Robust Portfolio Choice with External Habit Formation and Equilibrium Asset Prices

Transcription

1 Robust Portfolio Choice with External Habit Formation and Equilibrium Asset Prices Tong Suk Kim, Hyo Seob Lee This version: 1 Nov, Graduate School of Finance, KAIST, 87 Hoegiro, Dongdaemoon-gu, Seoul, , Republic of Korea, tskim@business.kaist.ac.kr Corresponding author, Graduate School of Management, KAIST, 87 Hoegiro, Dongdaemoon-gu, Seoul, , Republic of Korea,tosun@business.kaist.ac.kr

2 Robust Portfolio Choice with External Habit Formation and Equilibrium Asset Prices Abstract This paper examines optimal consumption and portfolio choice for the agent concerned about a worst-case scenario with respect to external habit formation. A robust investor more decreases stock investment as the volatility of consumption surplus increases more than an agent without model uncertainty. We theoretically derive the countercyclical uncertainty aversion. In Lucas equilibrium, the robust agent faces a higher market price of risk and a lower risk-free rate than the agent without model uncertainty. This paper provides more plausible parameter choices to explain both the equity premium puzzle and the low risk-free rate puzzle. JEL classification: G11; G12; C61 Keywords: Model uncertainty; external habit formation; equilibrium asset price; countercyclical uncertainty aversion;

3 1 Introduction We examine the optimal consumption and portfolio strategy with external habit formation in the model uncertainty framework. The representative agent optimally invests in the production market and decides her optimal portfolio for the risky asset. We derive the countercyclical uncertainty averse preference, which is disentangled from a risk-averse preference. Dynamic portfolio theory stems from Merton (1969, 1971), who assumes a constant investment opportunity set. Kim and Omberg (1996) and Liu (2007) solve for a dynamic optimal portfolio when the investment opportunity set is stochastic. Detemple and Karatzas (2003) and Detemple and Zapatero (1992) investigate portfolio choice with habit formation, assuming a constant investment opportunity set. Munk (2008) obtains optimal portfolio choice with habit formation and a stochastic opportunity set. Munk (2008) does not assume external habit formation and does not consider model uncertainty. External habit formation of Campbell and Cochrane (1999) is different from internal habit formation in that the individual s level of habit depends on aggregate past consumption not on the individual s past consumption. Specifically, Detemple and Karatzas (2003), Detemple and Zapatero (1992) and Munk (2008) assume that the individual s habit level follows a deterministic process. Unlike theses papers, we assume that the log consumption surplus, which plays the role of a state variable, follows a mean reverting Ornstein-Uhlenbeck process. This paper allows the agent to have both a reference model and an alternative model. Our agent worries about a pessimistic scenario caused by both distorted wealth dynamics and a state variable, that incorporates external habit formation to obtain time varying preferences. Using robust control theory, which is developed by Anderson, Hansen, Sargent (2003), we measure the discrepancy between the reference model and the worse case alternative model. Maenhout (2004, 2006) studies the dynamic portfolio and consumption rules in the framework of Anderson, Hansen, Sargent (2003). Assuming uncertainty about the return process of wealth dynamics, he finds a closed-form solution for the optimal portfolio strategy and estimates constant uncertainty aversion. Maenhout (2004) shows that robustness decreases 1

4 the demand for risky assets and obtains equilibrium ex-dividend stock price and the risk free rate. Maenhout (2006) solves the portfolio problem for the robust agent, who faces a mean reverting risk premium. These papers do not investigate the time varying uncertainty aversion caused by a state variable. 1. In external habit formation, we derive the state dependent model uncertainty aversion. Habit modeling has been widely used for explaining economic puzzles. 2 Campbell and Cochrane (1999) assume that individual s habit is determined by the history of aggregate consumption. They explain several asset pricing issues, such as the equity premium puzzle, the low risk free rate puzzle, procyclical variation of stock prices, countercyclical variation of stock price s volatility, and long-run predictability of excess stock returns. Li (2007) uses the continuous framework of Campbell and Cochrane (1999) and sets forth the possibility that countercyclical variation in risk aversion yields a procyclical risk premium. The present paper is also based on the continuous time framework of Campbell and Cochrane (1999). Unlike the Li (2007), by using model uncertainty in external habit formation, we show the countercyclical variation in uncertainty aversion 3. Our model provides more plausible parameters to explain both the equity premium and low risk-free rate puzzles. We also incorporate the Lucas style equilibrium ex-dividend stock price and risk-free rate under model uncertainty framework with external habit formation. Numerous studies attempt to explain both the Mehra and Prescott (1985) s equity premium puzzle and the Weil (1989) s low risk-free rate puzzle. With external habit formation, Campbell and Cochrane (1999) assume the special form of volatility of log consumption surplus, and they show a plausible explanation for the equity premium and risk-free rate puzzles. However, the Campbell and Cochrane (1999) s model requires a large volatility of log consumption surplus that cannot be supported by financial data. We use a small volatility of log consumption 1 In the economic sense, uncertainty aversion is discerned from risk aversion. The former is an aversive attitude about unknown distribution, the latter is an aversive attitude about known distribution 2 See Abel (1990), Constantinides (1990), Sundaresan (1989); these papers use a framework of external habit formation developed by Campbell and Cochrane (1999). 3 Rosenberg and Engle (2002) refer to countercyclical risk aversion, which is extracted from option prices. 2

5 surplus for the sake of plausible parameter choices to satisfy high equity premium and low risk-free rate. This paper proceeds as follows. In Section 2 and 3, we solve the portfolio choice problem with external habit formation, respectively without model uncertainty, and with model uncertainty. In Section 4, based on Lucas style equilibrium model, we provide the equilibrium market price of the risk and risk-free rate. In Section 5, we present numerical examples to support our theoretical result. In Section 6, we set forth our conclusion. 2 Optimal portfolio for external habit formation without model uncertainty 2.1 The model We consider a continuous-time financial market, which has one riskless asset and one risky asset in the infinite time horizon [0, ). A riskless asset follows the deterministic process with constant drift r, i.e., r represents interest rate. A risky asset follows the stochastic differential equation, where µ and σ are the drift and volatility, and B t is a standard Brownian motion on a probability space {Ω, F, P}. And {F t : t [0, )} is a filtration generated by the Brownian motion. ds t = µ t dt + σ t db t (1) The representative agent optimally decides consumption and portfolio strategy (π t, c t ). Consumption process c t is nonnegative and progressively measurable with respect to F t. The portfolio strategy process π t is F t measurable and square integrable, i.e., c L 2. 0 c t dt <, c t 0 (2) 3

6 0 π 2 t dt < (3) We denote the consumption surplus ratio S t (c t X t ) /c t, which is defined in Campbell and Cochrane (1999). In Campbell and Cochrane (1999), the log of the consumption surplus ratio follows a mean reverting process. In the continuous time framework, Campbell and Cochrane (1998) gives an identical solution to the discrete time case. We assume that the log consumption surplus ratio follows a mean reverting stochastic process and that its risky source is perfectly correlated with the risky source of a risky asset. 4 In the following equation, the volatility of log consumption surplus σ s is exactly equal to λ(s)σ of Campbell and Cochrane (1999) 5. ds t = k(s s t )dt + σ s db t, s(0) = s (4) Let us define the market price of risk, and the state-price density (or stochastic discount factor) respectively. θ = µ r { σ, H t = exp θb t 1 } 2 θ2 t (5) Without the intermediate income or endowment process, the wealth process evolves according to the following stochastic differential equation, which implies the investor s budget constraint. dw t = {r t W t + (µ t r t )π t W t c t }dt + π t σw t db t, W (0) = w (6) Controlling for consumption and portfolio weight on the risky asset, the investor faces the following maximization problem. 4 The market is incomplete when the Brownian motion of the log consumption surplus ratio is correlated with the Brownian motion of stock prices. 5 In Campbell and Cochrane (1999), λ(s) satisfies three properties. (1) the risk-free rate is constant; (2) habit is predetermined at the steady state; and (3) habit moves nonnegatively with consumption. 4

7 Problem 2.1. The agent maximizes a CRRA utility with the budget constraint. J(t, w, s) = sup E c,π [ t e βu (c u X u ) 1 γ ] du 1 γ s.t dw = {rw + (µ r)π u W c u }du + π u σw db, W (t) = w (7) ds = k(s s)dt + σ s db, s(t) = s 2.2 The solution Through a dynamic programming approach 6, we derive the Hamilton-Jacobi-Bellman equation. We define the indirect utility function J(t, w, s) = sup c,π E e βu (c u X u ) 1 γ du w [ t 1 γ t = ] w, s t = s. We then simplify the partial derivative of value function with subscripts. The Hamilton-Jacobi-Bellman equation is: { 0 = sup e βt (c X)1 γ c,π 1 γ + J w [rw + (µ r)πw c] + J s k(s s) + J t J wwπ 2 t w 2 σ 2 + J ws πwσσ s J ssσ 2 s } (8) The first order conditions for optimal consumption and optimal portfolio weight are: c t = e s (e βt J w ) 1 γ π t = J w θ J ww w σ J ws σ s J ww w σ (9) Conjecture that J(t, w, s) takes the form 6 Instead of the dynamic programming method, Cox and Huang (1989) and Karatzas, Lehoczky and Shreve (1987) use the martingale approach to solve the optimization problem. 5

8 βt w1 γ [ ] γ J(t, w, s) = e f(s, t) 1 γ where, f(s, t) = exp(a(t) + B(t)s + C(t)s 2 ) (10) Substitution of optimal consumption and optimal portfolio weight into equation (8) yields differential equation with respect to log consumption surplus ratio s and time t. Solving this differential equation with initial conditions w(t) = w, s(t) = s, we obtain the following value function and optimal consumption and portfolio choice. 7 Proposition 2.1. βt w1 γ [ ] γ J(t, w, s) = e f(s, t) = e βt w1 γ [ 1 exp(a(t) + B(t)s + 1 γ 1 γ 2 C(t)s2 ) ] γ where, A(t),B(t),C(t) is a solution of γe A(t)+(B(t) 1)s+ 1 2 C(t)s2 + (1 γ)r + γ(s s)[b(t) + C(t)s]κ β + [ A(t) + Ḃ(t)s + 1 2Ċ(t)s2 ] + γ(1 γ) [µ r 2 γσ + (B(t) + C(t)s)σ ] 2 s (11) + γ 2 [γ(b(t) + C(t)s)2 + C(t)] = 0 The optimal consumption and portfolio choice are c t = w f(s, t) π t = µ r γσ 2 + (B(t) + C(t)s) σ s σ Without intermediate consumption, the above differential equation takes the form of a Ricatti equation. Kim and Omberg (1996) and Liu (2007) provide a specific solution for 7 When both k(s s) and σ s are zero, the value function equals that of Merton (1969). 6

9 the Ricatti equation. Since the above differential equation has the exponential term, it is difficult to obtain the exact solution of A(t), B(t), C(t). Rather, we find a particular solution for the above differential equation assuming that A(t), B(t), C(t) is a constant. The following proposition shows a value function, optimal consumption and portfolio choice. Proposition 2.2. βt w1 γ [ J(t, w, s) = e exp( log(k1 ) s) ] γ 1 γ c t = K 1 w π t = µ r γσ 2 σ s σ (12) where, K 1 = r + β r γ γ 1 (θ γσ γ 2 s ) 2 + k(s s) 1 2 γσ2 s When both the drift of log consumption surplus and its volatility are zero, the optimal consumption and portfolio choice equal the solution in Merton (1969). Under general conditions, optimal consumption decreases as the volatility of exogenous consumption surplus 8. Optimal portfolio weight on risky assets is inversely proportional to the volatility of log consumption surplus. Also, the present high consumption surplus induces decreased next period s consumption, since the coefficient K 1 includes k(s s). External positive shock on current consumption compared to the investor s habit level provides an opportunity to increase reserves, which serves as preparation for uncertain future conditions. 8 Campbell and Cochrane (1999) states that the external consumption surplus plays the role of a state variable to represent market conditions. High log consumption surplus implies a bull market condition, and low consumption surplus implies a bear market condition increases. 7

10 3 Optimal portfolio strategy for external habit formation with model uncertainty In this section, we determine the optimal portfolio for external habit formation when the investor is aware of the model uncertainty. State-dependent external habit formation produces state-dependent uncertainty aversion. Based on Anderson, Hansen, Sargent (2003) and Maenhout (2004, 2006) we solve for the investor s problem with respect to external habit formation with model uncertainty. 3.1 The model Several differences from the problem presented in Section 2 are evident, in that the agent has an alternative worst-case wealth dynamics. The worst-case wealth dynamics dw a is distorted by the drift of wealth dynamics. Also the worst-case log consumption surplus process ds is distorted by the drift of log consumption surplus. In equation (13), h 1, h 2 are arbitrary random processes. dw = µ(w )dt + σ(w )db t = [rw + (µ r)π t W c t ]dt + π t σw db t dw a = [µ(w ) + σ(w ) 2 h 1 (W )]dt + σ(w )db t (13) ds = µ(s)dt + σ(s)db t = k(s s)dt + σ s db t ds a = [µ(s) + σ(s) 2 h 2 (s)]dt + σ(s)db t Given the maximization problem, the agent seeks to minimize relative entropy, which measures the distance between the reference model and the alternative model. Relative entropy is defined by the expectation of the log likelihood of the Radon-Nikodym derivative: 8

11 R(Q) = 1 [ 2 E e βu (σ(w )h 1 (W ) + σ(s)h 2 (s)) 2 du ] (14) t The agent faces the min-max expected utility problem. Problem 3.1. The agent maximizes a CRRA utility with the budget constraint. [ βu (c X)1 γ J(t, w, s) = inf sup E e h 1,h 2 1 γ c,π t ] du + 1 ψ R(Q) s.t dw a = [rw + (µ r)πw c + σ 2 h 1 ]du + πσw db, W (t) = w (15) ds a = [k(s s) + σ 2 sh 2 ]du + σ s db, s(t) = s In Problem 3.1 above, the agent controls the arbitrary function h = (h 1, h 2 ) to minimize entropy. The inverse of the coefficient for relative entropy, that is ψ 1, stands for the measure of uncertainty aversion. As the parameter ψ goes positive infinite, the agent increasingly doubts the reference wealth dynamics. Conversely, when parameter ψ goes to zero, the agent faces the problem without model uncertainty, which means that the agent assumes zero distortions about wealth dynamics The solution Using Bellman principle, we derive the Hamilton-Jacobi-Bellman equation with min-max formation. Taking first order conditions with respect to h, h satisfies h = (h 1, h 2 ) = ( ψj w, ψj s ) (16) 9 If parameter ψ equals zero, this problem is identical to the problem 2.1 9

12 Substituting h into the min-max Bellman equation, we obtain the following Bellman equation with general max formation: { 0 = sup e βt (c X)1 γ c,π 1 γ + J w [rw + (µ r)π t w c t ] + J s k(s s) + J t J wwπt 2 w 2 σ 2 + J ws π t wσσ s J ssσs 2 ψ [ ]} π 2 2 t σ 2 w 2 Jw 2 + 2σσ s π t wj w J s + σsj 2 s 2 (17) The first-order conditions for optimal consumption and optimal portfolio weight are c t = e s (e βt J w ) 1 γ J w π t = [J ww ψjw]w 2 θ σ J ws σ s [J ww ψjw]w 2 σ + ψj w J s σ s [J ww ψjw]w 2 σ (18) Conjecture that J(t, w, s) takes the form βt w1 γ [ ] γ J(t, w, s) = e g(s, t) 1 γ where, g(s, t) = exp(a(t) + B(t)s + C(t)s 2 ) (19) Similar to problem 2.1, substitution of optimal consumption and optimal portfolio weight into equation (17) gives the differential equation describing log consumption surplus ratio s at time t. For a homothetic portfolio weight 10, we define ψ = φ, where φ is constant. (1 γ)j Solving the differential equation with initial conditions w(t) = w, s(t) = s, we obtain the following value function, optimal consumption and portfolio weight Maenhout (2004, 2006) use this transformation to derive the homothetic portfolio choice. 11 When both k(s s) and σ s are zero, the value function equals that of Merton (1969). 10

13 Proposition 3.1. βt w1 γ [ ] γ J(t, w, s) = e g(s, t) = e βt w1 γ [ 1 exp(a(t) + B(t)s + 1 γ 1 γ 2 C(t)s2 ) ] γ where, A(t),B(t),C(t) is a solution of γe A(t)+(B(t) 1)s+ 1 2 C(t)s2 + (1 γ)r + γ(s s)[b(t) + C(t)s]κ β + [ A(t) + Ḃ(t)s + 1 2Ċ(t)s2 ] + γ 2 [γ(b(t) + C(t)s)2 + C(t)] (γ + φ)(1 γ) [µ r + 2 γσ + γσ s (γ + φ) (B(t) + C(t)s) γφσ s (1 γ)(γ + φ) (B(t) + C(t)s)] 2 γ (1 γ) [B(t) + C(t)s]2 = 0 (20) The optimal consumption and portfolio choice are c t = w g(s, t) πt = µ r (γ + φ)σ + γσ s 2 (γ + φ)σ (B(t) + C(t)s) γφσ s (B(t) + C(t)s) (1 γ)(γ + φ)σ We guess a particular solution for the above differential equation. Assuming the function g is independent of t, we find a candidate solution. Substitution of g into equations (18) and (19) produces the following value function, optimal consumption and portfolio choice. 11

14 Proposition 3.2. βt w1 γ [ J(t, w, s) = e exp( log(k2 ) s) ] γ 1 γ c t = K 2 w π t = µ r (γ + φ)σ 2 γσ s (γ + φ)σ + γφσ s (1 γ)(γ + φ)σ (21) where, K 2 = r + β r γ + k(s s) 1 2 γσ2 s + φ γ 2 1 γ σ2 s 1 1 γ 2 γ(γ + φ) { θ γσs + γφ 1 γ σ } 2 s When parameter φ equals zero, the consumption and portfolio weights are identical to the case without model uncertainty. Optimal consumption decreases as the constant parameter φ increases under the condition that γ is greater than 1. Optimal portfolio weight consists of three parts. The first term is myopic hedging demand, and the second term is intertemporal hedging demand. The third term is hedging demand for model uncertainty, and it is not included without model uncertainty stated in the previous section. If the parameter γ is greater than 1, the optimal portfolio weight is inversely proportional to the constant parameter φ, which measures degree of uncertainty aversion. The more the investor worries about a pessimistic scenario, the less investment in risky assets. 3.3 The countercyclical uncertainty aversion The parameter ψ = φ (1 γ)j represents an aversive attitude about model uncertainty. We define this parameter ψ as a state dependent uncertainty aversion 12. State dependent uncertainty aversion ψ is a function of log consumption surplus, which plays the role of a state variable. The following proposition shows that the derivative of uncertainty aversion with respect to s is negative under general conditions. 12 θ, γ is constant, and J = J(t, w, s) 12

15 Proposition 3.3. ψ s < 0 where, s s < 1 (22) When the economy is good, uncertainty aversion is at a minimum. When the economy is bad, the uncertainty aversion reaches a maximum. These countercyclical uncertainty aversion conditions are similar to the countercyclical risk aversion referred to in Rosenberg and Engle (2002). With empirical pricing kernels, they provide time-varying risk aversion throughout the business cycle. Fama and French (1989) show that risk premia are negatively correlated with the business cycle. Our main departure from these papers is that we focus on the investor s aversive attitude with respect to worst case scenario, and we theoretically derive the relationship between uncertainty aversion and a state variable. 4 Equilibrium asset price In equilibrium, our agent maximizes her expected utility and satisfies the market clearing condition 13. We assume that the endowment process or dividend process D t follows a geometric Brownian motion with drift µ d, and volatility σ d : dd t = µ d D t dt + σ d D t db t (23) By substitution of both optimal consumption c t and portfolio weight π t from equation (21), the wealth dynamics of equation (13) satisfy the following stochastic differential equation: 13 Market clearing condition implies that the optimal consumption process equals the endowment process. 13

16 dw w = [ r + (γ + φ)σd 2 + γσ d σ s γ 1 γ φσ dσ s r β r k(s s) + 1 γ 2 γσ2 s φ γ 2 1 γ σ2 s γ { θ γσs γφ 2 γ(γ + φ) 1 γ σ } 2 ] s dt + σd db (24) Optimal consumption c is expressed as a multiplication by wealth. Comparing the dc and dd processes, we obtain the equilibrium risk free rate and equilibrium market price of risk. Proposition 4.1. θ = (γ + φ)σ d + γσ s γ 1 γ φσ s r = β + γµ d + kγ(s s) 1 2 γ(γ + 1)σ2 d γ 2 σ d σ s 1 2 γ2 σ 2 s+ γ 2 1 γ φσ dσ s + φ 2 γ 2 1 γ σ2 s (1 + γ) φσd 2 2 (25) When parameter φ is zero, the market price of risk and the equilibrium risk-free rate equals that in Li (2007). If the risk aversion parameter γ is greater than 1, the market price of risk increases as the uncertainty aversion parameter φ increases. Li (2007) states that the volatility of log consumption surplus σ s is countercyclical. Without model uncertainty (φ = 0), the market price of risk is also countercyclical. With model uncertainty, the uncertainty aversion parameter φ interacts with the volatility of log consumption surplus σ s. When the economy is bad, the robust investor has a greater market price of risk relative to the good economy, since she faces the high volatility of log consumption surplus. If the risk aversion parameter is greater than 1, the higher the uncertainty aversion parameter φ, the lower the risk free rate. The robust investor worries about the pessimistic scenario, which directs money flow into savings. This precautionary saving reduces the equilibrium risk-free rate. 14

17 The equilibrium asset price can be obtained by the conditional expectation of discounted dividend stream: S t = E t [ where, t H s D s ds ] H t H t = exp ( θb t (r θ2 t )t ) (26) Using the equilibrium market price of risk and the risk free rate of equation (25), we derive the equilibrium ex-dividend stock process. Proposition 4.2. dp = ds t + D t dt S t = µ p dt + σ p db t where, µ p = β + γµ d + kγ(s s) 1 2 γ(γ + 1)σ2 d γ 2 σ d σ s 1 2 γ2 σ 2 s+ γ 2 1 γ φσ dσ s + φ 2 γ 2 1 γ σ2 s (1 + γ) φσd 2 + (γ + φ)σd 2 + γσ d σ s 2 γ 1 γ φσ dσ s (27) σ p = σ d In equilibrium, the portfolio weight on the risky asset of equation (21) equals 1, which implies that goods market clearing yields capital market clearing. The market price of risk is µ p r σ p = (γ + φ)σ d + γσ s γ 1 γ φσ s (28) When the uncertainty aversion parameter φ is zero, the equity premium is explained by the external habit model. Li (2007) interprets that the value of σ s is almost seven times the value of σ d to explain the equity premium puzzle by using external habit formation of Campbell and Cochrane (1999). However, for our robust agent with model uncertainty, the 15

18 value of σ s is sufficient to equal less than two times the value of σ d. As the third term of the equation shows that the model uncertainty interacts with external habit formation, even with small volatility of log consumption surplus ratio, the higher model uncertainty aversion produces a high equity premium. 5 Numerical examples 5.1 Optimal consumption and portfolio Figure 1 reports that the optimal consumption-wealth ratio decreases as the volatility of log consumption surplus increases. Since the volatility of log consumption surplus plays the role of a countercyclical state variable 14, the agent consumes less in recession and consumes more in boom. It is interesting that the higher the uncertainty aversion parameter, that is, the more the agent worries about a pessimistic situation, the lesser the agent optimally consumes. For example, given the 1% level of volatility of log consumption surplus, a robust agent with aversion parameter (φ = 10) has about a 20% smaller consumption wealth ratio than a non-robust agent with (φ = 0). [Figure 1 about here] Figure 2 shows the relationship between optimal consumption and consumption surplus. The optimal consumption-wealth ratio is a negative and convex function of consumption surplus. Regardless of the uncertainty aversion parameter, the agent has an identical relationship between optimal consumption and the volatility of log consumption surplus. When the agent faces a 10% higher level than the historical level of aggregate consumption, she has a 7% consumption wealth ratio. On the other hand, in the steady state with S = S, the agent has a 20% consumption-wealth ratio, which is three times greater than that of an agent with consumption surplus S = The agent has low volatility of log consumption surplus. In contrast, the agent has high volatility of log consumption surplus when the economy is at a trough. 16

19 [Figure 2 about here] Figure 3 and 4 illustrate the effect of consumption surplus and its volatility on the optimal portfolio of risky assets. Figure 3 shows that optimal portfolio weight on risky assets decreases as the volatility of log consumption surplus increases. This is related to precautionary saving. Interestingly, the robust agent requires higher precautionary saving demand than a non-robust agent. That is, the more the agent worries about a pessimistic situation, the lesser she invests in a risky asset. In equation (21), the optimal portfolio weight consists of three parts: myopic hedging demand, intertemporal hedging demand, and model uncertainty hedging demand. For the case γ > 1, high volatility of log consumption surplus negatively affects both the second and the third hedging demands. In particular, the large uncertainty aversion parameter raises model uncertainty hedging demand sharply. Figure 3 depicts that, given the 1% level of volatility of log consumption surplus, the optimal portfolio weight of an investor with (φ = 10) is 0.2, which is 10 times smaller than that of a non-robust investor with (φ = 0). Also, if a borrowing constraint does not hold, a robust investor with (φ = 10) optimally makes zero investment in risky asset when the volatility of consumption surplus exceeds 1.5%. Figure 4 plots a constant optimal portfolio, which is not affected by consumption surplus. [Figure 3 about here] [Figure 4 about here] 5.2 Equity premium and risk-free rate puzzles Campbell and Cochrane (1999) explain equity premium and risk-free rate puzzles that are developed by Mehra and Prescott (1985) and Weil (1989). They require a large volatility of log consumption surplus to satisfy both the equity premium puzzle and the risk-free rate puzzle. But a high volatility of log consumption surplus is not consistent with empirical analysis. 17

20 We argue that a low level of uncertainty aversion parameter is enough to solve both the equity premium and risk-free rate puzzles. We use the postwar data set that is used in Campbell and Cochrane (1999). Table 1 summarizes the parameter. [Table 1 about here] Campbell and Cochrane (1999) assume that the volatility of log consumption surplus is a function of a log consumption surplus. This restriction yields a negatively skewed distribution of log consumption surplus, which plays the role of a state variable. Without restriction on the volatility of log consumption surplus, we incorporate model uncertainty both in wealth dynamics and in the log consumption surplus process. Based on robust control theory from Anderson, Hansen, Sargent (2003), the drift of log consumption surplus could be distorted and be negatively related to the volatility of log consumption surplus. Our model captures the procyclical P/D ratio as shown in figure 5. The P/D ratio becomes high when the consumption surplus ratio and the volatility of log consumption surplus increases. As the model uncertainty aversion parameter increases, that is, the agent increasingly considers a worst case scenario, the P/D ratio increases. We suggest that the 2.5% volatility of log consumption surplus with model uncertainty(φ = 100) is enough to explain a 15 times P/D ratio. Without model uncertainty, satisfying the level of 15 times P/D ratio requires a 10% volatility of log consumption surplus. [Figure 5 about here] Based on propositions 4.1 and 4.2, we investigate the equilibrium expected return and risk-free rate. Regardless of model uncertainty, the expected return and and risk-free rate decreases as the consumption surplus increases. When the consumption surplus is low, that is, when the agent faces a bad state, the agent has the high level of discounted rate. The higher the level of model uncertainty parameter, the lower the risk-free rate and the larger the equity premium. As Figure 6 shows, the robust agent has a greater distance between expected return and risk-free rate. 18

21 [Figure 6 about here] We consider a feasible pair that consists of time preference and risk aversion parameters for a non-robust agent(φ = 0) and a robust agent(φ = 1, 5, 10). When the standard deviation of consumption growth equals the volatility of log consumption surplus, we plot the possible parameter pair (β, γ) according to the volatility of log consumption surplus, which satisfies the risk-free rate and market price of risk, respectively 1% and 6.7%. The top-left of Figure 7 shows the calibration of the non-robust agent, who should have a 10% level of volatility of log consumption surplus to satisfy the reasonable parameter choice(0 β 1, 0 γ 10). However, as the bottom of Figure 7 shows, 2% volatility of log consumption surplus could explain the feasible time preference and risk aversion parameter(0 β 1, 0 γ 10). [Figure 7 about here] 6 Conclusion We solve the optimal consumption and portfolio problem of external habit formation in model uncertainty. Our robust agent with model uncertainty decreases the investment on risky assets more than the agent without model uncertainty. Optimal investment in risky assets is determined by three factors. These factors include myopic hedging demand, model uncertainty hedging demand, and intertemporal hedging demand that is interacted with model uncertainty. The more aversive about model uncertainty and the higher the volatility of log consumption surplus, the less is the investment in risky assets. These two effects interact to decrease investment. Also, we derive the countercyclical uncertainty aversion parameter based on the above solution of an optimal portfolio. The better the economy, the lower the uncertainty aversion; the worse the economy, the greater is the uncertainty aversion produced. This implies that as the economy gets worse, the representative agent worries more about unexpected pessimistic 19

22 scenarios. Countercyclical uncertainty aversion might help to explain unexpected patterns of market crisis. Finally, we provide equilibrium asset price and risk free rate in the Lucas (1978) style equilibrium. In equilibrium, the robust agent faces higher market price of risk and lower riskfree rate than the agent without model uncertainty. Compared to Campbell and Cochrane (1999), our model provides more reasonable parameter choices to explain both the equity premium and the low risk-free rate puzzles. 20

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24 Kim, T. S., E. Omberg, Dynamic Nonmyopic Portfolio Behavior Review of Financial Studies, 9 (1996), George Li, Time-Varying Risk Aversion and Asset Prices Journal of Banking and Finance, 31 (2007), Jun Liu, Portfolio Selection in Stochastic Environments Review of Financial Studies, 20 (2007), Maenhout, P., Robust Portfolio Rules and Asset Pricing Review of Financial Studies, 17 (2004), Maenhout, P., Robust Portfolio Rules and detection error probabilities for a mean-reverting risk premium Journal of Economic Theory, 128 (2006), Mehra, R., and E.Prescott, The Equity Premium: A Puzzle Journal of Monetary Economics, 15 (1985), Merton, R. C., Lifetime Portfolio Selection: The Continuous-Time Case Review of Economics and Statistics, 51 (1969), Merton, R. C., Optimum Consumption and Portfolio Rules in a Continuous-Time Model Journal of Economic Theory, 3 (1971), Munk, C., Portfolio and Consumption Choice with Stochastic Investment Opportunities and Habit Formation in Preferences Journal of Economic Dynamics and Control, 32 (2008), Rosenberg, J. V., Engle, R. F., Empirical Pricing Kernels Journal of Financial Economics, 64 (2002), Sundaresan, Suresh M., Intertemporally Dependent Preferences and the Volatility of Consumption and Wealth Review of Financial Studies, 2 (1989), Weil, P., The Equity Premium Puzzle and the Risk-Free Rate Puzzle Journal of Monetary Economics, 24 (1989),

25 TABLE 1 Assumed Parameter Choices Parameter Variable Value log risk free rate r mean consumption growth g standard deviation of consumption growth σ mean reverting coefficient k 0.02 steady state of surplus consumption ratio S 0.05 curvature of power utility γ 2 time preference β

26 FIGURE 1 Optimal consumption and volatility of log consumption surplus φ=0 φ=5 φ=10 optimal consumption volatility of log consumption surplus Figure 1 reports the relationship between optimal consumption and volatility of log consumption surplus at steady state of log consumption surplus ratio(φ = 0, 5, 10, β = 0.1, r = , µ = 0.08, σ = 0.2, σ s = [0, 0.02], κ = 0.03, γ = 2, S = 0.05, w = 100). As the volatility of log consumption surplus increases, optimal consumption rate decreases. With higher uncertainty aversion parameter, the lower the agent has optimal consumption. 24

27 FIGURE 2 Optimal consumption and consumption surplus ratio φ=0 φ=5 φ=10 11 optimal consumption consumption surplus Figure 2 reports the relationship between optimal consumption and consumption surplus ratio. (φ = 0, 5, 10, β = 0.1, r = , µ = 0.08, σ = 0.2, s = [0, 0.12], κ = 0.03, γ = 2, S = 0.05, w = 100). As the consumption surplus ratio increases, optimal consumption rate decreases. With higher uncertainty aversion parameter, the lower the agent has optimal consumption rate. 25

28 FIGURE 3 Optimal portfolio and volatility of log consumption surplus φ=0 φ=5 φ= optimal portfolio volatility of log consumption surplus Figure 3 reports the relationship between optimal portfolio and volatility of log consumption surplus. (φ = 0, 5, 10, β = 0.1, r = , µ = 0.08, σ = 0.2, σ s = [0, 0.02], κ = 0.03, γ = 2, S = 0.05, w = 100). As the volatility of log consumption surplus increases, that is, as the economy becomes worse, the optimal investment in risky asset decreases. The higher uncertainty aversion parameter, the lower the level of optimal investment in risky asset. Without borrowing constraint, when the volatility of log consumption surplus exceeds a critical level, the robust investor optimally do zero investment in risky asset. 26

29 FIGURE 4 Optimal Portfolio and consumption surplus ratio optimal portfolio consumption surplus Figure 4 reports the relationship between optimal portfolio and consumption surplus ratio. (φ = 0, 5, 10, β = 0.1, r = , µ = 0.08, σ = 0.2, σ s = [0, 0.12], κ = 0.03, γ = 2, S = 0.05, w = 100). The optimal investment in risky asset is constant regardless of consumption surplus ratio. The model uncertainty does not affect the level of optimal investment in risky asset. 27

30 FIGURE 5 Consumption surplus ratio and P/D ratio 20 φ=0 φ=10 φ=100 σ s = φ=0 φ=10 φ=100 σ s = P/D ratio P/D ratio Consumption surplus ratio Consumption surplus ratio Figure 7 reports the relationship between consumption surplus ratio and P/D ratio. P/D ratio is an increasing function of a consumption surplus ratio. As the model uncertainty parameter increases, P/D ratio increases. With 10% volatility of log consumption surplus, non-robust investor faces around 15 level of P/D ratio at the level of 0.06 consumption surplus ratio. However, with 2.5% volatility of log consumption surplus, the robust investor(φ = 100) has around 15 level of P/D ratio at the level of 0.06 consumption surplus ratio. 28

31 FIGURE 6 Expected return, risk-free rate and consumption surplus ratio 0.25 φ=0 E[R] rf 0.25 φ=5 E[R] rf Expected Return, risk free rate Expected Return, risk free rate Consumption surplus ratio Consumption surplus ratio Figure 8 shows that both expected return and risk-free rate are decreasing function of consumption surplus ratio for the non-robust agent(left) and robust agent(right). The robust agent s equity premium between expected return and risk-free rate is bigger than that of non-robust agent. Also, equilibrium risk-free rate for the robust agent is lower than that of non-robust agent. 29

32 FIGURE 7 calibration of risk aversion and time preference 0.5 φ=0 0.5 φ=1 time preference 0 σ s =0.1 time preference 0 σ s =0.1 σ s =0.01 σ s = risk aversion risk aversion 0.5 φ=5 0.5 φ=10 time preference 0 σ s =0.04 σ s =0.1 σ s =0.01 time preference 0 σ s =0.02 σ s =0.1 σ s = risk aversion risk aversion Figure 7 reports the calibration of risk aversion and time preference. In the non-robust case(top-left), it requires high volatility of log consumption surplus to satisfy positive time preference and low risk aversion parameter. In the model uncertainty case of φ = 10(bottom-right), relative small volatility of log consumption surplus explains positive time preference and low risk aversion parameter. 30