Capital Asset Pricing With Price Level Changes

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JOURNAL OF FINANCIAL ANDQU NTIT TIVEANALYSIS

September 1976

CAPITAL ASSET PRICING WITH PRICE LEVEL CHANGES

Robert L. Hagerman and E, Han Kim

I. Introduction

Economists anti men of affairs have been interesteti in the trade-off between

risk and return for centuries, but only recently has any significant progress

been made in specifying this relationship. Sharpe [12], Lintner [7] and Mossin

[9] have developed a single-period mean-variance model of capital asset pricing

that explicitly incorporates risk. This model has been useful in examining

various economic questions, including the social discount rate, the theory of

capital structure, and capital budgeting.

Most of the major assumptions that are the basis of the capital asset pric-

ing model do not conform to what we observe in the real world. As a result,

several authors have investigated the effect of relaxing some of the basic

assumptions of this model. Their analysis suggests that the basic form of the

capital pricing model is robust with regard to its underlying assumptions.

Unfortunately, little attention has been given to how the model changes when

the assumption of a constant general-price-level is relaxed. Roll 10) has

incorporated price level changes in a capital asset pricing model while retain-

ing the assumption that a real risk-free interest rate exists. Since one of

the major problems associated with price-level changes is that they make nom

inally riskless assets risky in real terms, the purpose of this paper is to pre-

sent a capital asset pricing model and to illustrate difficulties involved in

testing it empirically when price levels change.

State University of New York at Buffalo and Ohio StateUniversity re-

spectively. The authors would like to acknowledge the helpful advice of Jerry

Baesel,Charles Haley, Edward Kane, Stanley Kon, and ananonyinousreferee.

For example, see Bailey and Jensen [1] and Rubinstein [11].

See Jensen [6] for a good summary of these studies.

After our study was almost completed we learned that Chen and Boness i4]


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II. Market Equilibrium in Real Terms

If price-level changes make all assets risky, it would seem that Black s

[2] capital asset pricing model which assumes no riskless borrowing or lendin

could be used to analyze the impact of price-level changes. In the appendix

we show that this is the case, but for expositional purposes we will develop

the model by beginning with the basic investor optimization problem. To do t

we assume that:

1. All investors are risk averse single-period maximizers of expected

utility of

eal

terminal wealth with quadratic utility functions.

2. All market participants have homogeneous expectations about the ex-

pected return and variance of return for each capital asset as well

as the expected value and variance of future changes in the general

price level.

3. The capital markets are perfect in the sense that all assets are

perfectly liquid, all investors are price-takers, there are no

transactions costs or taxes, and all information can be acquired

by all investors at a zero cost.

4.

The relative prices of all consumption goods are constant.

5. The supply of all assets is fixed.

6. Borrowing and lending is riskless in nominal but not in real terms.

7. Direct exchanges of consumption goods result in transaction costs.

These costs are eliminated by a central authority which issues token

money at the beginning of the trading period and redeems it at the

end of the period.4

This last assumption appears artificial and therefore requires some expl

tion. Typical models of a timeless exchange economy assume that one commodit

is selected as the numeraire. In a one-period model such an assumption resul

in changes in relative commodity prices, which is a subject we believe should

be studied separately in the context of the capital asset pricing model. The

alternative of including money as an asset would require the inclusion of the

dynamic considerations which cause individuals to hold money in a static mode

What assumption 7does,in essence, is to introduce into the model only the

transactions demand for money.

In this world individuals want to maximize the expected utility of real

terminal wealth subject to the constraint imposed on them by their initial

4

This trading period occurs before the capital markets clear since the


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weal th endow ment . The nominal terminal weal th of the i individual can be ex-

p r e s s e d a s :

(1) i =

^jj^ f

w h e r e

WV = the terminal nominal weal th of the i investor whi ch is a random

variable as denoted by the tilde,

S.. = investor i's dolla r holdings of asset j in terms of curren t dolla rs,

B.

= the dolla r amount the investor has borrow ed or len t,

R = one plu s the fixed nominal rate of return on investor borrowing o r

lending, and

R. = one plu s the nominal rate of retur n on the j ass et.

Equa tion (1) can be expressed in more re levant real ter ms by defining a, the

price level adjustment factor, as one divided by one plus the percentage change

in the general price-level and writing :

(2) W . = oW*? = i:s..R.+ B.R-

1 1 ^ ID D 1 f

w h e r e

R. = aR. whi ch is the real return on the j as se t, and

~ N

R^ = aR^ whi ch is the real return on individual bo rrowin g and l endi ng.

The investor's portfolio selec tion probl em can'be expressed as a simple

Lagrang ian constrained optimiza tion problem of the form:

(3) Max E[u(W, )] + L.(W. - ZS .. - B.)

{ S . . ^ B.} ^ ^ X . ., X

Where u() represents the utility of wealth function. Taking partial deriv a-

tives, denoted by prim es, with respe ct to the decision va riabl es, i.e., the

amou nt to hold of each capital asset and the amount to borrow or le nd, give s

the following first-order conditi ons:

(3a) E[ u^ (R.)] - L^ = 0 , j = 1,.. ., N

(3b) E[ u^ (R^)] - L = 0 .

These conditions imply that:


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Since the utility function is assumed to show risk aversion and to be quad rati

in the fo rm:

u. W. = W. - a.W^

1 1 1 1 1

this impli s

that:

(5) u. = I - 2a.W. .

1 1 1

Substituti ng (5) into (4) and rearranging giv es:

E(u. ) - . .

(6) [E(R.) - E(R,)] - ^ = Cov(W., R. - R )

D f 2a ^ 1 ] f

as a conditi on of personal equilibr ium.

Of primary concer n, howe ver, are the mark et equilibrium cond itio ns. Sin

all investors have the same expec tatio ns, equation (6) can be summed over all

investors to obtain:

(7) E(R.) = E(R ) + * , [C OV (J :W ., R . - R ) ] .

E( u. ) i ^ ^ ^

Equati on (2) can be summed over all investors whic h results i n:

(8) ZW. = R IS . + R,ZB. = R ZS .

^1 m j fi m j

since SS.R.

' m ~

~Ys

whic h is the value-w eighted return on all risky ass et s,

j

ZB^ = 0 since economy-wi de borrowin g and lending mus t net to zero ,

i and

S = rs whic h is the total initial mark et value of the j*^ ^ asset.

3 i ^J

Substituti ng equati on (8) into (7) yie lds :

(9) E(R.) = E ( R J + XCov(R , R. - R )

D . t m 3 f

wh r


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Now ec^ation (9) must hold for the expected return on the marke t portfo lio,

R , as well as the expected return on individual risky assets in equilibrium

m

and as a result

- E R ^ )

ov R , R - R,)

m m I

Thus equation (9) can be rewritten as:

Cov(R , R. - R )

(10) E(R.) = E ( R J + , *

J

r ^ [E(R ) -

3 ^ Cov(R , R - R^) ^

m m

T.

which is the equilibrium relationship between risk and real return when changes

in the general price level are introduced into the capital asset pricing mod el.

Ecjuation (10) shows that whe n general pri ce le vel mov emen ts ar e explicit ly

incorporated into an analysis of the risk-return relati onship, the expected

real return on any asset is still a linear function of its risk. What is dif -

ferent in this mod el, howev er, is the measure of risk. Instead of risk being

measured as the ratio of the covariance of the return on the asset with the

marke t return divided by the variance of the market return , risk is now measured

by the ratio of two cova rian ces. The differenc e between ecjuation (10) and the

standard capital ass et pricing model is due to the assumption that the real re-

turn on the borrowing and lending rate can vary stochastically. It is easy to

show that if the borrowing and lending rate is constant in real t erms , then

equation (10) collapses into the standard capital asset pricing mod el:

Cov(R,, R )

(11) E(R.) = R, +

z^

2L_ [E(R ) _ R ]

^ ^ Var(R ) ^

m

since

Cov(R, - R,, R ) Cov(R., R )

] t m _ 3 m

Cov(R - R,, R ) Var(R )

m I m m

if R- is a constant.

One interesting implication of equation (10) is that the expected real

return on any risky asset is not affected by the relationship between changes

in the general price level and the asset itself. This can be best seen by

writing e quation (10) as :


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What equation 12) illustrates is that none of the terms in the equation whi

are unique to the j asset are affected by price-level movements. The influ

of price-level changes is incorporated solely on the basis of how they affect

the return on the market portfolio. The intuitive reason for this is that, t

the extent price-level changes are unrelated to real market returns, price-

level changes are unsystematic risks which can be eliminated by diversificati

To illustrate this point, consider the case in which real market returns

are independent of price-level changes. This could occur if the price-level

gains and losses of individuals and firms cancel out when averaged over the

entire economy. In this case the measure of risk reduces to:

Cov R. - R,, R ) Cov R., R )

] t m _ j m

Cov R - R , R ) Var R )

m f m m

since

Cov R , R.) = R^ Cov R , a) = 0.

m t r m

Thus,under the assumption that real market returns are not correlated with

price-level changes, the equilibrium condition expressed in equation 10) is

identical to that of the traditional capital asset pricing model, i.e., ecjua-

tion 11), except that the expected real return on borrowing and lending re-

places the constant risk-free rate. The reason for this is that if price-

level changes are uncorrelated with market returns, investors can eliminate t

risks associated with price-level movements by simply holding the market port

folio.

In sum, then, the essential features of the standard capital asset

pricing model still hold even where the general price level is allowed to be

random variable if these movements are uncorrelated with the real returns on

the market.

III. Market Ec^uilibrium in Nominal Terms

Since nominal and not real returns are observed in the world, it is use-

ful to translate equation 10) into nominal returns. This could be easily do

in a formal sense by recalling that


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~~N ~ N ~~N

--N M - --K, - N C o v ( a R . - OR , aR )

(13) E ( o R ) = R^E(a) + [E(aR ) - E(a)R ] ir:^ > ^

^

.

^ C o v ( a R - a R , , a R )

m I m

Unf ortu na tel y, this equat ion involve s mul tipl icat ive rando m varieiiles

a n d

t h erefore woul d be d i f f i c u l t t o t e s t em piricall y .

Equation (13) ccm be simplified by u sing t h e p r o o f s d e v e l o p e d b y S t e v e n s

[13] assu ming ag ain that t h e real return o n t h e m a r k e t p o r t f o l i o i s n o t c o r r e -

la t e d with price-l e v e l cha n g e s a n d t h a ta , R . a n d R a l l h a v e normal marginal

d i s t r i b u t i o n s . T o d o

t h i s ,

first consider t h e risk term wh ich i s t h e ratio o f

t w o c o v a r i a n c e s . If the real market return is u n c o r re l a t e d w i t h p r i c e - l e v e l

c h a n g e s ,

t h e n t h e risk meas u re term in real t erms is the s o - c a l l e d be ta coef fi-

c i e n t

a s

discus s e d above . This risk meas ure

c a n b e

w r i t t e n

a s :

C o v ( R . , R ) C o v ( a R . , R )

j m _ i m

N ~

Var(R

)

Cov(aR

, R )

m mm

Using Stev en's [13] equation (9) this can be expanded to

(14a)

. t -

I

R, a )

m m

and again using Steven's [13] results this becomes

V

E ;)COV R;,

R ^

Var(R )

m m m m

which is the risk of the j capital asset expressed in nominal return s, and

N

will be referred to as B..

This allows us t o write equa tion (10) as :

(15) E(a R^ ) = R^ E(a ) + [E(aR^) - R^E(a)]B'? .

D f m f j

Ecjuation (15) still co nta ins mul tipl icativ e random variables,, but these can be

eliminated by substituting the definition of the expected val ue of the produc t

of two ran dom varieibles and writ ing :


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-M , Co v(a, R.) Cov( o, R )

(16) E R:^)=

. L - ^ [ : ^ E(? ) R^l

^ E(a) E(a) m f 3

which is the market equilibrium condition in nominal terms.

Even though we have made the simplifying assumption that the return on

the market portfolio is uncorrelated w ith price-level chang es, the resulting

asset-pricing model in nominal terms is not in the same form as the model in

real terms. This has obvious implications for empirical testing of this mod el

Most empiric al studies have tested the capital asset-pricing model expressed

inex antereal terms by using nominalex postreturn s. Equation (16) demo n-

strates that this may lead to an inappropriate specification of the model.

Equation (16) may also be used to explain the empirical results of B lack ,

Jensen , and Scholes [3] . Their evidence indicates that the measured relat ion-

ship between portfolio excess returns and portfolio betas ha s a positive inter

cept and a slope which is less than E(R ) - R^. Howeve r, accordinq to the

m f

traditional mod elwithoutprice-level c hange s, the intercept should be zero an

the slope should be equal to E(R^) - R^. One possible explanation is that the

capital asset-pricing mode lwithprice-level changes suggests risk should be

measured by the beta coefficient expressed in real terms (or equation (14b) if

the real market return is uncorrelated with price-level changes) while Black,

Jense n, and Scholes [3] used nominal returns to estimate beta . Fama and McBet

[5] have provided some evidence suggesting that beta coefficients measured in

nominal and real terms are approximately equal . Therefore we will assume that

the two are equal in the following analys is.

Consider first the term Cov( a, R^) in equation (16 ). If the real return

of a portfolio is uncorrelated with a, then this covariance must be neg ative.

If realized returns are on average equal to their expe ctations, this ne gative

covariance will be reflected in a positive intercept in the regression,

(17) R - R^ = a^ + a,6 + e ,

P f 0 1 p p

Cov(a , R^) ....

since a will equal z^ . The term Cov(a, R ) will also be negative

E(a) .

and the estimated slope coefficient will equal :


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model without price-level changes. Thus,equation 16) is consistent with the

Black, Jensen, and Scholes [3] overall results.

IV. Summary and Conclusion

A capital asset-pricing model which relates risk and return under condi-

tions of changing price levels has been developed in this paper. The resulting

model implies that price-level changes do not affect the expected real returns

on individual assets except through their impact on the return of the market

portfolio. If real market returns are independent of price-level movements,

the model is very much like the standard capital asset-pricing model expressed

in real returns. This version of the capital asset-pricing model does not,

however, resolve all the difficulties associated with changing price levels,

since we have assumed that the nominal default-free rate is determined outside

the model and that relative prices do not change. These limitations, however,

also apply to all other single-period capital asset-pricing models.

In addition, the model was converted into nominal returns by assuming

that price-level changes and the real market returns are uncorrelated. The re-

sulting equation illustrates the difficulty involved in using nominal returns

to test a model expressed in real returns. The same equation also provides a

possible explanation for the noted discrepancies between the empirical evidence

found by Black, Jensen, and Scholes [3] and the prediction of the traditional

capital asset-pricing model.


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REFERENCES

[1] Bailey, M., and M. C. Jensen. Risk and the Discount Rate for Public In-

vestments. InStudies in the Theory of Capital Markets edited by M. C.

Jensen.

New York: Praeger Publishers, 1972.

[2] Black, F. Capital Market Equilibrium with Restricted Borrowing. Journal

of Business vol. 45 (July

1972),

pp. 444-454.

[3] Black, F.; M. Jensen; and M. Scholes. The Capital Asset Pricing Model:

Some Empirical Tests. InStudies in the Theory of Capital Markets edited

by M. C. Jensen. New York: Praeger Publishers, 1973.

[4] Chen, A., and J. Boness. Effects of Uncertain Inflation on the Invest-

ment and Financial Decisions of a Firm. Journaiof Finance. Forthcoming.

[5] Fama, E. F., and J. D. MacBeth. Tests of the Multi-Period Two-Parameter

Model.

Journal of inancialEconomics vol. 1 (March

1974),

pp. 43-66.

[6] Jensen, M. C. Capital Markets: Theory and Evidence. Bell Journal of

Economics and Management Science vol. 3 (Autumn

1972),

pp. 357-398.

[7] Lintner, J. The Valuation of Risk Assets and the Selection of Risky In-

vestments in Stock Portfolio and Capital Budgets. The Review of Economics

and Statistics vol. 47 (February1965),pp.

13-37.

[8] Long, J. B., Jr. Stock Prices, Inflation, and Term Structure of Interest

Rates.

Journalof inancialEconomics vol. 1 (July

1974),

pp. 131-170.

[9] Mossin, J. Equilibrium in a Capital Asset Market. Econometrica vol.

34 (October

1966),

pp. 768-783.

[10] Roll, R. Assets, Money and Commodity Price Inflation under^Uncertainty.

Journal of Money Credit and Banking vol. 5 (November1973),pp. 903-923.

[11] Rubinstein, M. A Mean-Variance Synthesis of Corporate Finance. Journal

of Finance vol. 28 (March1973),pp. 167-181.

[12] Sharpe, W. F. Capital Asset Prices: A Theory of Market Ecjuilibrium under

Risk. Journal of Finance vol. 19 (September

1964),

pp. 425-442.

[13] Stevens, G. Two Problems in Portfolio Analysis: Conditional and Multi-

plicative Random Variables. Journal of inancialand Quantitative Analy-

sis

vol. 6 (December

1971),

pp. 1235-1250.


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APPENDIX

ProofofEquation (10) Using Black sNoRiskless BorrowingorLending Capital

Asset Pricing Model

Black[2] hasshown thatin aworld withnoriskless borrowingorlending

the expected real returnonanyasset will equal:

Cov(R,,R )

(A-1) E(R.)=E(R^)+ , "*

[E(R

)-

E(R

)]

^ Var(R) "* ^

m

where

E(R)=theexpected real returnontheminimum variance portfolio which

is uncorrelatedtothereturnonthemarket portfolio.

This relationship must also hold

for the

expected real returnon

the

default-

free borrowingorlendingandbysubtractionwe canobtain:

Cov(R.,R)-Cov(R, R )

(A-2) E(R.)-E(R-)= 3 E _ E EL.[E(R )-E(R)].

^ Var(R) " ^

m

This etjuation

can be

rewritten

as:

Cov R.

- R,, R )

Cov R

, R )

f

V

(A-3) E(R.)=

E(RJ

[E(R )-E(R^)

m z

We know, however, that

(A-4) E ( R j -E(R)= i^[E(R )-E(R)]

Var(Rj "

andbysubstituting equation (A-4) into (A-3)weobtain:

Cov R. - R , R )

A-5)

E R.) = E R ) + r , , [E R ) - E R )

Cov R - R , R )

Which i s the Scune as etjuation 10) in the t ext .


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Capital Asset Pricing With Price Level Changes

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