Risk-aversion and the term structure of real interest rates

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S. F. LeRoy / Risk-aversion

and term structure of real mterest rates

transition probabilities II = {r( j 1i)} = {prob( X,+ , = X’ 1X, = X’)), i, j = h, 1. It is shown in LeRoy and LaCivita (1981, eq. (9)) that real interest rates are given by

(2) Here r; is the real k-period interest rate if the state at the initial date is i( i = h, I), and X = u’(Z’)/u’(xh) is a composite measure of risk and risk-aversion such that X = 1 under risk-neutrality and X > 1 under risk-aversion. The rr,Jj 1i) are the k-period transition probabilities: TJ j 1i) = prob(Z,+, = Xl I X, = X’), calculated directly from the assumed one-period transition probabilities as in footnote 2 below. Eqs. (1) and (2) are easily understood. For either endowment state the real interest rate must be such that individuals receive an equal (expected) utility increment if they increase current consumption by a given small amount or if they invest an equal amount at the currently prevailing k-period real interest rate and consume the proceeds at maturity. Thus we must have

Setting i = h, I, dividing by z/(X’), using the definition of h and solving for the interest rate leads to (1) and (2). The model just presented contains no robust implication that long-term interest rates are systematically higher than short-term rates. For example, if rk( j 1i) = 4 for all i, j, k, then E(l

+?J=((l

+ $)”

+ (1 + 472

= P-k

for all X, ,B and k, so that interest rates on bonds of all maturities fluctuate in an unbiased fashion around the discount rate implicit in ,0. Also, if h E 1, so that there is little or no risk in the system (Xh = X’) or individuals are nearly risk-neutral (u” s 0), then from (1) and (2) we have that

(1 +$)“-,&“(l

-~Jllh)(h-

(1+~~)kzpk(l+7r,JhI/)(h-1)),

l)), sothat

357

S. F. L.eRoy / Risk -aversion and term structure of real rnterest rates

E(l

lt~J=7r(h)(l

The last equality that

+ ,),

+ 7r( I)( 1 + r;)” = p-“.

follows from the property

of two-state

Markov processes

~~(/Ih)~(h)=~~(I,h)=a,(h,/)=~~(hII)~(l), where a(h)

and r(I)

are the steady-state

(3) probabilities,

defined

by

~(h)=~r(hIh)n(h)+~~(hII)~(l),

for any k, and rk( j, i) is the probability of the joint event that the state is at t and j at t + k. We have shown that in special cases average interest rates are the same at all maturities; if these examples are interpreted as borderline cases, it follows that for general transition probabilities and risk-aversion, the average interest rate may either rise or fall with maturity. It should not be surprising that consumption asset pricing models have no robust implications for the average interest rate by maturity. That statistic has no direct relevance to individuals’ decision problems: in the absence of any information other than currently-determined interest rates, investors have no direct way to compare long-term bonds and short-term bonds. The appropriate comparison involves also the (unknown) price at which the longer bond is liquidated, or the (unknown) interest rate at which the shorter bond is rolled over. But this consideration suggests that the non-comparability problem can be circumvented if we base our analytical measures of the term structure of interest rates the term premia - on comparison of two alternative investment strategies with a common termination date rather than on comparison of two bonds with different maturities. Specifically, let us consider a j-period horizon, and assume first that an investor is comparing a j-period bond, which is necessarily risk-free at the horizon, and an i-period bond (i > j), which is to be sold at a random price at the horizon. Define the conditional term premia th(i, j) as the difference between the expected real return on the latter and the known real return on the former, assuming that the current state is h. We have i

(4)

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S. F. LeRoy / Risk -aversion and term structure of real interest rates

359

(198 1) have shown that in a production economy interest rates can be random under risk-neutrality, but in that case they also show that the expectations hypothesis generally does not obtain. Proposition 2. Zf interest rates are random the expectations never hold exactly.

hypothesis can

If the expectations hypothesis is identified with the proposition that t,,(i, j) = t,(i, j) = 0 for all i, j, it is easily shown that unless interest rates are non-random the proposition implies a violation of Jensen’s inequality [see LeRoy (1982), Cox, Ingersoll and Ross (1981), or a variety of earlier sources]. Proposition 3. The unconditional approaches unity.

term premia approach zero faster than X

In other words, we have lim (t(i,j)/A) x-1

=0

for all i,j

[to prove this, use 1’Hospital’s rule ad (3) above]. Thus if the expectations hypothesis is identified with the proposition that the average term premia are small relative to the movement of interest rates, we have that it is satisfied if the movement of real.interest rates is itself small. However, a stronger version of the expectations hypothesis, that (for example) at each state the k-period rate equals the product of (one plus) the relevant expected one-period rates, less one, requires that the stronger result that the conditional as well as the unconditional term premia vanish faster than X goes to one. Unfortunately, we have Proposition 4. The conditional than A approaches unity.

lim (th(i,j)/A) # 0,

x41

term premia

lim (t,(i,j)/x)

X-l

do not approach

zero faster

#0

so that this stronger restriction on the behavior of interest rates (which is frequently adopted in empirical studies) is not justified even under near risk-neutrality.

360

S. F. LeRoy / Risk-aversion

The following propositions est rates under non-vanishing Proposition 5. andA> 1.’

The conditional

and term structure of real interest rates

characterize the term premia risk and risk-aversion: term premia

of real inter-

are always positive for i > j

To understand this, observe that when an i-period bond is sold at t + j (i > j), if the high-endowment state occurs at t + j the (i - j)-period interest rate will be low, so that the real j-period return on the i-period bond is high when consumption is high. Thus from the consumption capital asset pricing model [see LeRoy (1982) and the papers cited there] it will be priced to yield a positive risk premium. Proposition 6. For i 1, the conditional term premia are positive if (I) the endowment state is positively autocorrelated, or (2) j - i is even, or both. Here again the proposition follows from the consumption CAPM. If the endowment is positively autocorrelated (rr( h 1h) + r( I ( I) > 1, implying 7r(hIh)>m(hII) and m(lII)>a(lIh)), or ifj-i is even, then if the high-endowment state occurs at t + i, the bond will be rolled over at a low interest rate, and it will probably also be the case that the high-endowment state will occur at t + j. Thus the sequence constitutes a negative-beta strategy. Since for i
References Cox, John C., Jonathan E. Ingersoll, Jr. and Stephen A. Ross, 1981, A reexamination of traditional hypotheses about the term structure of interest rates, Journal of Finance 36, no. 4, Sept., 769-799. LeRoy, Stephen F., 1982, Expectations models of asset prices: A survey of theory, Journal of Finance 37, no. 1, March, 185-217.

’ To prove this and the following proposition, use the facts that vk( h 1h) = n(h)+ &r(l) and ~~(11 I) = a(/)+ akn(h). Here (I is the second root of II (the first is, of course, unity), equal to?r(hIh)+vr(l(l)-1.

SF.

L.eRoy / Risk -aversion and term structure of real inrerest rales

LeRoy, Stephen F. and C.J. LaCivita, 1981, Risk-aversion prices, Journal of Business 54, no. 4, Oct., 535-547. Lucas, Robert E., Jr., 1978, Asset prices in an exchange 1426- 1446.

and

the dispersion

economy,

361

of asset

Econometrica

46,