The Role Of Risk Aversion And Intertemporal Substitution .

Dynamic Consumption and Portfolio Choice with Recursive Utility7characteristics of the risky and riskless assets. Thus, even in the case of constant investment opportunity set, the consumption policy depends on both risk aversion and the elasticity of intertemporalsubstitution.On the other hand, the optimal portfolio choice is independent of the investor’s intertemporalelasticity of substitution; the proportion of wealth the investor chooses to invest in the risky assetdepends only on the characteristics of the risky and riskless assets and her relative risk aversion.1This confirms the result in Svensson (1989).3.3Stochastic investment opportunity setWe now examine the effects of a stochastic investment opportunity set on optimal portfolio andconsumption decisions. We do this by extending to the case of recursive utility an example considered in Ingersoll (1987) for the case of expected utility. This is a simple three-date model withconsumption and portfolio decisions at t {0, 1} and bequest at t 2. Moreover, it is assumedthat at t 1 only a riskless asset is available while at t 0 there is also a risky asset that theinvestor can hold.In order to understand the results that follow, we start by considering the effect of a changein the investment opportunity set, through a change in the riskless interest rate, on consumption.This effect can be decomposed into “income” and “substitution” effects.Proposition 3 The substitution effect is given byρ1" 1 C1 ""a21 W1 β 1 ρ R 1 ρ R "J1 ρ(21)where1a1 [1 (βRρ ) 1 ρ ] 1 ,(22)and the income effect by(W1 C1 ) R 1 C1 W1 (1 a1 ) R 1 a1 . W1(23)1Note that in Proposition 2, by setting ρ α, we obtain the well known results for power utility and by takingappropriate limits, we obtain the results for logarithmic utility.

Dynamic Consumption and Portfolio Choice with Recursive Utility8The sum of these two effects is given by1dCρ a21 W R 1 (βRρ ) 1 ρ.dR1 ρ(24)Note that both the substitution and income effects of a change in the riskfree rate, R, onconsumption are independent of the relative risk aversion of the investor. The substitution effect isalways negative, whereas the income effect is always positive. This is intuitively clear: as the riskfreerate increases, future consumption becomes cheaper relative to current consumption; hence currentconsumption decreases (substitution effect). However, the increase in the riskfree rate increasesoverall wealth, which leads to an increase in current consumption (income effect). When ρ 0(elasticity of intertemporal substitution is greater than unity), the substitution effect is dominantand a rise in the riskfree rate leads to an decrease in current consumption. For the case whereρ 0 (elasticity of intertemporal substitution is less than unity), the income effect is dominantand as the riskfree rate rises, current consumption rises. When ρ 0, the income effect exactlyoffsets the substitution effect, which is a well-known result for the case of logarithmic preferences(ρ α 0). These results are similar to those in Weil (1990).We now restrict the model further in order to investigate how future changes in the investmentopportunity set affect demand for the risky asset. We deliberately define the model such thatdemand for the risky asset would be zero if the investment opportunity set were constant. Hence,the only demand for the risky asset arises from the desire for intertemporal hedging.Suppose that at t 0 the interest rate which will hold at t 1 is unknown. We assume thereare two equally probable realizations RD and RU with RD RU corresponding to two states Dand U , respectively. At t 0 the riskless asset offers a return R and at time t 1 the risky assetgives a return of 2R if the riskfree rate is RD and a return 0 if the riskfree rate is RU . The expectedreturn on the risky asset is the same as the riskless asset. Therefore, no single-period investorwould invest in any of the risky asset. However, multiperiod investors may hold the risky asset inorder to hedge against changes in the investment opportunity set.

9Dynamic Consumption and Portfolio Choice with Recursive UtilityProposition 4 The intertemporal hedging demand for the risky asset at time 0 is given by w1 (0) ρ 1 α(ρ 1) ρ 1 α(ρ 1)ρ(α 1)ρ(α 1)1 ρ1 ρ 1 βRU1 βRD1 ρ 1 ρβRD1 α(ρ 1)ρ(α 1) 1 ρ 1 ρβRU1 α(ρ 1)ρ(α 1)(25)Observe that the optimal portfolio weight depends on both the parameter controlling relativerisk aversion, α, and the parameter driving elasticity of intertemporal substitution, ρ. The intuition for why the portfolio weight depends on not just the risk aversion parameter but also onthe intertemporal substitution parameter is the following. Because the investment opportunityset is stochastic, the consumption-wealth ratio at time t 1, at 1 , and hence consumption, isalso stochastic; moreover, the variation in consumption depends on the elasticity of intertemporalsubstitution. From Cox and Huang (1989) we know that the role of the optimal portfolio is tofinance the optimal consumption desired by the investor. Given that the variation in consumptiondepends on intertemporal substitution, so must the optimal portfolio. On the other hand, whenthe investment opportunity set is non-stochastic, the consumption-wealth ratio, at 1 , and thereforeconsumption, are constant and so in this case the optimal portfolio also does not contain a hedgingterm that would depend on the parameter for intertemporal substitution.Proposition 5 The hedging demand for the risky asset at t 0 is strictly negative if and onlyif relative risk aversion is strictly less than unity and strictly positive if and only if relative riskaversion is strictly greater than unity. Hedging demand is zero if and only if relative risk aversionis unity.Thus, the above proposition shows that the sign of hedging demand for the risky asset dependsonly on the investor’s relative risk aversion. When relative risk aversion is unity (α 0), then thehedging demand for the risky asset is zero. This is a generalization of the result that logarithmicinvestors, for whom both relative risk aversion and the elasticity of intertemporal substitution areunity, have zero hedging demand.To understand why the sign of the hedging demand depends on the investor’s relative riskaversion, note that the first order condition for optimal portfolio choice isα Jt 1 (Wt 1 , t 1)Et(zi (t) R) 0. Wt 1(26)

10Dynamic Consumption and Portfolio Choice with Recursive UtilityEquation (26) shows that the optimal portfolio maximizes the conditional expected marginal utilityweighted by the excess return on the risky asset. The expression for the value function in equation(16) allows one to rewrite equation (26) as done in equation (15), which we reproduce below:α(ρ 1)ρα 1Et at 1 Zt (zi (t) R) 0.Writing the above expectation explicitly after substituting in the returns on the assets held in theportfolio, we obtain:[a1 (D)]Note that [a1 (D)]α(ρ 1)ρα(ρ 1)ρ[1 w1 (0)]α 1 [a1 (U )] [a1 (U )]α(ρ 1)ρα(ρ 1)ρ[1 w1 (0)]α 1if and only if RRA 1, while [a1 (D)]α(ρ 1)ρ(27) [a1 (U )]α(ρ 1)ρif and only if RRA 1, and the two terms are equal if and only if RRA 1. Hence, for the casewhere relative risk aversion is strictly less than unity, if w1 (0) 0 then the marginal utility ofwealth in the up (U ) state is lower than in the down (D) state. To hedge against this, the agentwill short the stock in order to increase marginal utility in the up state. On the other hand, forthe case where relative risk aversion is strictly greater than unity, if w1 (0) 0 then the marginalutility of wealth in the down state is lower than in the up state, and to hedge against this the agentwill go long the stock so that the marginal utility in the down state is increased.4ConclusionWe study the role of risk aversion and intertemporal substitution in the optimal consumptionand portfolio problem of an individual investor when the investment opportunity set is stochastic.Using a simple example that admits a solution in closed form we show that the consumption andportfolio decisions depend on both risk aversion and the elasticity of intertemporal substitution.Only for the special case where the investment opportunity set is constant is the optimal portfolioweight independent of the elasticity of intertemporal substitution, though even in this case theconsumption decision depends on both risk aversion and elasticity of intertemporal substitution.

11Dynamic Consumption and Portfolio Choice with Recursive UtilityProofsProof of Proposition 1We have the Bellman equation ρ/α 1/ραJ (Wt , t) sup (1 β) Ctρ β Et Jt 1(Wt 1 , t 1)Ct ,wtand the intertemporal budget constraint nwi (t) (zi (t) R) RW (t 1) I (t)i 1 I (t) Z (t) ,where Z (t) ni 1wi (t) (zi (t) R) R. Hence we can derive the first order conditions for optimality.The first order condition for consumption is given by ρ/α 1/ρ α(1 β) Ctρ β Et Jt 1(It Zt , t 1) 0. CtHence optimal consumption is given by#Ct 1 ρ 1 ρ/α 1β Jt 1 (Wt 1 , t 1)α 1αEt Jt 1 (It Zt , t 1)Et Jt 1 (Wt 1 , t 1)Zt.1 β Wt 1(28)The first order condition for the optimal portfolio is given by ρ/α 1/ρ α(1 β) Ctρ β Et Jt 1(It Zt , t 1) 0. witThis can be rewritten as Jt 1 (Wt 1 , t 1)α 1Et Jt 1 (Wt 1 , t 1)(zi (t) R) 0. Wt 1Now we consider the expression ρ/α 1/ραJ (Wt , t) (1 β) Ctρ β Et Jt 1(Wt 1 , t 1),whereWt 1 It Zt(29)

12Dynamic Consumption and Portfolio Choice with Recursive UtilityandIt Wt Ct ,Zt nwi (t) (zi (t) R) R.i 1We calculate the derivative J (Wt , t) Wt 1/ρ 1 1 ρ/α[(1 β) Ctρ(1 β) Ctρ β (Et J α (Wt 1 , t 1))ρ Wtρ/α β (Et J α (Wt 1 , t 1))].The above expression can be simplified to obtain J (Wt , t) Wt J 1 ρ (Wt , t) [(1 β) Ctρ 1 Ct Wtρ/α 1 β (Et J α (Wt 1 , t 1))Et J α 1 (Wt 1 , t 1) J (Wt 1 , t 1) Wt 1]. Wt 1 WtNow Wt 1 Wt (It Zt ) Wt It Zt Zt Wtn Iti 1 It It Ct Wt Ct Wt Zt wit Zt wit Wtn(zi (t) R) Iti 1 It It Ct Wt Ct Wt Ct wit Zt 1 Wt WtTherefore, using the first order conditions we can show that J (Wt , t) J (Wt 1 , t 1)ρ/α 1 βJ 1 ρ (Wt , t) [Et J α (Wt 1 , t 1)]Et J α 1 (Wt 1 , t 1)Zt . Wt Wt 1Recall that optimal consumption is given by equation (28). Hence,Ct 11ρ 11 J (Wt , t) ρ 11 J (Wt , t) ρ 1J(Wt , t) J (Wt , t) .1 β Wt1 β WtNow we can simplify the expression ρ/α 1/ραJ (Wt , t) sup (1 β) Ctρ β Et Jt 1(Wt 1 , t 1)Ct ,wt(30)

13Dynamic Consumption and Portfolio Choice with Recursive Utilityto obtain an equation for the value function, 1 J (Wt , t)J (Wt , t) (1 β)1 β Wt 1/ρρρ 1ραJ (Wt , t) β [Et J (Wt 1 , t 1)]We seek a trial solution of the formJ (Wt , t) Aht Wt ,where1A (1 β) ρhT 1to ensure that1J (WT , T ) (1 β) ρ WT B (WT , T ) .Therefore, J (Wt , t) Aht . WtHence, from equation (30)ρ1Ct (1 β) 1 ρ (Aht ) 1 ρ WtWe now simplify the last term in (31) ρ/ααβ [Et J α (Wt 1 , t 1)]ρ/α β Aα hαt 1 Wt 1αWt 1α(Wt Ct ) Ztα α ρ1 1 (1 β) 1 ρ (Aht ) ρ 1 Wtα Ztα Defineat ρ/αρ1(1 β) 1 ρ (Aht ) ρ 1ρ htρ 1Then(1 β) Ctρ (1 β) aρt Wtρandρ 11Aht at ρ (1 β) ρ .(31)

Dynamic Consumption and Portfolio Choice with Recursive Utility14Thus, we can write the value function as1ρ 1J (Wt , t) (1 β) ρ at ρ Wt .Hence ρ/ααβ Et Jt 1(Wt 1 , t 1) αραα β Et (Aht 1 ) Wt 1 αρ ααρ (ρ 1)α β Et at 1(1 β) ρ Wt 1 αρ αρ (ρ 1)α β (1 β) Et at 1Wt 1.Note that α ρ1αWt 1 1 (1 β) 1 ρ (Aht ) 1 ρ Wtα Zt α .Therefore αρα ρ/αρρ (ρ 1)αβ Et Jt 1(Wt 1 , t 1) β (1 β) (1 at ) Et at 1Zt α Wtρ .Then, equation (31) implies that(1 β)11 ρρ1 ρatρ1 ραρρρρρ (ρ 1) α αWt (1 β) at Wt β (1 β) (1 at ) Et at 1 ZtWt.We can simplify this expression to obtain1 & 1! ρ 1 ρα α ρ (ρ 1) α1 β Et (Zt ) at 1.%at We know that the first order condition for the optimal portfolio condition is Jt 1 (Wt 1 , t 1)α 1Et Jt 1 (Wt 1 , t 1)(zi (t) R) 0. Wt 1(32)This can now be simplified to obtain Et Ztα 1 hαt 1 (zi (t) R) 0.Therefore the first order condition for the optimal portfolio isα(ρ 1)Et Ztα 1 at 1ρ (zi (t) R) 0.

15Dynamic Consumption and Portfolio Choice with Recursive UtilityProof of Proposition 2Given that we assume the opportunity set is constant, at 1 is known at time t. Hence1ρ ! 1 ραa 1 1 βυa 1tt 1 ,(33)whereαυ Et (Zt ) .Note that because the opportunity set is constant, υ is independent of time. We can solve the differenceequation (33) to obtainρat 1 βυ α ρ (T t 1)/(1 ρ) ,1 βυ αwhere we have used the terminal conditionaT 1.We now only have one risky asset with two equally likely payoffs. Therefore we can simplify the optimalportfolio condition to obtain Et Ztα 1 (z1 (t) R) 0 α 1Et (w0 (t) R w1 (t) z1 (t))(z1 (t) R) 0 α 1Et ((1 w1 (t)) R w1 (t) z1 (t))(z1 (t) R) 0 α 1Et (R w1 (t) [z1 (t) R])(z1 (t) R) 0 1 α 1α 1[R w1 (t) (h R)](h R) [R w1 (t) (k R)](k R)2α 1[R w1 (t) (h R)]α 1(h R) [R w1 (t) (k R)]Hence, the optimal portfolio is given byw1 (t) R b (h R)1 b (h R)(R k)(R k) b1 bwhereb 1.1 αWe can also simplify the expression forαυ Et (Zt ) ,,(k R) 0 0

16Dynamic Consumption and Portfolio Choice with Recursive UtilityHenceαυ Et (w0 (t) R w1 (t) z1 (t))α Et [R w1 (t) (z1 (t) R)]1αα{[R w1 (t) (h R)] [R w1 (t) (k R)] }2 Simplifying the first term in the above expression yieldsR w1 (t) (h R) R R (h R)1 b R1 b (h R)(R k) b1 b b1 b (h R) (R k)1 b(R k)1 b (h R)1 b (h R) b (h R) (R k)(R k)1 b(R k) b R (h R) (h R)(R k)1 b R b(R k)1 b (h R)(R k)(h k)1 b (h R).1 b(R k)Similarly,R w1 (t) (k R) R R (k R)1 b R1 b (h R)(R k) b1 b1 b (k R) (h R)1 b1 b (h R) b (k R) (h R)(h R)1 b(R k)1 b (h R)(h R)(h k)1 b (h R)(R k) b1 b (R k)(R k) b R (h R) (h R)(R k)1 b R b(R k)1 b.Hence,υ 1 α1 αα1 b1 b (R k).R (h k) (h R)2

17Dynamic Consumption and Portfolio Choice with Recursive UtilityProof of Proposition 3The standard two-good cross-substitution Slutsky equation, which measures the change in the quantity ofthe i’th good, Qi , with respect to a change in the price of the j’th good, Pj , is""dQi Qi "" Qi "" Qj.dPj Pj "U W "Pi ,PjThe first term is the substitution effect and the second is the income effect. At t 1, Q1 C1 , Q2 W2 ,P1 1, and P2 R 1 . Substituting into the above expression and using dP2 /dR R 2 we obtaindC1dR"" C1 ""dP2 C1 "" RdP2 dC1 W2dR dP2dR R "J P2 W1 "R"" C1 "" 2 C1 "" 2 R R W2 R "J W1 "R""" C1 "" 2 C1 " W2 R R "J W1 "R""" C1 "" 1 C1 " (W C)R.11" R J W1 "R We know that1ρ 11ρ 1J (W1 , 1) (1 β) ρ a1 ρ W1 (1 β) ρ a1 ρ 1Hence1 1 J (1 β) ρ a1 ρ C1and1 1 1 a1 J1 (1 β) ρ a1 ρC1 . Rρ RFrom the Implicit Function TheoremWe know that" C1 "" J/ R ." R J J/ C11 & 1! ρ 1 ραα ρ (ρ 1) α1 β Et (Zt ) at 1.%at Now w1 (1) 0. Therefore Z1 R. Hencea 11 1 1C1 (1 β) ρ a1 ρ C1 .1! ρ 1 ραα ρ (ρ 1) α1 β E1 (R) a2

Dynamic Consumption and Portfolio Choice with Recursive Utility 1 ρ 1 ρα1 β (E1 (R) ) αbecause a2 11 1 (βRρ ) 1 ρWe calculate1 ρ 1 a1 R C11 1β) ρ a1 ρ1 J/ R J/ C1 ρ1 (1 β) ρ a1(1 1 C1 a1ρ a1 RC1 log a 11.ρ RFrom1ρ 1 ρa 1,1 1 (βR )we obtain log a 11 R 1 (βRρ ) 1 ρ1 1 R1 (βRρ ) 1 ρ1 ρρβ 1 ρ1 ρ 11 R1 ρ 1 (βRρ ) 1 ρ ρβ 1 ρ a1 R 1 ρ 1.1 ρρ1Henceρ J/ Ra1 C1 β 1 ρ R 1 ρ 1. J/ C11 ρ1Therefore we obtain the substitution effect"ρ1 C1 ""a21 W1 β 1 ρ R 1 ρ 1 , R "J1 ρwhere1ρ 1 ρa 1.1 1 (βR )The income effect is(W1 C1 ) R 1 C1 W1 (1 a1 ) R 1 a1 . W118

19Dynamic Consumption and Portfolio Choice with Recursive UtilityHence, the net effect isρa2 W1 β 1 ρ R 1 ρ 1 1 W1 (1 a1 ) R 1 a11 ρ ρ11β 1 ρ R 1 ρ2 1ρ 1 ρ (βR ) a1 W R1 ρ1dCdR1 a21 W R 1 (βRρ ) 1 ρ1 a21 W R 1 (βRρ ) 1 ρ 1 11 ρ ρ.1 ρNote that at time 1 , the investor’s value function is given by 1 ρ11ρJ (W1 , 1) (1 β) ρ 1 (βRρ ) 1 ρW1 . Proof of Proposition 4The first order condition for the optimal portfolio isα(ρ 1)Et Ztα 1 at 1ρ (zi (t) R) 0We know that 11 1 (βRρ ) 1 ρwith probability 1/2D 1a1 1ρ 1 ρ 1 (βRU)with probability 1/2Define a1 (D) a1 (U ) We now simplify the expression1 11 1ρ 1 ρ1 (βRD)ρ 1 ρ1 (βRU).α(ρ 1)ρα 1Et Zt at 1 (zi (t) R) 0α(ρ 1)ρα 1E0 a1Z0 (z1 (0) R) 0

20Dynamic Consumption and Portfolio Choice with Recursive Utility12# α(ρ 1) α(ρ 1)α 1α 1ρρa1(D) [(1 w1 (0)) R w1 (0) 2R](2R R) a1(U ) [(1 w1 (0)) R]( R)α(ρ 1)ρa1α 1(D) [(1 w1 (0)) R w1 (0) 2R]α(ρ 1)ρa1α(ρ 1)ρ(2R R) a1α 1(D) [(1 w1 (0))]α(ρ 1)α(ρ 1)a1ρ(α 1)α 1(U ) [(1 w1 (0)) R]α(ρ 1)ρ a1 0( R) 0α 1 0(U ) [(1 w1 (0)) R]α(ρ 1)a1ρ(α 1) (D) (1 w1 (0)) a1ρ(α 1) (U ) (1 w1 (0))α(ρ 1)α(ρ 1)α(ρ 1)(D) a1ρ(α 1) (U ) w1 (0) a1ρ(α 1) (U ) a1ρ(α 1) (D)Therefore w1 (0) 1 α(ρ 1)ρ(α 1)1 α(ρ 1)ρ(α 1)ρ 1 ρ1 (βRD)ρ 1 ρ1 (βRD) α(ρ 1) 1ρ(α 1)ρ 1 ρ 1 (βRU). α(ρ 1) 1ρ(α 1)ρ 1 ρ 1 (βRU ) Proof of Proposition 5The hedging demand for the risky asset is given by 1 w1 (0) 1ρ 1 ρ(βRD)1ρ 1 ρ1 (βRD) α(ρ 1)ρ(α 1) α(ρ 1)ρ(α 1) 1ρ 1 ρ(βRU) 1 α(ρ 1) 1ρ(α 1)ρ 1 ρ 1 (βRU)Note that this expression is strictly negative iffρ1 ρ 0 andα (ρ 1) 0 orρ (α 1)ρ1 ρ 0 andα (ρ 1) 0ρ (α 1)ρ1 ρ 0 andα (ρ 1) 0 ρ (α 1)ρ1 ρ 0 andα 0 1 αNote further that0 ρ, α 1 α(ρ 1)ρ(α 1)

Dynamic Consumption and Portfolio Choice with Recursive Utility21andρ1 ρ 0 andα (ρ 1) 0 ρ (α 1)ρ1 ρ 0 andα 01 αρ 0 or ρ 1 and 0 α 1Hence it is clear the hedging demand for the risky asset is strictly negative iff 0 α 1 and strictly positiveiff α 0. The case α 1 is excluded by our assumption in the definition of the utility function that α 1.This gives us Proposition 5.

Dynamic Consumption and Portfolio Choice with Recursive Utility22ReferencesCampbell, J. Y. and L. Viceira, 2002, Strategic Asset Allocation: Portfolio Choice for Long-TermInvestors, Oxford University Press.Chacko, G. and L. Viceira, 1999, Dynamic Consumption and Portfolio Choice with StochasticVolatility in Incomplete Markets, Working paper, Harvard University.Cox, J. and C. Huang, 1989, Optimum Consumption and Portfolio Policies When Asset PricesFollow a Diffusion Process, Journal of Economic Theory 49, 33-83.Dumas, B. and R. Uppal, 2001, Global Diversification, Growth, and Welfare with ImperfectlyIntegrated Markets for Goods, Review of Financial Studies 14, 227-305.Epstein. L. G. and S. E. Zin, 1989, Substitution, Risk Aversion, and the Temporal Behavior ofConsumption and Asset Returns: A Theoretical Framework, Econometrica 57, 937-969.Gio

The Role of Risk Aversion and Intertemporal Substitution in . the continuous time consumption-portfolio problem in terms of a backward stochastic differential . Let there be a riskless asset with return R, and denote the time-t price of the riskless security by P 0 (t).