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Money illusion, beta perceptions, and capital marketsegmentationJ. A. SCHNABEL aa Faculty of Management, University of Calgary , Calgary, Alberta, T2N 1N4, CanadaPublished online: 10 May 2007.

To cite this article: J. A. SCHNABEL (1982) Money illusion, beta perceptions, and capital market segmentation, InternationalJournal of Systems Science, 13:7, 789-798, DOI: 10.1080/00207728208926389

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INT, J, SYSTEMS SCI., 1982, VOL. 13, No.7, 789-798

Money illusion, beta perceptions, and capital marketsegmentation

.J. A. HCHNABELt

This paper' shows t Iru.t. in t-he contcxt of the capital asset. pr-icing model (CA P:\I). thecxiet cncc of money iflusiou on t.hc part of Horne investor» mduocs segmentation in thefinancial market. That is, investors who suffer' from money illusion would gravitatet owaa-ds a su bset of the seeurit.ie.... available in the market, whereas investors who donot suffer from money illusion would gravitate towards t.he complementary subset. ofsucut-itie«. This result» in the invalidut.ion of an irnportant. corollary of t.he CAP.:\I,i.e. all investor« hold t.ho market. portfolio.

Much of current finance theory invokes the equilibrium model of capitalmarkets derived by Sharpe (H)(J4), Lintner (1965), and Mossin (J966)-usuallyreferred to as the capital asset. pricing model (CAPM). The use of this modelhas been eriticized as intellectua.lly barren due to its two eritical assumptions ofperfection and equilibrium in capita.l markets, e.g. Vickers (1978), The aim ofthis paper is to buttress this contention by showing that. the presence of moneyillusion induces disequilibrium and segmentation in the capital market. whichin turn results in the invalidation of the capital asset pricing model.

rn the first. section of this paper the concept of money illusion is discussed.The second and third sections of this paper develop two different pricingrelationships, Associated with each of these pricing relationships is one of twoassumed investor groups: one comprised of investors who suffer from moneyillusion and the second comprised of investors who do not. suffer from moneyillusion. The fourth section of this paper then considers how the portfoliochoices of these two investor groups interact. It is shown that a form ofcapital market segmentation is induced, with every member of each groupfocusing on proper subsets of the universe of assets available, As no investorwould want to hold the market portfolio made up of all assets, the criticalseparation theorem, which states that every investor would want to hold such aportfolio, is not valid, Therefore, the capital asset pricing model fails to hold,

1. Money illusionMoney illusion refers to the systematic failure on the part of an economic

actor to take account of inflation or purchasing power risk in his economicbehaviour, This concept permeates much of economics, For example, theidea of a trade-off between inflation and unemployment, as em bodied in thePhillips curve, is grounded on the assumption that. ,tt least. some participants inthe labour market exhibit. this type of behaviour. In their studies of theaggregate consumption function, Branson and Klevorick (1969) have argued

Received II March 1982,t Faculty of Managcment, University of Calgary, Calgary, Alberta T2N IN4,

Canada,

0020-7721/82/1307 0789 ~03'OO © 1982 Taylor & Francis Ltd2N2

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790 J. A. Schnabel

where

tha.t some consumers arc subject to money illusion because of their finding thatthc price level yields an independent effect on real consumption expenditures.

In contrast to the foregoing, the literature on -the theory of finance hasalmost completely ignored the concept of money illusion. Two notableexceptions are the debtor-creditor hypothesis of Kessel (l956) and theModigliani and Cohn thesis (1979) regarding the capital market's undervaluationof shares. The essence of the debtor-creditor hypothesis is that firms con­sistently fail to anticipate inflation correctly and the rate of interest negotiatedin current loan agreements fails to allow sufficiently for the increase in thegencrallevel of prices that in fact occurs. Consequently, a firm loses purchasingpower as a result of holding monetary assets as the nominal value of theseassets, fixed by contracts or other institutional arrangements, remains the same,while the real value of currency declines. Similarly, a firm gains purchasingpower to the extent that it holds monetary liabilities in an inflationary period,in that it borrows currency with greatcr value than it subsequently repays tocreditors. Thus, money illusion results in debtors gaining and creditors losingin the presence of inflation. The Modigliani and Cohn thesis is related to thiscffect. They argue that investors incorrectly capitalize equity earnings at thenominal interest rate rather than the economically correct real rate of interest.In addition, they fail to take account of the gain which accrues to corporateshareholders due to the depreciation in thc real value of corporate liabilitieswhich tUC denominated in nominal terms. The result of these errors of moneyillusion is a systematic undervaluation of corporate stock.

Having provided some background on the concept of money illusion, thenext section derives a model of capital market equilibrium assuming that allinvestors suffer from money illusion.

2. A pricing relationship for investors who suffer from money illusionAll the assumptions of the capital asset pricing model are invoked except

for the presence of uncertain inflation and money illusion on the part of allinvestors. Thus, a single-period setting is invoked, multivariate normality ofall rates of return as well as the inflation rate is assumed, and all capital marketimperfections except those which induce money illusion are ignored. Given theassumption of universal risk aversion, all investors will choose mean-varianceefficient portfolios. As investors in this section are assumed to suffer frommoney illusion, they choose mean-variance of nominal, instead of real, returnefficient portfolios. It is shown that investors who suffer from money illusionwill price assets according to the following relationship

E(i')=rF+[E(i'M)-rFJ$j (1)

$. = cov (rj , i'JI)) a2(r M )

i'j, i'M' rF = nominal return on the jth asset, the market portfolio, and thenominally riskless asset, respectively. Tildes denote random variables., Rcturn' is used synonymous with "rute of return'. In general,nominal quantities will be denoted by lower case letters, real quantitiesby capital letters. The market portfolio is formed by an investment inevery asset in proportion to its market value.

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Money illusion and market segmentation 791

E( . ), a'( . ), cov ( " .) = expectation, variance, and covariance operators,respectively.

Equation (I) postulates a linear relationship between the equilibrium nominalreturn on an asset and a measure of its risk, Pj' which is similar to the betameasure of the capital asset pricing model. Equation (I) may be consideredthe security market line for investors who suffer from money illusion. Pj is thebeta of asset j as perceived by such investors.

Consider an investor, indexed by the letter i, who suffers from moneyillusion. His utility function has the stochastic value of his nominal terminalwealth as argument. Given the assumption of normality of the inflation rateand all uncertain rates of return, his investment activity is guided by the rule ofchoosing a mean-variance of nominal return efficient portfolio. Thus, the ithinvestor's portfolio selection problem may be modelled as

subject to

where

min xi'D.,.xix'

(2)

(3)

Xi = vector of wealth proportions invested in the nominally risky assetsavailable in the economy.

Il-r' D.r= mean vector and variance-covariance matrix of nominal returns onthe nominally risky assets.

e = vector of ones.

i'i = nominal return on the ith investor's portfolio.

To solve problem (2) the ith investor's Lagrangean is formed

Li=xiD.rxi- gi[Xi'll-r+ (l- x i'e)rF - E(i'i)]

where gi is a Lagrange multiplier. A necessary condition for the solution ofproblem (2) is that

This implies that

(4)

(5)

(6)

(7)

It is assumed that those investors who suffer from money illusion believe thatall investors are similarly inclined. Thus, the ith investor maintains that eqn.(.5) is valid for all investors in the economy.

Consider now the jth row of eqn. (5)

C -i) gi [E(- ) ]cov rj , r =2 rj -rF

Multiplying this equation through by the ratio of Wi' the wealth of the ithinvestor, to W = ~ lVi, the wealth of all investors in the economy, and aggre-

gating across all investors yields

cov (i'j' i'M)=[-~ ~ tvigiJ [E(i'j)-rF ]

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792 J. A. Schnabel

Multiplying this equation through by the ratio of ]11k' the market value of thekth asset, to ill = L M k> the market value of all assets in the economy, andsumming across all assets yields

2( - ) [I"" IVit] [E(-) JafM=2fw~i rU-fIt

This implies that

Substituting this into eqn. (7) yields

E(rj) = fit + [E(r.u) - fItJ~j

(8)

(9)

wherecov (rj , rJl)

a2(r .1I)

that is, eqn. (I), the security market line for investors who suffer from moneyillusion.

3. A pricing relationship for investors who do not suffer from money illusionThis section assumes that. the capital market is populated by investors who

do not suffer from money illusion. It is shown that these investors will priceassets according to the relationship

where

E(rj) = fit + [E(i'.u) - f F J.8/

cov (i'j, r.1/ ) - cov (rj, X)a2(i'.1I) - cov (r.u , ,\)

(10)

X= unoertain inflation rate.

Equation (10) also postulates a linear relationship between the equilibriumnominal return on an asset and a measure of its risk, f3/. f3/ is the beta ofasset .i as perceived by investors who do not suffer from money illusion.However, note that the beta risk measure presented in eqn. (10) differs fromthat presented in eqn. (I). While f3/ exhibits the impact of inflation on therisk of an asset, ~j ignores it. These results are not surprising given the corres­ponding capital market segments where these risk measures exist. Equation(4) may be considered the seeurity market line for those investors who do notsuffer from money illusion. Observe that the security market lines describedin eqns. (I) and (10) possess the same intercept, fit, the nominal return on thenominally riskless asset, and slope, [E(r.u ) - r J••J, the difference between thenominal returns on the market portfolio and the nominally riskless asset.Thus, the two security market lines differ only in terms of beta.

The approach adopted here for the proof of eqn. (10) is to make explicit arelationship implicit in the capital asset pricing model. Sharpe (1964),Lintner (1965), and Mossin (I !luu) have shown that, in the absence of moneyillusion, the following equilibrium relationship will hold

(Il)

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Money illusion and market segmentation 793

where{3.= cov (Rj , R,ll)

J a2(R.ll)

Rj , RJ J , Rj=real returns on the jth asset, the market portfolio, and thetruly riskless asset, respectively.

They derived eqn. (11) by assuming that investors have utility functions definedon inflation-adjusted terminal wealth and thus choose portfolios that are mean­variance of real return efficient. To apply eqn. (11) to the situation given inthis paper, i.e. a capital market characterized by the presence of some investorswho suffer from money illusion, one must assume that all investors who do notsuffer from money illusion believe that all investors similarly take account ofinflation in their economic calculations. Thus, investors who do not sufferfrom money illusion will maintain that eqn. (11) should hold.

Equation (11) is expressed in inflation-adjusted terms, However, as theanalysis in other parts of this paper is stated in nominal terms, this equationshould be translated into nominal terms.

Consider an asset, indexed by the subscript z, whose real return has a zerocovariance with the real return on the market portfolio. Applying eqn. (II)to this asset yields

Substituting this into eqn. (II), Black's (1972) gcneralizat.ion of the Sharpe­Lintner-Mossin model is obtained

(12)

(14)

As F indexes the nominally riskless asset and X denotes the uncertaininflation rate, observe that

Rj<'=rr- X

Rj=i'j- X

R.ll = i'JJ - X

These equations are strictly valid only if the returns are interpreted as con­tinuously compounded returns rather than returns over a discrete time period.Alternatively, these equations may be viewed as approximations where thecross-product 'terms involving the inflation rate and the nominal returns areignored.

Applying eqn. (12) to Rr results in

i'I i'I' i'I ,i'I cov (Rr . R.ll )E(J{~.) = E(J{,) + [E(J{.ll) - E(l(,) J a2(R ll)

which implies that

cov (X, RJ J )E(RF)=E(Rz)-[E(R.ll)-E(R,)] 2-

a (Rll )

Equation (14) may be subtracted from eqn. (12) to yield

E(Rj)-E(Rr)=[E(R.lI)-E(Rz)] [cov (Rj, ~~/;l;~S (X, RJI)J (15)

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7!l4·

or

r. A. Schnabel

thus

Equ.rtion (14) implies that

., " , ,,' [. eov (X, iJ..lJ)J i\ i\ i\ i\[b(1{.lJ) - jI,(1{zlJ 1+ a2(R.

lJ)= E(l{.lJ) - E(l{z) + E(l{z) - E(l{/,,) (18)

Applying eqn. (18) to (17) results in the following reformulation of eqn. (12) innominal terms

where

13.* = eov (i'j, i'.lJ) - eov (i'j~ X)} a 2V lJ) - eov (i'.lJ, ,\)

that. is, eqn. (10), which may be considered the security market line for investorswho do not suffer from money illusion.

4. Capital market segmentationSo far thc two assumed investor g!'Oups have been kept logically separated.

The cffccts induced by joining them together in an integrated capital rnarket.arc now examined.

Equations (1) and (10), developed in the preceding sections, exhibit thepricing relationships which would pertain for the two investor groups postulatedin this pnper. Observe that these security market lines differ only in terms ofthe relevant measure of beta risk. Given a specific securityj , the beta percep­tions of the two investor groups would differ in general.

Assume that Pj > f3/', i.e. the beta as perceived by the group that suffersfrom money illusion is greater than the beta perception of the group that doesnot suffer from money illusion. Referring to Fig. I, E(i'j) is defined as theexpected return on asset j' as given by eqn. (I) whereas E*(i'j) is defined as theexpected return on asset .i as given by eqn. (10). As the two security marketlines exhibit the same slope and intercept, it is implied thut E(i'j) > E*(I).Thus, investors who suffer from money illusion would require that the securityexhibit a higher yield than that required by investors who do not suffer frommoney illusion. Phrased another w:ty, the latter group of investors would viewthe asset as undervalued. Thus, only these investors would tend to purchasesecurities that exhibit this property. Figure I presents a pictorial depiction ofthis situation.

The reverse would be true when Sj < 13/. Only investors who suffer frommoney illusion would hold this asset as the yield requirement of investors whodo not suffer from money illusion would be more stringent, i.e. E*(i'j) > E(i'j)'

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1110ney illusion and market segmentation 795

expectedreturns

Figure 1.

expectedreturns

Figure 2.

betas

An asset considered undervalued bv investors who do not suffer frommoney illusio·;1.

betas

An asset considered undervalued bv investors who suffer from moneyillusion. •

An asset which displays this property would be considered undervalued byinvestors who suffer from money illusion. Thus, 10S Fig. 2 shows, only theseinvestors would tend to purchase a security with this property.

Finally, assets whose betas are such that Sj = f3/' would be purchased bvboth groups of investors as the required yields of both groups would be thesame, i.e. £(1) = E*(i). Each group would assess the same valuallon for thesesecurities. Thus, these securities would be acceptable to both groups ofinvestors.

The Appendix to this paper delineates the conditions which result in theabove three situa.tions : Sj > f3/, Sj < f3/' and Sj = f3/. It is notable that

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796 J. A. Schnabel

investors who suffer from money illusion would tend to gravitate toward shareswhose betas arc such that f3/ ;;, Sj' On the other hand, investors who do notsuffer from money illusion would tend to gravitate toward shares whose betasare such that Sj;;' f3;*. Thus, a type of capital market segmentation would beinduced-one in which different investors restrict themselves to differentsu bsets of the universe of securities available in the capital market. Such acapital market has been referred to by Rubinstein (1973) as 'weaklysegmented' .

What sort of equilibrium model would apply to the integrated market, i.e.across both investor groups? Unfortunately, neither the CAPM nor any of itsvariants would be representative. As Roll (1977) has observed, a criticalresult which underlies the CAPM is the optimality of the market portfolio. Tnfact, this is the one and only tangible implication of the CAPM. All otherresults are corollaries to this. Every investor would want to hold the marketportfolio along with some other portfolio. In the original CAPM context, thisother portfolio contains the riskless asset ulone whereas in the Black (1972)variation, this other portfolio is the minimum variance zero beta portfolio.Such results, known as separation theorems, are necessary and sufficientconditions for the CAPM to hold. However, the capital market segmentationdemonst.rutcrl above implies that no one would want to hold the marketportfolio. Everv investor would restrict, himself to a subset of the securitieswhich comprise t.he market portfolio. Thus, the presence of money illusioninvalidates the CAPM.

4. ConclusionThis paper has examined the impact of money illusion on the capital asset

pricing model. Two types of investors were postulated: those who sufferfrom money illusion and those who do not. Two types of security market lineswere derived, each one pertaining to one of these two groups. It was shownthat the lines share the same slope and intercept terms and that they differ onlyin terms of beta risk measures. Each group would view a different beta asrelevant in deriving the equilibrium valuation of a particular security.Because each gl"OUp has >1 different assessment of what the valuation of aspecific security should be, due to their differing beta perceptions, each wouldtend to specialize their investments in a subset of the securities available in thecapital market. Only shares whose nominal betas equal their real betas wouldbe purchasod by both gl"OUps. Thus, a form of capital market segmentationcalled wonk segmentation is induced. As the market portfolio is not held byany investor, the critical portfolio separation theorem, which underlies thecapital asset, pricing model and its variants, does not obtain.

AppendixComparison. between Sf and f3;*

This Appendix examines the relationship between Sj and f3/, i.e. the betameasures for the jth asset considered relevant by investors who suffer frommoney illusion and bv investors who do not suffer from money illusion,respectively.

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.Money illusion and market segmentation

Recall from eqns. (I) and (10) that Pi is defined by

p.= cov (ri , rM )

) a2(r.u )

797

(A 2)

whereas f3/ is defined by

f3* = cov (ri , i'.u) - cov (ri: X)) a2(r .l1) - cov Vu, .\)

Observe that, in the absence of inflation Pi and f3/ are identical. If inflation ispresent but forecasted with certainty, the same result obtains. Differencesbetween Pi and f3/ arise in the presence of stochastic inflation. As the abovedefinitions of Pi and f3/ show, an important quantity involved in the cornparisonbetween these two beta measures is the covariance between the nominal returnon an asset or portfolio and the inflation rate. Adopting the terminology ofChen and Boness (1975), this covariance term may be referred to as the asset'sor portfolio's ' inflation risk'. An asset or portfolio is said to be ' inflationpreferred' if its inflation risk is positive, , inflation-neutral' if its inflation riskequals zero, and' inflation averse ' if its inflation risk is negative.

From the definition of Pi and f3/, it is clear that if the market portfolio isinflation preferred and if the .ith asset is either inflat.ion neutral or inflationaverse, Pi will be less than f3/. If both the .ith asset and the market portfolioare inflation preferred and the market portfolio's inflation risk is less thana~Vu), then Pi will be less than f3;* if and only if

P> cov (ri' XJ (A 1)) cov (Fu, .\)

that is, if and only if Pi is greater than the asset's inflation risk relative to themarket portfolio's inflation risk. This may be shown by observing that, giventhe conditions of this proposition, Pi is less than f3/ if and only if

cov (ri , i'.\l) cov Vu, X»a2Vu) cov (ri , X)which in turn, is true if and only if inequality (A I) is true.

On the other hand, if both the .ith asset and the market portfolio areinflation preferred and the market portfolio's inflation risk is greater thana2 (r .u), then Pi will underestimate f3/ if and only if

f'i cov (ri , X)p.< -) cov (r.ll' .\)

that is, if and only if Pi is less than the asset's inflation risk relative to themarket portfolio's inflation risk. This may be shown by observing that, giventhe conditions of this proposition, Pi> f3/' if and only if

cov (ri , rJl) < a2(r .u) COV (r i , Xlwhich in turn is true if and only if inequality (A 2) obtains.

From the definitions of Pi and f3/ observe that if the market portfolio isinflation neutral, then Pi will exceed, equal, or fall short of f3/ if and only if the.ith asset is inflation preferred, inflation neutral, or inflation averse, respectively.

Finally, consider the situation when the market portfolio is inflation averse.If the asset is either inflation preferred or inflation neutral, Pi will exceed f3i*.

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798 Money illUliion and market segmentation

If the jth asset is also inflation averse, Pj will fall short of f3;* if and only if

f} cov (rj , X)Pj < _ (A 3)

COV (rM , .\)

that is, if and only jf the ratio of the asset's to the market portfolio's inflationrisk is greater than pj • This may be shown by observing that, given theconditions of this proposition, Pj < f3;* if and only if

cov (rj, rAI ) cov (rM , X) > 0 2(1',1/ ) cov (i'j' X)

which in turn is true jf and only if inequality (A 3) is true.The following table summarizes these results.

cov (1';, X)

>0 =0 <0

I

>0 + P; < f3;* P; < f3/

=0 P; > f3;* P;= f3;* P; < f3;*

I

<0 Pi> f3;* P; > f3;* +

+ Indeterminate

Comparison between P; and f3;* ..

REFERENCESBLACK, F., 1972, J. Bus., July, 444.BRANSON, W., and KLEVORICK, A., 1969, Am. econ. Rev., December, 832.CIlEN, A., and BONESS, A., 1975, J. Fin., May, 469.KESSEL, R., 1956, Am. Econ. Rev., March, 128.LINTNER, .J., 1965, Rev. Econ. Stat., February, 13.MODIOLIANI, :F., and COHN, R., 1979, Fin. Anal. J., March-April, 3.MOSSIN, J., 1966, Econometrica, October, 768.ROLL, R., 1977, J. Fin. Econ., January, 129.RUBINSTEIN, M., 1973, J. Fin. Quant. Anal., December, 749.SHARPE, W., 1964, J. Fin., September, 425.VICKERS, D., 1978, Financial Morkeu in the Capitalist Process (University of Pennsyl­

vania Press).

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