Consumption and Asset Prices with Recursive Preferences

CONSUMPTION AND ASSET PRICES WITH RECURSIVE PREFERENCES MARK FISHER AND CHRISTIAN GILLES Abstract. Weanalyzeconsumptionandassetpricingwithrecursivepreferences given byKreps{Porteusstochastic di(cid:11)erential utility(K{PSDU).Weshowthat utilitydependsontwostatevariables: currentconsumptionandasecondvariable (relatedtothewealth{consumptionratio)thatcapturesallinformationaboutfutureopportunities. Thisrepresentationofutilityreducestheinternalconsistency conditionforK{PSDUtoarestrictiononthesecondvariableintermsofthedynamics of a forcing process (consumption, thestate{price deflator, or thereturn onthemarketportfolio). Solvingthemodelfor(i)optimalconsumption,(ii)the optimalportfolio,and(iii)assetpricesingeneralequilibriumamountsto(cid:12)nding the process for the second variable that satis(cid:12)es this restriction. We show that the wealth{consumption ratio is the value of an annuity when the numeraire is changedfromunitsoftheconsumptiongoodtounitsoftheconsumptionprocess, and we characterize certain features of the solution in a non-Markovian setting. In a Markovian setting, we provide a solution method that it quite general and can be used to produce fast, accurate numerical solutions that converge to the Taylor expansion. 1. Introduction We solve for the dynamics of consumption, investment, and asset prices in a general-equilibrium, continuous-time stochastic model with a representative agent who has recursive preferences. The setting varies and determines what the main problem is. In an endowment economy, the dynamics of consumption is given and we solve for asset prices (the exchange problem); in a partial equilibrium setting, prices are given and we solve for the optimal consumption and investment plan (the planning problem); inaproductioneconomy, asetoflinear technologies isgivenand we solve forconsumption, investment andasset prices (theproduction problem). By focusing on the consumption{wealth ratio, we (cid:12)nd that these three problems are essentially equivalent, and we solve them all at once. Date: August 21, 1998. JEL Classi(cid:12)cation. G12. Key words and phrases. Recursive preferences, stochastic di(cid:11)erential utility, general equilibrium, optimal consumption, optimal portfolio, equity premium, term structure of interest rates, asset pricing. We thank Greg Du(cid:11)ee for useful conversations. The views expressed herein are the authors’ and do not necessarily reflect those of the Board of Governors of the Federal Reserve System or Bear, Stearns & Co. 1

2 MARK FISHER AND CHRISTIAN GILLES The recursive utility framework generalizes the standard time-separable power utility model, allowing the separation of risk aversion and intertemporal substitution. This framework was introduced by Epstein and Zin (1989), who analyze recursive preferences in a discrete-time setting, and Du(cid:14)e and Epstein (1992b), who develop a continuous-time formulation of Epstein and Zin’s class of recursive utility called stochastic di(cid:11)erential utility. We use a martingale approach to solve for the equilibrium, along the lines of Du(cid:14)e and Skiadas (1994), who show that the (cid:12)rst-order condition for optimality is equivalent to the absence-of-arbitrage conditions for asset prices|namely, that asset prices deflated by the state-price deflator are martingales. In addition, they provide a representation for the state-price deflator for the Kreps{Porteus stochastic di(cid:11)erential utility (K{P SDU) that we adopt here. We (cid:12)nd that utility depends on two state variables: current consumption and a second variable (the growth variable) that captures all information about future opportunities. This representation of utility depends on the homotheticity of K{ P SDU, and holds for the exchange problem as well as in economies with linear investment opportunities (covering both the case of the planning problem and that oftheproductionproblem). Equilibriuminthemodelreducestoacentralrestriction on the growth variable in terms of the dynamics of a forcing process. This forcing process can be either consumption (for the exchange problem), the real state{price deflator (for the planning problem), the return on the market portfolio (for the production problem), or something entirely di(cid:11)erent (for example, the state-price deflator expressed in an arbitrary numeraire). Solving the model for (i) optimal consumption, (ii) the optimal portfolio, and (iii) asset prices amounts to (cid:12)nding the process for the growth variable that satis(cid:12)es this restriction. Unless the elasticity of intertemporal substitution is unity, we can replace the growth variable with the wealth{consumption ratio. The homogeneity properties of the representative agent’s planning problem (homothetic preferences and linear technology) ensure that optimal consumption is proportional to wealth. We show that the optimal wealth{consumption ratio is the value of an annuity when the numeraire has been changed from units of the consumption good to shares in the consumption process. Thus, the wealth-consumption ratio is the value of an asset. As such, it must obey a standard absence-of-arbitrage condition. As a practical matter, the model is solved when we know how to obtain, analytically or numerically, an expression for the consumption-wealth ratio that satis(cid:12)es this condition. It is then straightforward to obtain expressions for the rate of interest and the price of risk|determined by the dynamics of the so-called state-price deflator|and other variables of interest. In order to focus on the role of preferences, it is convenient, in the spirit of Lucas (1978) (as well as Mehra and Prescott (1985) and Weil (1989)), to start with the exchange problem, in which the forcing process is consumption and we solve for the supporting prices, i.e., the state-price deflator. For the planning problem, we reverse the process, solving for the optimal consumption and investment plans using the state-price deflator as the forcing process. Finally, in the spirit of Cox, Ingersoll, Jr., and Ross (1985a) and Campbell (1993), we model technology, which we interpret as the return on the optimally

CONSUMPTION AND ASSET PRICES 3 invested wealth of the representative consumer. For this production problem, then, we solve for consumption and prices using the return on the market portfolio as the forcing process. Thechoiceofforcingprocessisnotlimitedtothethreealreadymentioned,namely consumption, the state-price deflator and the return on optimally invested wealth. It turns out, for example, that with the forcing process chosen as the product of the state-price deflator and wealth (which can be interpreted as the state-price deflatorwhenthenumeraireissharesinthewealthprocess)themodelcanbesolved algebraically. Although this choice does not correspond to any natural setting, it opens the path to generating examples with arbitrarily complex dynamics. Withanaturalforcingprocess,itisnotpossibleingeneralto(cid:12)ndaconsumptionwealth ratio that satis(cid:12)es the no-arbitrage condition, but progress is achieved by modeling the dynamics of the forcing process as driven by a (cid:12)nite set of Markovian state variables. In such a Markovian setting the no-arbitrage condition becomes a partialdi(cid:11)erentialequation(PDE)thatwewishtosolveforthewealth-consumption ratio as a function of the state variables (and time). Because this ratio is an annuity, its value is that of an integral of bond prices. In some circumstances, standard methods deliver exact solutions (numerically at least and sometimes even analytically) to the bond pricing problem, and we get the wealth-consumption ratio by numerical integration. In all other cases, we attack the annuity PDE directly and provide an approximate solution method that is quite general and can be used to produce fast, accurate numerical solutions that converge to the Taylor expansion of the exact solution. Much like standard bond pricing methods, our general solution method transforms the PDE into a set of simultaneous ordinary di(cid:11)erential equations (ODE) when the horizon is (cid:12)nite. A unique solution is guaranteed to exist, but only for horizons that are su(cid:14)ciently short. We solve the in(cid:12)nite-horizon problem by extendingthe(cid:12)nitehorizon andtakingalimit. Suchalimitdoesnotnecessarilyexist, but when it does, it is the solution of a set of algebraic equations. Relatedwork. Asnotedabove,Du(cid:14)eandEpstein(1992b)andDu(cid:14)eandSkiadas (1994) lay the groundwork for continuous-time modeling of recursive preferences. SchroderandSkiadas(1997) extendtheearlierworkinanumberofimportantways. They prove existence and uniqueness of solutions and address the relation between the (cid:12)rst-order conditions and optimality in a more general non-Markovian setting than has been treated previously, and we refer the reader to their paper regarding theseissues.1 Theyalsoprovidesomeclosed-formsolutionstotheplanningproblem in special cases that we also consider below. Du(cid:14)eandEpstein(1992a) derivetherepresentation forriskpremiainthesetting we adopt here. Both Du(cid:14)e and Epstein (1992a) and Du(cid:14)e, Schroder, and Skiadas (1997) solve one-factor models of the term structure in the special case where the 1By contrast, we propose a method that delivers candidate solutions for continuation utility and consumption. A martingale property must be checked for our proposed continuation utility to be valid, and the consumption process is only guaranteed to satisfy the (cid:12)rst-order condition for optimality. Schroder and Skiadas (1997) establish the su(cid:14)ciency of the (cid:12)rst-order condition for some, but not all, parameter values.

4 MARK FISHER AND CHRISTIAN GILLES dynamics of the state variable are introduced through the growth rate of consumption. Among other things, these papers address the how a change in the coe(cid:14)cient of relative risk aversion a(cid:11)ects the shape of the yield curve. Campbell (1993) linearizes the discrete-time model of Epstein and Zin (1991), and derives an approximate solution to the model in the homoskedastic case that is exact in the special case. We derive moregeneral conditions underwhich important aspects of Campbell’s solution are essentially exact, providing insight into the performance of his approximate solutions. In addition, we examine the approximate relations Campbell describes between the volatility of a perpetuity and the price of risk. Campbell’s model is used by Campbell and Viceira (1996) to study the planning problem. Outline. InSection2,weadoptanon-Markoviansettingtoanalyzethestructureof themodel. Weintroducetheutilityfunction(K{PSDU),forwhichwederiveatwostate-variablerepresentationinthecontextoftheexchangeproblem,therebysimplifying the model’s central restriction. The state variables independently capture the level andgrowth features of theendowmentprocess. Thewealth{consumption ratio depends in a simple way on the growth variable; the relation is one-to-one except in the case of unit elasticity of intertemporal substitution. We demonstrate that this ratio is the value of an annuity after the numeraire has been changed from units of consumption good to shares in the endowment itself. We then address the planning problem, for which the dynamics of the state{price deflator are given, and we show that after changing the level variable from consumption to wealth and presenting the problem as an exercise in dynamic programming, our representation for utility satis(cid:12)es the envelop condition identically. Next we turn to general equilibrium where we model technology and derive the restriction on the growth variable with respect to those dynamics. Finally we address the (cid:12)nite-horizon problem explicitly. In Section 3 we investigate the features of solutions to the model that can be inferred in the non-Markovian setting of Section 2. We unify the three restrictions on thegrowth variable in terms of the dynamics of a generic forcing variable. In the caseofunitelasticityofintertemporalsubstitutioncombinedwithhomoskedasticity, we show that the growth variable is a weighted average of expected future growth rates of the forcing variable. For other elasticities, we rely on the fact that the wealth{consumption ratio is an asset price (the value of an endowment annuity) to investigate the model. We show that the weak form of the expectations hypothesis as applied to the endowment term structure delivers useful results. In addition, we examine a number of limiting cases regarding the preference parameters. In Section 4, we present the PDEs that characterize the solution to the model in a Markovian setting. We solve by integration of bond prices when bond pricing methods deliver exact solutions, and we describe our generic power series solution methodthroughasequenceofexamples. Wealsoprovideanexamplethatillustrates that (i) thata solution mayfail to exist for an arbitrary(cid:12)nitehorizon and(ii) that, even when a solution exists for all (cid:12)nite horizons, it does not convergence as the horizon goes to in(cid:12)nity.

CONSUMPTION AND ASSET PRICES 5 In Section 5 we summarize briefly the contribution of this paper, and we discuss how the tools we provide can be brought to bear on asset pricing puzzles in future research. In a future version of this paper, we intend to include a numerical investigation to illustrate our method. In the meantime, we have included in an Appendix a complete Mathematica package that implements all aspects of our method. 2. The structure of the model Tastes: Stochastic di(cid:11)erential utility. Wenowintroducethepreferencesof the representativeagent,forwhichweadoptKreps{Porteusstochasticdi(cid:11)erentialutility (SDU). We present a value function for Kreps{Porteus SDU that is valid for the entire parameter space. Using this value function and the general representation for the SDU gradient given by Du(cid:14)e and Skiadas (1994), we obtain an explicit representation for the state-price deflator. We derive expressions for the interest rate and the price of risk in terms of this representation. AsexplainedbyDu(cid:14)eandEpstein(1992a) andDu(cid:14)eandEpstein(1992b), SDU (not just the Kreps{Porteus speci(cid:12)cation) can be represented by a pair of functions (f(cid:22);A(cid:22)) called an aggregator.2 Thefunctions f(cid:22)and A(cid:22)can beinterpreted as capturing separately attitudes toward intertemporal substitution and attitudes toward risk. Hypothetical experiments can beconducted, for example, by(cid:12)xingf(cid:22)and varyingA(cid:22) to studythe e(cid:11)ect of increasing riskaversion. Associated with theaggregator, there is a process V(cid:22)(t), called continuation utility, such that the value of the consumption plan fc(t) jt (cid:21) 0g is V(cid:22)(0). When SDU is well-de(cid:12)ned, the process for V(cid:22) is uniquely given by (cid:18) (cid:19) 1 dV(cid:22)(t) = −f(cid:22)(c(t);V(cid:22)(t))− A(cid:22)(V(cid:22)(t))k(cid:27) (t)k2 dt+(cid:27) (t) > dW(t); V(cid:22) V(cid:22) 2 for some (cid:27) (t). V(cid:22) To represent a given SDU, the aggregator is not unique. Importantly, there exists a normalized form (f;A) where A (cid:17) 0. Signi(cid:12)cant analytical simpli(cid:12)cation is achieved by using the normalized aggregator, although the convenient separation referredtoaboveislostsincebothaspectsofpreferencesarecombinedinf. Suppose thatwede(cid:12)neV(t) := (cid:7)(V(cid:22)(t)), where(cid:7)(v)isatwice-continuously di(cid:11)erentiableand strictly increasing transformation. Since only the ordinal properties of utility are of interest, the change of variables has no e(cid:11)ect on choices, but it changes the form of theaggregator (throughIt^o’slemma). Ifwechoose(cid:7) tosatisfy(cid:7) 00 (v)−A(cid:22)(v)(cid:7) 0 (v) = 0, then the new aggregator is (f;A), where A = 0 and f(c; v) is de(cid:12)ned implicitly in f(c;(cid:7)(z)) = (cid:7) 0 (z)f(cid:22)(c;z). With the new aggregator, we have dV(t) = (cid:22)V(t)dt+(cid:27)V(t) > dW(t); (2.1) where (cid:22)V(t)= −f(c(t);V(t)); (2.2) 2Ournotation reverses theroles of (f;A) and (f(cid:22);A(cid:22)).

6 MARK FISHER AND CHRISTIAN GILLES Whenever two processes c(t) and V(t) satisfy (2.2), then V(t) is the process for continuation utility corresponding to the consumption plan c(t).3 Using (2.1) and (2.2), we can also express recursive utility as (cid:20)Z (cid:21) T V(t) =E f(c(s);V(s))ds : (2.3) t t We are interested in both (cid:12)nite- and in(cid:12)nity-horizon settings. For the in(cid:12)nite horizon, we take the limit of (2.3) as T ! 1. The Riesz representation of the utility gradient for such preferences is given by:4 (cid:26)Z (cid:27) t G(t) := exp f (c(s);V(s))ds f (c(t);V(t)); (2.4) v c s=0 where f and f are the partial derivatives of f.5 Optimality of consumption rec v quires that the utility gradient be proportional to the state{price deflator. (See Appendix A for a discussion of the state{price deflator.) Note that the relative dynamics of the utility gradient are given by dG=G = df =f +f dt. c c v One of the aggregators for Kreps{Porteus SDU is6 (cid:18)v((c=v)1−1=(cid:17) −1) −γ f(cid:22)(c;v) = and A(cid:22)(v) = ; (2.5) 1−1=(cid:17) v where(cid:18),(cid:17),andγ areconstantparameters. AsshownbyDu(cid:14)eandEpstein(1992a), these preferences allow a disentangling of attitudes toward risk from attitudes toward intertemporal substitution. In our parameterization, (cid:18) > 0 is the rate of time preference, (cid:17) > 0 is the elasticity of intertemporal substitution and γ (cid:21) 0 is the coe(cid:14)cient of relative risk aversion. When γ(cid:17) = 1, Kreps{Porteus SDU specializes to standard time-separable preferences with power utility, characterized by indi(cid:11)erence toward the timing of resolution of uncertainty. (With (cid:17)γ > 1, the consumer prefers early resolution and with (cid:17)γ < 1, late resolution). That γ(cid:17) = 1 reduces to the case of standard preferences is more easily seen in terms of the utility gradient using the normalized aggregator, which we derive below. Du(cid:14)e and Epstein (1992b) show that preferences are homothetic if and only if there is an ordinally equivalent aggregator (f;A) satisfying (i) f is homogeneous of degree 1 and (ii) A is linearly homogeneous of degree −1. The aggregator (f(cid:22);A(cid:22)) given in (2.5) clearly satis(cid:12)es these conditions. To normalize the aggregator into the canonical form (f;0), use the transformation v1−γ −1 (cid:7)(v) = ; (2.6) 1−γ 3Du(cid:14)eand Lions (1992) address theexistence anduniquenessof V when cis modeled in termsof state variables (where c itself may be a state variable). 4Du(cid:14)eandEpstein(1992a) deriveaMarkovianversionof (2.4)usingtheBellman equation,while Du(cid:14)eand Skiadas(1994) derive (2.4) in a more general non-Markovian semimahr R tingale setting.i 5Marginal utility in the direction of the consumption process q is given by E T G(s)q(s)ds . t s=t See Du(cid:14)e(1996) for a discussion of the utility gradient. 6Thefunctionalformoff comesfromDu(cid:14)eandEpstein(1992a), p.418. Their(cid:26)is1−1=(cid:17), their (cid:11) is 1−γ,and their(cid:12) is (cid:18).

CONSUMPTION AND ASSET PRICES 7 producing (cid:18) (cid:19) (cid:16) (cid:17) c 1−1=(cid:17) (cid:18)V −1 V1=(1−γ) f(c;v) = ; V := 1+(1−γ)v: (2.7) 1−1=(cid:17) For each of the cases γ = 1 and (cid:17) = 1, the aggregator follows from taking a limit in (2.7).7 Du(cid:14)e and Epstein (1992a) use (cid:7)(v) = v1−γ=(1−γ). As a result we have 1+(1−γ)v where they have (1−γ)v in their normalized aggregator (p. 420). The advantage of our formulation is that we get the correct limit for γ = 1. Although (cid:7)(0) 6= 0 for our transformation, it will turn out that V is always positive for all positive values of γ and (cid:17). Continuation utility, the utility gradient, and the exchange problem. The (cid:12)rst problem we face is that of (cid:12)nding a representation for continuation utility and the utility gradient of an agent with K-P SDU and a given consumption process. This is the usual setting of an endowment economy. In this setting, the famous tree metaphor clari(cid:12)es some discussion, so we imagine for the time being that the endowment grows on identical trees, so normalized that in the current period each tree produces one unit of the good. In the current period, then, c denotes both current consumption and the number of trees. Wenowestablishanimportantresult. ForaconsumerwithK-PSDUpreferences, we can represent the state of the world with only two state variables: the level of currentconsumption,c(t),andanothervariable, (t),thatcaptures|conditionalon current consumption|all relevant information about future growth opportunities for consumption, i.e., about the productivity of a tree. As such, (t) depends on the dynamics of log consumption but not its current value. We can write the value of continuation utility as a deterministic function of the two state variables: V(t) = g(c(t); (t)). Note that we have two ways to measure the increase in utility from an increase in consumption: f (c;g(c; )) and g (c; ). The (cid:12)rst expression c c indicates the marginal utility of a unit of current consumption (holding the future valueofcontinuation utility(cid:12)xed),whilethesecondindicates themarginalutility of a tree. The ratio g (c; )=f (c;g(c; )) is the marginal rate of substitution between c c trees and current consumption. Given the homotheticity of preferences, this ratio must be independent of current consumption. De(cid:12)ne the function g (c; ) c h( ) := : (2.9) f (c;g(c; )) c Since (cid:25)(t) := h( (t)) is the value of a tree expressed in terms of consumption good and c(t) is the number of trees (as normalized in period t), we may interpret k(t) := (cid:25)(t)c(t) as the evaluation the consumer would give of his wealth, given his endowment, even in the absence of a market. Given the form of the normalized 7Wetakea particular interest in the case (cid:17)=1, which yields (cid:18) (cid:19) 1 f(c;v)=(cid:18)V log(c)− log(V) : (2.8) 1−γ

8 MARK FISHER AND CHRISTIAN GILLES aggregator, h( ) is independentof c if and only if 1+(1−γ)g(c; ) is homogeneous of degree 1 − γ in c, that is g(c; ) = (c1−γgb( ) − 1)=(1 − γ). Now, we have considerable freedom in the de(cid:12)nition of , because, at the cost of changing the form of gb, we can replace by any other variable that is in one-to-one relation with . The simplest choice is to de(cid:12)ne so that gb( ) = , but this choice complicates future expressions and is hard to interpret. Choosing gb( ) = 1−γ achieves the greatest simpli(cid:12)cation of future expressions, but our choice is slightly di(cid:11)erent and easier to interpret. By setting bg( ) = ( =(cid:18))(cid:17)(1−γ), it will turn out that is the marginalutilityofwealthcorrespondingtotheunnormalizedaggregator (f(cid:22);A(cid:22))(this aggregator is characterized by constant returns to scale, so that marginal utility of wealth is independent of wealth or current consumption). As a result, we have8 (c( =(cid:18))(cid:17))1−γ −1 g(c; ) = (2.10a) 1−γ (cid:25) = h( ) =(cid:18) −(cid:17) (cid:17)−1: (2.10b) Using (2.10a), we can write ( (cid:0) (cid:1)) (cid:17)(cid:18) ( =(cid:18))1−(cid:17) −1 f(c;g(c; )) = −(c( =(cid:18))(cid:17))1−γ : (2.11) 1−(cid:17) Note that (cid:0) (cid:1) (cid:17)(cid:18) ( =(cid:18))1−(cid:17) −1 lim = (cid:18) log( =(cid:18)): (cid:17)!1 1−(cid:17) The partial derivatives f (c;v) and f (c;v) evaluated at v = g(c; ) are given by c v f (c;g(c; )) = (cid:18)γ(cid:17) 1−γ(cid:17)c −γ (2.12a) c (cid:18) (cid:19) ( (cid:0) (cid:1)) 1 (cid:17)(cid:18) ( =(cid:18))1−(cid:17) −1 f (c;g(c; )) = −(cid:18)− −γ : (2.12b) v (cid:17) 1−(cid:17) We obtain the Riesz representation of the utility gradient by inserting (2.12) into (2.4). With standard preferences, (cid:17) = 1=γ, the utility gradient specializes to G(t) = (cid:18)e −(cid:18)tc(t) −γ as expected. Another benchmark case is γ = 1, for which f (c;g(c; )) = (cid:18)(cid:17) 1−(cid:17)c −1 and f (c;g(c; )) = (cid:18)(cid:17) 1−(cid:17) = (cid:25) −1. c v Thusfar in our discussion of g and , we have not distinguished between in(cid:12)niteand (cid:12)nite-horizon problems. The (cid:12)nite horizon imposes the boundary condition (cid:25)(T) = 0. This condition cannot be met when (cid:17) = 1 with the normalization adopted. At the end of this section, we will explicitly address the (cid:12)nite-horizon problem. For the present, we treat only the in(cid:12)nite-horizon case. A necessary condition for g(c(t); (t)) to be the continuation utility corresponding to the endowment, equation (2.2) requires that the drift of g(c(t); (t)) equal −f(c(t);g(c(t); (t))). This requirement produces the consistency condition that 8Asasserted above, V =1+(1−γ)g(c; )=(c( =(cid:18))(cid:17))1−γ is always positive.

CONSUMPTION AND ASSET PRICES 9 must satisfy in order that g(c(t); (t)) be continuation utility: (cid:0) (cid:1) (cid:17)(cid:18) ( (t)=(cid:18))1−(cid:17) −1 1 = (cid:22)e (t)+(cid:17)(cid:22)e (t)+(1−γ) k(cid:27) (t)+(cid:17)(cid:27) (t)k2; (2.13) 1−(cid:17) c 2 c where all the terms are implicitly de(cid:12)ned by9 dlog(c(t)) = (cid:22)e (t)dt+(cid:27) (t) > dW(t) c c dlog( (t)) = (cid:22)e (t)dt+(cid:27) (t) > dW(t): Our choice of g has allowed us to cancel (c(t)( (t)=(cid:18))(cid:17))1−γ from both sides of (2.13). If solves equation (2.13), then g as given in (2.10a) is continuation utility, R t > provided (cid:27)V(s) dW(s) is a martingale, where s=0 (cid:8) (cid:9) (cid:27)V(t) = (c(t)( (t)=(cid:18))(cid:17))1−γ (cid:27) c (t)+(cid:17)(cid:27) (t) : (See Proposition 3 in Schroder and Skiadas (1997) for a proof.) Thus, whenever a solution to the underlying problem exists, the solution to (2.13) provides it. However, the solution to (2.13) does not provide the solution to the underly problem unless the volatility of g is well behaved. We now seek the interest rate and price of risk that support the endowment. To support a consumption plan, prices must be aligned with marginal rates of substitution, which in the present context means that the state-price deflator m(t) must becolinear with the utility gradient G(t). In accord with (A.1), we note (from applying It^o’s lemma to G(t)) that in this case the short rate r and the price of risk (cid:21) are given by 1 (1−γ)(1−γ(cid:17)) 1 1 r(t) = (cid:18)+ (cid:22)e (t)+ k(cid:27) (t)+(cid:17)(cid:27) (t)k2 − k(cid:21)(t)k2 (2.14a) c c (cid:17) (cid:17) 2 2 (cid:21)(t) = γ(cid:27) (t)+(γ(cid:17)−1)(cid:27) (t); (2.14b) c wherewehave used(2.13) to eliminate the termin curlybrackets fromf (c;g(c; )) v in (2.12b). As a by-product, (cid:22)e has been eliminated as well from the expression for the interest rate. Consequently, (cid:27) is the only aspect of we will need for asset pricing. Note that (cid:27) enters the price of risk with a sign that depends on whether early or late resolution of uncertainty is preferred. Again, with (cid:17) = 1=γ, (2.14) specializes to the expected expressions for r and (cid:21) under the C{CAPM: r(t)= (cid:18)+γ(cid:22)e (t)−1 γ2k(cid:27) (t)k2 and(cid:21)(t)= γ(cid:27) (t). ThisisconsistentwithTheorem c 2 c c 2(a) (under condition I) in Schroder and Skiadas (1997). Usingtheutilitygradientasstate-pricedeflator,wecanpriceassets. Forexample, an asset that pays a continuous dividend at the rate of one unit of good per year (a 9Weuse thefollowing notational convention. If z(t) is explicitly strictly positive, then(cid:22) , (cid:22)e and z z (cid:27) refertothequantitiesimplicitlyde(cid:12)nedindz(t)=z(t)=(cid:22) (t)dt+(cid:27) (t)>dW(t)anddlog(z(t))= z z z (cid:22)e (t)dt+(cid:27) (t)>dW(t),implying(cid:22)e (t):=(cid:22) (t)−1k(cid:27) (t)k2. ThestatevariablesX(t)insection4 z z z z 2 z arenotnecessarilypositive. Forthesevariables,wewritedX(t)=(cid:22) (t)dt+(cid:27) (t)>dW(t),sothat X X (cid:22) (t) refers to thedrift of X (in level) and (cid:27) toits volatility. X X

10 MARK FISHER AND CHRISTIAN GILLES real consol) is valued at (cid:20)Z (cid:21) Z 1 G(s) 1 E ds = p(t;s)ds; (2.15) t G(t) s=t s=t where p(t;s)= E [G(s)=G(t)] is the value at time t of a zero-coupon bond that pays t one unit of consumption at time s. We can also (cid:12)nd the value of the endowment, which can be interpreted as the consumer’s wealth: (cid:20)Z (cid:21) 1 G(s) k(t) = E c(s)ds : (2.16) t G(t) s=t We note that the right-hand side of (2.16) is marginal utility in the direction of the endowment. Earlier, we claimed that wealth is given by k(t) = h( (t))c(t) = (cid:18) −(cid:17) (t)(cid:17)−1c(t), where h( ) := g (c; )=f (c;g(c; )). The two notions of wealth c c are, of course, the same. To see this, let rV(c;c 0 ;t) be the Gateaux derivative of V(t) evaluated at the endowment and in the direction of process c 0 (t). Then, our de(cid:12)nitionofatreeimpliesthatg (c(t); (t)) = rV(c;c;t), andthedesiredresultfolc lows from the Riesz representation of rV(c;c;t) given in Du(cid:14)e and Skiadas (1994) (note that with equation (2.12) substituted in, (2.4) is the Riesz representation of rV(c;c;0)). It is instructive to examine this result from a slightly di(cid:11)erent angle. To do this, it is convenient to de(cid:12)ne (cid:25)(t) := k(t)=c(t), where k(t) is de(cid:12)ned by (2.16), so that we need to show that (cid:25)(t) = h( (t)). Dividing both sides of (2.16) by c(t) produces (cid:20)Z (cid:21) 1 G (s) e (cid:25)(t) = E ds ; (2.17) t G (t) s=t e where G (t) = G(t)c(t). Equation (2.17) shows that (cid:25)(t) is the value of a consol e (an endowment consol) where the state{price deflator is given by rG (t). Formally, e G (t) is the state{price deflator where the numeraire has been changed from units e of the consumption good to units of the endowment process. (See AppendixA for a R T discussionofchangingnumeraires.) Given(2.17)wecanwrite(cid:25)(t) = p (t;s)ds, s=t e where p (t;s) = E [G (s)=G (t)] is the value at time t of a zero-coupon bond that e t e e pays one unit of the endowment at time s. > Let the dynamics of (cid:25)(t) be given by d(cid:25)(t)=(cid:25)(t) = (cid:22) (t)dt+(cid:27) (t) dW(t). Be- (cid:25) (cid:25) cause (cid:25)(t) is the value of an asset (when measured in endowment units), the drift of (cid:25) will be determined by the martingale property of deflated gains: 1 > (cid:22) (t)+ = r (t)+(cid:21) (t) (cid:27) (t); (2.18) (cid:25) e e (cid:25) (cid:25)(t) where r (t) and (cid:21) (t) follow from applying Ito^’s lemma to G (t): e e e (cid:18) (cid:19) 1 r (t) =r(t)− (cid:22)e (t)+ k(cid:27) (t)k2 +(cid:21)(t) > (cid:27) (t); (2.19a) e c c c 2 (cid:21) (t) =(cid:21)(t)−(cid:27) (t): (2.19b) e c

CONSUMPTION AND ASSET PRICES 11 Substituting (2.14) into (2.19) produces (cid:26) (cid:27) 1−(cid:17) 1 1 r (t) = (cid:18)+ (cid:22)e (t)+(1−γ) k(cid:27) (t)k2−(cid:17)(1−γ(cid:17)) k(cid:27) (t)k2 (2.20a) e c c (cid:17) 2 2 (cid:21) (t) = (γ−1)(cid:27) (t)+(γ(cid:17)−1)(cid:27) (t): (2.20b) e c For (cid:17) = 1, r (t) = (cid:18). With this constant interest rate, (2.18) implies (cid:25)(t) = 1=(cid:18) e as asserted in (2.10b). For (cid:17) 6= 1, the assertion in (2.10b) that (cid:25)(t) = (cid:18) −(cid:17) (t)(cid:17)−1 implies 1 (cid:22) (t) = ((cid:17)−1)(cid:22)e (t)+ k((cid:17)−1)(cid:27) (t)k2 (2.21a) (cid:25) 2 (cid:27) (t) = ((cid:17)−1)(cid:27) (t): (2.21b) (cid:25) Substituting (2.20) and (2.21) into (2.18) produces (2.13), which establishes the internal consistency of our assertion. Terminology. Forlackofbetterterms,wewillrefertor and(cid:21) astheendowment e e interest rateandtheendowmentpriceofrisk, respectively, todistinguishthemfrom the real interest rate and real price of risk, r and (cid:21). As will become evident, these constructs have important applications beyond an endowment economy. Optimal consumption and portfolio choice: The planning problem. In this section, we consider the problem of the optimal investment of wealth. When the agent has the recursive preferences assumed here, this problem is analyzed by Campbell and Viceira (1996) in a discrete-time setting, and by Du(cid:14)e and Epstein (1992a) and Schroder and Skiadas (1997) in a continuous-time setting. In the previous section, the consumption process was given, and so there was no question of the optimality of consumption. Current opportunities were given by current consumption and future opportunities were determined by the dynamics of consumption. Nevertheless, we were able to (cid:12)nd wealth|the value of the endowment process. In this section, we change perspective: Consumption is no longer given exogenously. Current opportunities are given by current wealth, k(t), and futureopportunitiesare determined bythe state{price deflator (as reflected in asset prices) and current consumption which decreases the amount to invest. The second state variable, , summarizes all relevant information aboutfutureopportunities as reflected in the dynamics of the state{price deflator, namely the interest rate, r(t), and theprice of risk, (cid:21)(t). Inthis setting, we willsolve for theoptimal consumption and investment plans. The investment opportunity set can be characterized by n risky securities with dynamics of the form d(cid:30) (t) i > = (cid:22) (t)dt+(cid:27) (t) dW(t): (2.22) (cid:30) (t) (cid:30)i (cid:30)i i The expected return on security i is determined by the absence-of-arbitrage condition > (cid:22) (t) =r(t)+(cid:27) (t) (cid:21)(t): (2.23) (cid:30)i (cid:30)i

12 MARK FISHER AND CHRISTIAN GILLES (The dynamics of the risky assets reflect the reinvestment of any dividends paid.) In addition there is the money-market account (MMA): d(cid:12)(t) = r(t)dt: (cid:12)(t) A portfolio can be characterized by a vectoPr of weights, (cid:11)(t), for the risky securities andaweight(cid:11) (t)fortheMMA,suchthat n (cid:11) = 1. Let(cid:6) bethematrixwhose 0 i=0 i (cid:30) > > i-th columnis(cid:27) andde(cid:12)neM := ((cid:22) ; ::: ; (cid:22) ) and (cid:11):= ((cid:11) ; ::: ; (cid:11) ) . The (cid:30)i (cid:30) (cid:30)1 (cid:30)n 1 n value of a portfolio evolves as follows: Xn d(cid:30)(t) d(cid:12)(t) d(cid:30) (t) i > = (cid:11) (t) + (cid:11) (t) = (cid:22) (t)dt+(cid:27) (t) dW(t); 0 i (cid:30) (cid:30) (cid:30)(t) (cid:12)(t) (cid:30) (t) i i=1 where (using (2.23)) > > (cid:22) (t) = (cid:11) (t)r(t)+M (t) (cid:11)(t) = r(t)+(cid:21)(t) (cid:27) (t) (cid:30) 0 (cid:30) (cid:30) and (cid:27) (t) = (cid:6) (t)(cid:11)(t): (2.24) (cid:30) (cid:30) Wealth evolves according to d(cid:30)(t) dk(t) = k(t) −c(t)dt: (2.25) (cid:30)(t) On the optimal path, continuation utility is given by (2.10a) and the utility gradient by (2.12a), where must satisfy (2.13). Optimality of the consumption process requires that the utility gradient be aligned with the state{price deflator: G(t) = (cid:11)m(t), for some positive constant (cid:11), so that equations (2.14) are satis(cid:12)ed. We assume here that this (cid:12)rst-order condition is also su(cid:14)cient for the optimality of the solution. This is an important caveat, because su(cid:14)ciency has not been proved for all parameter values. The most complete results so far can be found in Schroder and Skiadas (1997). Using these equations to eliminate (cid:22)e (t) and (cid:27) (t) from (2.13), c c we can reduce these optimality conditions to the following restriction: (cid:0) (cid:1) (cid:26) (cid:27) (cid:18) ( (t)=(cid:18))1−(cid:17) −1 1 (cid:18)+ = r(t)+ k(cid:21)(t)k2 +(cid:22)e (t) 1−(cid:17) 2 (cid:18) (cid:19) 1−γ 1 + k(cid:21)(t)+(cid:27) (t)k2: (2.26) γ 2 Thesolution to (2.26) for given r and (cid:21)can then beusedin (2.14) to solve for the dynamicsof optimal consumption, (cid:22)e (t)and(cid:27) (t). Hereafter, weidentify theutility c c gradient with the state{price deflator (ignoring the constant of proportionality). The next step is to show that the dynamics of optimal consumption imply the dynamics of the optimal portfolio. Since (cid:30)(t) is the value of an asset, its drift is determined by its volatility (conditional on r and (cid:21)): > (cid:22) (t) =r(t)+(cid:21)(t) (cid:27) (t): (2.27) (cid:30) (cid:30)

CONSUMPTION AND ASSET PRICES 13 Thus the portfolio problem is reduced to solving for (cid:27) (t). To establish the link (cid:30) between (cid:30) and c we can use (2.25) and k(t) = (cid:25)(t)c(t) to produce (cid:18)Z (cid:19) t (cid:30)(t) = c(t)(cid:25)(t) exp (cid:25)(s) −1ds : (2.28) s=0 Applying It^o’s lemma to (2.28) and matching drifts and di(cid:11)usions yields10 1 (cid:22)e (t) = (cid:22)e (t)+(cid:22)e (t)− (2.29a) (cid:30) c (cid:25) (cid:25)(t) (cid:27) (t) = (cid:27) (t)+(cid:27) (t): (2.29b) (cid:30) c (cid:25) Recall that(cid:25)(t) = (cid:18) −(cid:17) (t)(cid:17)−1, sothat(cid:27) (t) = ((cid:17)−1)(cid:27) (t). Together with(2.14b), (cid:25) we have established11 (cid:18) (cid:19) (cid:18) (cid:19) 1 1−γ (cid:27) (t) = (cid:21)(t)+ (cid:27) (t): (2.30) (cid:30) γ γ Any solution (cid:11) to (cid:27) (t) = (cid:6) (t)(cid:11)(t) is a solution to the portfolio problem. The (cid:30) (cid:30) matrix (cid:6) is l(cid:2)n, where l is the number of Brownian motions that determine the (cid:30) state of information and n is the number of linear activities. For simplicity, we assume that (cid:6) is of full rank. If l > n, it is impossible to hedge against all sources (cid:30) of risk in the economy. In this case, if necessary through a Choleski decomposition of the Brownian motions that rede(cid:12)nes the matrix (cid:6) and all volatilities but keeps (cid:30) > all covariances unchanged, one can assume without loss of generality that (cid:6) = (cid:16) (cid:17) (cid:30) (cid:6) (cid:3)> 0 , where (cid:6) (cid:3) is an n(cid:2)n invertible matrix. In terms of the new Brownian (cid:30) (cid:30) (cid:16) (cid:17) motions, we have (cid:27) > = (cid:27) (cid:3)> (cid:27) 0> . If (cid:27) 0 6= 0, the optimal investment problem has (cid:30) (cid:30) (cid:30) (cid:30) 0 no solution. So we assume (cid:27) = 0. If n > l, then it is possible to drop activities (cid:30) that are not needed in the optimal portfolio, keepin(cid:16)g only(cid:17)l such activities. By renumberingactivitiesifnecessary,wecanwrite(cid:6) = (cid:6) (cid:3) 0 andweset(cid:27) (cid:3) = (cid:27) .12 (cid:30) (cid:30) (cid:30) (cid:30) We can now write the solution in the general case as (cid:18) (cid:19) (cid:18) (cid:19) (cid:11) (cid:3) (t) = (cid:6) (cid:3)−1(t)(cid:27) (cid:3) (t)= 1 (cid:6) (cid:3)−1(t)(cid:21) (cid:3) (t)+ 1−γ (cid:6) (cid:3)−1(t)(cid:27) (t) (2.31) (cid:30) (cid:30) γ (cid:30) γ (cid:30) (cid:3) using (2.30). In this equation, (cid:21) corresponds to the n (cid:12)rst components of (cid:21) (under the new Brownian motions) if l > n and is equal to (cid:21) otherwise.13 This result is consistent with Theorem 2(c) (under conditions I and II) in Schroder and Skiadas (1997). 10Note that(cid:22) (t)=(cid:22)e (t)+ 1k(cid:27) (t)k2. (cid:30) (cid:30) 2 (cid:30) 11This expression has appeared in Campbell and his citations. 12Someactivitiesmayslipinandoutoftheoptimalportfolio. Attheexpenseofsomenotationfor keeping track of which activities are in the optimal portfolio at time t, we do not need to assume that theset of activities in theportfolio neverchanges. 13De(cid:12)ning(cid:21)(cid:3)(t)=(cid:6)(cid:3)−1(t)>(M (t)−r(t)),we can eliminate (cid:21)(cid:3) from (2.31): (cid:30)(cid:18) (cid:19) (cid:30) (cid:18) (cid:19) (cid:11) (cid:3) (t)= 1 (cid:6) (cid:3)−1(t)(cid:6) (cid:3)−1(t) > (M (t)−r(t))+ 1−γ (cid:6) (cid:3)−1(t)(cid:27) (t): γ (cid:30) (cid:30) (cid:30) γ (cid:30)

14 MARK FISHER AND CHRISTIAN GILLES We see that when γ = 1, (cid:11)(t) = (cid:6) (cid:3)−1(t)(cid:21) (cid:3) (t)=γ. This component is the so- (cid:30) called \myopic" component of portfolio demand, in the terminology of Campbell and Viceira (1996). The other component constitutes a hedge against changes in investment opportunities. The (cid:12)rst component is easily found without knowledge of the consumption plan, but evaluation of the second component requires such knowledge (through (t)). The production problem. In a representative-agent general equilibrium, we interpret k(t) as the value of the capital stock and d(cid:30)(t)=(cid:30)(t) as the return on the aggregate investment portfolio|i.e., the return on the market portfolio. At this level of analysis, we ignore the portfolio allocation problem, except to require zero net investment in the money-market account, treating this as an economy with a single investment opportunity. We can think of (cid:30)(t) itself as the value of a portfolio where the consumption \dividends" are continuously reinvested. In this role, we refer to (cid:30) as the capital account. If we wish, we may think of this economy as a production economy, where the return on the capital account is the result of linear production technology subject to random shocks as in Cox, Ingersoll, Jr., and Ross (1985a). The state variables in this case are k(t) and (t), where (t) impounds information regarding the dynamics of technology. In this case, we need to ensure that the interest rate and the price of riskbeproperlyrelated to the dynamics of technology. We can achieve this by using (2.27) and (2.30) to eliminate r and (cid:21) from (2.26): (cid:0) (cid:1) (cid:18) ( (t)=(cid:18))1−(cid:17) −1 1 (cid:18)+ = (cid:22)e (t)+(cid:22)e (t)+(1−γ) k(cid:27) (t)+(cid:27) (t)k2: (2.32) 1−(cid:17) (cid:30) 2 (cid:30) Having solved (2.32) for given (cid:22)e and (cid:27) we can use (2.27) and (2.30) to solve (cid:30) (cid:30) for r and (cid:21) and then use (2.14) to solve for (cid:22)e and (cid:27) . c c Relationtostochasticcontrol. Thetraditionalapproachtosolvingtheconsumption{investment problem is to apply the stochastic control method. In this approach,weassumewecanwriteoptimalconsumption(thepolicyfunction)andoptimizedutility(thevaluefunction)intermsofthestate variables: c(t) = C(k(t); (t)) and V(t) = j(k(t); (t)), where j(k; ) := g(C(k; ); ). In our case, the envelop condition, j = f , delivers the form of the policy function. Given the de(cid:12)nition of k c j, we have j = C g , and so the envelop condition implies C = f =g = 1=h as k k c k c c established in (2.9). We conclude that C(k; ) = k=h( ). Therefore we can write14 (k )1−γ −1 C(k; ) = (cid:18)(cid:17) 1−(cid:17)k and j(k; ) = : (2.33) 1−γ Once we have the form of the policy function, Bellman’s principle of optimality in essence turns the stochastic control problem into a \recursive utility" problem for the value function: (cid:22)V(t) = −f (cid:3) (k(t); (t)); where f (cid:3) (k; ) := f(C(k; );g(C(k; ); )) (2.34) 14Closely related to C and j in (2.33) are the functions (A.1) and (A.2) in Giovannini and Weil (1989).

CONSUMPTION AND ASSET PRICES 15 and (cid:22)V(t) is the drift of j(k(t); (t)). If we take (cid:30) as the forcing variable (for example), then (2.34) is equivalent to (2.32). Finite horizon and (cid:17) 6= 1. When the horizon is (cid:12)nite, the wealth{consumption ratio, (cid:25)(t), is the valueof an annuity, which goes to zero as thehorizon goes to zero. We can rewrite the absence-of-arbitrage condition for (cid:25)(t) given in (2.18) as > (cid:22)(cid:22) (t)+1= r (t)(cid:25)(t)+(cid:21) (t) (cid:27)(cid:22) (t); subject to (cid:25)(T) = 0: (2.35) (cid:25) e e (cid:25) We have expressed the dynamics of the (cid:25) in (2.35) in absolute terms: d(cid:25)(t) = > (cid:22)(cid:22) (t)dt+(cid:27)(cid:22) (t) dW(t). Given(2.10b),theboundaryconditionrequires todepend (cid:25) (cid:25) on the horizon: ( 0 if (cid:17) >1 lim (t) = t!T 1 if (cid:17) <1. Even though behaves badly as the horizon approaches, (cid:25) itself is well-behaved and, as long as (cid:17) 6= 1, we can use (2.35) as the restriction to solve for (cid:25) and then we can (cid:12)nd by inverting (cid:25) = (cid:18) −(cid:17) (cid:17)−1. Therefore, for (cid:17) 6= 1, we can simply reinterpret the equations from our previous analysis of the in(cid:12)nitehorizon problem. The case (cid:17) = 1 is more complicated. Finite horizon and (cid:17) = 1. For (cid:17) = 1, the boundary condition (cid:25)(T) = 0 cannot be satis(cid:12)ed without changing the way c enters g(c; ). For this case we de(cid:12)ne15 (cid:16) (cid:17) G(c; ;(cid:28)) := g cq((cid:28)); ; where q((cid:28)):= 1−e −(cid:18)(cid:28): Note that G(c; ;1) = g(c; ). Using G, we have G (c; ;(cid:28)) q((cid:28)) c (cid:25) = H((cid:28)) = = ; (2.36) f (c;G(c; ;(cid:28))) (cid:18) c which agrees with Theorem 2(b) (under condition II) in Schroder and Skiadas (1997). Using G(c(t); (t);T −t) in place of g(c(t); (t)), the restriction (cid:22)V+f = 0 becomes 1 (cid:18) log( (t)=(cid:18)) =q(T −t)(cid:22)e (t)+(cid:22)e (t)+(1−γ) kq(T −t)(cid:27) (t)+(cid:27) (t)k2; c c 2 (2.37) subject to (T) = (cid:18). Note that (2.13) as (cid:17) ! 1 and (2.37) as T ! 1 converge to the same restriction. The partial derivatives of f(c;v) evaluated at v = G(c; ;(cid:28)) are f (c;G(c; ;(cid:28))) =(cid:18)γ 1−γc −Q((cid:28)) (2.38a) c n o f (c;G(c; ;(cid:28))) = −(cid:18)−(cid:18)(1−γ) log( =(cid:18))−e −(cid:18)(cid:28) log(c) ; (2.38b) v 15The expression cq can be seen in the term structure example in Du(cid:14)e and Epstein (1992a) for example.

16 MARK FISHER AND CHRISTIAN GILLES where Q((cid:28)) = γ + (1 − γ)e −(cid:18)(cid:28). With standard preferences, γ = 1, the utility gradient specializes to G(t) = (cid:18)e −(cid:18)tc(t) −γ as expected. In the general case, the interest rate and price of risk are given by (cid:18) (cid:19) 1 r(t)= (cid:18)+(cid:22)e (t)+ −Q(T −t) k(cid:27) (t)k2 +(1−γ)(cid:27) (t) > (cid:27) (t) (2.39a) c c c 2 (cid:21)(t) = Q(T −t)(cid:27) (t)+(γ −1)(cid:27) (t); (2.39b) c where (as before) we can have used (cid:22)V +f = 0 to eliminate both and (cid:22)e from the interest rate. Again, with γ = 1, (2.39) specializes to the expected expressions for r and (cid:21) under the C{CAPM: r(t)= (cid:18)+(cid:22)e (t)− 1k(cid:27) (t)k2 and (cid:21)(t) = (cid:27) (t). To c 2 c c rea(cid:14)rm the internal consistency, note that substituting (2.39) into (2.19) produces r (t) = (cid:18). With this constant interest rate, (2.35) implies (cid:25)(t) = q(T − t)=(cid:18) as e asserted by (2.36). Comparing (2.14) with (2.39), a discontinuity is evident for the (cid:12)nite-horizon case. Suppose (cid:22)e and (cid:27) are constant, so that (cid:27) = 0. For (cid:17) 6= 1 the price of risk is c c constant at γ(cid:27) , while for (cid:17) = 1 the price of risk depends on the horizon, moving c toward(cid:27) ast !T. Asweshowbelowinthecontextofaspeci(cid:12)cplanningproblem, c the consumption process (viewed as a function of the preference parameters) is discontinuous at (cid:17) = 1 whenever the data are continuous, but only in the (cid:12)nitehorizon case. In the in(cid:12)nite-horizon, the discontinuity disappears and in any case, the wealth-consumption ratio (cid:25)(t) is continuous. Turning to optimal consumption and optimal portfolio, we can use (2.39) to eliminate (cid:22)e and (cid:27) from (2.37) produces c c (cid:18) (cid:19) 1 (cid:18) log( (t)=(cid:18)) = (cid:22)e (t)+q(T −t) r(t)+ k(cid:21)(t)k2 −(cid:18) 2 1−γ 1 + kq(T −t)(cid:21)(t)+(cid:27) (t)k2: (2.40) Q(T −t) 2 In this case, the optimal portfolio weights must satisfy (cid:18) (cid:19) (cid:18) (cid:19) 1 1−γ (cid:27) (t) = (cid:21)(t)+ (cid:27) (t): (2.41) (cid:30) Q(T −t) Q(T −t) Finally, using (2.27) and (2.41) to eliminate r and (cid:21) from (2.40), we have 1 (cid:18) log( (t)=(cid:18)) = (cid:22)e (t)+q(T −t)((cid:22)e (t)−(cid:18))+(1−γ) kq(T −t)(cid:27) (t)+(cid:27) (t)k2: (cid:30) (cid:30) 2 (2.42) 3. Analysis of the model in a non-Markovian setting Thus far, we have considered three forcing processes: consumption, the state{ price deflator, and technology, each within a speci(cid:12)c context: endowment economy, optimal consumption in a partial equilibrium, and production economy. Having established the equilibrium relationships among all three processes, we are now free to model whichever forcing process we (cid:12)nd convenient and (based on the solution for related to that process) infer the dynamics of the other two processes. For

CONSUMPTION AND ASSET PRICES 17 example, we can choose consumption as the forcing process and infer the dynamics of technology that would generate that consumption process. It is useful to recognize that there are other ways in which one can introduce dynamicsintothemodel(inadditiontothethreechoiceslistedabove). Wemayinfact start by choosing and it dynamics. By itself, this is insu(cid:14)cient to solve the model (in the sense of being able to determine the dynamics of consumption, etc.). Examining (2.32), for example, weseethat ifwealso model(cid:27) , then(2.32) determines (cid:22)e (cid:30) (cid:30) andwecan infactsolve themodelfortheother processesof interest. Thisapproach is tantamount to modeling the deflated value of wealth, m (t) = m(t)k(t), which k can be interpreted as a state{price deflator after a change of numeraire:16 Indeed, from the dynamics of m and k given in (2.14) and (2.25), we see that dm (t) k = −(cid:18)(cid:17) (t)1−(cid:17)dt+(1−γ)((cid:27) (t)+(cid:27) (t)) > dW(t); (cid:30) m (t) k where we have used (2.30) to eliminate (cid:21) from the volatility of m . Instead of k modeling(cid:27) in conjunctionwith , wecan modeleither (cid:21)or (cid:27) . Thecorresponding (cid:30) c restriction can then be solved for r or (cid:22)e . Although this approach to modeling the c dynamics may seem unnatural, it has the advantage of o(cid:11)ering immediate solutions for equilibrium relationships among variables of interest. Nevertheless, one may choose to model one of the three forcing processes, in which case a solution for must be found. The challenge to using K{P SDU preferences is simply (cid:12)ndingthe process that solves the proper restriction identically given a forcing process. In this section, we will go as far as we can in a non-Markovian setting. Below we will have more to say on this subject after we adopt a Markovian structure. Before proceeding, though, it is convenient to unify all three solutions. To that end, we denote the forcing variable y and its dynamics dlog(y(t)) = (cid:22)e (t)dt+(cid:27) (t) > dW(t), where y is either y y consumption (c), the capital account ((cid:30)), and the inverse of the state{price deflator (1=m). y(t) (cid:22)e (t) (cid:27) (t) d d d d d ((cid:28)) " = d +d y y 0 1 2 3 4 1 2 c(t) (cid:22)e (t) (cid:27) (t) (cid:18) 1 −1 1−γ 0 1 1 −γ c c (cid:17) (cid:17) m(t) −1 r(t)+ 1k(cid:21)(t)k2 (cid:21)(t) (cid:17)(cid:18) 1−(cid:17) 1 −1 −1 Q((cid:28)) 1 −(cid:17) 2 γ γ (cid:30)(t) (cid:22)e (t) (cid:27) (t) (cid:17)(cid:18) 1−(cid:17) 1−γ −1 1 2−(cid:17)−γ (cid:30) (cid:30) Table 1. The coe(cid:14)cients of Equations (3.2) and (3.11) in terms of the preference parameters. Q((cid:28)) = γ+(1−γ)e −(cid:18)(cid:28). 16The interest rate|measured in unitsof wealth|is the consumption{wealth ratio, 1=(cid:25).

18 MARK FISHER AND CHRISTIAN GILLES Unit elasticity of intertemporal substitution and in(cid:12)nite horizon. Given (cid:17) = 1, each of (2.37), (2.40), and (2.42) can be written, for (cid:12)nite T, as (cid:22)e (t)= (cid:18) log( (t)=(cid:18))−q(T −t)((cid:22)e (t)+d (cid:18)) y 3 (1−γ) − kq(T −t)(cid:27) (t)+(cid:27) (t)k2; (3.1) 2d (T −t) y 4 where d and d ((cid:28)) are given in Table 1. For T = 1, (3.1) specializes to 3 4 1 (cid:22)e (t) = (cid:18) log( (t)=(cid:18))−(cid:22)e (t)−d k(cid:27) (t)+(cid:27) (t)k2 −d (cid:18); (3.2) y 2 y 3 2 whered isgiveninTable1. Inthiscase,aslongasC = −d 1 k(cid:27) (t)+(cid:27) (t)k2−d (cid:18) 2 2 2 y 3 is constant, the solution to (3.2) is (t) =(cid:18) exp((cid:16)(t)−C=(cid:18)); (3.3) from which it follows that (cid:22)e (t)= (cid:22) (t) and (cid:27) (t) = (cid:27) (t); (3.4) (cid:16) (cid:16) > where d(cid:16)(t) = (cid:22) (t)dt+(cid:27) (t) dW(t). Substituting (3.3) and (3.4) into (3.2) pro- (cid:16) (cid:16) duces the restriction that (cid:16) must satisfy: (cid:22) (t)= (cid:18)(cid:16)(t)−(cid:22)e (t): (3.5) (cid:16) y Nowde(cid:12)ne(cid:16)(t)asaweightedaverageofexpectedgrowthratesoftheforcingprocess y: Z (cid:18)Z (cid:19) 1 s (cid:16)(t):= (cid:18)e −(cid:18)(s−t) E [(cid:22)e (u)]du ds: (3.6) t y s=t u=t It^o’s lemma applied to (3.6) delivers (3.5) and Z (cid:18)Z (cid:19) 1 s (cid:27) (t) = (cid:27) (cid:16) (t)= (cid:18)e −(cid:18)(s−t) (cid:27)b (cid:22)e y (t;u)du ds; (3.7) s=t u=t where (cid:27)b (cid:22)e y (t;s) is the volatility of E t [(cid:22)e y (s)]. C is constant under either of two conditions. The (cid:12)rst is γ = 1, which (with (cid:17) = 1) produces log utility. The second condition obtains when (i) (cid:27)b (cid:22)e y (t;u) is a deterministic function of u−t, so that (cid:27) (t) and, therefore, (cid:27) (t) is constant, and (cid:16) (ii) (cid:27) (t) is constant. The most important feature of the solution is the expression y for(cid:27) (t) = (cid:27) (t)given in(3.7), becauseitis(cid:27) (t)thatcontributes toriskpremia.17 (cid:16) Non-unit elasticity of intertemporal substitution. We turn to analyzing the model when (cid:17) 6= 1. 17Campbell (1993) derives a similar result for the case of y = (cid:30) in a discrete-time version of this model,wherehereferstok(cid:27) k2 as\newsaboutthediscountedvalueofallfuturemarketreturns."

CONSUMPTION AND ASSET PRICES 19 The utility gradient in terms of observables. As long as (cid:17) 6= 1, we can replace the unobservable with the observable (cid:30) in the utility gradient, following Epstein and Zin (1991). Using (2.28) with (cid:25)(t) = (cid:18) −(cid:17) (t)(cid:17)−1, we can solve for (cid:18)Z (cid:19) t (cid:18)(cid:17) (s)1−(cid:17) (t) = exp ds (cid:18) −(cid:17)=(1−(cid:17))c(t)1=(1−(cid:17))(cid:30)(t) −1=(1−(cid:17)): 1−(cid:17) s=t Substituting this expression for (t) into (2.12a) produces m(t) = G(t) = (cid:18)(cid:14)e −(cid:18)(cid:14)tc(t) −(cid:14)=(cid:17)(cid:30)(t)(cid:14)−1; (3.8) where the parameter (cid:14) is de(cid:12)ned by 1−γ (cid:14) := : 1−1=(cid:17) Applying It^o’s lemma directly to (3.8) yields a convenient representation for the short rate r(t) and the price of risk (cid:21)(t) in terms of the observable dynamics of the growth rate of consumption and the return on the market portfolio (cid:30): 1 r(t) = ((cid:14)=(cid:17))(cid:22)e (t)+(1−(cid:14))(cid:22)e (t)+(cid:18)(cid:14)− k(cid:21)(t)k2; (3.9a) c (cid:30) 2 (cid:21)(t) = ((cid:14)=(cid:17))(cid:27) (t)+(1−(cid:14))(cid:27) (t): (3.9b) c (cid:30) Note that (cid:14) = 1 (i.e., (cid:17)γ = 1) delivers standard preferences and the C{CAPM. By contrast (cid:14) = 0 (i.e., γ = 1) delivers an intertemporal CAPM, where risk premia are determined by the covariance with the market portfolio. These loci are plotted in Figure 1 (where γ is plotted on the vertical axis against (cid:17) on the horizontal axis), along with (cid:17) = 1 and a fourth locus that we will discuss after some discussion of the consol equation. The annuity equation. When (cid:17) 6= 1, we can make progress by focusing on solving (2.35), which describes (cid:25)(t) as the price of an annuity and which we repeat here as > (cid:22)(cid:22) (t)+1 = r (t)(cid:25)(t)+(cid:21) (t) (cid:27)(cid:22) (t); (3.10) (cid:25) e e (cid:25) subject to (cid:25)(T) = 0, where r (t) and (cid:21) (t) are given in (2.19). e e By modeling r and (cid:21) directly, we can solve for \endowment bond" prices and e e hence (cid:25). On the other hand, we may choose instead to model the dynamics of either (cid:30) or c or m directly. Based on equations (2.19) and (3.9), and using (2.28) and k(t) = (cid:25)(t)c(t), we can write18 (cid:13) (cid:13) r e (t)= d 0 +d 1 (cid:22)e y (t)+d 1 d 2 1 k(cid:27) y (t)k2 +("=d 1 ) 1 (cid:13) (cid:13) (cid:13) (cid:27)(cid:22) (cid:25) (t) (cid:13) (cid:13) (cid:13) 2 (3.11a) 2 2 (cid:25)(t) (cid:27)(cid:22) (t) (cid:21) (t)= −d (cid:27) (t)−("=d ) (cid:25) ; (3.11b) e 2 y 1 (cid:25)(t) 18Note that lim t!T k(cid:27)(cid:22) (cid:25) (t)=(cid:25)(t)k2 =0, since Z Z T T (cid:27)(cid:22) (t)= p (t;s)(cid:27) (t;s)ds and (cid:25)(t)= p (t;s)ds; (cid:25) e pe e s=t s=t where (cid:27) (T;T)=0 and p (T;T)=1. pe e

20 MARK FISHER AND CHRISTIAN GILLES g 2 1 h 1 2 Figure 1. The coe(cid:14)cient of relative risk aversion, γ, versus the elasticity of intertemporal substitution, (cid:17). The shaded areas show where (cid:14) = (1−γ)=(1−1=(cid:17)) < 0. whered dependsonlyon(cid:17)andd dependsonlyonγ. (SeeTable1fortheparticular 1 2 values.) Given(cid:22)e and(cid:27) ,ifwecansolve(3.10)for(cid:25),theremainingremainingdrifts y y and di(cid:11)usions can be found via the relations established above.19 Recall that the endowment deflator, which has r as interest rate and (cid:21) as price e e of risk, is given by m (t)= m(t)c(t), which, in light of (3.8), is e m (t) =(cid:18)(cid:14)e −(cid:18)(cid:14)tc(t)1−(cid:14)=(cid:17)(cid:30)(t)(cid:14)−1: (3.12) e Note that the conditions for " = 0 in Table 1 are the same as 1 − (cid:14)=(cid:17) = 0 and (cid:14)−1 = 0, which lead one or the other of c(t) and (cid:30)(t) to be absent from m (t). We e discuss these two cases. Case 1: (cid:14) = 1, i.e., γ = 1=(cid:17). This locus is plotted in Figure 1 as the rectangular hyperbola of standard preferences. (Recall that this is the dividing line between preference for early versus late resolution of uncertainty.) In this case, the endowment deflator depends only on consumption: m (t) = (cid:18)e −(cid:18)tc(t)1−γ. As a consequence, an e 19It turns out that other drifts and di(cid:11)usions can always be expressed in terms of (cid:22)e , (cid:27) , and (cid:27) y y (cid:25) without (cid:22)e . See(2.14). (cid:25)

CONSUMPTION AND ASSET PRICES 21 exactsolution tothemodelispossiblewhentheforcingprocessisc. Moreover, (2.19) can be used to replace (cid:22)e (t) and (cid:27) (t) with r(t) and (cid:21)(t), so that an c c exact solution is also possible when the forcing process is 1=m. In both cases, " = 0 in (3.11). Case 2: (cid:14) = (cid:17), i.e., γ+(cid:17) = 2. This locus is plotted in Figure 1 as the diagonal line. In this case, the endowment deflator depends only on the capital account: m (t) = (cid:18)(cid:30)(t)(cid:17)−1. e As a consequence an exact solution to the model is possible when the forcing process is (cid:30). In this case, "= 0 in (3.11). Deterministic endowment interest rates. The terms in (3.11) involving (cid:27) (cid:25) present challenges to solving the model, and as such we refer to them as \nuisance" terms. However, there are two circumstances under which the nuisance terms will not be present: (i) when r is deterministic, (cid:27) (cid:17) 0, and (ii) when " = 0. We e (cid:25) deal with the (cid:12)rst case immediately and the second case later in this section and in the next section where we introduce Markovian state variables. We treat the case where the nuisance terms are present in the next section as well. Therearetwowaysforr (t)tobedeterministic. First,if(cid:17) = 1(i.e.,d = 0),then e 1 r (t) = (cid:18), and (3.10) devolves to (cid:25)(t) = q(T−t)=(cid:18). This equation provides no other e information and we solve (3.1) for (t) as above. Second, if (cid:22)e (t) + d2 k(cid:27) (t)k2 y 2 y is deterministic, then (cid:27) (t) = 0 so that r (t) is deterministic, even when y(t) is (cid:25) e stochastic (that is, (cid:27) 6= 0). Below we show an analytical solution to (3.10) for a y special case where the drift and volatility of log(y(t)) are each deterministic but nontrivial; the next example shows the solution in the constant case. Constant investment opportunity set with (cid:17) 6= 1. When (cid:22)e and (cid:27) are constant r y y e is constant as well. In this case, the solution to (3.10) is 1−e −re(T−t) (cid:25)(t) = ; (3.13) r e so that (cid:27) = 0. Setting y(t) = 1=m(t), for example, so that r and (cid:21) are constant, (cid:25) we see that (3.11) implies (cid:18) (cid:19) (cid:18) (cid:19) 1 1 r = (cid:17)(cid:18)+(1−(cid:17)) r+ k(cid:21)k2 and (cid:21) = 1− (cid:21): e e 2γ γ Now, from (2.19), we get k(cid:21)k2 (cid:21) (cid:22) (t)= (cid:17)(r−(cid:18))+(1+(cid:17)) and (cid:27) = : c c 2γ γ Schroder and Skiadas (1997) call this case the constant investment opportunity set, and derive the solution through a di(cid:11)erent approach in their Theorem 2.20 Note that when (cid:25)(t) is deterministic, is deterministic as well, from (2.10b). We have seen in the context of the exchange problem (when the consumption process is (cid:12)xed) that the equilibrium interest rate and price of risk exhibits a discontinuity at (cid:17) = 1 in the (cid:12)nite horizon case. We now show that in the context 20Their (cid:12) is (cid:18), their γ is 1−1=(cid:17), and their (cid:11) is (cid:14)−1 if (cid:17)6=1 and 1−γ otherwise.

22 MARK FISHER AND CHRISTIAN GILLES of the planning problem with a constant opportunity set, the consumption process exhibits a discontinuity at (cid:17) = 1. The discontinuity of consumption at (cid:17) = 1. When (cid:17) = 1, and r and (cid:21) are constant, the solution to (2.40) is (cid:18) (cid:19) k(cid:21)k2 (cid:18) log( (t)=(cid:18)) = q((cid:28)) r−(cid:18)+ 2γ (cid:26) (cid:18) (cid:19) (cid:27) k(cid:21)k2 k(cid:21)k2 −e −(cid:18)(cid:28) (cid:28) (cid:18) r−(cid:18)+ +log(Q((cid:28))) ; 2γ2 2γ2 where (cid:28) := T − t. This result, then, implies that (cid:27) (t) = 0, and we obtain the dynamics of consumption from (2.39): k(cid:21)k2 (cid:21) (cid:22) (t)= r−(cid:18)+ and (cid:27) = : c c Q((cid:28)) Q((cid:28)) Since Q((cid:28)) di(cid:11)ers from γ except when the horizon is in(cid:12)nite, contrasting this case with that of (cid:17) 6= 1 reveals a discontinuity in the dynamics of consumption. This discontinuity disappears when the horizon is in(cid:12)nite. Note, however, that (cid:25)(t) is given by (3.13) even when (cid:17) = 1, in which case r = (cid:18), so that the wealth consumpe tion ratio is continuous in the preference parameters even when the consumption process is not. The relation between the volatility of and the volatility of forecast revisions. For the in(cid:12)nite-horizon case when (cid:17) = 1, we showed that if (cid:27) (t) is y constant and (cid:27)b (cid:22)e y (t;u) is a deterministic function of u − t, then log( (t)) equals a weighted average of expected growth rates of y (plus a constant), where the weights are exponentially declining. As a consequence, the volatility of equals the weighted average of revisions to expected growth rates of y. In this section, we show that essentially the same relation between the volatilities holds when (cid:17) 6= 1 (under the same conditions for (cid:27) y (t) and (cid:27)b (cid:22)e y (t;u)) as long as " = 0. In particular, for y = (cid:30) or y = 1=m, the volatility of still equals the volatility of a weighted average of revisions to expected growth rates of y (where the weights are roughly exponentially declining). Fory = c, thevolatility of isproportionaltotheaverage of the revisions, with proportionality constant (cid:17). These results are a consequence of the weak form of the expectations hypothesis of the term structure of interest rates where the numeraire is taken to be units of the endowment process. Let p(t;s) be the value at time t of a zero-coupon bond that pays one unit of an arbitrary numeraire at time s. De(cid:12)ne the yield to maturity as follows: y(t;s) := −log(p(t;s))=(s−t). Without loss of generality we can write R s E [r(u)]du−(cid:12)(t;s) y(t;s) = u=t t ; (3.14) s−t

CONSUMPTION AND ASSET PRICES 23 for some process (cid:12)(t;s) such that y(t;t) (cid:17) r(t). Let the dynamics of (cid:12)(t;s), r(t), and E [r(u)] be given by t > d(cid:12)(t;s) = (cid:22) (t;s)dt+(cid:27) (t;s) dW(t) (cid:12) (cid:12) > dr(t) = (cid:22) (t)dt+(cid:27) (t) dW(t) r r dE [r(u)] = (cid:27)b (t;u) > dW(t): t r The strong form of the expectations hypothesis implies (cid:12)(t;s) (cid:17) 0, while the weak form implies (i) (cid:27) (t;s) (cid:17) 0 and (ii) (cid:22) (t;s) is a deterministic function of s−t. (cid:12) (cid:12) Su(cid:14)cient conditions for the weak form to hold are (cid:27) and (cid:21) constant. Given the r weak form, we have (cid:18)Z (cid:19) (cid:0) (cid:1) s > dlog(p(t;s)) = r(t)+(cid:22) (t;s) dt− (cid:27)b (t;u)du dW(t): (cid:12) r R u=t 1 Let $(t) := p(t;s)ds be the value of the perpetuity. Applying Ito^’s lemma to s=t > this de(cid:12)nition of $(t) produces d$(t)=$(t) = (cid:22) (t)dt+(cid:27) (t) dW(t), where $ $ Z ( (cid:13)Z (cid:13) ) (cid:22) $ (t) = r(t)− 1 + 1 w(t;s) (cid:22) (cid:12) (t;s)+ 1 (cid:13) (cid:13) (cid:13) s (cid:27)b r (t;u)du (cid:13) (cid:13) (cid:13) 2 ds (3.15a) $(t) 2 s=t u=t Z (cid:18)Z (cid:19) 1 s (cid:27) (t) = − w(t;s) (cid:27)b (t;u)du ds; (3.15b) $ r s=t u=t and where w(t;s) := p(t;s)=$(t). At this point, we apply the foregoing to the endowment term structure. We assume(i)" = 0, (ii)(cid:27) y (t)isconstant, and(iii)(cid:27)b (cid:22)e y (t;u)isadeterministicfunction of u−t. These conditions are su(cid:14)cient to ensure that (cid:27) and (cid:21) are constant and re e the weak form holds. Given these conditions, we can write (3.15b) for r(t) = r (t) e as Z (cid:18)Z (cid:19) 1 s (cid:27) (t) = − w (t;s) (cid:27)b (t;u)du ds; (3.16) (cid:25) e re s=t u=t wherew (t;s) := p (t;s)=(cid:25)(t).21 Theprecedingconditionsalsoensurethat(cid:27)b (t;u) = e e re d 1 (cid:27)b (cid:22)e y (t;u), where (cid:27)b (cid:22)e y (t;u) is the volatility of E t [(cid:22)e y (u)]. Therefore, given (cid:27) (t) = (cid:27) (t)=((cid:17)−1), we can write (3.16) as (cid:25) (cid:18) (cid:19)Z (cid:18)Z (cid:19) d 1 s (cid:27) (t) = 1− 1 (cid:17) w e (t;s) (cid:27)b (cid:22)e y (t;u)du ds: (3.17) s=t u=t For y = (cid:30) and y = 1=m, d =(1−(cid:17)) = 1, while for y = c, d =(1−(cid:17)) = (cid:17). Comparing 1 1 (3.17) with (3.7) (in which (cid:27) (t) = (cid:27) (t)), we see there is a close relationship (cid:16) between (cid:27) (t) and (cid:27) (t) even when (cid:17) 6= 1 as long as the expectations hypothesis (cid:16) holds. 21 Notethat if theterm premium were stochastic, we would have Z (cid:18)Z (cid:19) 1 s (cid:27) (t)= w (t;s) (cid:27)b (t;u)+(cid:27) (t;u)du ds: (cid:25) e re (cid:12)e s=t u=t

24 MARK FISHER AND CHRISTIAN GILLES Recall that when y = 1=m, (cid:22)e (t) = r(t)+ 1k(cid:21)(t)k2 and (cid:27) (t) = (cid:21)(t), where r y 2 y is the real interest rate. Our assumption that (cid:27) is constant implies that d(cid:22)e (t) = y y dr(t), so that (cid:27)b (cid:22)e y (t;u) = (cid:27)b r (t Z ;u). Thus fo (cid:18) r Z y = 1=m, (3.17 (cid:19) ) is 1 s (cid:27) (t) = w (t;s) (cid:27)b (t;u)du ds: (3.18) e r s=t u=t On the other hand, the volatility of a real perpetuity is given by (3.15b) where r is interpreted as the real interest rate. Therefore we have established a close relationship between (cid:27) and −(cid:27) . They di(cid:11)er only by the weights: w (t;s) versus $ e w(t;s). However, even when (cid:17) = 1, the weights do not converge: When (cid:17) = 1, the endowment perpetuity weights are w (t;s) = (cid:18)e −(s−t)(cid:18) (see (3.7)), while the real e perpetuity weights are determined by r and (cid:21) and need not bear any particular relation to w (t;s). Nevertheless, whenever consumption remains constant we have e m proportionaltom,sothattherealandendowmenttermstructuresareidentical. e This situation occurs with (cid:17) = 0 and γ = 1 and does not requirehomoskedasticity. (See the discussion on limit cases below.)22 When y = (cid:30), (3.17) becomes Z (cid:18)Z (cid:19) 1 s (cid:27) (t) = w e (t;s) (cid:27)b (cid:22)e (t;u)du ds: (3.19) (cid:30) s=t u=t This expression sheds some light on an approximation in Campbell (1993). Relying on the relation between and (cid:16) that holds when (cid:17) = 1 (see (3.7)), Campbell approximates (cid:27) with a weighted average of forecast revisions of y = (cid:30) in a ho- (cid:30) moskedastic model. He compares the solutions based on this approximation with full numerical solutions to the discrete-time version of our model. Equation (3.19) shows that Campbell’s approximation is essentially exact not just for (cid:17) = 1, but also for γ +(cid:17) = 2: Along both lines the nuisance terms are absent. The fact that the result holds on more than one line in the parameter space helps explain why Campbell’s approximation is as good as it is over such a wide range of parameter values. Limit cases. Inthis section weanalyze themodelat thelimitvalues of theparameter space for intertemporal substitution and risk aversion. We consider the cases where (cid:17) and γ are zero or in(cid:12)nity. We can identify certain features of the solution evenincaseswherewecannotsolvethemodelentirely. Wemusttakecare, however: For a given limit, there may be no equilibrium for arbitrarily chosen dynamics of a given forcing variable. By examining the limiting values of d and d in Table 1, 1 2 we can rule out certain combinations. Case 1: (cid:17) = 0. This corresponds to extreme aversion toward intertemporal substitution. As indicated in Table 1, this is consistent with solving for equilibrium when the forcing process is either (cid:30) or 1=m; but it is inconsistent with solving for equilibrium in an endowment economy (d and " are both in(cid:12)nite). In this 1 case, (2.13) provides a restriction on the consumption process that must be 22Campbell (1993) discusses therelation between the real and endowment perpetuities.

CONSUMPTION AND ASSET PRICES 25 satis(cid:12)ed. One might think that (cid:17) = 0 would lead to choose a constant rate of consumption. Instead, we see that (cid:22)e = (γ −1) 1 k(cid:27) k2. It is interesting c 2 c to note that if in addition γ = 1, the CAPM case, then log consumption is a martingale. Case 2: (cid:17) = 0 and γ = 1. Supposetheagent is extremely riskaverse as well as unwillingto substitute. Table 1 indicates that this is possible only if y = 1=m (there is no equilibrium with arbitrary technology or endowment). In this case, (2.13) can only be satis(cid:12)ed if (cid:22)e = (cid:27)e = 0, which means that c(t) is a constant determined by c c the initial wealth. Not surprisingly (2.30) shows that the optimal portfolio is determined entirely by the hedging component: (cid:27) (t) = −(cid:27) (t). Since (cid:30) consumption is constant, r = r and (cid:21) = (cid:21), so (cid:25)(t) is the value of a real e e annuity as well as that of an endowment annuity. Finally, given (cid:25)(t) = 1= (t) when (cid:17) = 0, we have −(cid:27) (t) = (cid:27) (t). Therefore the optimal portfolio is a real (cid:25) annuity in this case (a real perpetuity in the in(cid:12)nite-horizon case).23 Case 3: (cid:17) = 0 and γ = 0. Now, the forcing process must be y = (cid:30). The agent is unwilling to substitute consumption across periods but perfectly willing to substitute across states of nature (risk neutral). The state-price deflator is m(t) = c(t)=(cid:30)(t), and from (2.13), we infer that c(t) is a martingale, a fact that has far-reaching implications in the in(cid:12)nite-horizon case. Because c(t) is a positive martingale, it converges. If technology is such that a constant and strictly positive asymptotic consumption flow is feasible, then consumption might converge to suchavalue. Otherwise, theconsumerasymptotically exhaustshiswealth and consumption converges to zero, for any value of the rate of time preference (cid:18). For example, suppose that (cid:22) is constant (even if (cid:22)e and (cid:27) are not). Then (cid:30) (cid:30) (cid:30) the solution is 1=(cid:25)(t) = (t) =r(t) = (cid:22) , (cid:21)(t)= 0, and (cid:27) (t) = (cid:27) (t) = (cid:27) (t). (cid:30) c k (cid:30) If (cid:27) (t) does not go to zero, then c(t) and k(t) both do almost surely (though, (cid:30) of course, not in the L norm). 1 Case 4: γ = 0. Thisistheriskneutralitycase,andTable1suggeststhatthereisnosolution when y = 1=m. In fact, there is a solution, but in general it involves setting c(t) = 0 often, a corner solution that does not satisfy our system of equations (our solution method assumes positivity of log consumption, and thus rules out corner solutions). An important point to make, however, is that risk neutrality of agents does not imply that the price of risk, (cid:21)(t), is zero. In light of equation (2.14b), (cid:21)= −(cid:27) and thus the condition (cid:27) = 0 must hold for the price of risk to vanish. But (cid:27) = 0 requires (cid:17) = 1 in addition to γ = 0. With standard preferences, of course, γ = 0 implies (cid:17) = 1, so that in such models risk neutrality indeed implies (cid:21)(t) = 0. (See Case 6 below.) Forexample,lety = (cid:30). If(cid:17) = 2then" =0,inwhichcase,r (t) = 2(cid:18)−(cid:22) (t) e (cid:30) and (cid:21) (t) = −(cid:27) (t). Therefore, as long as (cid:22) is not deterministic (and the e (cid:30) (cid:30) 23This point is made by Campbell and Viceira (1996).

26 MARK FISHER AND CHRISTIAN GILLES average value of (cid:22)e is less than 2(cid:18)), (cid:27) 6= 0. With (cid:17) = 2, equation (2.21) (cid:30) (cid:25) shows that (cid:27) (t) = (cid:27) (t), so that (2.14b) implies (cid:21)(t) = −(cid:27) (t) 6= 0. (cid:25) (cid:25) Case 5: (cid:17) = 1. The forcing variable must be consumption. In this case = (cid:18), since lim (t) = lim (cid:18)(cid:17)=((cid:17)−1)(cid:25)(t)1=((cid:17)−1) = (cid:18) (cid:17)!1 (cid:17)!1 regardless of the value of (cid:25)(t). Since " = γ, we have, equating (3.11) with (2.19), (cid:21)(t) = γ((cid:27) (t)−(cid:27) (t)). c (cid:25) Case 6: (cid:17) = 1 and γ = 0. ThiscaseinheritsallofthepropertiesofCase5. Withγ = 0," = 0,theprice of risk vanishes, (cid:21)(t)= 0, and, from (3.9a), the interest rate is determined by the rate of time preference, r(t) = (cid:18). As a result, the expected rate of return on any asset is equal to (cid:18), and the yield curve is flat at that level. 4. A Markovian setting In this section we introduce state variables and provide explicit solutions to the model. Modeling the dynamics of the forcing process. We suppose there are d Markovian state variables X driving the exogenous process y, where y is either (cid:30), c, or 1=m. The joint dynamics of X and y are given by 0 1 0 1 0 1 > dX(t) (cid:22) (X(t)) (cid:27) (X(t)) @ A @ X A @ X A = dt+ dW(t); dlog(y(t)) (cid:22)e (X(t)) (cid:27) (X(t)) > Y Y (cid:0) (cid:1) > > whereW = W ;W isal-dimensionalvectoroforthonormalBrownianmotions, x y with W (l−1)-dimensional andW scalar. Thedimensionsof(cid:27) (x)and(cid:27) (x)are x y X Y respectively l(cid:2)d and l(cid:2)1. We assume that the last column of (cid:27) X (x)(cid:0) > is a vecto(cid:1)r > > of zeros, so that the state variables are not a(cid:11)ected by W : (cid:27) (x) = (cid:6) (x) 0 , y X X where (cid:6) (x) is (l−1)(cid:2)d.24 Note that even if the state variables are deterministic X ((cid:27) (x) (cid:17) 0), y can bestochastic; acompletely deterministic economy wouldrequire X (cid:27) (x) (cid:17) 0 as well. Y The PDE. We assume that the horizon is (cid:12)nite, treating the case of an in(cid:12)nite horizon as a limit. We distinguish two cases. First case: (cid:17) 6= 1. We transform the annuity equation (3.10) into a PDE in terms of a Markovian annuity function (cid:5)(x;(cid:28)) such that where (cid:25)(t) = (cid:5)(X(t);T −t). Let (cid:5) , (cid:5) , and (cid:5) denote the obvious partial derivatives, and de(cid:12)ne an operator x xx (cid:28) 24Thisassumptioniswithoutlossofgenerality. Itsimplyallowsforthepossibilitythatthereexists a shock, W , that a(cid:11)ects y but not X. y

CONSUMPTION AND ASSET PRICES 27 F(x;(cid:28);@(cid:1)) operating on the space of candidate solutions so that25 (cid:18) (cid:19) 1 F(x;(cid:28);@(cid:5)) := 1+(cid:22)(cid:22) (x;(cid:28))− d +d (cid:22)e (x)+d d k(cid:27) (x)k2 (cid:5)(x;(cid:28)) (cid:5) 0 1 Y 1 2 Y 2 1 k(cid:27)(cid:22) (x;(cid:28))k2 > (cid:5) +d (cid:27) (x) (cid:27)(cid:22) (x;(cid:28))+("=d ) ; (4.1) 2 Y (cid:5) 1 2 (cid:5)(x;(cid:28)) where h i 1 (cid:22)(cid:22) (x;(cid:28)) = (cid:5) (x;t)(cid:22) (x)+ tr (cid:27) (x) > (cid:27) (x)(cid:5) (x;(cid:28)) −(cid:5) (x;(cid:28)) (cid:5) x X X X xx (cid:28) 2 (cid:27)(cid:22) (x;(cid:28)) = (cid:5) (x;(cid:28))(cid:27) (x): (cid:5) x X The PDE to solve is F(x;(cid:28);@(cid:5)) = 0 subject to (cid:5)(x;0) = 0; (4.2) which is a linear PDE if the nuisance term is zero and quasi-linear otherwise. Second case: (cid:17) = 1. We Markovianize (3.1) after changing variable from to (cid:4) through (t) = exp((cid:4)(X(t);T −t)). For this case, the de(cid:12)nition of F in (4.1) is replaced by F(x;(cid:28);@(cid:4)) := (cid:18)((cid:4)(x;(cid:28))−log((cid:18)))−(cid:22)(cid:22) (x;(cid:28))−q((cid:28))((cid:22)e (x)+d (cid:18)) (cid:4) Y 3 (1−γ) − kq((cid:28))(cid:27) (x)+(cid:27)(cid:22) (x;(cid:28))k2; (4.3) Y (cid:4) 2d ((cid:28)) 4 where h i 1 (cid:22)(cid:22) (x;(cid:28)) = (cid:22) (x) > (cid:4) (x;(cid:28))+ tr (cid:27) (x) > (cid:27) (x)(cid:4) (x;(cid:28)) −(cid:4) (x;(cid:28)) (cid:4) X x X X xx (cid:28) 2 (cid:27)(cid:22) (x;(cid:28)) = (cid:27) (x)(cid:4) (x;(cid:28)): (cid:4) X x In this case, the PDE is F(x;(cid:28);@(cid:4)) = 0 subject to (cid:4)(x;0) = log((cid:18)); (4.4) which is a linear PDE. A general solution method. Except in a few special cases, closed-form analytic expressionsforthefunctions(cid:5) and(cid:4) willnotbeavailable. (Wewilldiscusssomeof thosespecialcasesbelow.) Wedescribeasolutionmethodthatisquitegeneral. The method generates an approximation to the solution that can be made arbitrarily accurate. In the limit, the method generates the Taylor series representation of the solution. Themethodapplies to boththe(cid:17) 6= 1 and(cid:17) =1 cases with anynumberof state variables. For expositional simplicity, however, we describe the method with a single state variable. The generalization is obvious. Markovianbond-pricingtechniquesthatdecomposeaPDEintoasystemofODEs can be applied to (4.3), as in Du(cid:14)e and Epstein (1992a), Du(cid:14)e, Schroder, and Skiadas(1997), andSchroderandSkiadas(1997). Thesetechniquescanalsobeapplied indirectly to (4.1) through the PDE for endowment bond prices when the nuisance 25Thenotation@(cid:1)occurringinF isshortforallthepartialderivativeoperatorsthatappearinthe PDE.

28 MARK FISHER AND CHRISTIAN GILLES termisabsent(i.e.,whenthelasttermin(4.1)isidenticallyzero). Thesetechniques decompose a PDE into a system of ODEs by expressing the log of bond prices as a Taylor polynomial.26 They have typically been applied where the Taylor expansion of the solution is (cid:12)nite-order (it is (cid:12)rst-order in the Du(cid:14)e-Kan exponential-a(cid:14)ne class). Our method extends these techniques to the case where the nuisance term is present in (4.1), producing an in(cid:12)nite system of (cid:12)rst-order ODEs. Truncating this system produces an approximation to the solution. But by including more equations in the truncated system, the solution can be made arbitrarily accurate. Bond-pricing techniques, including our extension, owe their tractability to the fact that the boundary condition is independent of the state variables. For bond prices the condition is Pe(x;0) = 1, while for the value of an annuity the condition is (cid:5)(x;0) = 0. The implicit boundary condition for the value of a perpetuity is lim (cid:28)!1 Pe(x;(cid:28)) = 0 which again is independent of the state variables. To apply our method, the following functions of the state variable, called collectively the data, must be real analytic (cid:22) (x); (cid:27) (x) > (cid:27) (x); (cid:27) (x)(cid:27) (x); (cid:22)e (x); and (cid:27) (x) > (cid:27) (x): (4.5) X X X X Y Y Y Y This assumption is su(cid:14)cient to guarantee that there exists a unique real analytic solution (cid:5)(x;(cid:28)) for any x in the domain of the functions in the data and any (cid:28) in a neighborhood of 0 (Cauchy{Kowaleskaya theorem, see Rauch (1991, Chapter 1)). Therefore, treating (cid:28) as a parameter, we can write the solution as a Taylor series X1 (cid:5)(i)(x ;(cid:28)) (cid:5)(x;(cid:28)) = A ((cid:28))(x−x )i; where A ((cid:28)) (cid:17) 0 ; (4.6) i 0 i i! i=0 where each A is analytic in (cid:28). This representation is guaranteed to exist in a i neighborhood of (cid:28) = 0, that is, when the horizon is short enough. The derivatives (cid:5) (x;(cid:28)) and (cid:5) (x;(cid:28)) can be computed term-by-term in (4.6). In addition x xx X1 (cid:5) (x;(cid:28)) = A0 ((cid:28))(x−x )i; (cid:28) i 0 i=0 where A0 ((cid:28)) denote the derivative with respect to (cid:28), and the boundary condition i can be written (cid:5)(x;0) = 0 =) A (0) = 0; for i= 0;1;2;:::: i With the solution (cid:5) expanded as in (4.6), de(cid:12)ne F(x;(cid:28)) := F(x;(cid:28);@(cid:5)). Then, F is itself real analytic, so that we can also write it as a Taylor series: X1 F(i)(x ;(cid:28)) F(x;(cid:28)) = 0 (x−x )i; (4.7) 0 i! i=0 whereF(i)(x ;(cid:28))denotesthei-thderivativeofF withrespecttox. Theuniquesolu- 0 tion (cid:5)(x;(cid:28)) is characterized by(cid:5)(x;0) = 0and F(x;(cid:28)) (cid:17) 0, so thatF(i)(x ;(cid:28)) = 0 0 26See Du(cid:14)eand Kan (1996) for a discussion of this techniqueas applied to the exponential-a(cid:14)ne class of term-structure models. Other models such as those in Constantinides (1992) and Rogers (1997) can besolved using thesame techniques.

CONSUMPTION AND ASSET PRICES 29 for i = 0;1;2;:::. These conditions producea system of (cid:12)rst-order ODEs in the coe(cid:14)cient functions that (along with the initial conditions) uniquely determine them. For our approximate solution method, we (cid:12)x n and we (cid:12)nd a polynomial approximation of order n to (cid:5), written as Xn (cid:5) (x;(cid:28)) = a ((cid:28))(x−x )i; where a (0) = 0 for i = 0;1;::: ;n. n ni 0 ni i=0 We donotclaim that a = A , so(cid:5) is notnecessarily thetruncation of the Taylor ni i n series of (cid:5). De(cid:12)ning a ((cid:28)) := 0 for i>n, the coe(cid:14)cients a (for i(cid:20) n) are found ni ni by replacing A by a for i = 0;1;::: ;1 in F(x;(cid:28)), and then solving the system i ni of equations F(i)(x ;(cid:28)) = 0 for i = 0;1;::: ;n. Nevertheless, as a consequence of 0 the assumed analyticity, lim n!1 a ni = A i for all i. When our method is applied to (4.4), it delivers exact solutions with a (cid:12)nite number of terms when the data in (4.5) are polynomial. We illustrate our method with a sequence of examples. Bond pricing examples. We begin with the case (cid:17) 6= 1 and " = 0. In this case we can write F(x;(cid:28);@(cid:5)) =1+(cid:5) (x;(cid:28)) > (cid:22)b (x)−(cid:5) (x;(cid:28))−R(x)(cid:5)(x;(cid:28)) x X (cid:28) h i 1 > + tr (cid:27) (x) (cid:27) (x)(cid:5) (x;(cid:28)) ; (4.8) X X xx 2 where 1 R(x) =d +d (cid:22)e (x)+d d k(cid:27) (x)k2 (4.9a) 0 1 Y 1 2 Y 2 (cid:22)b (x) =(cid:22) (x)+d (cid:27) (x) > (cid:27) (x): (4.9b) X X 2 Y X To evaluate our solution method, it is useful to have an alternative method. In the present case, one approach to solving (4.2) is to solve the related PDE for endowment zero-coupon bond prices. Let Pe(x;(cid:28)) be the price of an endowment bond with maturity (cid:28), (so p (t;s) = Pe(X(t);s−t), in the notation of section 2). e The PDE for bond prices is (cid:18) (cid:19) (cid:20) (cid:18) (cid:19)(cid:21) Pe(x;(cid:28)) > 1 Pe (x;(cid:28)) Pe(x;(cid:28)) R(x)= x (cid:22)b (x)+ tr (cid:27) (x) > (cid:27) (x) xx − (cid:28) ; Pe(x;(cid:28)) X 2 X X Pe(x;(cid:28)) Pe(x;(cid:28)) (4.10) subject to the boundary condition Pe(x;0) = 1. The solution for (cid:5) then is Z (cid:28) (cid:5)(x;(cid:28)) = Pe(x;s)ds: (4.11) s=0 Note that (cid:5) (cid:28) (x;(cid:28)) = Pe(x;(cid:28)). A necessary condition for lim (cid:28)!1 (cid:5)(x;(cid:28)) to be well-de(cid:12)ned is lim (cid:28)!1 Pe(x;(cid:28)) = 0. The Taylor expansion for the bond around x =x is given by 0 X1 Pe;(i)(x ;(cid:28)) Pe(x;(cid:28)) = 0 (x−x )i: 0 i! i=0

30 MARK FISHER AND CHRISTIAN GILLES where Pe(i)(x ;(cid:28)) denotes the i-th derivative with respect to x. Given (4.11) we 0 have Z (cid:28) Pe(i)(x ;s) A ((cid:28)) = 0 ds: i i! s=0 Viewing the annuity as a flow of zero-coupon bonds, then, provides a solution method. In the present case, this method produces all the coe(cid:14)cients of the Taylor expansion of (cid:5). Many Markovian models of bond prices are of the following form: XN log(Pe(x;(cid:28))) = − B ((cid:28))(x−x )i; i 0 i=0 where N is (cid:12)nite. For the exponential-a(cid:14)ne class, where N = 1, the Taylor expansion for the price of a bond is X1 (−1)iB ((cid:28))i Pe(x;(cid:28)) = e −B0((cid:28)) 1 (x−x )i: 0 i! i=0 Even though log(Pe(x;(cid:28))) has a Taylor expansion of (cid:12)nite order, Pe(x;(cid:28)) does not. In this case, the Taylor coe(cid:14)cients for the annuity are given by Z (cid:28) (−1)iB (s)i A ((cid:28)) = e −B0(s) 1 ds: (4.12) i i! s=0 Equation (4.12) will serve as a benchmark in the examples that follow. Deterministic state variables, (cid:17) 6= 1. Consider the following example. The forcing variable y is stochastic, driven by a single Brownian motion, and its drift and di(cid:11)usion depend on a single deterministic state variable, X, in such a way that we can write R(x) = R +R (x−X(cid:22)): 0 1 The \risk-adjusted" dynamics of X are given by: (cid:22)b (x) = (cid:20)(X(cid:22) −x) and (cid:27) (x) = 0: X X This is an exponential-a(cid:14)ne model where, choosing to expand around x = X(cid:22), 0 (cid:0) (cid:1) B ((cid:28)) = R (cid:28) and B ((cid:28)) = (R =(cid:20)) 1−e −(cid:20)(cid:28) : (4.13) 0 0 1 1 In this case we have (cid:18) (cid:19) A ((cid:28)) = (−1)i(R 1 =(cid:20))i Xi (−1)j i 1−e −(j(cid:20)+R 0)(cid:28) ; (4.14) i i! j j(cid:20)+R 0 j=0 (cid:0) (cid:1) where i is the binomial coe(cid:14)cient. This provides an exact solution for (cid:5). j Turning to our power series approximation method, write F(x;(cid:28);@(cid:5)) = 0 as (cid:5) (x;(cid:28)) =1+(cid:20)(X(cid:22) −x)(cid:5) (x;(cid:28))−R(x)(cid:5)(x;(cid:28)) (4.15) (cid:28) x

CONSUMPTION AND ASSET PRICES 31 Substituting into (4.15) the Taylor expansions of (cid:5) and its partial derivatives, we then can write F(x;(cid:28)) = 0 as X1 A0 ((cid:28))+ A0 ((cid:28))(x−X(cid:22))i = 0 i i=1 X1 n o 1−R 0 A 0 ((cid:28))− R 1 A i−1 ((cid:28))+(i(cid:20)+R 0 )A i ((cid:28)) (x−X(cid:22))i: i=1 We see that the PDE decomposes into a system of linear ODEs with constant coe(cid:14)cients, each corresponding to a condition of the form F(i) = 0: A0 ((cid:28)) = 1−R A ((cid:28)) 0 0 0 (4.16) A0 i ((cid:28)) = −R 1 A i−1 ((cid:28))−(i(cid:20)+R 0 )A i ((cid:28)) for i = 1;2;:::: Subject to A (0) = 0 for i = 0;1;2;:::, system (4.16) has the unique solution given i in (4.14). Followingoursolutionmethod,wechoosentospecify(cid:5) (x;(cid:28))andsetF(i)(x;(cid:28)) = n 0 for i = 0;1;::: ;n and a = 0 for i > n. This produces the following system of ni n+1 ODEs: a 0 ((cid:28)) = 1−R a ((cid:28)) n0 0 0 (4.17) a 0 ni ((cid:28)) = −R 1 a n;i−1 ((cid:28))−(i(cid:20)+R 0 )a ni ((cid:28)) for i = 1;2;::: ;n: System (4.17) has a unique solution subject to a (0) = 0 for i = 0;1;2;::: ;n. ni Comparing (4.16) with (4.17) we see that a ((cid:28)) (cid:17) A ((cid:28)) for i = 0;1;2;::: ;n for ni i 0 all n. This occurs because a does not depend on a for j > i, and is not the ni n;j general case as we show below. Increasing the horizon. TheCauchy{Kowaleskaya theoremguarantees theexistence of a solution, but only in a neighborhood of (cid:28) = 0. As the horizon increases, the value of the annuity converges to the value of a perpetuity, but only if the latter is well-de(cid:12)ned. Asu(cid:14)cientcondition forthevalueofaperpetuitytobewell-de(cid:12)ned is thatthein(cid:12)nite-horizonasymptoticforwardrateexistsandispositive. Theforward rate function is given by @log(Pe(x;(cid:28))) f(x;(cid:28)) := − = R +e −(cid:20)(cid:28) R (x−X(cid:22)): 0 1 @(cid:28) For lim (cid:28)!1 f(x;(cid:28)) to be well-de(cid:12)ned and positive, we need R 0 > 0 and (cid:20)> 0. When the value of a perpetuity is well-de(cid:12)ned, the time derivative in the PDE vanishes as (cid:28) !1, in which case lim A0 ((cid:28)) = 0 for i= 0;1;2;:::: (cid:28)!1 i Therefore in the limit (4.16) becomes a system of linear algebraic equations: 0 = 1−R A (1) (4.18a) 0 0 0 = R 1 A i−1 (1)+(i(cid:20)+R 0 )A i (1) for i = 1;2;:::: (4.18b)

32 MARK FISHER AND CHRISTIAN GILLES In the in(cid:12)nite-horizon case, then, there is no di(cid:11)erential equation to solve; the solution is found by solving a set of algebraic equations. In the present case, the form of the general term can be found directly from (4.18) or by specializing (4.14): (−1)iRi A (1) = Q 1 : (4.19) i i j(cid:20)+R j=0 0 The power series for the perpetuity can be written as e(cid:12)(x)Γ((cid:11);0;(cid:12)(x)) (cid:5)(x;1) = ; (4.20) (cid:20)(cid:12)(x)(cid:11) where Z (cid:12) (cid:11) := R =(cid:20); (cid:12)(x) := −R (x−X(cid:22))=(cid:20); and Γ((cid:11);(cid:31);(cid:12)) := t(cid:11)−1e −tdt; 0 1 t=(cid:31) where Γ((cid:11);(cid:31);(cid:12)) is the generalized incomplete gamma function.27 Stochastic state variable. We extend the previous example by allowing a non-zero volatility for the statpe variable, though we still assume the nuisance terms are zero. We let (cid:27) (x) = s &(x), where &(x) := (1 −(cid:11)) +(cid:11)x. This volatility function X X encompasses both the Gaussian ((cid:11) = 0) and square-root ((cid:11) = 1) models. To simplifynotation, wekeep thesameriskadjusteddriftforX. Inthiscase, theseries representation for (cid:5)(x;(cid:28)) in F(x;(cid:28);@(cid:5)) produces the following in(cid:12)nite system of (cid:12)rst-order linear ODEs with constant coe(cid:14)cients: A0 ((cid:28)) =1−R A ((cid:28))+s2 &(X(cid:22))A ((cid:28)) 0 0 0 X 2 A0 i ((cid:28)) =−R 1 A i (cid:0) −1 ((cid:28))−(i(cid:20)+R 0 )A i ((cid:28)) (cid:1) (4.21) +s2 (cid:11)c A ((cid:28))+&(X(cid:22))c A ((cid:28)) for i= 1;2;::: X i i+1 i+1 i+2 P where c = i j. Note that A depends on A and A . Conditional on A i j=1 i i+1 i+2 0 and A , we can solve for all of the rest of the functions. 1 We treat A and A as unknown|they are part of the solution. With a re- 0 1 ni placing A , our method generates a system of ODEs that is formally identical to i the (cid:12)rst n + 1 equations in (4.21) except that a = 0 for i > n|in particular, ni a = a = 0. The result is a system of n+1 linear (cid:12)rst-order ODEs with n;n+1 n;n+2 27For comparison with expressions in Campbell (1993) note that the (cid:12)rst-order approximation to log((cid:5)(x;1)) is R log((cid:5)(x;1))(cid:25)−log(R )− 1 (x−X(cid:22)); 0 (cid:20)+R 0 whichshowsthat(cid:5)(x;1)|aweightedaverageofexponentials|isapproximatelyexponentialitself around X(cid:22).

CONSUMPTION AND ASSET PRICES 33 constant coe(cid:14)cients:28 a 0 ((cid:28)) = 1−R a ((cid:28))+s2 &(X(cid:22))a ((cid:28)) n0 0 n0 X n2 a 0 ni ((cid:28)) = −R 1 a n (cid:0) ;i−1 ((cid:28))−(i(cid:20)+R 0 )a ni ((cid:28)) (cid:1) +s2 (cid:11)c a ((cid:28))+&(X(cid:22))c a ((cid:28)) for i = 1;2;::: ;n−2 X i n;i+1 i+1 n;i+2 (cid:0) (cid:1) a 0 n;n−1 ((cid:28)) = −R 1 a n;n−2 ((cid:28))−((n−1)(cid:20)+R 0 )a n;n−1 ((cid:28))+s2 X (cid:11)c n−1 a n;n ((cid:28)) a 0 nn ((cid:28)) = −R 1 a n;n−1 ((cid:28))−(n(cid:20)+R 0 )a nn ((cid:28)) (4.22) System (4.22) (along with the initial conditions a (0) = 0) has a unique solution. ni For (cid:12)nite n, however, a ni 6= A i , although lim n!1 a ni = A i for all i. We can use (4.12) to investigate the convergence of the factor loadings in this case. For simplicity, let (cid:11) = 0 so that &(x) (cid:17) 1. In this case, B ((cid:28)) is as given in (4.13), while 1 (cid:0) (cid:1) s2 R2 2(cid:20)(cid:28) −3+e −(cid:20)(cid:28) −e −2(cid:20)(cid:28) B ((cid:28)) = −R (cid:28) + X 1 : 0 0 4(cid:20)3 Given values for R , R , (cid:20), X(cid:22), and s , one can compare a ((cid:28)) with A ((cid:28)). What 0 1 X ni i onediscoversisthatas(cid:20)approacheszero, holdingn(cid:12)xed,theapproximationa to n;i the annuity factor loadings A worsens. This is due to the fact that A and A i n+1 n+2 become larger and larger as (cid:20) !0, making the approximation a = a = 0 n;n+1 n;n+2 worse and worse. Of course as long as endowment bond prices can be computed directly (as in this example), the series coe(cid:14)cients for the endowment annuity can be computed by integrating the series coe(cid:14)cients for the bond prices. A nuisance term. Adding a nuisance term to the PDE eliminates the possibility of usingbondprices to solve for the value of theannuity, since the endowment interest rate and the price of endowment risk depend on the volatility of the annuity. (See (3.11).) Nevertheless, our method allows us to solve the PDE for the value of the annuity. The presence of the nuisance term introduces nonlinear terms into the system of ODEs without otherwise changing the character of the problem or its solution. The additional terms for the (cid:12)rst two ODEs in (4.21) are ! 1 1 &(X(cid:22))A ((cid:28))2 ("=d )s2 2 1 and 1 X 2 A ((cid:28))2 0 ! 1 −&(X(cid:22))A ((cid:28))3 1 A ((cid:28))2+2&(X(cid:22))A ((cid:28))A ((cid:28)) ("=d )s2 1 + 2 1 1 2 : 1 X 2 A ((cid:28))3 A ((cid:28))2 0 0 Our solution method proceeds as before, truncating the system and setting a = 0 n;i for i= n+1;n+2;:::. The system of ODEs for the functions a ((cid:28)) has a unique ni solution (given the initial conditions) that converges on the true solution to the PDE as n !1. As noted above, as the rate of mean reversion decreases, the accuracy of the Taylor coe(cid:14)cients decreases (holding n (cid:12)xed). When the nuisance term is present, we cannot compute endowment bond prices directly. Nevertheless, we can improve 28Some equations may need modi(cid:12)cation for n<3.

34 MARK FISHER AND CHRISTIAN GILLES the accuracy as follows. We can solve a linear PDE (that is related to the quasilinear PDE in a purely formal way) for pseudo endowment bond prices. The Taylor expansion for these pseudo bond prices can be integrated to provide better approximations for a and a . In particular, we solve (4.10) where Pe(x;(cid:28)) n;n+1 n;n+2 is replaced by P b e(x;(cid:28)), the value of a pseudo endowment bond (since " 6= 0). If b the data are such that the Taylor expansion for log(Pe(x;(cid:28))) is (cid:12)nite order, we b can compute the Taylor expansion for Pe(x;(cid:28)) and integrate it to obtain values for a and a to be used in place of zero in the system of ODEs. This procen;n+1 n;n+2 dure delivers the exact Taylor coe(cid:14)cients if the nuisance term is absent. Moreover our numerical investigation indicates that when the nuisance term is present, this procedure delivers signi(cid:12)cant improvement. An example where (cid:17) = 1. We now turn to solving (4.4). For this example, we assume y = c, so that F(x;(cid:28);@(cid:4)) := (cid:18)((cid:4)(x;(cid:28))−log((cid:18)))−(cid:22)(cid:22) (x;(cid:28))−q((cid:28))(cid:22)e (x) (cid:4) C 1 −(1−γ) kq((cid:28))(cid:27) (x)+(cid:27)(cid:22) (x;(cid:28))k2: (4.23) C (cid:4) 2 For this example, let (cid:22) (x) = (cid:20)(X(cid:22) −x), (cid:22)e (x) = x, X C 0 1 0 1 p p s s @ xA @ c1A (cid:27) (x)= &(x) ; and (cid:27) (x) = &(x) ; X C 0 s c2 where (as before) &(x) = (1−(cid:11))+(cid:11)x.29 In this example the data are (cid:12)rst-order polynomials in x, so the solution is also a (cid:12)rst-order polynomial (cid:4)(x;(cid:28)) = A ((cid:28))+A ((cid:28))(x−X(cid:22)): 0 1 The system of ODEs is A0 ((cid:28)) =(cid:18)(log((cid:18))−A ((cid:28)))+q((cid:28))X(cid:22) +&(X(cid:22))D 0 0 A0 ((cid:28)) =q((cid:28))−((cid:20)+(cid:18))A ((cid:28))+(cid:11)D; 1 1 subject to A (0) = log((cid:18)) and A (0) = 0, where 0 1 (cid:0) (cid:1) 1 D := (1−γ) (q((cid:28))s )2+(q((cid:28))s +s A ((cid:28)))2 : c1 c2 x 1 2 When (cid:11) 6= 0 and γ 6= 1, explicit analytical expressions for A ((cid:28)) and A ((cid:28)) are 0 1 di(cid:14)cult to obtain. (Numerical solutions are easy in any case.) When (cid:11) = 0, we have (cid:18) (cid:19) (cid:24)((cid:28)) (cid:18) A ((cid:28)) = ; where (cid:24)((cid:28)) = 1−e −(cid:18)(cid:28) 1+q((cid:28)) : (4.24) 1 (cid:20)+(cid:18) (cid:20) 29With(cid:11)=1,thisexampleisessentiallythetermstructureexampleinDu(cid:14)eandEpstein(1992a), whilewith(cid:11)=0itisessentiallythetermstructureexampleofDu(cid:14)e,Schroder,andSkiadas(1997) without thesignal extraction problem. (The signal extraction problem putsa trend ins .) x

CONSUMPTION AND ASSET PRICES 35 Note that (cid:24)(0) = 0 and (cid:24)(1) = 1. Given that (cid:27) = (cid:27)(cid:22) = A (cid:27) , equation (4.24) (cid:4) 1 X implies 0 1 s x (cid:24)(T −t) (cid:27) (t) = @(cid:20)+(cid:18) A : (4.25) 0 The term structure and the equity premium. Continuing with this example where (cid:11) = 0, we investigate the term structure of interest rates and the premium on the capital account. Inserting (4.25) into (2.39), we have (cid:18) (cid:19) 1 s s (cid:24)(T −t) r(t)= (cid:18)+x(t)+ −Q(T −t) (s2 +s2 )+(1−γ) c1 x 2 c1 c2 (cid:20)+(cid:18) 0 1 s (cid:24)(T −t) Q(T −t)s +(γ −1) x @ c1 A (cid:21)(t)= (cid:20)+(cid:18) : Q(T −t)s c2 When(cid:17) = 1, (cid:25)(t) = q(T−t)=(cid:18) is deterministic, sothat (cid:27) (t) = (cid:27) (t). Becausewe k c also know from (2.25) that (cid:27) (t) = (cid:27) (t), we see that the premium on the capital (cid:30) k account is (cid:21)(t) > (cid:27) (t) = Q(T −t)k(cid:27) (t)k2+(γ −1)(cid:27) (t) > (cid:27) (t): (cid:30) c c The average slope of the term structure at the origin depends on the sign of the > > average of (cid:21)(t) (cid:27) (t) = (cid:21)(t) (cid:27) (t). (A negative sign produces a positively sloped r X term structure.) In this case we have s s (cid:24)(T −t) (cid:21)(t) > (cid:27) (t)= Q(T −t)(s2 +s2 )+(γ −1) c1 x (cid:30) c1 c2 (cid:20)+(cid:18) and s2(cid:24)(T −t) (cid:21)(t) > (cid:27) (t) = (cid:21)(t) > (cid:27) (t) = Q(T −t)s s +(γ−1) x : r x c1 x (cid:20)+(cid:18) Let us consider two cases. First, suppose s = 0, so that the expected growth c1 > rate of consumption is uncorrelated with the actual growth rate ((cid:27) (t) (cid:27) (t) = 0). x c In this case, the premium on the capital account is unambiguously positive and increases with γ, while the slope of the average yield curve depends on γ (the slope is positive if γ < 1, negative otherwise). An increase in γ reduces both the interest rateandtheaverage slopeoftheyieldcurveattheorigin. Second,supposeγ = 0, so that the agent is risk neutral and has a preference for late resolution of uncertainty. Suppose also that the horizon is in(cid:12)nite; in this case, the sign of the premium on the capital account is determined by the sign of the covariance between expected andactualgrowthratesofconsumption,whiletheyieldcurveunambiguouslyslopes upward on average at the origin.

36 MARK FISHER AND CHRISTIAN GILLES Existence and convergence: An example. Although theCauchy{Kowaleskaya theorem guarantees a unique solution exists in the neighborhood of (cid:28) = 0, it does not guarantee the existence of a solution for all (cid:12)nite (cid:28), not does it guarantee, even when a solution exists for all (cid:12)nite (cid:28), that the solution converges as (cid:28) ! 1. To begin to address these issues, consider the case where the nuisance term is absent. In this case, r (t)= R (X(t)) and (cid:21) (t) = (cid:3) (X(t)), where e e e e R (x) = d +d (cid:22)e (x)−d2k(cid:27) (x)k2 and (cid:3) (x) = d (cid:27) (x); e 0 1 Y 1 Y e 1 Y where we have used d = −d . If 2 1 R (x); (cid:22) (x)−(cid:27) (x) > (cid:3) (x); and (cid:27) (x) > (cid:27) (x) e X X e X X are all a(cid:14)ne in x, then we have an exponential-a(cid:14)ne model of endowment bond prices. For example, suppose there are three Brownians and two independent state variables, each of which has dynamics given by p dX (t) = (cid:20) (X(cid:22) −X (t))dt+s & (X )dW (t); i i i i i i i i where30 & (x) = (1−(cid:11))X(cid:22) +(cid:11)x, so that i i 0 1 p 0 1 s & (x ) 0 (cid:20) (X(cid:22) −x ) B B 1 1 1 p C C (cid:22) X (x) = @ (cid:20) 1 (X(cid:22) 1 −x 1 ) A and (cid:27) X (x)= B @ 0 s 2 & 2 (x 2 )C A : 2 2 2 0 0 Now further suppose that we are solving the planning problem. We choose a one-factor model of the term structure and allow the second factor to a(cid:11)ect the equity market. In particular, r(t)= R(X(t)) and (cid:21)(t) = (cid:3)(X(t)), where (cid:16) p p (cid:17) > R(x) = x 1 and (cid:3)(x) = q1 &1(x1) 0 q2 &2(x2) : s1 s2 Note that (cid:16) (cid:17) > > (cid:27) X (x) (cid:3)(x) = q 1 & 1 (x 1 ) 0 ; ensuring that X does not a(cid:11)ect real bond prices. Turning to the PDE for (cid:17) 6= 1, 2 note that (cid:22)e (x) = R(x)+ 1k(cid:3)(x)k2 and (cid:27) (x) = (cid:3)(x). Y 2 Y First consider the case where (cid:11)= 0. In this case (cid:0) (cid:1) 1 R (x) = (1−(cid:17))x +(cid:17)(cid:18)+(cid:17)(1−(cid:17)) X(cid:22) q2=s2+X(cid:22) q2=s2 : e 1 2 1 1 1 2 2 2 Although the parameters of the dynamics of the second state variable appear, the second state variable itself does not. In fact, the model devolves to a single-statevariable model in this case. We observe that if R (X(cid:22)) (cid:20) 0 then the solution will e not converge, since the value of a perpetuity is not de(cid:12)ned in this case. There is however a more telling condition. We can solve for the asymptotic rate. If it is not 30The de(cid:12)nition of &((cid:1)) here is slightly di(cid:11)erent from thede(cid:12)nition in theprevious section.

CONSUMPTION AND ASSET PRICES 37 positive, the solution will not converge. The asymptotic forward rate can be shown to be (cid:0) (cid:1) (1−(cid:17))2X(cid:22) s2+2(cid:20) q R (X(cid:22))− 1 1 1 1 : e 2(cid:20)2 1 Consider the parameters be given in Table 2 for example.31 With these parameters the solution will not converge for (cid:17) > 1:6218. i X(cid:22) (cid:20) s2 q i i i i 1 3=100 1=15 1=50 −11=75 2 1=100 3=500 1=2500 −1=50 Table 2. Parameter values. Now consider the case where (cid:11) = 1. In this case, we have (cid:0) (cid:1) (cid:0) (cid:1) (1−(cid:17)) (cid:17)q2+2s2 x (1−(cid:17)) (cid:17)q2 x R (x)= (cid:17)(cid:18)+ 1 1 1 + 2 2 : e 2s2 2s2 1 2 Here we have the following problem. If one or both of the coe(cid:14)cients on the state variables in R (x) is negative, there may not be a solution for all (cid:12)nite (cid:28). In the e present case, we run into problems as soon as (cid:17) > 1. When a nuisance term is present we cannot determine in advance the regions of existence andconvergence becausethe endowmentinterest rate andprice of riskare notknownabsentthesolutionforthevalueoftheendowmentannuity. Nevertheless, we believe the considerations are essentially the same. In closing, we note that our preliminarynumericalinvestigations indicatethatwhen(cid:11)= 1solutionsdonotexist for all (cid:12)nite horizons when (cid:17)γ ’ 1.32 5. Conclusion Summary. In a nutshell, we we have made two complementary but independent contributions to asset pricing theory underrecursive preferences|the (cid:12)rsttheoretical and the second numerical. On the theoretical front, we present a representation of continuation utility that reduces the general-equilibrium problem to a bond pricing problem. On the numerical front, we extend the class of Markovian models for which wecan (cid:12)ndthetermstructureof interest rates in termsof thestate variables. For any parameter values and any process for the forcing variable, we reduce the solution (as long as a solution exists) to that of (cid:12)nding bond prices under a derived process for the interest rate and price of risk, via a standard PDE. But this 31Theseparametervaluesareforillustrativepurposesonly;wehavenotattemptedtocalibratethe model to actual data. Nevertheless, the parameters for X are representative of the estimates of 1 Brown and Schaefer (1996) for the real term structure. 32AsnotedintheIntroduction,weintendtoincludeanumericalinvestigationinafutureversionof this paper. In themeantime, we have included a complete Mathematica package that implements of method.

38 MARK FISHER AND CHRISTIAN GILLES transformation of the original problem into that of a term-structure problem would only carry us so far, if we did not know how to solve the resulting PDE. Du(cid:14)e and Kan (1996) show how to solve the bond pricing PDE for the class of exponentiala(cid:14)ne term-structure model by breaking it into a (cid:12)nite set of ordinary di(cid:11)erential equations (one for each of the state variables, plus one for the constant term) that can be numerically solved in a fraction of a second on any modern computer. We show that as long as the dynamics of the forcing variables are su(cid:14)ciently smooth (but not necessarily a(cid:14)ne) the equilibrium-based annuity PDE also breaks down into a set of ODEs, though the set may be in(cid:12)nite. Solving all of these equations providesanexactsolution,whilesolvingfora(cid:12)nitenumberprovidesanapproximate solution. Further research. Notwithstanding some existing claims, we are not yet convinced that the framework of recursive preferences cannot rationalize the major asset-price puzzles. In our view, a complete and thorough examination has been hamperedbythelackoftractabletoolsthatwouldallowanunrestrictedexploration. Most of the model-building work in this area has been conducted in a discrete-time setting where the number and, perhaps more importantly, the nature of shocks has been limited. Moreover, there has been the tendency to equate the horizon for the short-term risk-free interest rate with the sampling frequency of the consumption growth data. Therefore, in our view, the empirical results that bear on the ability of a model such as we have focused on in this paper to resolve asset pricing puzzles is mixed. Ontheonehand,CochraneandHansen(1992) provideevidencethatrecursivepreferences are consistent with the unconditional moments of the data. On the other hand, the attempts to build models consistent with the conditional moments have failed thus far. For example, Weil (1989) reaches the negative conclusion that the equity premium puzzle remains for reasonable values of the preference parameters. In his setting for the exchange problem, however, the shocks are homoskedastic. Yet heteroskedasticity, for example, might prove important for resolving the equitypremium puzzle, which stems from the inability of the model to generate su(cid:14)cient volatility of the wealth-consumption ratio. If the term premium in (3.14) is stochastic rather than deterministic, then the volatility of a perpetuity dependson the volatility of the term premium as given in footnote 21. Depending on the covariance between endowment interest rate and the term premium, (cid:27)b (t;u) > (cid:27) (t;u), re (cid:12)e the relative variance of the wealth-consumption ratio, k(cid:27) (t)k2, may be bigger in (cid:25) an economy with state-dependent volatilities than in an economy in which the expectations hypothesis holds for the endowment term structure. Our continuous-time formulation of the model provides signi(cid:12)cant advantages over discrete-time formulations in that we have available both the tools of stochastic calculus that allow us to manipulate expressions and the solution techniques for di(cid:11)erential-equations. Any serious calibration attempt, however, will require care in matching moments with the data. Although matching moments for asset prices may be fairly straightforward, matching moments for consumption growth will require some e(cid:11)ort. In particular, we have no closed-form solutions for consumption

CONSUMPTION AND ASSET PRICES 39 growth-rates over discrete time periods in general. These must be computed from simulations given the parameters of the instantaneous dynamics. Yet this very dif- (cid:12)culty provides some additional hope that the model may (cid:12)t tolerably well, since the link between the instantaneous dynamics and the (cid:12)nite-horizon dynamics may contain the needed flexibility. We cannot claim that the framework of recursive preferences will turn out to be compatible with all of the major known asset-pricing puzzles. But we can claim to have provided some tools that will be prove useful for a thorough study of the question. In future research, we intend to pursue this study. Appendix A. The absence of arbitrage The state-price deflator. We adopt the stochastic framework studied in Du(cid:14)e (1996), to which we refer the reader for all omitted details. We restrict attention to a Brownian environment, by which we mean that we are given a l-dimensional vectoroforthonormalBrownianmotions, W(t),de(cid:12)nedona(cid:12)xedprobabilityspace, and the (cid:12)ltration is that generated by W(t). In other words, the information that agents have at time t is that contained in the path of W(s) for s < t. We assume theexistence of astate-price deflator, which follows astrictly positive It^o process m(t) that we write as: dm(t) = −r(t)dt−(cid:21)(t) > dW(t); (A.1) m(t) > where \ " denotes the transpose, r(t) is the instantaneous rate of interest and (cid:21)(t) is the price of risk. Observe that we are free to model r(t) and (cid:21)(t) independently, as long as a solution to (A.1) exists.33 A state-price deflator m(t) guarantees that asset prices are free of arbitrage possibilities. Thepriceof anyasset (expressedin agiven unitof account) is determined by the formula that its deflated gain is a martingale. To see what this means, consider an asset with cumulative dividend D(t) and value V(t), both It^o processes. For simplicity of exposition, assume that V(t) is strictly positive and that D(t) is locally riskless, so their processes can be written as: dV(t) > = (cid:22) (t)dt+(cid:27) (t) dW(t); and dD(t) = Z(t)dt; V V V(t) where Z(t) is the flow of dividends. The gain is the sum of the asset’s value and its cumulative dividend, G(t) := V(t)+D(t), while the deflated gain is G(t)m(t). To say that G(t)m(t) is a martingale is equivalent to saying that the price process V(t) obeys (cid:20)(cid:18) (cid:19) Z (cid:18) (cid:19) (cid:21) m(T) T m(s) V(t)= E V(T)+ Z(s)ds ; (A.2) t m(t) m(t) s=t 33In the example given in Cox, Ingersoll, Jr., and Ross (1985b), the solution to (A.1) does not exist.

40 MARK FISHER AND CHRISTIAN GILLES foranyT > t,whereE standsfortheexpectationconditionalontime-tinformation. t A direct implication of the pricing equation (A.2) is the no-arbitrage condition: > (cid:22) (t)+(cid:16)(t) =r(t)+(cid:21)(t) (cid:27) (t); (A.3) V V where (cid:16)(t):= Z(t)=V(t) is called the dividend rate. Changing the numeraire. Two kinds of assets play important roles in the sequel: zero-coupon bondsand assets that pay a continuous flow of dividends forever. Starting with zero-coupon bonds, let p(t;T) denote the price at time t of the bond paying one unit of account at time T. According to the pricing formula (A.2), the terminal condition p(T;T) = 1 implies (cid:20) (cid:21) m(T) p(t; T)= E ; t m(t) so that the term structure theory reduces to the problem of producing conditional forecasts of the state-price deflator. Turning to the case of an asset that pays a continuous dividend flow Z(t) forever, the pricing formula (A.2) implies that its price process obeys (cid:20)Z (cid:21) 1 m(s) V(t) = E Z(s)ds ; (A.4) t m(t) s=t assuming lim T!1 E t [m(T)V(T)] = 0. We now show that a change of numeraire transforms V(t) into the price of a consol. Changing numeraire is thus often convenient, not only to study both real and nominal (or foreign and domestic) yield curves, but also to turn many other asset pricing problems into term structure problems. To illustrate, let b(t;T) denote the value of a claim to the single strictly positive payment of S(T) at time T, so that (cid:20) (cid:21) m(T) b(t;T) = E S(T)] : t m(t) We can think of this asset as a zero-coupon bond that makes its payment in a di(cid:11)erent \currency." De(cid:12)ne b (t;T) := b(t;T)=S(t) to be the value of the bond S in the new currency units, and let m (t) := m(t)S(t); then, the pricing equation S above becomes (cid:20) (cid:21) m (T) S b (t;T) = E ; (A.5) S t m (t) S which we recognize as the value of a zero-coupon bond when m (t) replaces m(t) S as the state-price deflator. In general, given the choice of any strictly positive It^o process S(t) as the new numeraire, m (t) := m(t)S(t) de(cid:12)nes a new state-price deflator. Gains that are S measured in the new units, G(t)=S(t), and deflated by the new deflator are martingales. Since m (t) is a state-price deflator, its drift and di(cid:11)usion are (minus) the S short rate, r , and price of risk, (cid:21) , in the new units, and we are free to model r S S S and (cid:21) independently (as long as the It^o process they de(cid:12)ne exists). Moreover, we S are free to model independently any two of the processes m(t), m (t), and S(t), S

CONSUMPTION AND ASSET PRICES 41 leaving the third process to inherit the dynamic properties of the other two from the de(cid:12)nition m = mS. S If the original units are those of a consumption good and S(t) is the real value of a unit of currency (so that 1=S(t) is the price level), then b (t;T) given in equation S (A.5) is thenominalvalueof azero-coupon currency-denominatedbond,andb(t;T) is its real value. Weapplythischange-of-numerairetechniquetoin(cid:12)nitely-livedassetsintwoways. In the (cid:12)rst case, we set S(t) = V(t), the strictly positive price process for an asset with dividend flow Z(t). The new state-price deflator is m (t) := m(t)V(t), whose V dynamics, by Ito^’s lemma and the absence-of-arbitrage condition (A.3), are dm V (t) = −(cid:16)(t)dt−((cid:21)(t)−(cid:27) (t)) > dW(t); (A.6) V m (t) V so that the short rate and price of risk are r (t) = (cid:16)(t) and (cid:21) (t) = (cid:21)(t)−(cid:27) (t). V V V In the second case, we set S(t) = Z(t), so that the new state-price deflator is m (t) := m(t)Z(t) (here we require that the dividend Z(t) be strictly positive). Z The value of the asset in the new units is V (t) := V(t)=Z(t) = 1=(cid:16)(t). Let b (t;T) Z Z denote the value (in Z units) of a zero-coupon bond paying Z(T) (i.e., one Z unit) at time T. With the foregoing de(cid:12)nitions equation (A.4) produces (cid:20)Z (cid:21) Z 1 1 1 m (s) Z = V (t) = E ds = b (t;s)ds: (A.7) Z t Z (cid:16)(t) m (t) s=t Z s=t Thus, the inverse of the dividend rate, 1=(cid:16)(t), is the value of a consol, which has a unit dividend flow and a yield of (cid:16)(t) (measured in the new units). Appendix B. Mathematica code Here is the Mathematica package used to compute the PDE solutions in the paper. (* :Title: AnnuitySolve.m *) (* :Context: AnnuitySolve‘ *) (* :Author: Mark Fisher *) (* :Summary: In an economy where the representative agent has recursive preferences (Kreps-Porteus stochastic differential utility), the optimal wealth-consumption ratio is the value of an annuity that satisfies a quasi-linear 2nd-order PDE. This package provides tools to compute the series solution to the PDE. *) (* :Package Version: 1.0 (August 1998) *) (* :Mathematica Version: 3.0 *) (* :Sources:

42 MARK FISHER AND CHRISTIAN GILLES Fisher, M. and C. Gilles (1998) "Consumption and asset prices with recursive preferences." Photocopied, Federal Reserve Board. *) (* :Discussion: NAnnuitySolve is designed to compute the series solution to a quasi-linear 2nd-order PDE in terms of an unknown function A[x,t], where the data are real analytic and where the boundary condition is of the form A[x,0] == C, for C constant. For an annuity, C == 0, while for a zero-coupon bond, C == 1. The Cauchy-Kowaleskaya theorem guarantees the existence of a unique real analytic solution in the neighborhood of t == 0. The solution provided by NAnnuitySolve is of the form Sum[a[i][t](x-x0)^i, {i, 0, order}] where the a[i][t] are functions of t. The boundary condition is specified indirectly via the option InitialCondition -> C, which imposes the condition a[0][0] == C (along with a[i][0] == 0 for i >= 1). For exponential-polynomial bond prices, the option FunctionalForm -> Exp allows one to solve for Log[A[x,t]] subject to a[0][0] == 0. *) (* :Requirements: Utilities‘FilterOptions‘ *) (* :Examples: For examples, see the notebook AnnuitySolveExamples.nb, which is available from the author upon request. *) BeginPackage["AnnuitySolve‘", {"Utilities‘FilterOptions‘"}] AnnuitySolve::usage = "AnnuitySolve[pde, A, t, {x, x0}, order] calculates a symbolic series solution for the pde to the order specified, where A is an unknown function of x and t. It expands a polynomial of factor loadings in t around the point x0. Multiple state variables can be specified as in AnnuitySolve[pde, A, t, {x, x0}, {y, y0}, order]. As specified, AnnuitySolve returns a pure function; if A is replaced by A[outargs], AnnuitySolve returns the function evaluated at outargs. AnnuitySolve has the options FunctionalForm and InitialCondition. In addition, options can be passed to DSolve." NAnnuitySolve::usage = "NAnnuitySolve[pde, A, {t, min, max}, {x, x0}, order] calculates a numerical series solution for the pde to the order specified, where A is an unknown function of x and t. It expands a polynomial of factor loadings in t around the point x0. Multiple state variables can be specified as in AnnuitySolve[pde, A, {t, min, max}, {x, x0}, {y, y0}, order]. As specified, NAnnuitySolve returns a pure function; if A is replaced by A[outargs], NAnnuitySolve returns the function evaluated at outargs. NAnnuitySolve has the options FunctionalForm, InitialCondition, FactorLoadingSymbol, PolynomialOrderDifferential, and DifferentialLoadings. In addition,

CONSUMPTION AND ASSET PRICES 43 options can be passed to NDSolve." BondToAnnuitySeries::usage = "BondToAnnuitySeries[bond, t, {x, x0}, order] takes an expression for zero-coupon bond prices as a function of x and t and returns series coefficients (as functions of t that have been symbolically integrated) up to the order specified for the associated annuity. Multiple state variables can be specified as in BondToAnnuitySeries[bond, t, {x, x0}, {y, y0}, order]. Options can be passed to Integrate." NBondToAnnuitySeries::usage = "NBondToAnnuitySeries[bond, {t, min, max}, {x, x0}, order] takes an expression for zero-coupon bond prices as a function of x and t and returns series coefficients (as functions of t that have been numerically integrated) up to the order specified for the associated annuity. Multiple state variables can be specified as in NBondToAnnuitySeries[bond, {t, min, max}, {x, x0}, {y, y0}, order]. The output is designed to be used with the option DifferentialLoadings (in conjunction with PolynomialOrderDifferential) for NAnnuitySolve. Options can be passed to NDSolve." AbsValuePDE::usage = "AbsValuePDE[pde, A, soln] returns the absolute value of the deviations of the PDE from zero (as a pure function), where the function A has been replaced by the trial solution. The function is useful for determining how well the PDE is satisfied. AbsValuePDE[pde, A[args], soln] returns the function evaluated at args. " FunctionalForm::usage = "FunctionalForm is an option for AnnuitySolve and NAnnuitySolve. The default setting is FunctionalForm -> Identity. The setting FunctionalForm -> Exp can be used to solve for exponential-polynomial bond prices." InitialCondition::usage = "InitialCondition is an option for AnnuitySolve and NAnnuitySolve. It specifies the value of the zero-order factor loading function at t = 0. The default setting for NAnnuitySolve is InitialCondition -> 10^-100, which avoids division by zero when the PDE is quasi-linear." PolynomialOrderDifferential::usage = "PolynomialOrderDifferential is an option for NAnnuitySolve. The default setting is PolynomialOrderDifferential -> 0. This option is used in conjunction with the option DifferentialLoadings. Typically for this purpose the setting would be PolynomialOrderDifferential -> 2." DifferentialLoadings::usage = "DifferentialLoadings is an option for NAnnuitySolve. The default setting is DifferentialLoadings -> {}. This option is used in conjunction with the option PolynomialOrderDifferential. Typical use would involve computing exponential-polynomial bond prices (using NAnnuitySolve with

44 MARK FISHER AND CHRISTIAN GILLES FunctionalForm -> Exp), and then calling NBondToAnnuitySeries, the output of which would be used as the DifferentialLoadings." FactorLoadingSymbol::usage = "FactorLoadingSymbol is an option for NAnnuitySolve. It is used to coordinate passing the output of NBondToAnnuitySeries to NAnnuitySolve via the option DifferentialLoadings. The default setting is FactorLoadingSymbol -> $a, which need not be changed unless there is a symbol conflict." $a::usage = "$a is the symbol for the factor loadings." MakeCoefficients::usage = "MakeCoefficients[order, n] is an auxiliary function, called by other functions." MakePolynomial::usage = "MakePolynomial[xvars, t, coeffs] is an auxiliary function, called by other functions." Begin["‘Private‘"] AnnuitySolve::badargs = "The arguments to the function in the PDE are not properly specified." Options[AnnuitySolve] = {FunctionalForm -> Identity, InitialCondition -> 0} AnnuitySolve[pde_Equal, A_Symbol, t_Symbol, xargs:{_Symbol, _}.., order_Integer?Positive, opts___?OptionQ] := Module[{ff, ic0, fls, dopts, x, x0, n, arglist, args, coeffs, a, poly, polyeqn, seriesargs, le, initconds, odes}, {ff, ic0} = {FunctionalForm, InitialCondition} /. {opts} /. Options[AnnuitySolve]; fls = FactorLoadingSymbol /. {opts} /. Options[NAnnuitySolve]; dopts = FilterOptions[DSolve, opts]; {x, x0} = Transpose[{xargs}]; n = Length[x]; arglist = Union @ Cases[pde, f_[A][x__] | A[x__] :> {x}, Infinity]; args = First @ arglist; If[Length[arglist] != 1 || Union[args] =!= Union[Join[x, {t}]], Message[AnnuitySolve::badargs]; Return[$Failed]]; coeffs = MakeCoefficients[order, n, fls]; poly = MakePolynomial[x - x0, t, coeffs]; polyeqn = (Subtract @@ pde) /. A -> Function @@ {args, ff @ poly}; seriesargs = Sequence @@ Thread[{x, x0, order}]; le = List @@ LogicalExpand[Series[polyeqn, seriesargs] == 0]; initconds = Thread[ Through[coeffs[0]] == Prepend[Table[0, {Length[le] - 1}], ic0] ]; odes = Join[le, initconds]; Function @@ {args, ff @ poly /.

CONSUMPTION AND ASSET PRICES 45 ( First @ DSolve[odes, coeffs, t, Evaluate[dopts]] )} ] AnnuitySolve[pde_Equal, A_Symbol[outargs__], t_Symbol, xargs:{_Symbol, _}.., order_Integer?Positive, opts___?OptionQ] := AnnuitySolve[pde, A, t, xargs, order, opts][outargs] Options[NAnnuitySolve] = {FunctionalForm -> Identity, InitialCondition -> 10^-100, PolynomialOrderDifferential -> 0, DifferentialLoadings -> {}, FactorLoadingSymbol -> $a} NAnnuitySolve[pde_Equal, A_Symbol, range:{t_Symbol, tmin_?NumericQ, tmax_?NumericQ}, xargs:{_Symbol, _?NumericQ}.., order_Integer?Positive, opts___?OptionQ] := Module[{ff, ic0, diff, diffloads, fls, ndopts, x, x0, n, arglist, args, allcoeffs, coeffs, poly, polyeqn, seriesargs, le, initconds, odes}, {ff, ic0, diff, diffloads, fls} = {FunctionalForm, InitialCondition, PolynomialOrderDifferential, DifferentialLoadings, FactorLoadingSymbol} /. {opts} /. Options[NAnnuitySolve]; ndopts = FilterOptions[NDSolve, opts]; {x, x0} = Transpose[{xargs}]; n = Length[x]; arglist = Union @ Cases[pde, f_[A][x__] | A[x__] :> {x}, Infinity]; args = First @ arglist; If[Length[arglist] != 1 || Union[args] =!= Union[Join[x, {t}]], Message[AnnuitySolve::badargs]; Return[$Failed]]; allcoeffs = MakeCoefficients[order + diff, n, fls]; coeffs = Select[allcoeffs, ( Max @@ # ) <= order &]; diffloads = Select[diffloads, ( Max @@ #[[1]] ) > order &]; poly = MakePolynomial[x - x0, t, allcoeffs] /. diffloads; polyeqn = (Subtract @@ pde) /. A -> Function @@ {args, ff @ poly}; seriesargs = Sequence @@ Thread[{x, x0, order}]; le = List @@ LogicalExpand[Series[polyeqn, seriesargs] == 0]; initconds = Thread[ Through[coeffs[0]] == Prepend[Table[0, {Length[le] - 1}], ic0] ]; odes = Join[le, initconds]; Function @@ {args, ff @ poly /. ( First @ NDSolve[odes, coeffs, range, Evaluate[ndopts], StartingStepSize -> 10^-6, MaxSteps -> 10^6] ) } ] NAnnuitySolve[pde_Equal, A_Symbol[outargs__], range:{t_Symbol, tmin_?NumericQ, tmax_?NumericQ}, xargs:{_Symbol, _?NumericQ}.., order_Integer?Positive, opts___?OptionQ] :=

46 MARK FISHER AND CHRISTIAN GILLES NAnnuitySolve[pde, A, xargs, order, range, opts][outargs] BondToAnnuitySeries[bond_, t_Symbol, xargs:{_Symbol, _}.., order_Integer?Positive, opts___?OptionQ] := Module[{fls, iopts, x, x0, n, seriesargs, ser, sercoeffs, g, loadings}, fls = FactorLoadingSymbol /. {opts} /. Options[NAnnuitySolve]; iopts = FilterOptions[Integrate, opts]; {x, x0} = Transpose[{xargs}]; n = Length[x]; seriesargs = Sequence @@ Thread[{x, x0, order}]; ser = Series[bond, seriesargs]; sercoeffs = SeriesTable[ser, order, n]; loadings = Integrate[sercoeffs /. t -> s, {s, 0, t}, Evaluate[iopts]]; Thread[MakeCoefficients[order, n, fls] -> (Function[t, #]& /@ loadings)] ] NBondToAnnuitySeries::badbond = "The expression for the value of the bond is not numeric." NBondToAnnuitySeries[bond_, range:{t_Symbol, tmin_?NumericQ, tmax_?NumericQ}, xargs:{_Symbol, _?NumericQ}.., order_Integer?Positive, opts___?OptionQ] := Module[{fls, ndopts, x, x0, n, seriesargs, ser, sercoeffs, loadings, g}, fls = FactorLoadingSymbol /. {opts} /. Options[NAnnuitySolve]; ndopts = FilterOptions[NDSolve, opts]; {x, x0} = Transpose[{xargs}]; If[!NumericQ[bond /. Thread[x -> x0] /. t -> (tmax + tmin)/2], Message[NBondToAnnuitySeries::badbond]; Return[$Failed]]; n = Length[x]; seriesargs = Sequence @@ Thread[{x, x0, order}]; ser = Series[bond, seriesargs]; sercoeffs = SeriesTable[ser, order, n]; loadings = (g /. First @ NDSolve[{g’[t] == #, g[0] == 0}, g, range, Evaluate[ndopts], StartingStepSize -> 10^-6, MaxSteps -> 10^6])& /@ sercoeffs; Thread[MakeCoefficients[order, n, fls] -> loadings] ] AbsValuePDE[pde_Equal, A_Symbol, soln_] := Module[{arglist, args}, arglist = Union @ Cases[pde, f_[A][x__] | A[x__] :> {x}, Infinity]; If[Length[arglist] != 1, Message[AnnuitySolve::badargs]; Return[$Failed]]; args = First @ arglist; Function @@ {args, Abs[Subtract @@ pde] /. A -> Function @@ {args, soln}}]

References 47 AbsValuePDE[pde_Equal, A_Symbol[outargs__], soln_] := AbsValuePDE[pde, A, soln][outargs] (* auxiliary functions *) MakeCoefficients[order_, n_, sym_:$a] := Module[{b}, Flatten @ Table[sym @@ Array[b, n], Evaluate[ Sequence @@ Thread[{Array[b, n], 0, order}] ] ] ] MakePolynomial[xvars_List, t_, coeffs_List] := Apply[Times, xvars^# & /@ (coeffs /. $a -> List), {1}].Through[coeffs[t]] (* this is a kludge; SeriesCoefficient[series, {n1, n2, ...}] doesn’t work *) SeriesTable[series_, order_, n_] := Flatten @ Fold[ Map[Table[SeriesCoefficient[#, i], {i, 0, order}]&, #1, {#2}]&, series, Range[0, n - 1] ] End[] EndPackage[] References Backus, D., A. Gregory, and S. Zin (1989). Risk premiums in the term structure: Evidence from arti(cid:12)cal economies. Journal of Monetary Economics 24, 371{ 399. Brown, R. and S. Schaefer (1996). Ten years of the real term structure: 1984{ 1994. Journal of Fixed Income March. Campbell, J. Y. (1993). Intertemporal asset pricing without consumption data. The American Economic Review 83, 487{512. Campbell,J.Y.(1996). Understandingriskandreturn.Journal of Political Economy 104(2), 298{345. Campbell, J. Y. and L. M. Viceira (1996). Consumption and portfolio decisions when expected returns are time varying. Working Paper 5857, NBER. Cochrane, J. H. and L. P. Hansen (1992). Asset pricing explorations for macroeconomics. In O. Blanchard and S. Fischer (Eds.), NBER Macroeconomics Annual, pp. 115{182. Cambridge MA: MIT Press. Constantinides, G. M. (1992). A theory of the nominal term structure of interest rates. Review of Financial Studies 5(4), 531{552. Cox, J. C., J. E. Ingersoll, Jr., and S. A. Ross (1985a). An intertemporal general equilibrium model of asset prices. Econometrica 53(2), 363{384. Cox, J. C., J. E. Ingersoll, Jr., and S. A. Ross (1985b). A theory of the term structure of interest rates. Econometrica 53(2), 385{407.

48 References Du(cid:14)e, D. (1996). Dynamic Asset Pricing Theory (second ed.). Princeton, New Jersey: Princeton University Press. Du(cid:14)e, D. and L. G. Epstein (1992a). Asset pricing with stochastic di(cid:11)erential utility. Review of Financial Studies 5(3), 411{436. Du(cid:14)e, D. and L. G. Epstein (1992b). Stochastic di(cid:11)erential utility. Econometrica 60(2), 353{394. Appendix C with Costis Skiadas. Du(cid:14)e,D.andR.Kan(1996). Ayield-factor modelofinterestrates.Mathematical Finance 6(4), 379{406. Du(cid:14)e, D. and P.-L. Lions (1992). PDE solutions of stochastic di(cid:11)erential utility. Journal of Mathematical Economics 21, 577{606. Du(cid:14)e, D., M. Schroder, and C. Skiadas (1997). A term structure model with preferences for the timing of uncertainty. Economic Theory 9, 3{22. Du(cid:14)e, D. and C. Skiadas (1994). Continuous-time security pricing: A utility gradient approach. Journal of Mathematical Economics 23, 107{131. Epstein, L. G. and S. E. Zin (1989). Substitution, risk aversion, and the temporal behaviorbehaviorofconsumptionandassetreturns: Atheoreticalframework. Econometrica 57, 937{969. Epstein, L. G. and S. E. Zin (1991). Substitution, risk aversion, and the temporal behavior of consumption and asset returns: An empirical analysis. Journal of Political Economy 99(2), 263{286. Giovannini, A. and P. Weil (1989). Risk aversion and intertemporal substitution in the capital asset pricing model. Working Paper 2824, NBER. Lucas, Jr., R. E. (1978). Asset prices in an exchange economy. Econometrica 46, 1429{1445. Mehra, R. and E. C. Prescott (1985). The equity premium: A puzzle. Journal of Monetary Economics 15(2), 145{161. Rauch, J. (1991). Partial di(cid:11)erential equations. New York: Springer-Verlag. Rogers, L. C. G. (1997). The potential approach to the term structure of interest rates and foreign exchange. Mathematical Finance 7, 157{176. Schroder, M. and C. Skiadas (1997). Optimal consumption and portfolio selection with stochastic di(cid:11)erential utility. Working Paper 226, Kellogg Graduate School of Management, Northwestern University. Weil, P. (1989). Theequity premiumpuzzle and the risk-free rate puzzle. Journal of Monetary Economics 24, 401{421. (Mark Fisher) Monetary Affairs, Board of Governors of the Federal Reserve System, Washington, DC (Mark Fisher, as of August 31, 1998) Research Department, Federal Reserve Bank of Atlanta, 104 Marietta St., Atlanta, GA 30303 E-mail address: mark.fisher@atl.frb.org (Christian Gilles) FAST Group, Bear, Stearns & Co., 245 Park Ave., New York, NY 10167 E-mail address: cgilles@bear.com