WORKING PAPER NO. 14c
CAPITAL MARKET EQUILIBRIUM
AND THE TERM STRUCTURE OF INTEREST RATES
Fischer Black
Gradua1Je School of Business
University of Chicago
Revised November, 1972
CAPITAL MARKET EQUILIBRIUM
AND THE TERM STRUCTURE OF INTEREST RATES
INTRODUCTION
Most of the literature on the term structure of interest rates has made use
of models that are inconsistent with capital market equilibrium. Various
market imperfections are assumed that make individual ~ssuers of debt securities
or holders of debt securities prefer one maturity to another. Hicks [lJ, for
example, assumes that the supply of bonds is fixed, or constrained, and that
investors' demands for bonds of different maturity cause rates of return to
vary with maturity. Even given a fixed supply, he does not consider investors'
holdings of bonds in a context with their holdings of other assets. Kessel [2J
assumes that liquidity differences make bonds of different maturity attractive
to different investors, and also fails to think of bonds as held in portfolios
containing other assets. Modigliani and Sutch [3J assume that both borrowers
and lenders have various institutional reasons for preferring oue maturity to
another, and that the interaction of supply and demand schedules as a function
of rate and maturity determines the term structure. None of these writers
explains adequately why intermediaries do not spring up who can convert the
maturities that borrowers prefer into the. maturities that lenders prefer at
zero economic cost.
While market imperfections may be important in explaining certain aspects of
the observed term structure, we would certainly like to have a theory of the
term structure that holds when there are no market imperfections. The effects
of market imperfections can then be explored in the context of this theory.
Roll [4] has begun the development of a theory of the term structure that is
consistent with capital market equilibrium, by applying the capital asset
pricing model developed by Treynor [5J, Sharpe [6J, Lintner [7J, and Mossin [8J
to bond prices. Unfortunately, he is not able to obtain either a complete
theory of the term structure that is consistent with the capital asset pricing
model, or an adequate empirical test of the thedry.
The capital asset pricing model states that under certain assumptions, the
expected return on any capital asset for a single period will satisfy:
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(1)
(One of the problems in Roll's analysis is that he states the capital asset
pricing model incorrectly.) The symbols in equation (1) are defined as follows:
'0
R
m
The return on asset i for the period. The return is the
change in the price of the asset, plus any dividends, interest,
or other distributions, divided by the price of the asset at
the start of the period.
The return on the market portfolio of all assets taken together.
The return on a riskless asset for the period.
The "market sensitivity" of asset i. It is equal to the
slope of the regression line relating R.
~
and R m
'0
The market portfolio used in defining R should in theory include all assets:
m
common stocks, bonds, real estate, privately held businesses, human capital,
and so on. There is reason to believe, however, that the return on a market
portfolio of common s.tocks alone, or of common stocks plus bonds, will be very
highly correlated with the return on the theoretical market portfolio that
contains all assets. Thus equation (1) will hold approximately if a market
portfolio of common stocks, or of common stocks plus bonds, is used instead of
the theoretical market portfolio.
A riskless asset is a pure discount bond (with no coupons), that has no default
risk, that matures at the end of the period. When the period is very short,
the rate of inflation will be known in advance, and the asset will be riskless
in both real and nominal terms. The return on any asset, however, t. meant to
be a nominal return, not a real return.
The market sensitivity of an asset is a measure of that part of the asset's risk
that cannot be diversified away by combining it with other assets in a portfolio.
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In market equilibrium, it is only this part of the asset's risk that influences
its expected return. This is very important in analyzing bonds, because all or
almost all of the risk in a long term bond can be diversified away. Thus in
market equilibrium, most or all of the risk in a bond has no effect on its
expected return. The market sensitivity of an asset is defined algebraically
as follows:
(2) ~ ~ ~ coV(Ri,R )/var(R ) m m
The assumptions that are generally used in deriving equation (1) are as follows:
(a) All investors have the same opinions about the possibilities of
various end-of-period values for all assets. They have a common
joint probability distribution for the returns on the available
assets.
(b) The common probability distribution describing the possible
returns on the available assets is joint normal.
(c) Investors choose portfolios that maximize their expected end-ofperiod
utility of wealth, and all investors are risk averse.
(Every investor's utility function on end-of-period wealth
increases at a decreasing rate as his wealth increases.)
(d) An investor may take a short position of any size in any asset,
including the riskless asset. An investor may borrow or lend
any amount he wants at the riskless rate of interest.
(e) There are no taxes, transactions costs, or other market imperfections.
The length of the period for which the model applies is not specified. The
assumptions of the model, however, make sense only if the period is taken to
be infinitesimal. For any finite period, the distribution of possible returns
on an asset is likely to be closer to lognormal than normal; in particular, if
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the distribution of returns is normal, then there will be a finite probability
that the asset will have a negative value at the end of the period.
The model is derived as a single period model; but in principle it is easy tD
generalize to a multiperiod model. We can simply say that it must apply to
each successive infinitesimal period in time.
The riskless asset, in this context, may be thought of as a savings account
with an interest rate that fluctuates continually. If an investor has a short
position in the riskless asset, then he has what amounts to a demand loan with
a fluctuating interest rate.
Given the assumptions underlying the capital asset pricing model, it must apply
to all assets, including bonds of different maturities. We will start our
analysis, then, by assuming that equation (1) applies to all bonds, when the
length of the period is taken to be infinitesimal.
HOLDING PERIOD RETURNS
The return il:i on a bond in equation (1) is a "holding period return," and
has little to do with the bond's yield to maturity. Two bonds that mature at
different times, or have different coupons, or have different amounts of
default risk may have different yields to maturity, but if they have the same
S's , they will have the same expected holding period returns.
If we regress the return on an index of government bonds against the return on
a market portfolio containing both stocks and bonds, we find that the 8 of the
bond index is very near zero. Given the uncertainty in our estimates of B ,
we cannot reject the hypothesis that the S's on all government bonds are equal
to zero. It will be convenient, and a good approximation to reality, to assume
that the S's of all bonds that have no default risk are equal to zero.
This means that the expected return on any bond free of default risk is equal to
the return R
f
on the riskless asset. The expected return on such a bond is
independent of its maturity.
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RISK PREMIUMS
Assuming that the S's of all bonds are zero, there should be no risk premiums.
Investors should be indifferent to the greater risk in long term bonds, because
it is risk that can be diversified away.
MARKET SEGMENTATION
Under the assumptions of the capital asset pricing model, there is no reason for
market segmentation. An investor cares only about the expected return and risk
of his portfolio. Indeed, as shown by Sharpe ~6] and others, every investor
holds the same portfolio of risky assets, including long term bonds, and mixes
it with a long or short position in the riskless asset to suit his risk
preferences. Financial intermediaries and firms are completely indifferent to
the term structure of their assets and liabilities, because the investors who
hold their shares can offset any undesirable characteristics of these shares
by taking appropriate long or short positions in bonds of different maturity.
TRANSACTIONS COSTS
Let us assume that the transactions costs are lower on short maturity bonds
than on long maturity bonds, as observed by Malkiel [10]. Then Kessel [2]
and others have argued that the returns on shorter maturity bonds will be
lower than the returns on longer maturity bonds, because shorter maturity
bonds have value as liquid assets. They can be sold at low cost to meet
sudden needs for cash. Given a choice of holding a short maturity bond or
a long maturity bond offering the same return, an investor will choose the
short maturity bond because of its greater liquidity.
But an investor does not have to hold a short maturity bond to be able to sell
a short maturity bond to meet a sudden need for cash. He can always sell short:
that is, he can borrow at the short term rate. A lender "ill be happy to take
his long term bond as"collateral, and to lend him a very large percentage of
its value. It is true that transactions will be made by buying and selling
the shortest maturity bonds (the riskless asset), but this need have no effect
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on an investor's holdings of risky assets.
CONCLUSIONS
Under conditions of market equilibrium, the expected returns on bonds of all
maturities will satisfy equation (1). If the risk in a bond is entirely
independent of the risk in the market portfolio, then the expected returns
on bonds of all maturities will all be equal to the return on an ideal savings
account. There can be neither risk premiums, nor habitat premiums, nor liquidity
preffilliums in long maturity bonds.
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ACKNOWLEDGEMENTS
Helpful comments and suggestions were provided by Myron Scholes, Jack Treynor,
Michael Jensen, and Franco Modigliani.
This work was supported in part by the Ford Foundation.
REFERENCES
1. Hicks, John R. Value and Capital. Oxford: Oxford University Press, 1946.
2. Kessel, Reuben. The Cyclical Behavior of the Term Structure of Interest
Rates. New York: National Bureau of Economic Research, 1965.
3. Modigliani, Franco, and Richard M. Sutch. "Innovations in Interest Rate
Policy." American Economic Review 56 (May, 1966): 178-197.
4. Roll, Richard. "Investment Diversification and Bond Maturity." Journal
of Finance 26 (March, 1971): 51-66.
5. Treynor, Jack L. "Toward a Theory of Market Value of Risky Assets."
Unpublished memorandum, 1961.
6. Sharpe, William F. "Capital Asset Prices: A Theory of Market Equilibrium
Under Conditions of Risk." Journal of Finance 19 (September, 1964): 425-442.
7. Lintner, John. "The Valuation of Risk Assets and the Selection of Risky
Investments in Stock Portfolios and Capital Budgets." Review of Economics
and Statistics 47 (February, 1965): 13-37.
8. Mossin, Jan. "Equilibrium in a Capital Asset Market." Econometrica 34
(October, 1966): 768-783.
9. Meiselman, David. The Term Structure of Interest Rates. Englewood Cliffs,
N.J.:Prentiae Hall, 1962.
10. Malkiel, Burton G. The Term Structure of Interest Rates. Princeton:
Princeton University Press, 1966.