the preceding sections. Furthermore, it is even able to optimize these functions under nearly arbitrary side conditions, in particular also under non- linear side conditions. This aspect opens some further fields of application. We will give some examples.

4.1. Multiple-currency portfolios

There is no problem with embedding the possibility of multi-currency portfolios in the TA system. It is much more difficult to start the system with adequate scenario probabilities. Of course, the currency relations have to be part of the scenarios.

4.2. Cash-flow control

Some investors, e.g. pension funds, have a particular need for a steady cash flow from their portfolio. It is rather easy to formulate such requirements as side conditions.

4.3. Depreciation and losses

Losses for a portfolio could lead to depreciations if the value of the bonds drops below its book value. In our model, we try to avoid losses by the use of suitable risk measures. However, in the real world the rule is ‘no return without risk’, i.e. under a typical scenario distribution losses cannot be completely excluded. But we can implement conditions which help to avoid losses in the book-value if the investor is not willing to have an effect of losses on his balance sheets. On the other hand, an investor may like to have losses in book-values once there are losses at all. This might reduce his tax duties. All these features (risk of book-losses, limitation of book-losses under all scenarios, ‘only book-losses, if any’) can be built into our model.

4.4. Income taxes

Often it may be useful to incorporate various taxes into the model. In multi-currency portfolios different taxes in different countries may apply. Although tax tables usually do not represent smooth functions, TA has no difficulties with this point.

4.5. Transaction costs

Transaction costs are often neglected in mathematical concepts ‘for simplicity.’ In reality they may be essential. If the investor’s opinion on scenario probabilities changes, it is usually not possible to exchange the complete portfolio from an efficient solution in the former model to an efficent portfolio in the new one. However, the reinvestment of returns can carefully


respect the revised model conceptions. Again, these aspects can be implemented in our computational model.

5 . OPTIMIZATION

For the optimization of different utility functions we used a TA algorithm as discussed in Reference 3. TA is an iterative local exchange procedure which acts here on the set of portfolios respecting the imposed constraints. One starts with an arbitrary feasible initial portfolio (say, that consisting of bonds with the highest return), i.e. a portfolio respecting the imposed constraints. In each step one changes the current portfolio slightly into a new feasible one. One compares the ‘qualities’ of the two portfolios, i.e. the two values of the objective function (utility, risk, ...) for the two portfolios. Afterwards a decision is made whether the new portfolio is ‘acceptable’ or not. If the new portfolio is acceptable it serves as the ‘old’ portfolio for the next step. If it is not acceptable, the algorithm proceeds with a new change of the old portfolio.

5.1. TA algorithm for minimization

choose an initial portfolio choose an initial THRESHOLD T > 0 Opt: choose a new portfolio which is a stochastic small

perturbation of the old portfolio compute AE := quality (new portfolio)- quality (old portfolio) IF A E < T

IF a long time no decrease in quality or too many iterations

IF the threshold is too low to promise further improvements

THEN old portfolio := new portfolio

THEN lower THRESHOLD T

THEN stop GOT0 Opt

In case of portfolio optimization, exchange steps are fictitious shifts in the portfolio:

(a) buying bonds for cash (b) selling bonds against cash (c) exchanging bonds for other bonds (selling and buying for approximately the same

Any of these exchange steps includes two assets (bonds or cash). The exchange trials may follow a predetermined order or be chosen randomly. It is examined after every exchange trial whether the risk or utility is acceptable. TA accepts every new portfolio which is not much worse than the old one.

The convergence of this procedure is assured, and for a given tolerance the number of iterations grows polynomially in the input size.’ However, in any practical experience with TA we found that TA converges using a rather small number of exchange steps. It is sufficient to choose a running time proportional to the number of unknowns, i.e. the number of different assets. TA has already been successfully applied to extremely large optimization problems like chip placement. We used TA here on about 100,000 unknowns and got much better placements than by any other procedure.

amount)


Principally, TA does not necessarily stop with a proved global optimum. Therefore, there may be the problem of trust in the results of TA. In our applications, TA gives excellent solutions which differ only neglectibly from the global optimum (in cases where the optimum is known the problem is easy). In problem areas with integer conditions, TA usually gives results about one per cent away from the optimum. Note that standard software is not available for any non-linear problem with integer conditions. If the optimization models deal with conditions like round-lot constraints, there is no standard algorithm to solve this problem. However, TA is not affected by integer constraints. In the implementation of TA, only the exchange steps have to be defined in such a manner that only admissible portfolios are considered.

We have implemented two TA-variants. One acts simply on the set of feasible portfolios. The other is also allowed to accept infeasible portfolios and the ‘degree of unfeasibleness’ is embedded as penalty in the objective function. The penalty is chosen in such a way that TA ends with a feasible portfolio, For some risk measures this approach seems to be more efficient.

We now give a specific TA algorithm for portfolio choice. This algorithm enables us to compute efficient frontiers for different utility functions as well as concrete portfolios under certain constraints.

5.2. The TA algorithm for portfolio choice

The TA algorithm being described in what follows is essentially an exchange algorithm. For the design of a local search algorithm we have to define what a ‘small perturbation’ of a current solution should mean. The idea is that one exchanges a fixed fraction between two x- variables. In our model x I may stand for the cash position. Then a change 1 -+ j represents the buying and j --t 1 the selling of bond j . A change j + k represents the selling of bond j and buying of bond k. In this case differences caused by different prices of the bonds are settled with the cash position.

The objective function used for the portfolio optimization is essentially the utility function of the investor, for example one of the functions studied in former sections.

The TA algorithm for portfolio choice is as follows:

choose an initial feasible portfolio ( x l , ..., xm) with expected return F 2 rd

choose an initial THRESHOLD T > 0 initialize the array fraction of size k FOR i := 1 TO k DO BEGIN

FOR j : = 1 TO steps DO BEGIN

UNTIL ( a z b ) A ( a = 1 V x, 2 fraction ( i ) ) DO BEGIN

END IF ( a = 1) THEN BEGIN

choose randomly a, b € 11, . . . , rn)

try to buy fraction (i) of bond b from cash compute AE := quality (new configuration)- quality (old configuration) IF A E i T THEN old configuration := new configuration

END

170 G . DUECK AND P . WINKER

IF (b = 1) THEN BEGIN

try to sell fraction ( i ) of bond b for cash compute A E := quality (new configuration)- quality (old configuration) IF A E c T THEN old configuration := new configuration

END IF ( a z 1 ~b # 1) THEN BEGIN

try to exchange fraction ( i ) between bonds a and b compute A E := quality (new configuration)- quality (old configuration) IF AE < T THEN old configuration := new configuration

END compute new THRESHOLD

ENDDO ENDDO

The parameters fraction and steps define parameters for the running time of the optimization, i.e. more ‘steps’ and more different ‘fractions’ lead to a higher quality of the results. Once the initial parameters are fixed and a feasible portfolio has been chosen, the algorithm runs k rounds. In any round it chooses steps times two different bond types (one of it may be the cash position) and tries to exchange a fixed amount of these types in the current portfolio. The new portfolio is accepted if it is not ‘much’ worse than the old one regarding the given utility function. The threshold is lowered proportionally to the improvement in the construction of the portfolio.

6 . AN APPLICATION TO PORTFOLIOS O F BUNDS

As an example of computing according to different conceptions of risk, we considered portfolios consisting of bonds of the Federal Republic of Germany (Bundesrepublik Deutschland), which are commonly called ‘bunds’. In order to obtain results, a planning horizon of 52 weeks was chosen. For simplicity we excluded bunds expiring in less then 52 weeks and bunds with maturity after 2001. We were left with the problem of optimizing portfolios out of 71 different bunds.

With a TA algorithm as described above, we could optimize different risk functions under fairly comparable conditions. Given a required return and the form of the risk function (if necessary including a fixed reference point) a portfolio of minimum risk (maximum geometric mean for the Cobb-Douglas function) was searched for.

Because of the high performance of this TA application, it was possible to obtain segments of the efficient frontier of the set of feasible portfolios.

The investor in our model was provided with some capital of amount C . The investor’s borrowing was restricted to 1% of C . As a further incidental condition, the expected return of the portfolio should be greater than a given one, i.e. we minimize the risk under condition

The computational results already seen in Figure 1, as well as the following ones, are based

The segments of efficient frontiers for different utility functions given in Figures 1 and 5

rd.

on the bond prices of March 5, 1990.

resulted from the scenario distribution given in Table 11.


1025 10.50 10.75 11.00 11H 11.10 11.75 1Z.W Rhrn

Figure 1. Efficient frontier (variance)

1016 10.50 10.n 11.00 1111 11.60 11.76 I200 1 dun

Figure 2. Particular bonds in the optimal portfolio

1225

I

11

For March 1990 this table might describe a reasonable assumption. The discussions on the expenses for the reunification of the Federal Republic of Germany and the German Democratic Republic led to a strong increase of interest rates in the first two months of 1990. Most experts assumed that this dramatic reaction of the bond markets was exaggerated. It was expected that interest rates would drop slowly during the year.

Figure 2 is affiliated to Figure 1. It represents the development of some particular German government bonds in the optimal portfolios when rd is increased.

In Figure 3 we will see that the efficient frontier shown in Figure 1 is not as ‘smooth’ as it seems to be at a first glance. In fact, it consists of rather linear segments of increasing gradient.


1119 11s IIA I I ~ III IIM IIY I Ik

Figure 3. Detail of Figure 1

#.DO 929 P.60 0.75 10.00 10.!6 10.10 10.76 t1.00 112S 11.60 11.71 1!.00 nhm

Figure 4. Efficient frontier (variance)

The curve proceeds from one linear segment to another if the fraction of particular bonds in the efficient portfolios changes sharply. In Figure 3 there is such a point at x-coordinate 11.9. One bond has reached its maximum, one is raising from zero, one is falling to zero. These ‘phase transitions’ are responsible for non-smooth points in our efficient frontier curves. For illustration we focus on detail of Figure 1.

In the following figures we will give segments of the efficient frontiers for different risk measures. Figures 4, 6 and 7 have been calculated for the variance riv, whereas the efficient frontier in Figure 5 has been calculated for the generalized semi-variance rir with reference point 10.5%. Every point on the efficient frontier corresponds to a near-to-optimum portfolio with an expected return equal to its x-coordinate and smallest risk with respect to the chosen risk measure.

Of course, we might be interested in the quality of these portfolios under other risk measures. The dotted lines in Figures 4-7 indicate the ‘risk’ of these portfolios in terms of an alternative risk measure (semi-variance in Figures 4 and 6, variance in Figure 5 and geometric


11.00 1115 11.w t1 .n I Z P O 1115 dmll

Figure 5. Efficient frontier (semi-variance (10.5%))

mean in Figure 7). We call this risk the affiliated one. It becomes clear that efficient portfolios for a given risk measure in general will not be efficient for alternative risk measures.

From Figure 4 and Figure 5 we see that variance and semi-variance with reference point 10.5% show a quite different behaviour. A much better insight is provided by a look at the portfolios which give rise to the efficient frontier.

Let us focus on the point 11.5%. We compute portfolios with minimum risk for different risk measures and with the side conditions that the expected return ? is greater or equal to 11.5% and that the investor’s borrowing is restricted by 1 % of the invested capital C. The results are shown in the Tables 111-VII.

Optimizing the geometric mean under the incidental conditions described above, we obtained a portfolio containing only one type of bond (6.375 Bund v.86) and with a borrowing

Table 111. Portfolio A optimized for variance

Bond Per cent of invested capital

8.250 Bund v.83 IV 73.33 7 .000 Bund v.85 26-87

Cash -20111*80 DM

Table 1V. Portfolio B optimized for semi-variance (10 * 5%)


6-000 Bund v.78 I1 66- 10 I-OOO Bund v.85 3.75 6.375 Bund v.86 30.16

Cash -378.65 DM


Table V. Portfolio C optimized for exponential risk function


8.250 Bund v.83 I1 3.55 8.250 Bund v.83 IV 61 -67 8.250 Bund v.84 6-72 7.000 Bund v.84 1-27 7.000 Bund v.85 26.80 ~

Cash - 1122.68 DM

Table VI. Portfolio D optimized for semi- exponential risk function (10.5%)


6.000 Bund v.78 I1 65.14 7.000 Bund v.85 6.93 6.375 Bund v.86 27.92

Cash 754.40 DM

Table VII. Some characteristics for Portfolios A-D

Utility risk Portfolio A Portfolio B Portfolio C Portfolio D

Expected return 11 -501 11.500 11.502 11.500 Variance 1 a4746 2.8293 1 .4924 2.7852 Semi-variance (10.5%) 0.1403 0.0928 0 - 141 1 0.0929 Geometric mean 1.11517 1.11495 1.11498 1.11494

Table VIII. Scenario Distribution 2

Scenario 1 2 3 4 5

Probability 0.05 0.15 0.3 0.3 0.2

Years to maturity

0 1 2 3 4 5 6 7 8 9

10

7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0%

8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8%

10.0%

8.5% 8.7'70 8.9% 9.1% 9.3% 9.5% 9.7% 9.9%

10.1% 10.3% 10.5%

9.0% 9.2% 9.4% 9.6% 9.8%

10 - 0% 10.2% 10 * 4% 10.6% 10*8Vo 11 .O%


too0 9.050 WOO L1M I100 9.tIo t.3W * Figure 6 . Efficient frontier (variance) for scenario 2

extended near to the maximal value. The geometric mean for this portfolio is 1.12375, the expected return 12-286%, the risk in terms of variance is 6.8271 and the risk in terms of semi- variance with fixed reference point at 10.5% is 0.2358. With this example it is evident that in general different risk measurements lead to quite different optimal portfolios with quite different characteristics.

We also computed segments of the efficient frontier for different utility functions for the scenario distribution 2, which is shown in Table VIII. It represents an investor’s view such as ‘I think the interest rate will very probably increase.’

As an example we show in Figure 6 a segment of the efficient frontier for the mean-variance analysis. Regarding the affiliated semi-variances we see once more that minimizing one risk measure may not give good results for other risk measures.

In Figure 7 we recognize again Figure 6 now with the affiliated geometric means instead of

Figure 7. Efficient frontier (variance) for scenario 2 (detail)


loo0 1.160 $100 $160 LW dun

Figure 8. Particular bonds in the optimal portfolio

the affiliated semi-variances. Both of the curves change their behaviour in the proximity of 9-13 on the x-axis. If one chooses a variance-optimized portfolio in this area one may very well trust in this choice because the geometric mean is only moderately increasing to the right of this point. Portfolios might be considered to be really well chosen if their quality is satisfactory for many risk measures. Of course, we could also easily implement optimizers that minimize risk 1 under the same constraint risk2 < ... .

Figure 8 is affiliated to Figure 7. It represents the development of some particular bunds in the optimal portfolios when rd is increased. Here one can directly observe the ‘phase transition’ in the proximity of 9.13 on the x-axis. A similar phase transition is to be seen at 9.23 (cf. Figures 6 and 8).

Table IX. Portfolio E optimized for variance


10.750 Bund v.81 26.12 7.500 Bund v.83 73-84

Cash 3444-00 DM

Table X. Portfolio F optimized for semi-variance (9.5%)


7-500 Bund v.83 111 14.64 7.500 Bund v.83 85.36

Cash 189.80 DM


Table XI. Portfolio G optimized for exponential risk function

Bond

10.250 Bund v.81 0.87 10.750 Bund v.81 24.86 7.500 Bund v.83 74.24

Per cent of invested capital

Cash 3890.30 DM

Table XII. Portfolio H optimized for semi- exponential risk function (9.5%)


7-500 Bund v.83 111 14-47 7.500 Bund v.83 85.49

Cash 4606.40 DM

Table XIII. Some characteristics of Portfolios E-H

Utility risk Portfolio E Portfolio F Portfolio G Portfolio H

Expected return 9.750 9.750 9-750 9-750 Variance 0.6589 0.8370 0-6636 0.8388 Semi-variance (9.5%) 0.0700 0.0638 0 * 0700 0-0639 Geometric mean 1 -09745 1 * 09747 1 *09745 1 * 09744

In Tables X-XIII, we present some optimized portfolios for scenario distribution 2. In this example, 9.5% has been chosen as the reference point for semi-variance and rd = 9.75. With the investor's borrowing restricted to 1% of C, we obtained the portfolios shown for different utility functions.

Optimizing the geometric mean under the incidental conditions described above, we have obtained a portfolio containing only one type of bund (7.500 Bund v.83) with geometric mean 1.0964, expected return 9.866 and variance 1 - 1056. The risk in terms of semi-variance with fixed reference point at 9.5% is 0.0699.

7. CONCLUSION

From our studies with different risk measurements it became clear that there is no universal risk measure, Under different risk measures the optimal portfolios differ considerably in their qualities.

I There is a strong need for many different optimizers for various risk functions. 1


A portfolio manager should work on a good expert opinion (‘What is the interest rate in the near future?’). He should know what amount of what kind of risk he or his client is willing to accept. A portfolio manager should not sit down and create an excellent portfolio. This is a purely algorithmic, tedious, difficult, monotonous and lengthy work: This is the computer’s job.

It would be interesting to provide experts in bond markets with a scenario distribution and to ask them for good portfolios in some given risk sense. Of course, a TA algorithm should generate better results. However, what really is the difference in US$? Is an excellent expert able to find a very-near-to-optimum solution or not? We doubt it. Up to now, we only have experience of human solutions in industrial areas (e.g. job scheduling, chip placement). Here, machine solutions are usually 5% to 30% better than solutions provided by human experts.

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