Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets

Transcription

1 Dynamic Consumption and Portfolio Choice with Stochastic Volatility in Incomplete Markets George Chacko and Luis M. Viceira First draft: October 1998 This draft: January 2002 Chacko: Harvard University, Graduate School of Business Administration, Boston MA Tel , Viceira: Harvard University, Graduate School of Business Administration, Boston MA 02163, and NBER. Tel , We want to thank an anonymous referee, John Campbell, Pascal Maenhout, Robert Merton, Rachel Pownall, Enrique Sentana, Raman Uppal and seminar participants at HBS, CEMFI, the NBER, the XVII Latin American Meeting of the Econometric Society and the 1999 Conference of the European Finance Association for their comments and suggestions. This paper is a revised version of Working Paper 7377 of the National Bureau of Economic Research.

2 Abstract This paper examines the optimal consumption and portfolio choice problem of long-horizon investors who have access to a riskless asset with constant return and a risky asset ( stocks ) with constant expected return and time varying volatility. Markets are not complete, and investors have recursive preferences defined over intermediate consumption. The paper obtains a solution to this problem which is exact for investors with unit elasticity of intertemporal substitution of consumption, and approximate otherwise. The optimal demand for stocks includes an intertemporal hedging component whose sign depends on the sign of the correlation between volatility and stock returns, and on whether investors coefficients of relative risk aversion is above or below one. It is negative when this correlation is negative and risk aversion coefficients exceed one, and its magnitude increases with correlation, with the variance of shocks to volatility and also with their persistence, provided that their rate of decay is larger than the investors subjective discount rate. The paper also obtains an approximate solution when expected stock excess returns are an affine function of volatility. Estimates of the volatility process for the US stock market show a negative and large correlation; however, shocks to volatility do not appear to be variable and persistent enough to generate large intertemporal hedging demands. Empirical comparative statics exercises show that optimal portfolio demand barely changes with investors elasticity of intertemporal substitution, and that the magnitude of intertemporal hedging is relatively more sensitive to changes in the variance of volatility than to changes in persistence or correlation. JEL classification: G12.

3 1 Introduction There is strong empirical evidence that the conditional variance of asset returns, particularly stock market returns, is not constant over time. Bollerslev, Chou and Kroner (1992), Campbell, Lo and MacKinlay (1997, Chapter 12), Campbell, Lettau, Malkiel and Xu (2001) and others review the main findings of the ample econometric research on stock return volatility: Stock return volatility is serially correlated, and shocks to volatility are negatively correlated with unexpected stock returns. Changes in volatility are persistent (French, Schwert and Stambaugh 1987, Campbell and Hentschel 1992). Large negative stock returns tend to be associated with increases in volatility that persist over long periods of time. Stock return volatility appears to be correlated across markets over the world (Engle, Ito and Lin 1990, Ang and Bekaert 1999). While there is an abundant literature exploring the pricing of assets when volatility is time varying, there is not much research exploring optimal dynamic portfolio choice with volatility risk. This situation is unfortunate, because Samuelson (1969) and Merton (1969, 1971, 1973) have shown that time variation in investment opportunities imply optimal portfolio strategies for multi-period investors that can be different from those of single-period, or myopic, investors. Multi-period investors value assets not only for their short-term risk-return characteristics, but also for their ability to hedge consumption against adverse shifts in future investment opportunities. Thus these investors have an extra demand for risky assets that reflects intertemporal hedging. Intertemporal hedging is not only conceptually interesting; it is also empirically relevant. Recent research summarized in Campbell in Viceira (2002) has found that 1

4 intertemporal hedging is quantitatively important in light of the observed predictable variation in both interest rates and equity premia in the US (Balduzzi and Lynch 1997, Barberis 2000, Brandt 1998, Brennan, Schwartz and Lagnado 1996, 1997, Campbell and Viceira 1999, 2001, Campbell, Chan and Viceira 2002). This paper, as well as concurrent work by Liu (2000, 2001) that we discuss below, explores systematically optimal portfolio choice with volatility risk in a continuoustime setting. 1 We solve for the optimal consumption and portfolio choice of longhorizon investors when there is predictable variation in stock market return volatility, 2 and use these solutions to evaluate the importance of volatility risk for intertemporal hedging in the US stock market. Investors have two assets available for investment, a riskless asset and a risky asset ( stocks ). We assume that the return on the riskless asset and that the expected return on stocks are both constant. We also assume that stock return precision, the reciprocal of volatility, follows a mean-reverting, square-root process. Volatility is instantaneously correlated with stock returns. We allow for this correlation to be imperfect. As a result we work in a setting where markets are not complete. Itô s Lemma implies a process for stock return volatility that inherits the properties of the process for precision. We work with precision instead of volatility for mathematical 1 Lynch and Balduzzi (2000) have also addressed tangentially the implications of time-varying volatility for portfolio choice in their study of optimal portfolio rebalancing with stock return predictability and transaction costs. They find that allowing for return heteroskedasticity can have important effects on the optimal portfolio rebalancing behavior of long-horizon investors. 2 The term volatility is somewhat vague, and it is used in the literature sometimes as meaning variance and sometimes as meaning standard deviation. Throughout this paper, though, when we use the term volatility, we mean variance. 2

5 convenience. 3 Note that our assumptions about precision or volatility capture the main stylized empirical facts about stock market volatility. We consider investors whose preferences over consumption are described by the Duffie and Epstein (1992) recursive utility function. Recursive utility is a convenient generalization of power utility that separates relative risk aversion from the elasticity of intertemporal substitution of consumption (Epstein and Zin 1989, 1991), which under power utility are inextricably linked by a single parameter. Under these assumptions, we derive analytic expressions for the optimal consumption and portfolio policies. These expressions are exact when investors have unit elasticity of intertemporal substitution of consumption, and they are approximate otherwise. We find that, when expected stock excess returns are constant, the optimal portfolio demand for stocks is a linear function of precision. When changes in volatility are correlated with stock returns, and investors have coefficients of relative risk aversion different from one, optimal portfolio demand includes an intertemporal hedging component which is also linear in precision. The sign of intertemporal hedging demand depends on the sign of this correlation, and on whether the coefficient of relative risk aversion is larger or smaller than one. It is negative when the correlation is negative, and the coefficient is larger than one. A negative correlation implies that stocks then do to worse when there is an increase in risk. This reduces the willingness of conservative investors to hold stocks in their portfolios. We also show that the magnitude (in absolute value) of intertemporal hedging demand increases with the size of the instantaneous correlation between changes in 3 Portfolio problems require very often working with precision rather than with volatility itself. One example is the mean-variance allocation to risky assets, which is linear in precision. 3

6 precision or volatility and stock returns, and with the variance of shocks to precision or volatility. When the rate at which shocks to precision decay is larger than the rate at which investors discount future utility, an increase in their persistence also leads to an increase in the absolute magnitude of intertemporal hedging demand. We also consider a model where the expected excess return on stocks is an affine function of volatility. In that case, we find that both total optimal portfolio demand and intertemporal hedging demand are affine functions of precision. Interestingly, the intercept and the slope of intertemporal hedging demand have opposite signs, reflecting the fact that increases in volatility are at least partially compensated by changes in the expected return on investors portfolios. We conduct an empirical calibration of these models using monthly U.S. stock returns from 1926 to 2000, and annual stock returns from 1871 to Consistent with previous empirical evidence, we estimate a large instantaneous negative correlation between changes in volatility and stock returns in both samples. This generates negative intertemporal hedging demand for stocks for investors with coefficients of relative risk aversion larger than one. However, volatility is not persistent and variable enough in our monthly sample to generate sizable intertemporal hedging demands. By contrast, volatility is considerably more persistent and variable in the annual sample, which generates significantly larger intertemporal hedging demands. Nevertheless, the size of these intertemporal hedging demands is still modest compared with the size of intertemporal hedging demands generated by the estimated variation in interest rates and equity premia in the U.S. One of the contributions of this paper is to expand the set of known exact analytical solutions to continuous-time intertemporal portfolio problems. Kim and Omberg 4

7 (1996) solve analytically for the optimal portfolio rule of investors who maximize power utility defined over terminal wealth and choose between a riskless asset and a risky asset whose expected return follows a mean-reverting process. Wachter (2002) extends this solution to the case where investors have utility defined over intermediate consumption, provided that shocks to expected returns are instantaneously perfectly correlated with shocks to realized returns or, equivalently, that markets are complete. Brennan and Xia (2001) and Wachter (2000) consider models with time-varying interest rates in a complete markets setting. They derive exact solutions for investors with power utility defined, respectively, over terminal wealth and over intermediate consumption. Schroder and Skiadas (1999) and Fisher and Gilles (1998) also explore the implications of complete markets for optimal consumption and portfolio choice for investors with recursive utility. This paper provides an exact solution for the case of utility defined over intermediate consumption which does not require assuming that markets are complete. The assumption of perfect capital markets is not empirically plausible in a variety of settings, including the setting of this paper, given the evidence about the imperfect correlation between shocks to volatility and unexpected stock returns. This exact solution requires though that investors have unit elasticity of intertemporal substitution of consumption. This assumption is also difficult to justify on empirical grounds, because the existing estimates of this elasticity from aggregate and disaggregate data are well below one (Hall 1988, Campbell and Mankiw 1989, Campbell 1999, Vissin-Jorgensen 2001). However, our calibration exercise suggests that this assumption is not particularly constraining if one is interested only in dynamic portfolio choice. This exercise shows that optimal portfolio allocations are very 5

8 similar across a wide range of values for the elasticity of intertemporal substitution of consumption. Working in discrete time, Campbell and Viceira (1999, 2001, 2002) and Campbell, Chan and Viceira (2002) also reach similar conclusions in their analysis of optimal consumption and portfolio choice with time variation in expected returns and interest rates. 4 In two papers closely related to ours, Liu (2000, 2001) examines the optimal allocation to stocks when stock return volatility is stochastic. Both papers provide exact analytical solutions in an incomplete markets setting for investors with power utility defined over terminal wealth, and specifications of stochastic volatility which are slightly different from the ones in this paper. Liu (2000) considers the Heston (1993) and Stein and Stein (1991) models of stochastic volatility, in which volatility follows a mean-reverting process and stock returns are a linear function of volatility. These models imply a Sharpe ratio of stocks that is increasing in the square root of volatility, and a ratio between expected stock excess returns and stock return volatility the mean-variance allocation to stocks that is constant. Our model where we assume that expected stock returns are an affine function of volatility have similar implications for the Sharpe ratio and the mean-variance allocation to stocks in the special case where we constrain the intercept of the affine function to be zero. Liu (2000) also considers a model that includes both interest rate risk and volatility risk. A calibration of this model to US data arrives at conclusions similar to ours regarding the relatively modest size of intertemporal hedging demands generated by volatility risk. Finally, Liu (2001) considers a general class of stochastic volatility models that nests our basic specification with constant expected returns. 4 Their analytical solutions are also exact for investors with unit elasticity of intertemporal substitutionofconsumption,uptoadiscrete-time approximation to the log return on wealth. 6

9 The paper is organized as follows. Section 2 states the dynamic optimization problem, Section 3 presents an exact solution to the problem in the case with unit elasticity of intertemporal substitution. Section 3 also presents some comparative statics results. Section 4 explains the continuous-time approximate solution method that allows us to solve the problem when the elasticity of intertemporal substitution differs from unity, and states the solution implied by the method. Section 5 explores the solution to the problem when expected excess returns are an affine function of volatility. Section 6 calibrates the model to monthly U.S. stock market data and explores the empirical implications of stochastic volatility for portfolio choice. Section 7 discusses some alternative approximate solution and issues related to the accuracy of the approximate analytical solution. Finally, Section 8 concludes. 2 The Intertemporal Consumption and Portfolio Choice Problem 2.1 Investment opportunity set We assume that wealth consists of only tradable assets. Moreover, to keep the analysis simple,weassumeinthispaperthatthereareonlytwotradableassets. Oneofthe assets is riskless, with instantaneous return db t = rdt. B t The second asset is risky, with instantaneous total return dynamics given by r ds t 1 = µ + dw s, (1) S t y t 7

10 where S t is the value of a fund fully invested in the asset that reinvests all dividends, and y t is the instantaneous precision of the risky asset return process and 1/y t is the instantaneous variance. Equation (1) implies that the expected excess return on the risky asset over the riskless asset (µ r) is constant over time we relax this assumption in Section 5. However, the conditional precision of the risky asset return varies stochastically over time, and this induces time variation in investment opportunities. We assume the following dynamics for instantaneous precision: dy t = κ(θ y t )dt + σ y t dw y. (2) Precision follows a mean-reverting, square-root process with reversion parameter κ > 0, and long-term mean and variance equal to θ and σ 2 θ/2κ, respectively (Cox, Ingersoll, and Ross, 1985). In order to satisfy standard integrability conditions, we assume that 2κθ > σ 2. The stochastic process for precision implies a mean-reverting process for the instantaneous volatility v t 1/y t. Applying Ito s Lemma to (2) we find that proportional changes in volatility follow a mean-reverting, square-root process: dv t = κ v (θ v v t ) dt σ v t dw y, (3) v t where θ v =(θ σ 2 /κ) 1 and κ v = κ(θ σ 2 /κ) κ/θ v.itisconvenienttonotehere that the unconditional mean of instantaneous volatility is approximately equal to: E[v t ] 1 θ θ 2 κ. (4) This follows from taking expectations of a second-order Taylor expansion of v t 1/y t around θ. σ 2 Since we have assumed that the expected return on the risky asset is constant, equation (4) is also the unconditional variance of the risky asset return. 8

11 We also assume throughout the paper that the shocks to precision are correlated with the instantaneous return on the risky asset, with dw y dw S = ρdt. This in turn implies that proportional changes in volatility are correlated with stock returns, with instantaneous correlation given by µ dvt Corr t, ds µ t = Corr t dy t, ds t = ρ. v t S t S t This model for stock returns and precision or volatility can capture the main stylized empirical facts about stock return volatility, in particular its mean-reversion and negative correlation with stock returns. It also implies that proportional changes in volatility are more pronounced in times of high volatility than in times of low volatility. Another important implication of this model of changing risk is that the ratio of the expected excess return on the risky asset to its variance is a linear function of the state variable. This model assumption greatly facilitates solving the dynamic optimization problem that we present below. It is important however, to remark that the Sharpe ratio of the risky asset in this model is not a linear function of the state variable, but a square-root function. Thus this model is not mathematically equivalent to a model where volatility is constant and the expected excess return on the risky asset changes stochastically in a mean-reverting fashion, as in Kim and Omberg (1996) or Campbell and Viceira (1999). 9

12 2.2 Investor preferences and dynamic optimization problem Investor s preferences are described by a recursive utility function. Recursive utility is a generalization of the standard, time-separable power utility model that separates relative risk aversion from the elasticity of intertemporal substitution of consumption. Power utility restricts the elasticity of intertemporal substitution of consumption parameter to be the inverse of the relative risk aversion coefficient, while conceptually these two parameters need not be related to one another. Epstein and Zin (1989, 1991) derive a parameterization of recursive utility in a discrete-time setting, while Duffie and Epstein (1992a, 1992b) and Fisher and Gilles (1998) offer a continuous-time analogue. We adopt the Duffie and Epstein (1992b) parameterization: J = Et Z t f (C s,j s ) ds, (5) where f(c s,j s ) is a normalized aggregator of current consumption and continuation utility that takes the form f (C, J) = β 1 1 ψ Ã (1 γ) J C ((1 γ) J) 1 1 γ! 1 1 ψ 1, (6) β > 0 istherateoftimepreference,γ > 0 is the coefficient of relative risk aversion and ψ > 0 is the elasticity of intertemporal substitution. Power utility obtains from (6) by setting ψ =1/γ. The normalized aggregator f(c s,j s ) takes the following form when ψ 1: f (C, J) =β (1 γ) J log (C) 1 log ((1 γ) J). (7) 1 γ 10

13 The investor maximizes (5) subject to the intertemporal budget constraint dx t =[π t (µ r)x t + rx t C t ]dt + π t X t r 1 y t dw s, (8) where X t represents the investor s wealth, π t is the fraction of wealth invested in the risky asset and C t represents the investor s instantaneous consumption. 3 An Exact Solution with Unit Elasticity of Intertemporal Substitution of Consumption Building on the work of Merton (1969, 1971, 1973), Giovannini and Weil (1989), Campbell and Viceira (1999, 2001), and Campbell, Chan, and Viceira (2002), we show in this section that it is possible to find an exact solution to the intertemporal optimization problem (5)-(8) when investors have unit elasticity of intertemporal substitution of consumption. Merton s (1969, 1971, 1973) work characterizes the general solution to intertemporal optimization problems with time-varying investment opportunities, and finds explicit analytical solutions in certain special cases. In particular, Merton shows that for log utility investors (γ = ψ 1), it is optimal to behave myopically, ignoring intertemporal hedging considerations. They behave myopically in the sense that the dynamic problem for these investors is equivalent to a sequence of independent single-period problems, for which there is a known explicit solution. Giovannini and Weil (1989) consider investors with recursive utility. They show that for investors with unit elasticity of intertemporal substitution (ψ =1)itisopti- 11

14 mal to behave myopically regarding the consumption decision, but that intertemporal considerations still enter their portfolio decision if their relative risk aversion is different from one. Unfortunately, they do not explicitly characterize the optimal portfolio rule for these investors. In this section we derive an exact analytic solution that allows us to fully characterize portfolio choice under time-varying investment opportunities for investors with ψ =1. Section 4 presents an approximate analytic solution for the general case in which ψ is not restricted to one Bellman equation The optimization problem given by (5)-(8) has one state variable, the precision of the risky asset return or, equivalently, the volatility of the risky asset return. Therefore, the value function of the problem (J) depends on financial wealth (X t )andthisstate variable. The Bellman equation for this problem is ½ 0 = sup f (C s,j s )+[π t (µ r)x t + rx t C t ]J X + 1 π,c 2 π2 Xt 2 J 1 XX + κ(θ y t )J y y t + 1 ¾ 2 σ2 J yy y t + ρσπ t X t J Xy, (9) where f (C, J) is given in (7) and subscripts on J denote partial derivatives. 5 Campbell and Viceira (1999, 2001) derive an analytical expresion for the optimal portfolio rule in a discrete-time setting. However, their solution is only approximate, because it is based on a discrete-time approximation to the log return on wealth. In continuous time, however, we can derive an exact expression for the log return on wealth which is linear in portfolio shares. 12

15 The first-order conditions for this equation are C t = β (1 γ) J J X, (10) π t = J X X t J XX (µ r) y t J Xy X t J XX ρσy t. (11) Equation (10) results from the envelope condition, fc = J X, from which the optimal consumption rule obtains once the value function is known. Equation (11) shows that the optimal portfolio share in the risky asset has two components. The first one is proportional to the risk premium times the inverse of the coefficient of relative risk aversion in the indirect utility function. This is the optimal demand for risky assets we find in single-period models, or in multi-period models with constant investment opportunities. For this reason it is called myopic demand. The second component is Merton s intertemporal hedging demand. It depends on instantaneous rates of change of the value function, the instantaneous variance of the state variable, and the instantaneous correlation between changes in the state variable and the risky asset. By inspection of (11), it is immediately seen that the hedging component of portfolio demand is non-zero unless σ =0(constant investment opportunities), ρ =0 (no hedging value in risky asset), or J Xy =0. This last equality obtains when γ =1 (Merton 1969, 1971, 1973, Giovannini and Weil 1989). Note, however, that equations (10) and (11) do not represent a complete solution to the model until we solve for J(X t,y t ). Substituting the first-order conditions into (9) and rearranging gives the Bellman 13

16 equation: 0 = f (C (J),J) J X C (J) 1 (J X ) 2 2 J XX (µ r) 2 y t J XJ Xy ρσ (µ r) y t J XX +J X X t r 1 (J Xy ) 2 ρ 2 σ 2 y t + J y κ(θ y t )+ 1 2 J XX 2 J yyσ 2 y t, (12) where C(J) denotes the expression for consumption resulting from (10). We now guess a solution of the form J(X t,y t )=I(y t ) X1 γ t 1 γ. (13) Substituting this solution into the Bellman equation and simplifying yields the following ordinary differential equation (ODE): 0 = (logβ 1) βi 1 (µ r)2 βi log I + Iy t + ri 1 γ 2γ ρσ (µ r) + I y y t + ρ2 σ 2 γ 2γ This ODE has a solution of the form (I y ) 2 I y t γ I σ 2 yκ(θ y t )+ 2(1 γ) I yyy t. (14) I =exp{ay t + B}. (15) Substitution of (15) into (14) leads to two algebraic equations for A and B: where aa 2 + ba + c = 0, (16) (1 γ)(β log β + r β) βb + κθa = 0, (17) a = b = c = σ 2 γ 1 ρ 2 + ρ 2, 2γ (1 γ) (18) ρσ(µ r) β + κ γ 1 γ, (19) (µ r)2. 2γ (20) 14

17 The first equation is a quadratic equation in A, and the second equation is linear in B given A. For general parameter values, the equation for A has two roots. We show in Appendix A that these roots are always real and have opposite signs when γ > 1, andtheyarerealandhavethesamesignwhenγ < 1 provided that γ > σ(µ r)(2ρ + σ(µ r)/ (β + κ)) (β + κ)+σ(µ r)(2ρ + σ(µ r)/ (β + κ)). (21) This condition also implies that both roots are positive. The existence of real roots is a necessary (but not sufficient) condition for the existence of a solution to the problem. We still need to determine which root delivers the correct solution to the model. We show in Appendix A that only one of these roots ensures that the limit of the solution as γ 1 equals the well-known solution in the special case of log utility (γ = ψ =1), for which A = B =0,andthevalue function is simply log(x t ) (Merton, 1969, 1971, 1973). Convergence to the known solution is obtained by selecting the root associated with the positive root of the discriminant of the quadratic equation (16) when γ > 1, and the negative root of the discriminant when γ < 1. This selection implies that A<0 when γ > 1, anda>0 when γ < 1; or,equivalently,thata/(1 γ) > 0. The alternative selection leads to a solution that diverges from the known solution in this special case. 6 6 We would like to show that the alternative selection also implies that the unconditional expectation of the value function is not bounded, hence violating the standard transversality condition. However, we have not been able to show this analytically, because there is no closed form expression for the expectation of a exponential function of a square root process. 15

18 3.2 Optimal policies We now state the complete solution, and discuss some of its most important properties: Proposition 1 When ψ =1, there is an exact analytical solution to problem (5)-(8) with value function given by J (X t,y t )=exp{ay t + B} X1 γ t 1 γ. (22) This value function implies the following optimal consumption and portfolio rules: C t X t = β, (23) and π t = 1 γ (µ r) y t + µ 1 1 ( ρ) σay t, (24) γ where A A/(1 γ) > 0, anda and B are given by the solution to the system of equations (16)-(17). Proof. The value function and its coefficients follow immediately from (14) and the ensuing discussion. The optimal policies follow immediately from direct substitution of the value function (22) and its derivatives into the first order conditions (10) and (11). Proposition 1 shows that for investors with unit elasticity of intertemporal substitution, the optimal log consumption-wealth ratio is invariant to changes in volatility and it is equal to their rate of time preference. For these investors, the income and 16

19 substitution effects on consumption produced by a change in the investment opportunity set exactly cancel out, and it is optimal for them to consume a fixed fraction of her wealth each period. For this reason this consumption policy is usually termed myopic. Equation (24) shows the optimal portfolio rule. This rule has two components. The first component is myopic portfolio demand. The second component is intertemporal hedging demand. Myopic portfolio demand is a linear function of instantaneous precision the reciprocal of instantaneous volatility, the risk premium and the reciprocal of relative risk aversion. We have argued in our discussion of the first order condition (11) for optimal portfolio choice that a myopic rule is always the optimal portfolio demand for investors with unit coefficient of relative risk aversion (logarithmic investors) and, when investment opportunities are constant, the optimal demand for all other investors. We can easily verify these claims using equation (24), by noting that the second component of portfolio demand is zero when γ =1or σ =0. Equation (24) shows that intertemporal hedging demand is also a linear function of current precision. Thus both components of demand are linear in precision, which implies that their ratio is independent of the current level of precision or volatility. This is the result of returns being instantaneously correlated with proportional changes in volatility rather than with absolute changes in volatility. The intertemporal hedging component of portfolio demand depends on all the parameters that characterize preferences and the investment opportunity set. The sign of the intertemporal hedging demand depends on the sign of the correlation between unexpected returns and changes in volatility ( ρ) andthesignof(1 1/γ). 17

20 When this correlation is negative ( ρ < 0), intertemporal hedging demand is negative for investors with γ > 1, and positive for investors with γ < 1. Investors who are more risk averse than logarithmic investors have a negative hedging demand for the risky asset because it tends to do worse when there is an increase in risk. On the other hand, investors who are more aggressive than logarithmic investors have a positive intertemporal hedging demand for the risky asset; they are willing to trade off worse performance when volatility is high for extra performance when volatility is low. Intertemporal hedging demand is zero when ρ =0. In that case investors cannot use the risky asset to hedge changes in risk. Intertemporal hedging demand is also zero when investors are infinitely risk averse. This follows from the fact that lim γ A =0. For these investors, the optimal overall allocation to the risky asset is zero, since the myopic component of portfolio demand is also zero when γ. 3.3 Comparative statics We can use the explicit solution for the optimal policies given in Proposition 1 to conduct some comparative statics exercises. In particular, we want to determine the effect on intertemporal hedging demand of changes in σ, κ and ρ. To this end it is useful to rewrite here the intertemporal component of total portfolio demand: µ π h t = 1 1 ( ρ) σay t. (25) γ First, we consider the effect on π h t of changes in σ. From equation (25), we have that π h t / σ is proportional to (σa)/ σ. Appendix B shows that (σa)/ σ > 0, which implies that an increase in the instantaneous variance of risk leads to an increase 18

21 in the absolute magnitude of intertemporal hedging demand. Second, we consider the effect on π h t of changes in κ. From equation (25), we have that π h t / κ is proportional to A/ κ. Appendix B shows that A/ κ < 0, which implies that a reduction in κ leads to an increase in the absolute magnitude of intertemporal hedging demand. However, since a reduction in κ is equivalent to an increase in both the persistence and the unconditional variance of the process for precision, 7 it is difficult to tell from this result whether an increase in the persistence of shocks to precision leads to an increase in the absolute magnitude of intertemporal hedging demand. To determine the effect of changes in persistence on intertemporal hedging demand we need to consider compensated changes in κ that leave the unconditional variance of precision unchanged. This requires examining the sign of (σa)/ κ. Appendix B shows that (σa)/ κ and (β κ) have the same sign when we hold the unconditional mean and variance of precision constant. Thus a compensated reduction in κ i.e., a compensated increase in the persistence of the shocks to precision increases the absolute magnitude of intertemporal hedging demand only when β < κ that is, when the rate at which investors discount future utility of consumption is smaller than the rate at which shocks to precision die out. Our estimates of the rate of decay of shocks to precision in Section 6 which are also consistent with previous estimates in the literature suggest that this inequality is likely to hold empirically for plausible values of the rate of time preference of investors. Finally, we are also interested in the effect on intertemporal hedging of changes in the instantaneous correlation between risky asset returns and precision on intertempo- 7 Recall from Section 2.1 that the unconditional variance of precision is equal to σ 2 θ/2κ. 19

22 ral hedging demand. Equation (25) implies that π h t / ρ is proportional to (ρa)/ ρ, and Appendix B shows that this derivative is positive when γ > 1. That is, for investors who are more conservative than a logarithmic investor, an increase in the this instantaneous correlation improves the ability of this asset to hedge changes in risk. However, the sign of this derivative is ambiguous when γ < 1. 4 An Approximate Solution for the General Case We now address the general case, where the investor s elasticity of intertemporal substitution of consumption can take any value. When ψ is not necessarily equal to one, both optimal portfolio choice and consumption react to changes in the state variable. The general case is interesting for two reasons. First, it is empirically relevant, since estimates of ψ available from both aggregate data and disaggregate data on individual investors suggest that ψ is below one (Hall 1988, Campbell and Mankiw 1989, Campbell 1999, Vissin-Jorgensen 2001). Second, it nests as a special case the time-additive power utility case standard in the literature. Since γ =1/ψ with power utility, the ψ =1case does not nest power utility unless we restrict ourselves to the special case of log utility where γ =1/ψ =1. Unfortunately, there is no exact analytical solution to the model in the general case. However, we show in this section that we can still find an approximate analytical solution to the problem. This solution provides strong economic intuition about the nature of optimal portfolio choice with time-varying risk, and converges to the known 20

23 exact solution in some special cases. We argue in section 7 that, for all other cases, it is reasonably accurate, even for values of the elasticity of intertemporal substitution far below one, in line with the findings of Campbell (1993), Campbell and Koo (1997), and Campbell et al. (2002) for the case in which risk premia and interest rates vary over time. 4.1 Bellman equation When ψ is not restricted to one, the Bellman equation for the problem is still given by equation (12). The first order condition for portfolio choice is still given by (11), but the first order condition for consumption resulting from the envelope condition fc = J X is different, because the aggregator takes a different form, given in (6). The first order condition for consumption is now given by: C t = J ψ X 1 γψ [(1 γ) J] 1 γ β ψ. (26) After plugging (11) and (26) into the Bellman equation (12), and guessing that J(X t,y t )=I(y t )X 1 γ t /(1 γ), we obtain the following ODE : 0 = 1 1 ψ βψ I 1+ 1 ψ + ρ2 σ 2 2γ 1 γ + ψ 1 ψ (µ r)2 βi + Iy t + 2γ ρσ(µ r) I y y t + ri γ (I y ) 2 y t + 1 I t 1 γ I σ 2 yκ(θ y t )+ 2(1 γ) I yyy t. (27) We can further simplify this equation by making the transformation I = H 1 γ 1 ψ, which gives the following non-homogeneous ODE: 0 = β ψ H 1 + ψβ + (1 ψ)(µ r)2 y t 2γ 21 ρσ(µ r)(1 γ) H y γ H y t

24 +r (1 ψ)+ ρ2 σ 2 (1 γ) 2 2γ (1 ψ) µ µ + σ2 1 γ 2 1 ψ +1 Hy H µ Hy H 2 y t σ2 2 2 y t H y H κ(θ y t) H yy H y t. (28) Unfortunately, equation (28) is a non-linear ODE in H whose analytical solution is unknown except in three special cases: log utility (γ = ψ 1); constant investment opportunities (κ, σ =0); and power utility, provided that the state variable is perfectly instantaneously correlated with the risky asset return so that markets are complete. The first two cases are well-known from Merton s (1969, 1971, 1973) work. The third case has been explored by Wachter (1998) for a model whose state variable is the expected excess return. Wachter s result also holds in our model and, in general, in any model with isoelastic preferences and perfect correlation between the state variable and the risky asset return. 8 With ρ =1, equation (28) becomes a nonhomogeneous version of the Gauss hypergeometric ODE, which has a closed-form solution in terms of the confluent hypergeometric function (see Polyanin and Zaitsev 1995, p.143). 9 Unfortunately, the assumption of perfect correlation between changes in volatility and asset returns is not empirically plausible. For example, in Section 6 we estimate that for the US market this correlation is large, but still far from perfect. suggests that we should consider the general case. This In the general case, the nonlinear ODE (28) has no exact analytical solution. Nevertheless, it is still possible to find an approximate analytic solution based on a 8 In concurrent work with this paper, Liu (1998) shows a similar result. 9 It is worth noting here that this solution has a rather abstruse mathematical form, from which it is very dificult to obtain any useful economic insights. 22

25 log-linear expansion of the consumption-wealth ratio around its unconditional mean. Campbell (1993), Campbell and Viceira (1999, 2001), and Campbell, Chan, and Viceira (2002) have used an identical approximation to solve for optimal intertemporal portfolio and consumption problems. However, while they work in discrete-time and use the approximation to linearize the log budget constraint, we work here in continuous-time and use it to linearize the Bellman equation. We can view this approach as a particular class of the perturbation methods of approximation described in Judd (1998), where the approximation takes place around a particular point in the state space the unconditional mean of the log consumption-wealth ratio. We start by noting that the envelope condition (10) implies β ψ H 1 =exp{c t x t }, where c t x t =log(c t /X t ). Therefore, using a first-order Taylor expansion of exp{c t x t } around E[c t x t ] (c x) we can write where h 1 =exp{c x}, andh 0 = h 1 (1 log h 1 ). β ψ H 1 h 0 + h 1 (c t x t ), (29) Substituting (29) for β ψ H 1 in the first term of (28), it is easy to see that the resulting ODE has a solution of the form H =exp{a 1 y t +B 1 }. This solution implies a value function of the form J (X t,y t )=exp ½ µ ¾ 1 γ X 1 γ t (A 1 y t + B 1 ) 1 ψ 1 γ. (30) The approximate ODE leads to two algebraic equations for A 1 and B 1 : a 1 A b 1 A 1 + c 1 = 0, (31) h 0 h 1 [B 1 ψ log β] ψβ r(1 ψ)+κθa 1 = 0, (32) 23

26 where a 1 = µ σ2 1 γ γ 1 ρ 2 + ρ 2, 2γ 1 ψ (33) b 1 = (1 γ)ρσ(µ r) (h 1 + κ), γ (34) c 1 = (1 ψ)(µ r)2. 2γ (35) The analysis of the quadratic equation (31) for A 1 is parallel to the analysis of the quadratic equation (16) for A in the ψ =1case, so that we simply state here the properties of A 1 derived from this analysis. First, A 1 /(1 ψ) is independent of ψ given h 1. Second, comparison of equations (33)-(35) with equations (18)-(20) shows that A 1 /(1 ψ) and A/(1 γ) are non-negative identical functions of h 1 and β, respectively. 10 Third, when γ > 1, the discriminant of equation (31) is always positive and the roots of the equation are real and have opposite sign; when γ < 1, the discriminant can have either sign but, if it is positive, the roots of the equation are real and have the same sign. However, only the positive square root of the discriminant ensures in both cases that the approximate solution approaches the exact solution when ψ = Optimal policies We now state the approximate solution in the following proposition: 10 This equivalence is also apparent from comparing the value function (22) in the ψ =1case and the value function (30) in the general case. 24

27 Proposition 2 When ψ 6= 1, there is an approximate analytical solution to problem (5)-(8) with value function given by (30). The optimal consumption and portfolio rules implied by this value function are C t X t = β ψ exp { A 1 y t B 1 }, (36) and π t = 1 γ (µ r) y t + µ 1 1 ( ρ) σa 1 y t, (37) γ where A 1 A 1 /(1 ψ) > 0, anda 1 and B 1 are given by the solution to the system of equations (31)-(32). A 1 does not depend on ψ except through the loglinearization coefficient h 1,anditreducestoA in Proposition 1 when h 1 = β. Proof. The value function follows immediately from (28), (29) and (31)-(32). The optimal policies follow from (30) and the first order conditions (11) and (26). We have already shown above that A 1 A 1 /(1 ψ) and A A/(1 γ) are non-negative identical functions of h 1 and β, respectively. The approximate solution depends on the loglinearization coefficient h 1,whichis itself endogenous, since h 1 exp {E[log(C t /X t )]}. However, Proposition 2 shows that we can still derive a number of properties of the solution without solving explicitly for h 1,usingthefactthatitliesbetweenzeroandone. Wenowcommentonsome of these properties, and leave for Section 6 the description of a simple procedure to compute numerical values for h 1 and the optimal policies. Proposition 2 shows that the optimal log consumption-wealth ratio is a linear function of instantaneous precision. Since A 1 /(1 ψ) < 0, the consumption-wealth ratio is a decreasing monotonic function of volatility for investors whose intertemporal 25

28 elasticity of consumption ψ is smaller than one, while it is an increasing function of volatility for investors whose elasticity is larger than one. This property reflects the comparative importance of intertemporal income and substitution effects of volatility on consumption. To understand this, consider the effect on consumption of an unexpected increase in volatility. This increase implies a deterioration in investment opportunities, because returns on the risky asset are now more volatile, while its expected return is the same. A deterioration in investment opportunities creates a positive intertemporal substitution effect on consumption because the investment opportunities available are not as good as they are at other times but also a negative income effect because increased uncertainty increases the marginal utility of consumption. For investors with ψ < 1, theincomeeffect dominates the substitution effect and they reduce their current consumption relative to wealth. For investors with ψ > 1, the substitution effect dominates, and they increase their current consumption relative to wealth. Proposition 2 also characterizes optimal portfolio demand in the general case. This proposition implies that optimal portfolio demand in the ψ 6= 1case is qualitatively analogous to optimal portfolio demand in the ψ =1case. This follows immediately from direct comparison of equations (37) and (24). These equations are identical, except for the positive coefficients A 1 and A. Section 6 shows that, for empirically plausible characterizations of the process for precision, these coefficients are very close, which implies that the effect of ψ on optimal portfolio choice is quantitatively small. Campbell and Viceira (1999, 2001) and Campbell, Chan, and Viceira (2002) show a similar result in models with time variation in risk premia and interest rates. 26

29 The comparative statics results for intertemporal hedging demand shown in Section 3 for the ψ =1case also extend to the general case, provided that we hold h 1 i.e., the mean log consumption-wealth ratio constant. This follows from the fact that coefficients A 1 and A are identical positive functions of h 1 and β, andboth h 1 > 0 and β > 0. However, h 1 is itself a highly non-linear function of σ, κ, andρ, the parameters of interest. Therefore, to fully evaluate the effect on intertemporal hedging demand of changes in these parameters we need to allow h 1 to change. This is very difficult to do analytically, but it is very easy to do numerically using the analytical expressions for the optimal policies. We conduct these comparative statics exercises in Section 6. Finally, we want to note that an important feature of the approximate solution is that it delivers the exact expression for the optimal policies in the special cases of log utility (γ = ψ 1), unit elasticity of intertemporal substitution, and constant investment opportunities (κ, σ =0and v t v). Appendix C shows this convergence result. 5 Consumption and Portfolio Choice When Expected Excess Returns Covary with Volatility The analysis of optimal consumption and portfolio choice with time-varying risk in Sections 3 and 4 assumes that expected excess returns are constant. A natural extension of this analysis is to replace the assumption of constant expected excess returns on the risky asset with another that allows for the expected excess return on the risky 27

30 asset to change linearly with volatility or the reciprocal of precision: dst Et rdt = α 1 + α 2 v t = α 1 + α 2. (38) y t S t When α 2 > 0, this functional form implies that an increase in volatility increases both risk and the expected excess return on the risky asset. When α 1 =0,thismodel reduces to the model in Sections 3 and 4. To derive the optimal policies under this new assumption we follow the same method as in Section 4. We describe here the main steps of the derivation, and provide full details in Appendix D. Guessing the same functional forms for J(X t,y t ) and I(y t ) as in Section 4, the Bellman equation for this problem simplifies to an ODE in H(y t ). This ODE has a closed form solution, provided that we make the approximation β ψ H 1 h 0 + h 1 (c t x t ). 11 The solution takes the form H =exp{a 1 y t + A 2 log y t + B 2 }, which implies a value function of the form J (X t,y t )=exp ½ µ ¾ 1 γ X 1 γ t (A 1 y t + A 2 log y t + B 2 ) 1 ψ 1 γ, where A 1 and A 2 solve two independent quadratic equations and B 2 solves an equation which is linear, given A 1 and A 2.The quadratic equation for A 1 depends on α 1, but it does not depend on α 2 except through h 1.Thisequationisidenticaltoequation (31) in Section 4, except that α 1 replaces (µ r). Thus the analysis of equation (31) in Section 4 is also valid here. This analysis implies that A 1 /(1 ψ) < 0. Thequadratic equation for A 2 depends on α 2, but does not depend on α 1 except through h 1.The analysis of this equation also implies that A 2 /(1 ψ) > When we substitute h 0 + h 1 (c t x t ) for β ψ Ht 1 in the Bellman equation, we still need to do a further approximation of log y t =logv t around its conditional mean. 28

31 The approximation implies the following optimal policies: Proposition 3 The optimal consumption and portfolio rules when Et[(dS t /S t ) rdt] =α 1 + α 2 /y t are C t X t = β ψ exp { A 1 y t A 2 log y t B 2 }, (39) and π t = 1 γ (α 1y t + α 2 )+ µ 1 1 ( ρ) σ (A 1 y t + A 2 ), (40) γ where A 1 A 1 /(1 ψ) > 0, anda 2 A 2 /(1 ψ) < 0. BothA 1 and A 2 do not depend on ψ except through the loglinearization constant h 1. Proof. See appendix D. Proposition 3 shows that both the myopic component and the intertemporal hedging component of portfolio demand are affine functions of precision. Since A 2 depends on α 2,butnotonα 1, the intercept of the intertemporal hedging component captures the effect of compensated changes in volatility on intertemporal hedging. On the other hand, since A 1 depends on α 1, but not on α 2, the slope captures the effect of uncompensated changes in volatility. These effects always have opposite signs, offsetting partially or totally each other. In particular, Appendix D shows that a negative instantaneous correlation between excess returns and volatility implies that the sign of the intercept of the intertemporal hedging component is always the same as the sign of the difference (γ 1), regardless of the sign of α 2, and that the sign of the slope is always opposite to the sign of this difference. 29

32 The special case α 1 =0and r f =0is helpful to understand why the sign of the intercept of the hedging component is independent of the sign of α 2.Whenα 1 =0, both the myopic component and the intertemporal hedging component of optimal portfolio demand are constant over time, 12 and the sign of α 2 determines the sign of the optimal holdings of the risky asset. They are positive when α 2 > 0, and negative when α 2 < 0. Since the Sharpe ratio for the risky asset is α 2 vt, it follows that the Sharpe ratio of the investor s portfolio is increasing in volatility, irrespective of the sign of α Therefore, the investor always sees a negative shock to volatility as a worsening in investment opportunities. But a negative correlation between shocks to volatility and realized excess returns on the risky asset also implies that negative shocks to volatility tend to occur simultaneously with positive realized excess returns on the risky asset. An investor with γ > 1 will have a positive intertemporal hedging demand for the risky asset, because it tends to pay when investment opportunities worsen and the marginal utility of consumption is high. By contrast, an investor with γ < 1 is less concerned about insuring against bad states of the world, and prefers assets that tend to pay when investment opportunities are good. This investor will have a negative hedging demand for the risky asset. This explains why the sign of hedging demand is independent of the sign of α Appendix D shows that α 1 =0implies that A 1 =0, while A 2 is not necessarily zero. 13 The Sharpe ratio of the investor s portfolio is equal to πα 2 v t / π 2 v t =sign(π)α 2 vt = α 2 v t, since π has the same sign as α 2. 30