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Working Paper 78-2
UNCERTAIN INFLATION, SYSTEMATIC RISK, AND THE
CAPITAL ASSET PRICING MODEL
Thomas A. Lawler
Federal Reserve Bank of Richmond
February, 1978
The views expressed here are solely those of
the author and do not necessarily reflect the
views of the Federal Reserve Bank of Richmond.
1
1.
Introduction
The Sharpe-Lintner two-parameter Capital Asset Pricing Model
(CAPM) hasbeen
the basis for a extraordinary amount of theoretical
and empirical work.
As originally developed, the CAPM did not
explicity account for the effect of uncertain inflation on asset
prices.
Moreover, almost all major empirical studies of the CAPM have
employed nominal rates of return in testing the model.
The recent
experience of high and variable inflation has led a number of economists
to question the validity of a model such as the CAPM which assumes
that portfolio decisions are based on nominal instead of real returns.
Indeed, Lintner
(1975, p.11) notes that an asset-pricing model which
makes this assumption "is misspecified unless the risk premium on all
individual stocks is invariant with respect to inflation."
Thus any
test of the CAPM which utilizes nominal rates of return may be
misspecified,
and the instability of estimated parameters may be the
result of such misspecification.
The purpose of this paper is to examine the effect a change in
purchasing power risk may have on an asset's "beta coefficient"
(or
systematic risk) calculated using nominal rates of return when the
response of an asset's nominal rate of return to unexpected price
level changes varies across assets.
It is shown that this differential
response can cause an asset's beta coefficient to be unstable when
the degree of inflation uncertainty varies over time.
2
2.
The Model
The following notation is used in deriving the model.
'iii= the nominal return on asset i, with mean E(zi) and variance 02(zi)
'i;,= the nominal return on the market portfolio, with mean E(&s) and
variance A2(;im).
IT = the rate of inflation, with mean E(y) and variance ~~(7).
R f = the riskless nominal interest rate.
cov = covariance.
Tildes (-) denote random variables.
In the traditional CAPM, which ignores purchasing power risk,
the equilibrium relationship between the expected nominal return on
any asset i and the expected nominal return on the market portfolio is
given by
(1)
E(?;i) = Rf(l-Bi) + BiE(k)t
where pi = COV(Zi,a?i)/G2(&).
Purchasing power risk is introduced here
by assuming that the relationship between unexpected changes in the
inflation rate and unexpected changes in an asset's nominal return is given
(2)
iTimE
= bi [?-E(T)] + ?i
where bi = COV(';~~,7) /a2 (7) , E(Fi)
It
=
0, and COV
(TrYi)
=
0.
is obvious that the various bi 's will vary across assets.
For example,
an asset yielding a constant nominal rate of return will have a bi = 0,
while an asset yielding a constant real rate of return will have a bi = 1.
The following relationships can be derived from (2) (realizing that
(2) also applies to the market portfolio):
3
=
bibmc2 (7) + COV(TiFm) ;
(34
COV CiTi
,~)
(3b)
(7) + a2(rm,.
a2&I = bm202
and
Assuming that the variance of the inflation rate ~~(7) is a valid
measure of purchasing power risk, one can evaluate the effect of
a change in purchasing power risk on the slope coefficient of the
traditional CAPM equation by calculating 3f3/ao2(?).
Using (3a) and (3b), one finds that
02(e?&)bibm-COV(Li;i,W&)bm2
(5) aBi/acr2(Z =
.g (<)I L
Equation
.
(5) can be rewritten to obtain
bm (bi-Bi&)
(6) aBi/aa2 (7) =
o2 6&J .
Obviously, from (6) @i will not change if b,=O--i.e., if the
nominal return on the market portfolio is invariant with respect
to inflation.
If bm # 0, then (6) can be rewritten
(7) aBi/ao2 CT) =
o2 t-&J
I
which implies that
(84
bi
af3i/atr2(rr)
> 0 if g > @i ;
m
(8b)
a(3i/aa2(7r)< 0 if '$ < Pi ;
m
and
(8~)
b*
aBi/ao2 (IT)= 0 if 1 = Bi .
b,
4
Thus Bi will only be stable during periods of changing purchasing
power risk if a) b,=O; or b) Bi=bi/bm*
In the case where b,#O,
this means that Bi will only be stable if the change in Ri relative
to Rm in response to an unanticipated change in inflation is equal
to the change in Ri relative to Rm when no purchasing power risk
exists.
This condition will not in general hold for all assets.
in general
Thus if b,#O and 02(?) is not constant over time, then Bi will not be stable over time.
Recent evidence suggests that bm<O for a number of widely
used proxies for the market portfolio
(Nelson (1976), Bodie (1976)).
Moreover,it is apparent that ~~(7) has not been stable in recent
years.
This implies that the traditional CAPM equation is
misspecified,
and that any instability found in estimates of the
CAPM parameters over time may
power risk.
be the result of changing
purchasing
REFERENCES
Bodie, Zvi. "Common Stocks as a Hedge against Inflation,"
Journal of Finance, 31(May 1976), 459-470.
Chen, A.H. and Boness, A.J. "Effects of Uncertain Inflation
on the Investment and Financing Decisions of the Firm,"
Journal of Finance, 30(May 1975), 469-483.
Friend, I., Landskroner, Y., and Losq, E. "The Demand for
Risky Assets under Uncertain Inflation," Journal of Finance,
31(December, 1976), 1287-1297.
Lintner, John. "The Valuation of Risk Assets and the Selection
of Risky Investments in Stock Portfolios and Capital Budgets,"
The Review of Economics and Statistics, 47(February 1965), 13-37.
"Inflation and Security Returns," Journal of Finance
30(May 1475) , 259-280.
"Inflation and Rates of Return on Common
Nelson, Charles.
Stocks," Journal of Finance, 31(May 1976), 471-483.
Sharpe, William F. "Capital Asset Prices: A Theory of Market
Equilibrium under Conditions of Risk," Journal of Finance,
191September 1964), 425-442.
Siegel, J. and Warner, J. "Indexation, the Risk-Free Asset,
and Capital Market Equilibrium," Journal of Finance,
32(September 1977), 1101-1107.
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