Ren M Stulz.

Asset pricing and expected inflation online

. (page 1 of 2)
Online LibraryRen M StulzAsset pricing and expected inflation → online text (page 1 of 2)
Font size
QR-code for this ebook


HD28
.M414



Si



,nuir-



WORKING PAPER
ALFRED P. SLOAN SCHOOL OF MANAGEMENT



Asset Pricing and Expected Inflation
Rene M. Stulz
September 1985

Working Paper #1706-85



•„..;^



/ KH



. R R I *1



MASSACHUSETTS

INSTITUTE OF TECHNOLOGY

50 MEMORIAL DRIVE

CAMBRIDGE, MASSACHUSETTS 02139



Asset Pricing and Expected Inflation
Rene M. Stulz
September 1985

Working Paper #1706-85



Asset Pricing and Expected Inflation



/



Rene" M. Stulz*



January 1985



Fourth revision, September 1985



Visiting Associate Professor, Sloan School of Management, M.I.T. I am
grateful for useful comments from the participants of seminars at the
University of Toronto, the Ohio State University, the University of Minnesota
and at the European Finance Meetings in Bern, Switzerland. I also thank K.C.
Chan and Bill Schwert for useful discussions, and an anonymous referee for
useful comments.



-

NOV 1985

i



Abstract

This paper provides an equilibrium model in which expected real returns
on common stocks are negatively related to expected inflation and money
growth. It is shown that the fall in real wealth associated with an increase
in expected inflation decreases the real rate of interest and the expected
real rate of return of the market portfolio. The expected real rate of return
of the market portfolio falls less, for a given increase in expected
inflation, when the increase in expected inflation is caused by an increase in
money growth rather than by a worsening of the investment opportunity set.
The model has empirical implications for the effect of a change in expected
inflation on the cross-sectional distribution of asset returns and can help to
understand why assets whose return covaries positively with expected inflation
may have lower expected returns. The model also agrees with explanations
advanced by Fama (1981) and Geske/Roll (1983) for the negative relation
between stock returns and inflation.



1_. Introduction

The negative relation between stock returns and (a) expected inflation,
(b) changes in expected inflation and (c) unexpected inflation, has been
extensively documented. Fama (1981) brings forth a large amount of
evidence to justify the hypothesis that an unexpected increase in the growth
rate of real activity not only causes an increase in stock prices, but also a

decrease in the price level and a decrease in inflation because of its impact

2
on money demand. Geske/Roll (1983) strengthen this hypothesis by showing

that a decrease in economic activity increases the Federal budget deficit and

that part of the deficit is monetized. Their analysis implies that a decrease

in economic activity brings about an increase in the expected growth rate of
the money supply and, hence, a larger increase in inflation than if there were
no negative relation between real activity and the growth rate of the money
supply. While the work of Fama (1981), Nelson (1979) and Geske/Roll (1983)
provides a convincing explanation of the relation between stock returns and
changes in expected inflation and the price level, it does not really explain
"the most puzzling result of all" (Schwert (1981), p. 28), which is that ex
ante real returns on common stocks are negatively related to ex ante expected
inflation. Indeed, one would expect information about future economic
activity to be capitalized in stock prices and to have no effect on ex ante
real stock returns, i.e., on how investors value real cash flows.

Day (1984) and LeRoy (1984) provide valuable analyses which build on
Fama (1981) and show through the money demand equation that a negative
relation between expected inflation and the expected real cash flows from
invested capital can occur in efficient capital markets. However, their
models take the expected real cash flows from invested capital to be ex-
ogeneously given. Consequently, they are unable to explain why expected



- 2 -



stock returns, as opposed to expected real cash flows, are negatively related
to expected inflation. They convincingly show, however, that when money
demand is positively related to real cash flows, a decrease in expected real
cash flows increases expected inflation for constant expected money growth.

Fama/Gibbons (1982) provide an hypothesis which suggests a negative
relation between endogenously determined ex ante returns on risky assets and
expected inflation. They argue that an increase in economic activity in-
creases not only inflation but also desired capital expenditures and hence
increases the demand for funds to be invested in production. In this case,
expected real rates of return must increase to equate investment and saving
when an improvement in economic activity increases desired investment more
than saving. While Fama/Gibbons (1982) provide a story which fits the facts,
they mainly explain the relation between the real rate of interest and
economic activity as they do not analyze explicitly the relation between
expected inflation and the cross-sectional distribution of ex ante returns of
risky assets.

This paper presents an equilibrium model in which the endogeneously
determined expected real rate of return of the market portfolio of risky

assets and the real rate of interest are negatively related to the level of

3
expected inflation. In this model, an increase in expected inflation,

irrespectively of its origin, decreases the real wealth of the households

because it increases the opportunity cost of real balances and hence decreases

the households' holdings of real balances. To keep all the stock of capital

invested in production, the households' desired holdings of nominal assets,

i.e., default-free nominal bonds and cash, must fall by the same dollar amount

as their desired holdings of real balances. This can only be achieved by a

fall in the real rate of interest which makes investments in nominal assets



- 3 -



less attractive relative to investments in production. With a constant real
rate of interest, there would be excess demand for nominal assets as the
relative investment proportions of risky assets are not affected by a fall in
real wealth. Because of decreasing absolute risk aversion, the decrease in
real wealth caused by the increase in expected inflation leads households to
choose a portfolio of investments in production with a lower mean and variance
of return.

In this paper, the expected real rate of return of the market portfolio
falls less if expected inflation increases because of an increase in expected
money growth than if it increases because of a worsening of the production
investment opportunity set (defined as the joint distribution of the returns
of investments in production technologies). This follows from the fact that
the portfolio households hold after an increase in expected money growth
cannot be held if, instead, the production investment opportunity set has
become less favorable. We show that, when expected inflation increases
because of a worsening of the investment opportunity set, the expected real
rate of return on the market portfolio of risky assets may fall by more than
the real rate of interest. While our analysis provides a theoretical found-
ation for a negative regression coefficient of stock returns on expected
inflation, the absolute value of the estimate of this coefficient presented by
Fama/Schwert (1977) is probably too large to be explained by the effects
discussed in this paper.

The paper proceeds as follows. We present the model in Section 2.
Section 3 shows the effect of an increase in expected inflation on the
expected real rate of return of the market portfolio for a given production
investment opportunity set. In Section 4, we discuss the effect of a change
in the production investment opportunity set on the expected excess return of



- 4 -



the market portfolio. While Sections 2 through 4 present our results when
households have a logarithmic utility function of consumption services,
Section 5 shows that when households have a coefficient of relative risk
aversion which exceeds one, they try to hedge against unanticipated changes in
expected inflation. This means that assets whose return is positively
correlated with changes in expected inflation have a lower expected return
than predicted by the security market line. Consequently, our model offers a
possible explanation for the empirical findings of Chen/Roll/Ross (1983) and
Loo (1984) that expected returns on risky assets are negatively related to
their covariance with expected inflation. Section 5 also uses our model to
discuss Fama's (1981) money demand argument. Section 6 offers some concluding
remarks.

2. The Model.

The economy studied in this paper resembles closely the economy studied

in Cox/Ingersoll/Ross (1985), except that households find it useful to hold

4
cash balances in equilibrium. Only one commodity is produced. Each

household can invest in n-1 constant returns to scale production technologies,
a nominal default-free bond and a real default-free bond. Financial assets
are assumed to trade on perfect markets, i.e., each household takes prices as
given, there are no taxes or transaction costs and no restrictions on short-
sales. There is no outside supply of bonds. As all households are assumed to
be the same, expected returns must be such that no household chooses to hold
bonds. There is a government which changes the money stock by purchases and
sales of the commodity and is assumed to make no transfers to households.

To complete the description of the investment opportunity set, we need
to specify the distribution of the real rate of return of the various assets



- 5 -



households can hold. The instantaneous return of an investment of k. in the
i-th production technology is given by:

dk = u k dt + o k dz i = 2, ..., n (1)

1 K. 1 K 1 K .

where y and o are taken to be constant in this section and the next, and

k i K i
dz^ is the increment of a standard Wiener process. The instantaneous nominal

k i
rate of return of the safe nominal bond R is a function of the expected rate

of change of the price of money tt and cannot be assumed to be constant. It

will be more convenient to work with the price of money rather than with its

inverse, the price level. Expected inflation is equal to minus the expected

rate of change of the price of money plus its variance. By definition, the

instantaneous real rate of return on a nominal bond is Rdt + dir/iT per unit

of time. While the dynamics for tt are solved for explicitly in Section 4,

we assume here that tt follows:

dTT = y TTdt + a TTdz (2)

TT TT TT

where o is constant and u changes over time according to:

dy = yy dt + o y dz (3)

TT TT Y " Y

where Y and a are assumed to be constant for simplicity. We assume that
the government chooses the dynamics of the money supply so that tt follows
equation (2), y^ follows equation (3) and the price of money is always
expected to fall (i.e., expected inflation is always positive). The
instantaneous real rate of return on the safe real bond is r and it is a
function of y like R. All households are assumed to have the same real
wealth w, which is the sum of the per capita stock of capital k and real
balances m. Each household is infinitely-lived and maximizes the following
expected utility function of lifetime consumption:

CD

E / e" pT [alnc(T) + (l-a)lnm(T ) ]dx (4)

t



- 6 -



where c(x) is the household's consumption of the commodity and m(x) is its
holdings of real balances. One possible justification for including real
balances in the utility function is that households hold real balances to
reduce the nonpecuniary costs associated with their trips to the bank to get
cash to shop for the commodity.

Let n^w be a household's holdings of nominal assets and let n.w >
be a household's investment in the i-th technology. With this notation, the
flow budget constraint of the household is given by:

dTT n dk i
dw = n (Rdt - rdt)w + Z n ( — rdt)w + rwdt - cdt - Rmdt (5)

i=2 i

The household solves for its portfolio by maximizing (4) subject to
(5). Let n be the nxl vector of shares of real wealth invested in risky
assets. Solving for n, we get:

n = V -1 (p - r.l) (6)

where V is the inverse of the nxn variance-covariance matrix V of real
returns on investments, y_ is the nxl vector of expected real returns on
risky assets and 1 is a nxl vector of ones. The portfolio held by households
is the market portfolio of the capital asset pricing model of Sharpe (1965)
and Lintner (1964), except that here it is endogenously determined.

The solution of the portfolio selection problem yields the sum of the
household's investments in real balances and nominal bonds. To obtain the
household's real balances, one must use the properties of the utility function
given by equation (4). An infinitely-lived household which maximizes (4)
spends pwdt per unit of time on consumption services. As the expenditure
share of real balances is (1-a), this household spends (l-a)pwdt per
unit of time to acquire the services of money. Hence, this reasoning implies
that:

m = [(l-a)pw]/R (7)



- 7 -



In equilibrium, households hold no bonds, so that n,w - m.

It immediately follows from the solution of the household's portfolio

choice problem that, as in the logarithmic utility version of the Cox/

Ingersoll/Ross (1985) model, the expected real returns must satisfy the

following equation:

y - r = a (8)

i i,w

where o. is the instantaneous covariance between the real rate of return
i,w

of the i-th risky asset and the rate of growth of real wealth. However,
whereas in Cox/Ingersoll/Ross (1985) the distribution of the rate of growth of
real wealth depends only on the investment opportunity set and the constraint
that all wealth is invested in production, here the distribution of the rate
of growth of real wealth must be such that the available stock of capital is
wholly invested in production and that real balances held satisfy the
condition given by equation (7).

3. Expected returns and expected inflation.

In this section, we study the effect of an increase in expected infla-
tion under the assumption that the production investment opportunity set is
constant. To simplify the analysis, we first assume that unanticipated
changes in the price of money are uncorrelated with the returns of each
investment in production processes. In this case, the expected real rate of
return of the market portfolio of investments in production is:

u v -1 (y - r.l )

y = Tfj. ~ 6 =-

1 V (y - r.l )
— e— e — e — e

where the subscript e denotes the fact that the vectors and the matrix have
been multiplied by an nxn matrix H which is the identity matrix except for
zeros everywhere in the first row. Inspection of equation (9) shows that a



- 8 -



change in expected inflation can affect y, only through its effect on the
real rate of interest r. Consequently, we first analyze the effect of an
increase in expected inflation on the real rate of interest.

For a given real rate of interest and given holdings of real balances,
the asset pricing equation derived in Section 2, i.e., equation (8), implies
that an increase in expected inflation increases the nominal rate of interest
R by the same amount. The money demand equation, i.e., equation (7), shows
that, for given consumption of the commodity, an increase in the nominal rate
of interest decreases the holdings of real balances and hence decreases the
price of money. A fall in real balances decreases real wealth, so that the
nominal rate of interest is negatively related to real wealth. With constant
relative risk aversion and a constant investment opportunity set, households
invest a constant fraction of their wealth in production processes. Hence,
for a constant real rate of interest, a fall in real wealth decreases the
households' investment in production processes, which implies that part of the
stock of capital is no longer invested. Instead, households want to invest
part of their wealth in nominal bonds which are in zero net supply to keep the
fraction of their wealth invested in nominal assets constant, as their
holdings of real balances fell proportionately more than real wealth. To
insure that the whole stock of capital is invested in production, the fall in
real wealth must therefore be accompanied by a fall in the real rate of

Q

interest which makes investments in production more attractive relative to
investments in nominal bonds. We can now state the first result of this paper:

Theorem 1 : When the expected real rate of return on the market portfolio of
investments in production is given by equation (9), the instantaneous co-
variance of changes in the expected real rate of return of the market



- 9 -



portfolio with changes in the real rate of interest is positive.
Proof : See Appendix.

This result can be understood easily if one uses a familiar geometric
tool. Figure 1 reproduces the efficient frontier of investments in production.
For a real rate of interest r , the market portfolio has an expected real
rate of return y, . An increase in expected inflation makes households
poorer, so that they no longer want to invest the same dollar amount in the

portfolio with expected return y, . The real rate of interest falls to

2
r to induce households to keep the dollar amount invested in production

2
constant. For the new real rate of interest r and the new risk premium

required by households, the tangency portfolio has an expected real rate of

2 2 1

return equal to y, , such that y, < y, . We can now state our second

result:



Corollary 1 : An unanticipated increase in expected inflation decreases the
expected real rate of return of the market portfolio and increases the price
of production risk given by (y, - r)/a .

Note that the price of production risk would correspond, in this model,

9
to the empirical measure of the Sharpe-Lintner price of risk. An increase

in the price of production risk implies that the empirical Sharpe-Lintner

security market line becomes steeper. Hence, the expected real rate of return

of low risk assets is affected more by a change in expected inflation than the

expected real rate of return on high risk assets. In fact, a security which

has a high enough beta with respect to reference portfolios with expected real

1 2
returns y and y has an expected return which increases with expected



- 10 -

inflation. Notice however that while the two reference portfolios must have
positively correlated real returns, the beta of most securities will change as
expected inflation increases.

To find out how our results are affected when we remove the assumption
that unanticipated changes in the price of money tt are uncorrelated with the
returns of investments in production processes, one can use equation (8) to
obtain:

u, = r + o. = r + ( - )o. + ( - )a. „ (10)

k k,w w k w k,TT

where a, is the instantaneous covariance between the rate of growth of

k,TT

the capital stock and the rate of growth of it. So far, we have assumed that

o is equal to zero. In this case, an increase in expected inflation

2
increases k/w; hence, for given u, and o, , r must fall when expected

inflation increases. As long as this result holds, the value of a, v does

not affect our results. Notice that m/w is an increasing function of m.

Hence, as long as a, is not too high, the results of this section hold.

2
As, empirically, a does not seem to exceed half of a , our results

hold for plausible values of a, (see Fama/Gibbons (1983)). However,

k,TT

2
if a, were to exceed o, , our results would be reversed because, in this

case, real balances would be very risky and a decrease in real balances held
by households would decrease the variance of the rate of growth of real

wealth. Notice also that the Increase in the price of risk is a decreasing

2
function of a, and that, when o, = o, , an increase in expected

k,TT k,TT k

inflation does not affect the price of risk.

The results of this section provide a theoretical foundation for the
existence of a negative relation between expected inflation and expected stock
returns. However, Fama/Schwert (1977) obtain an estimate for the regression



- 11 -



coefficient of stock returns on the nominal rate of interest larger than five
in absolute value. Such an estimate is much too large to be explained by the
present analysis. It is obvious from Figure 1 that our analysis implies that
the expected rate of return on the market portfolio has to fall by less than
the real rate of interest. While some authors argue that the magnitude of the
regression coefficient estimate obtained by Fama/Schwert (1977) might be
misleading, most empirical studies have nevertheless presented results
which indicate that an increase in expected inflation is likely to be
associated with a fall in expected stock returns which is too large to be
explained by the analysis presented so far. Therefore, we now turn to an
extension of our analysis which can explain a larger absolute value for the
regression coefficient of stock returns on expected inflation.

4 . Changes in the production investment opportunity set.

In this section, we extend the model of Section 2 to allow for changes

in the production investment opportunity set and investigate the relation

between expected inflation and expected asset returns when the change in

expected inflation is caused by a worsening of the production investment

opportunity set. To model changes in the production investment opportunity

set, we assume first that the expected real rate of return on an investment in

the i-th technology is equal to u, .x for all i's, where x is a state

variable which changes stochastically over time. The dynamics of x are left

unspecified except that x always takes positive values and follows a diffusion

process. The variance of the real rate of return of investments in production

technologies is left unchanged from Section 2, so that the instantaneous

output of the i-th technology is given by:

dk, = u, .xk.dt + o, .k.dz, ., i = 2, . .. , n (11)

i ki i ki i ki



- 12 -



with equation (11) instead of equation (1), all the results of Section 2 still
hold, except for the fact that all y^.'s are multiplied by x.

We first consider the effect of an unanticipated fall in x assuming that
the expected rate of growth of the money stock increases so that expected
inflation is left unchanged. To simplify the discussion, we assume that
unanticipated changes in the price of money are uncorrelated with output. In
this case, equation (9) can be rewritten as:

• -1
xu V [xu - r.l ]

Mk = -r* - e — =si (12)

ieY. [xy e - r.l e ]
The effect of a fall in x on u, and r is given by the following theorem:



Theorem 2 . For a constant expected rate of change of the price of money, an
unexpected fall in x decreases the expected real rate of return of the market
portfolio of investments in production u, and the real rate of interest
r. Furthermore, u, falls by more than r and the price of production risk,
(y k -r)/a , falls.
Proof . See Appendix.

A fall in x decreases investment in production for a constant real rate of
interest. Hence, r must fall to keep the stock of capital Invested. However,
the fall In x also shrinks the mean-variance efficient frontier of investments
in production as the expected excess return of the i-th asset falls more than
the expected excess return of the j-th asset whenever u, . > Pi-j* This fact
induces households to hold a less risky portfolio, as such portfolios become
relatively more advantageous. To keep the price of production risk unchanged,
r would have to fall by more than the expected rate of return of the minimum
variance portfolio of investments in production. Hence, with a constant price



- 13 -



of production risk, the minimum variance portfolio becomes more attractive
relative to risk-free investments. For households to keep investing a
constant fraction of their wealth in production, the risk-free rate must fall
to the same extent as the expected rate of return of the minimum variance
portfolio. However, in this case, r falls by less than the expected rate of
return of the market portfolio. If r falls by the same amount as the expected
real rate of return of the minimum variance portfolio, the price of production
risk must fall because the slope of the positively-shaped segment of the
efficient frontier of investments in production has fallen everywhere as a
result of the decrease in x. This means that, as the investment opportunity


1

Online LibraryRen M StulzAsset pricing and expected inflation → online text (page 1 of 2)