Around and around: The expectations hypothesis

We show how to construct arbitrage-free models of the term structure of interest rates in which various expectations hypotheses can hold. McCulloch (1993) provided a Gaussian non-Markovian example of the unbiased expectations hypothesis (U{EH), thereby contradicting the assertion by Cox, Ingersoll, and Ross (CIR, 1981) that only the so-called local expectations hypothesis could hold. We generalize that example in three ways: (i) We characterize the U{EH in terms of forward rates; (ii) we extend this characterization to a class of expectations hypotheses that includes all of those considered by CIR; and (iii) we construct stationary Markovian and non-Gaussian economies. The building block is a maturity-dependent vector that travels around a circle at a constant speed as maturity increases. Around and around: The expectations hypothesis Introduction In one form or another, the expectations hypothesis has played a central role in the analysis of the term structure of interest rates. Perhaps the most common form of the expectations hypothesis is the so-called unbiased expectations hypothesis (U{ EH) that asserts that forward rates equal the conditional expectations of future spot rates, but other forms exist as well. Cox, Ingersoll, and Ross (1981) (CIR) characterized a number of mutually incompatible forms of the expectations hypothesis, including, besides the U{EH, the local expectations hypothesis (L{EH), under which the expected rate of return on all zero-coupon bonds (on all assets, in fact) equals the short-term risk-free rate. Of the various expectations hypotheses they considered, they claimed that only the L{EH was consistent with general equilibrium in continuous-time models.1 McCulloch (1993) provided a counter-example to CIR's claim. His example is in the spirit of Heath, Jarrow, and Morton (1992) (HJM), in the sense that it does not admit a representation in terms of a nite number of Markovian state variables. Indeed, McCulloch suggests that CIR's claim may be true within the framework of an economy with a nite number of Markovian state variables. As we show, it is not. There is a weak version of the U{EH according to which forward rates are biased predictors of future spot rates, but the bias, or term premium, is a constant that only depends on the forecast horizon. In this weak version, a regression of future spot rates on current forward rates has a slope coe cient equal to unity, but the intercept is unrestricted.2 To construct an arbitrage-free model of the yield curve in which the weak U{EH holds, it is su cient to let all volatilities be constant|the Gaussian case.3 By contrast, our objective is (in part) to construct models of the strong version of the U{EH. In this paper, we rst generalize McCulloch's example by putting the U{EH explicitly into the HJM framework, focusing on absence-of-arbitrage conditions rather than building from a general equilibrium model.4 We then extend the analysis to a class of expectations hypotheses, parametrized by a scalar q, that includes as special cases those considered by CIR. Finally, we show how to construct Markovian examples: We start with two-state-variable stationary Markov economies that are Gaussian| that is, volatilities are nonrandom. Then, we show how to construct non-Gaussian 1They extended their claim to discrete-time models where interest rates are continuously compounded. 2See, for example, Campbell and Shiller (1991), who tested, and rejected, a weak version of the U{EH. 3See Campbell (1986) for a one-factor, general-equilibrium example. 4In Appendix A we show how to interpret the results from a general equilibrium perspective. 1 twoand three-state-variable models. The two-state-variable Gaussian model and the three-state-variable non-Gaussian model (but not the two-state-variable nonGaussian model) are members of exponential-a ne class of term structure models characterized by Du e and Kan (1993). The building block for the construction of any of our examples is a vector-valued function (t; T ), where t is the current time and T is the maturity date. As long as, for any xed t, (t; T ) lies on a sphere, the expectations hypothesis holds. If, in addition, the path of (t; t+ ) describes a circle on the sphere and goes around that circle at a constant speed as increases, then the model is Markovian. 1 The U{EH in an HJM setting In this section, which serves to introduce the analysis, we generalize McCulloch's example by characterizing the U{EH in terms of the HJM absence-of-arbitrage restriction. Let P (t; T ) denote the price at time t of a default-free zero-coupon bond that pays one unit of account at time T . Assume that, at any time t, P (t; T ) is a di erentiable function of T , and de ne the instantaneous forward rate as f(t; T ) := @ @ T log[P (t; T )]; (1) which, of course, implies log[P (t; T )] = Z T s=t f(t; s) ds; (2) and de ne the short rate at time t, which is assumed to exist, as r(t) := lim T!t f(t; T ): Let Et[ ] be the conditional expectation operator.5 Following CIR, we de ne the U{EH as follows: Forward rates are the conditional expectation of future spot rates; i.e, f(t; T ) = Et[r(T )]: In the HJM approach to modeling the term structure, the primitives are (i) an initial yield curve ff(0; t) j t > 0g, (ii) the process for the market price of risk 5The concept of equivalent martingale measures hinders rather than facilitates the analysis of expectations hypotheses. As a result, we make no use of it and all expectations are taken with respect to the physical measure. 2 (t),6 and (iii) the volatility of forward rates. We restrict attention to economies in which forward rates are di usions driven by a d-dimensional vector W (t) of standard Brownian motions. Let the process for forward rates be7 df(t; T ) = f(t; T ) dt+ f(t; T )> dW (t); (3) where f (t; T ) is a d-dimensional vector of forward-rate volatilities. Note that the market price of risk, (t), is also a d-dimensional vector, and that it could be a random process, as could f(t; T ) and f(t; T ). It is now straightforward to derive the relationship between expected future short rates and current forward rates. Since r(T ) = f(T; T ) = f(t; T ) + R T s=t df(s; T ), we can write Et[r(T )] = f(t; T ) + Z T s=tEt[df(s; T )] = f(t; T ) + Z T s=tEt[ f(s; T )] ds: De ne the forward rate premium as follows: (t; T ) := f(t; T ) Et[r(T )] = Z T s=tEt[ f (s; T )] ds: (4) From (4), it is evident that if forward rates are unbiased predictors of future spot rates, then forward rates are martingales: f(t; T ) 0. If, in addition, an ergodic yield curve exists, then it is at. An upward sloping ergodic yield curve would require an expected decrease in forward rates on average. It might seem easy, then, to construct examples of the U{EH: simply choose processes for forward rates (3) with f(t; T ) 0. The problem is that doing so arbitrarily might introduce arbitrage opportunities. The HJM absence-of-arbitrage condition speci es the drift of forward rates as8 f(t; T ) = f (t; T )> (t) + Z T s=t f (t; s) ds! (5) for all 0 t T . Equation (5) shows that we must be able to write forward rate drifts in terms of their volatilities and the market price of risk. It follows directly that the HJM characterization of the U{EH is f (t; T )> (t) + Z T s=t f (t; s) ds! = 0: (6) 6See Appendix A for a discussion of the market price of risk. 7We use ( )> to denote transposes of vectors and matrices. 8We derive this expression in Appendix A. This was rst shown by HJM; see also Du e (1996), p. 151 or Hull (1993), p. 398{401. The form of our restriction di ers from the form that Du e and Hull give because in their presentations the drift is risk adjusted, while here f (t; T ) is not. 3 The modeling challenge, then, is to nd f (t); f (t; T )g pairs that satisfy (6). To meet this challenge, it is convenient to de ne (t; T ) := (t) + Z T s=t f(t; s) ds: (7) Note that 0(t; T ) = f (t; T ), where we de ne F 0(t; T ) := @ @T F (t; T ) for any function F ( ; ). Using (7), we can write (6) as 0(t; T )> (t; T ) = 0: (8) Any function (t; T ) that satis es (8) has constant length: k (t; T )k = k (t; t)k. Two comments are in order. First, note that when d = 1, this condition can be satis ed only if (t; T ) is a constant function of its second argument, in which case f (t; T ) 0, which means there is no uncertainty. Thus in order for (8) to hold when interest rates are stochastic, there must be at least two Brownians. Second, note that (8) does not restrict how (t; T ) behaves as a function of its rst argument: In particular, (t; T ) can be a stochastic process. We can restate the key relationship between (t; T ) and the U{EH as follows: If (t; T ) is a rotation of (t; t), then the U{EH is satis ed in an arbitragefree way. We now have a simple recipe for constructing arbitrage-free models of the U{EH: (i) choose (t; T ) such that (t; t) is some random process and (t; T ), for T > t, is a rotation of (t; t); (ii) de ne (t) = (t; t), and (iii) de ne f (t; T ) = 0(t; T ). McCulloch (1993) constructed an economy in which the U{EH holds.9 In McCulloch's example there are two sources of risk, so that d = 2. He chose10 (t; t+ ) = a p2 e e 2 ; 1 e >; where a = pg0 in his notation. Note that k (t; t+ )k = k (t; t)k = a. Clearly, as increases, (t; t+ ) turns continuously from (t; t) = (a; 0)> to (t; t+1) = (0; a)> going a quarter of the way around the circle over the in nite horizon. Finally note that McCulloch's example is Gaussian since (t; T ) is deterministic. 2 A class of expectations hypotheses In this section, we generalize the results from the previous section to encompass an entire class of expectations hypotheses. For this purpose, we will need to refer to the 9Frachot and Lesne (1994) noted that such an example could be constructed easily by exploiting equation (6). 10We reversed the order of McCulloch's Brownian motions for comparison with what follows. 4 process for zero-coupon bonds:11 dP (t; T ) P (t; T ) = P (t; T ) dt+ P (t; T )>dW (t): (9) From (2), note the following relation between the volatility of bond prices and that of forward rates P (t; T ) = Z T s=t f (t; s) ds: (10) CIR characterized four versions of the expectations hypotheses: the U{EH, the L{EH, the Yield-to-Maturity Hypothesis (YTM{EH), and the Return-to-Maturity Hypothesis (RTM{EH). They showed that the U{EH and the YTM{EH are identical in continuous time.12 CIR went on to show that|after imposing absence-of-arbitrage conditions|the three independent expectations hypotheses could be characterized in the following way: P (t; T )> (t) = q 2k P (t; T )k2; (11) where q = 8<: 0 under L-EH, 1 under YTM/U-EH, and 2 under RTM-EH. Equation (11) provides an equilibrium (or absence-of-arbitrage) characterization of the expectations hypotheses. Moreover, it shows that the three hypotheses are mutually inconsistent unless P (t; T ) 0. Although CIR only considered q 2 f0; 1; 2g, we allow q to be an arbitrary real number, and we refer to (11) as the q{expectations hypothesis (q{EH). CIR referred to (11) in making their claim that only the L{EH could hold in a continuous-time general equilibrium model. Clearly, the L{EH has a special status, since q = 0 implies (t) is orthogonal to P (t; T ) but imposes no other restriction; with (t) 0, for example, the L{EH is always satis ed and P (t; T ) is unrestricted. For any other value of q, by contrast, if (t) is orthogonal to P (t; T ), then P (t; T ) = 0. We can recast (11) in terms of forward rates by di erentiating both sides with 11In such a process, we call P the proportional drift of P ; the drift of P is P P , of course. 12CIR showed that the two hypotheses are identical in discrete time too if interest rates are continuously compounded. 5 respect to T , using (10), and rearranging: f (t; T )> (t) + q Z T s=t f (t; s) ds! = 0: (12) We see that (6) is a special case of (12) with q = 1. It is convenient to generalize the de nition of (t; T ): De ne (t; T ) implicitly by (t) = q (t; t) (13) and f (t; T ) = 0(t; T ); (14) so that q (t; T ) = (t) + q R T s=t f(t; s) ds. Using (13{14), we can write (12) as q 0(t; T )> (t; T ) = 0: (15) Equation (15) is satis ed automatically if q = 0. If q 6= 0, (15) reduces to (8), in which case the comments that follow (8) apply here to the generalized de nition of (t; T ). The recipe for constructing arbitrage-free models of the q{EH is this: (i) choose (t; T ) such that (t; t) is some random process and (t; T ), for T > t, is a rotation of (t; t); (ii) de ne (t) = q (t; t), and (iii) de ne f (t; T ) = 0(t; T ). For example, with McCulloch's (t; T ), we could choose (t) = q (t; t) for any q.13 Finally, note that we can restate the q{EH in terms of either forward rate drifts or term premia. Using (13{14), we can rewrite the no-arbitrage condition (5) as f(t; T ) = 0(t; T )> (q 1) (t; t) + (t; T ) : For q 6= 0, (15) implies f(t; T ) = (q 1) 0(t; T )> (t; t); (16) which in turn, in view of (4), holds if and only if (t; T ) = (1 q) Z T s=tEt[ 0(s; T )> (s; s)] ds: (17) In what follows, we assume for convenience that (8), (16), and (17) hold even when q = 0. 13The underlying general equilibrium would look di erent of course. See Appendix A. 6 3 Markovian models McCulloch (1993) established decisively that the unbiased expectations hypothesis is consistent with general equilibrium. But he left open the possibility that expectations hypotheses may be inconsistent with general equilibrium in the nite-state Markovian world analyzed by CIR. We settle this issue by exhibiting twoand three-state variable stationary Markov economies. The construction allows for the volatility of bond prices to be stochastic. The trick is to make (t; T ) proceed around a circle at a constant pace, producing an in nite number of cycles. In all of the examples we develop below, the processes for the short rate and its drift share the following structure: dr(t) = x(t) dt+ ! (t) dW2(t) (18) and dx(t) = y(t) dt !2 (t) dW1(t): (19) Note that the di usion for r(t) depends only on W2(t), while the di usion for its drift x(t) depends only on W1(t), which is orthogonal to W1(t). As we will see, this seemingly capricious ordering of Brownians follows from our canonical representation for (t). To facilitate the analysis of the examples, we de ne a pair of functions that we will use repeatedly and for which (q; !; ; z) is a vector of xed parameters: Y (r; ) := !2 r + (q 1) z2 2 and Fq(r; x; ) := + (r ) cos[! ] + x sin[! ] ! + (q 1) z2 (1 cos[! ]): A two-state-variable model Consider the following example, where d = 2 and (t; T ) has constant norm z and turns at constant angular velocity !: (t; t+ ) = z C(!; ) ; (20) where C(!; ) = cos[! ] sin[! ] : We prove below that this choice for (t; T ) leads to processes for the short rate r(t) and its drift x(t) of the form (18{19), where (t) = z and y(t) = Y (r(t); z) = !2 ( r(t)): 7 This is a model of the yield curve in which the two state variables r(t) and x(t) form a Markovian vector. The short rate is stationary and has unconditional mean equal to , while its drift x(t) is also stationary and its unconditional drift is zero. Volatilities are constant, and therefore the model is Gaussian. Finally, the market price of risk is given by (t) = q (t; t) = q z 0 : With linear drifts, constant volatilities, and a constant price of risk, the model belongs to the exponential-a ne class introduced by Du e and Kan [1993]. Its solution for forward rates isf(t; t+ ) = Fq(r(t); x(t); ): (21) Using the methods described in Fisher and Gilles (1996), it is possible to verify that the conditional expectation of the short rate is Et[r(t+ )] = F1(r(t); x(t); ): (22) Clearly, this forecast is independent of the value of q, as it must be given that the process for (r(t); x(t)) is independent of q. But the value of q a ects the market price of risk, and therefore the shape of the yield curve given in (21). From these equations, it is clear that the term premia are given by (t; t+ ) = f(t; t + ) Et[r(t+ )] = (q 1) z2 (1 cos[! ( )]); which agrees with the term premia under the q{EH as given in equation (17). In particular, under the unbiased expectations hypothesis (q = 1) all term premia vanish. The example is the canonical Gaussian model This example has a two-dimensional Markovian state vector with deterministic volatility|the Gaussian case. In Gaussian models, term premia are nonrandom functions of maturity. It would be interesting to nd out how to construct non-Gaussian models of the q{EH, because in such models term premia change randomly. We construct such examples by generalizing the canonical example. Before turning to the issue of non-Gaussian models, however, we prove two results about the canonical example, which clearly show that it is the place whence to generalize. First, we show that there exists no one-state variable model of the q{EH, Gaussian or not. This is simply because, under the q{EH, the univariate process for the short rate cannot be Markovian (all proofs appear in Appendix B). 8 Proposition 1 If the q{EH holds and the short rate r(t) is not deterministic, then its univariate process is not Markovian. Second, we show that any Gaussian model of the q{EH with a two-dimensional Markovian state vector amounts to a renormalization of the canonical example. Proposition 2 Suppose that the q{EH holds in a model with two Markovian state variables in which bond prices have deterministic volatilities. Then there exist constant scalars , !, and z such that, perhaps after changing the basis for the vector of Brownian motions (thus a ecting the representation of the processes, but not the form of the yield curve): (t; T ) has the form shown in (20); the processes for the short rate and its drift have the form shown in equations (18) and (19), with (t) = z and y(t) = Y (r(t); z); and the initial yield curve has the form f(0; ) = Fq(r(0); x(0); ): (23) Proposition 2 asserts that in a Gaussian and Markovian economy (with two state variables), the q{EH implies that (t; T ) keeps turning around the circle at constant angular velocity, !. It also speci es x(t), the drift of r(t), as linearly independent of r(t), and y(t), the drift of x(t), as a translation of r(t) with coe cient !2, and independent of the value of x(t) itself. The short rate is stationary with unconditional mean equal to , while its drift is also stationary with unconditional mean equal to zero. Because the model is Markovian, the initial time has no particular signi cance, and equation (23) for the \initial" yield curve delivers the form of the generic yield curve, which indeed agrees with the solution that follows from solving a Du e-Kan model, as we did to get (21). But while the Du e-Kan method requires solving a simultaneous system of three Ricatti diferential equations, we obtained the initial yield curve in the proof of Proposition 2 by solving a single second-order di erential equation. Clearly, yield curves can be at; in fact the yield curve is at if and only if r(t) = + (q 1) z2 and x(t) = 0: Fq( + (q 1) z2; 0; ) = + (q 1) z2: There is always an ergodic yield curve that is obtained by setting r(t) and x(t) at their respective unconditional means, and 0: Fq( ; 0; ) = + (q 1) z2 (1 cos[! ]): 9 The ergodic yield curve is thus the same as the at yield curve under the U{EH (q = 1), but in other cases it is a sine wave. Non-Gaussian models We now turn to the non-Gaussian case. The simplest way to generalize the canonical Gaussian example is to suppose that (t; t)|which is porportional to the market price of risk (t)|is an Ito process. To do this without increasing the number of state variables, replace equation (20) by (t; t+ ) = (t)C(!; ) where (t) is some function of r(t) and x(t). There result non-Gaussian two-state variable Markov models of the yield curve in which the q{EH holds. Although this strategy works well and delivers closed-form expressions for bond prices, checking that the q-hypothesis holds may not be easy in practice, because we do not have closed-form expressions for the conditional forecasts of the state variables. For this reason, we also introduce a three-state variable non-Gaussian model in which we know how to compute both bond prices and conditional forecasts. Proposition 3 Let (t; t + ) = (t)C(!; ), where C(!; ) is as in (20) and (t) = (r(t); x(t)), for any function ( ; ) (with the restriction that the implied stochastic processes for r(t) and x(t) have a solution). Suppose also that 2(t) has an unconditional mean z2. Pick a constant and initial conditions r(0) and x(0), and choose the following initial yield curve f(0; ) = Fq(r(0); x(0); ): Then: the resulting yield curve model is Markovian with a two-dimensional state vector (r(t); x(t)), as well as non-Gaussian if (t) is random, and it satis es the qexpectations hypothesis; the processes for the short rate and its drift have the form shown in equations (18) and (19), with y(t) = Y (r(t); (t)); at any time t, the yield curve is f(t; t+ ) = Fq(r(t); x(t); ): (24) We see that bond prices are independent of (t) and depend on the other two state variables r(t) and x(t) exactly as they do in the corresponding Gaussian model. The only di erence between the Gaussian and the non-Gaussian models is the distribution of these state variables; therefore yield curves of a given shape do not occur with the same frequency in both models. 10 In the non-Gaussian model, the drift of x(t) depends on 2(t) (except when q = 1), which complicates the task of making conditional forecasts. If 2(t) were a linear function of x(t) and r(t), then the model would be in the exponential-a ne class, and we would know how to compute conditional forecasts. Unfortunately, in our two-factor model there is no guarantee that either the interest rate or its drift can stay positive (in fact, the mean of x(t) equals zero), and no linear combination of these variables is guaranteed to stay positive. Therefore, 2(t) cannot be a linear function of (r(t); x(t)). We can get around this problem by adding a third, independent state variable, however. The following three-state-variable model belongs to the exponential-a ne class. Proposition 4 Set d = 3. Let (t; t + ) = (t)C (!; ), where C extends the function C given in (20) by adding a third component which identically equals zero; let the process for 2(t) satisfy d 2(t) = k (z2 2(t)) dt+ (t) dW3(t); and let f(0; ) = Fq(r(0); x(0); ), so that the initial yield curve is as in Proposition 3. Then: the resulting yield curve model is Markovian with a three-dimensional state vector (r(t); x(t); 2(t)), as well as non-Gaussian, and it satis es the q-expectations hypothesis; the processes for the short rate and its drift have the form shown in equations (18) and (19), with y(t) = Y (r(t); (t)); at any time t, the yield curve is f(t; t+ ) = Fq(r(t); x(t); ): The bond price formula in the three-state-variable model is identical to that in the two-state-variable model. The only di erence is that, because the former model belongs to the exponential-a ne class, it is possible to obtain closed-form solutions for the conditional forecasts of all state variables (as well as their conditional variances). The state variable of most interest, of course, is the short rate itself, for which we have: Et[r(t+ )] = F1(r(t); x(t); ) + (1 q)!2 k2 + !2 2(t) z2 e k cos[! ] + k sin[! ] ! ! : It can then be veri ed that the forward premium is (t; t+ ) = f(t; t+ ) Et[r(t+ )] 11 = (q 1) z2(1 cos[! ]) !2( 2(t) z2) k2+!2 e k cos[! ] + k sin[! ] ! : It can be further veri ed that, because Et[ (t+ s)2] = z2 + e ks (t)2 z2 ; the above expression for the term premium agrees with (17), which in the present case reduces to (t; t + ) = (q 1)! Z s=0Et[ (t+ s)2] sin[!( s)] ds: 4 Concluding remarks We have shown that the expectations hypothesis is compatible with general equilibrium even in Markovian settings. The models we have been able to construct, however, will not help to rehabilitate the expectations hypothesis. Rather, they show how implausible the hypothesis is. The main reason all expectations hypotheses are implausible in Markovian settings is that they share the same process for the short rate, which implies that the forecast of the short rate path is a sine wave with nondampening amplitude. Of course, forward rates behave in the same way (they are equal to the short rate forecast under the unbiased expectations hypothesis). In a non-Markovian setting, yield curves and forecasts of the path of the short rate can look more reasonable. Note that in his example, McCulloch did not exhibit a yield curve. In fact, McCulloch's example is compatible with any initial yield curve.14 The expectations hypothesis imposes restrictions only on the dynamics of the yield curve. Given the initial yield curve and its dynamics, it is in principle possible to reconstruct future yield curves for any path of the set of Brownian motions. But because there is no nite set of variables that summarizes the state of the economy, we cannot say what a typical yield curve looks like. At rst blush, it may seem that the Markovian models have the potential to represent the cyclical behavior of interest rates prior to the existence of the Federal Reserve. Unfortunately, the models cannot be made to reasonably approximate that sort of cyclical behavior. The problem is that the pre-Fed cycle occurs in absolute time, while the cycles in the Markovian models occur in relative time. In other words, there is no way to make summer (for example) be a high (or low) rate season on average. As a nal observation, we suspect that no equilibrium model of the expectations hypothesis, Markovian or non-Markovian, can guarantee the non-negativity of the 14Initial yield curves in our examples are determined only by the condition that the model is Markovian, as the proofs of the propositions make clear. 12 short rate. This is certainly true in McCulloch's example and all of our examples. Such a feature makes the expectations hypothesis a poor benchmark for nominal rates. The reason for the inability to keep the short rate positive is simple. If the short rate is to stay positive, its volatility must be small enough and its drift must be positive whenever its level is close to zero. But in all our examples, the drift of the short rate is independent of the short rate itself, and therefore will not always point in the right direction when the rate is small.