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Robust consumption and portfolio rules with time-varying model confidence
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Robust consumption and portfolio rules with time-varying model confidence Bong-Gyu Jang a, Seungkyu Lee a,∗, Byung Hwa Lim b a b
Department of Industrial and Management Engineering, POSTECH, India Graduate School of Financial Engineering, University of Suwon, South Korean
a r t i c l e
i n f o
Article history: Received 11 January 2016 Revised 24 March 2016 Accepted 15 May 2016 Available online xxx JEL codes: C61 G11 G12
a b s t r a c t This paper investigates robust optimal consumption and portfolio rules for an Epstein-Zin type investor who is concerned about model misspecification. We propose a semi-explicit solution for the generalized problem of [Hansen, L., Sargent, T., 2001. Robust control and model uncertainty. The American Economic Review 91 (2), 60–66.]. Numerical results show that the optimal behaviors change dramatically according to the investor’s confidence level on the estimated model and that the elasticity of intertemporal substitution in consumption can affect investment ratio. In addition, we show how the investor decides her optimal behaviors for the worst-case scenario. © 2016 Elsevier Inc. All rights reserved.
Keywords: Model misspecification Robust control Portfolio selection Stochastic differential utility Elasticity of intertemporal substitution Consumption
1. Introduction This paper presents a generalized form of the robust optimal consumption and investment problem of Hansen and Sargent (2001) and provides a semi-explicit solution to this problem. We generalize the problem of Hansen et al. (2006) with the recursive utility of Epstein and Zin (1989), so our problem and numerical algorithm can be applied to all problems which need to disentangle the elasticity of intertemporal substitution (EIS) effect from the risk aversion effect. In addition, our problem provides how the Epstein-Zin type investors in the problem of Hansen and Sargent (2001) and Hansen et al. (2006) choose their worst-case scenario of financial markets. We are interested in investigating the effect of ambiguity aversion in consumption and investment behaviors. Our representative investor is assumed to choose a worst-case probability measure because she is concerned about model misspecification (estimation error in the expected return of a risky asset), or because she is ambiguity-averse. The investor uses optimal consumption and investment strategies which can be financed even under the worst-case probability measure. The effect of ambiguity aversion in consumption and portfolio choices has been widely studied. The seminal paper of Gilboa and Schmeidler (1989) proposes an atemporal max-min expected utility to explain the (Ellsberg, 1961) paradox.
∗
Corresponding author. E-mail addresses: [email protected] (B.-G. Jang), [email protected] (S. Lee), [email protected] (B.H. Lim).
http://dx.doi.org/10.1016/j.frl.2016.05.012 1544-6123/© 2016 Elsevier Inc. All rights reserved.
Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Following this seminal paper, Hansen, Sargent, and their colleagues connect the max-min expected utility problem with the risk sensitivity problem by means of a penalty term (or the relative entropy). (See Hansen and Sargent, 2001; Hansen et al., 2006; Anderson et al., 2003; Barillas et al., 2009, and Hansen and Sargent, 2011.) These studies commonly consider a robust optimal consumption and investment problem with a family of alternative probability measures constrained via relative entropy. Although robust optimization techniques and algorithms have been extensively developed in static or discrete-time problems (Ben-Tal and Nemirovski, 1998; Glasserman and Xu, 2013), the advance in continuous-time robust optimization has been relatively negligible. This is because the derived differential equations are heavily nonlinear so they are hard to solve even by a numerical method. In order to cope with this difficulty, many researchers (Liu et al., 2005; Branger et al., 2012; Anderson et al., 2009) have adopted the artificially modified robust optimization model of Maenhout (2004) or the recursive multiple-priors utility of Epstein and Schneider (Epstein and Schneider, 2003; Easley and O’Hara, 2010; Easley et al., 2014; Epstein and Schneider, 2008; Garlappi et al., 2007; Illeditsch, 2011; Uppal and Wang, 2003). Using those models, they explained several stylized facts in finance with ambiguity aversion. Although ambiguity aversion has become an interesting research subject in finance, no one has provided an exact solution for a robust optimal consumption and investment problem of Hansen and Sargent (2001) with Epstein-Zin type, or even CRRA type, utilities. We consider a continuous-time problem with one risk-free asset and one risky asset. The problem has an investor who does not fully trust the reference probability measure. For the whole life span of the investor, she faces the problem of selecting optimal consumption and investment strategies with concern about the estimated model. The main objective of our study is to formulate a problem with a time-varying model confidence level, which is measured by a continuation entropy, and to present a simple and efficient numerical algorithm. Our problem is directly related to the traditional robust optimization. We show numerical results of our problem with real market data. Our numerical results are interesting because we can simultaneously investigate the EIS effect and the ambiguity effect. There are two interesting observations related to the EIS in consumption. First, depending on the EIS level of the investor, the optimal consumption can monotonically increase or decrease in the investor’s confidence level. Second, the optimal asset demands could vary across a range of EIS even under a constant investment opportunity. Furthermore, we present how the investor determines her worst-case scenario: the adjusted market price of the risk due to the model misspecification and the sensitivity of the investor’s confidence level to market risk. 2. The problem We consider a financial market with a riskless asset St0 (or a bond) and a risky asset St (or a stock). The prices of the two assets are evolved by the following equations:
dSt0 = rSt0 dt,
dSt = μSt dt + σ St dBt , where r is the risk-free interest rate, μ is the expected rate of return of the stock, and σ is the volatility of the stock. Let ct ≥ 0 and π t be the consumption level and the stock-to-wealth ratio of an investor, respectively. The investor’s wealth process is evolved by
dwt = {(r + πt (μ − r ))wt − ct }dt + wt πt σ dBt ,
with w0 = w.
(1)
The standard Brownian motion {Bt } is defined on a probability space (, P, F ). We call the probability measure P the reference measure (or reference model) and assume it is obtained from historical market data. Now, consider an equivalent probability measure Qh as a perturbed measure (or perturbed model), under which the process B˜t satisfying
dB˜t = dBt − ht dt for all t ≥ 0, is a standard Brownian motion for a given Ft -adapted process ht . We assume that ht lies on the interval [−(μ − r )/σ , 0] for all t ≥ 0, which implies that the absolute value, |ht |, of the negative premium occurring from model misspecification cannot exceed the stock market risk premium, (μ − r )/σ . This is quite natural because an investor who is most pessimistic about the current stock market condition will not participate in the market, because she thinks there might be no risk premium in the stock market. It is well known that the exponential martingale
zt := exp 0
t
hu d Bu −
t 0
2
hu du , for all t ≥ 0, 2
is the Radon-Nikodym derivative of the perturbed probability measure Qth with respect to the reference probability measure Pt or, equivalently, is dQth /dPt , where Pt (or Qth ) is the restriction of a given probability measure P (or Qh , respectively) on Ft . Hansen and Sargent (2001) quantify the distance between the perturbed measure Qh and the reference measure P by Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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introducing the relative entropy
R(Q ) := E δ
∞
h
dQh dQth exp(−δt ) t log dt dPt dPt
0
= Eh δ
∞
exp(−δt ) log zt dt = E
0
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∞
h 0
2
ht exp(−δt ) dt , 2
Eh [·]
where E[·] and are the expectations with respect to P and Qh , respectively. This relationship and the lower bound of ht lead us to an upper bound of the relative entropy,
( μ − r )2 . 2δσ 2
R ( Qh ) ≤
With our notation, we can restate the constraint robust control problem of Hansen and Sargent (2001): to find the value function of an investor with a utility function of UHS ,
v(w, η ) =
min Eh
max
(c,π )∈A(w ) Qh ∈Q(η )
∞ 0
exp(−δt )U HS (ct )dt ,
subject to Eq. (1), where
A(w ) = {(ct , πt )|wt ≥ 0
for all t ≥ 0},
and
Q(η ) = {Q ∈ Q|R(Q ) ≤ η}. h
h
Here, Q(η ) is the set of all perturbed measures and η is interpreted as the maximum magnitude of the admitted pertur∗ bation of the stock return distribution from the reference model. Obviously, a perturbed measure Qh ∈ Q(η ) representing ∗ the worst-case scenario exists such that R(Qh ) = η. The investor permits a few models close to the reference model as ∗ candidates of the worst-case scenario if R(Qh ) (or η) is small, whereas she takes lots of perturbed models into considera∗ ∗ h tion if R(Q ) (or η) is large enough. Therefore, we can think of R(Qh ) as the investor’s zero-time confidence level on the ∗ h reference model. As extreme cases, R(Q ) = 0 reflects the case where the investor fully trusts the reference model, and ∗ R(Qh ) = (μ − r )2 /(2δσ 2 ) can be thought of as the case where she does not trust the reference model at all. We reformulate the above problem by employing a new state variable et , continuation entropy, which measures the investor’s confidence level at every time t. Although the motivation for adopting the continuation entropy is similar to the work of Hansen and Sargent (2011), the derivation on the continuation entropy dynamics is based on the definition of the relative entropy, as in Hansen et al. (2006) and Skiadas (2003). We consider the initial continuation entropy e0 to be the ∗ maximum level of the admitted confidence level, e0 := R(Qh ) = η, and derive the dynamics of the continuation entropy as ∗ follows. Skiadas (2003) extend the 0-time relative entropy to t-time relative entropy Rth , ∗
Rth = Eh
∗
∞
t
exp(−δt )
2
ht∗ dt Ft . 2
Skiadas (2003) explicitly shows that the dynamics of the relative entropy is given by ∗
dRth =
∗
δ Rth −
2
ht∗ 2
dt + d (martingale under Qh ).
Therefore, we assume that, as time unfolds, the continuation entropy et grows at the subject discount rate δ and is consumed with the extent of the t-time consumption of the relative entropy ht2 /2. The relative entropy can be consumed at any time by an investor who is anxious about model misspecification. The drift of the continuation entropy, therefore, should be δ et − ht 2 /2. We represent the dynamics of the continuation entropy with the following form:
det =
δ et −
2
ht 2
dt + gt dB˜t
with e0 = e.
(2)
The volatility gt of the continuation entropy et can be interpreted as the sensitivity of the investor’s confidence in the reference model with respect to market shocks and we take gt as a control variable. This implies that we allow the investor’s sensitivity of confidence in the reference model to fluctuate so that she chooses the worst-case market scenario keeping the possibility of such fluctuation in mind. Hansen et al. (2006) also state that the process gt should be a control variable for allocating the continuation entropy across various future states because the minimizer should choose gt to allocate entropy over the next moment. Following (Skiadas, 2003), we do not impose any restriction on gt . In fact, the presence of this additional control variable gt leads to exactly the same Hamilton-Jacobi-Isaacs’ (or Isaacs’) equation as Hansen et al. (2006) do. Note again that the derivation of the continuation entropy is initiated from the original idea of the relative entropy. Because the range of the relative entropy is from 0 to (μ − r )2 /(2δσ 2 ), we only consider the control variables ht and gt which let the continuation entropy et stay in the same range, [0, (μ − r )2 /(2δσ 2 )]. Furthermore, because we extend the investor’s 0-time confidence level (the relative entropy) to the investor’s on-time confidence level (the continuation Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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entropy), we can take the market data-fitted level of the relative entropy as the initial value of the continuation entropy and also use the detection error probabilities technique developed by Anderson et al. (2003) to calibrate the initial value of the continuation entropy. The investor has a recursive type utility preference. We take the normalized stochastic differential utility (SDU) of Epstein and Zin (1989). The formal representation of this recursive utility is
Ut = Et
∞
0
δ csρ − (αUs )ρ /α ds . ρ (αUs )(ρ /α )−1
Here, δ is the subjective discount rate, α = 1 − γ (0 = α ≤ 1) for the coefficient of relative risk aversion (RRA) γ , and ρ = 1 − 1/ψ (0 = ρ ≤ 1) for the coefficient of EIS ψ . Now we can state our problem thoroughly. Problem 2.1. Find the value function
V (w, e ) := max
min Eh
c,π ∈A(w ) h,g∈B (e )
∞ 0
δ ctρ − (αV (wt , et ))ρ /α dt , ρ (αV (wt , et ))(ρ /α )−1
subject to Eqs. (1) and (2), where
A(w ) = {(ct , πt )|wt ≥ 0 for all t ≥ 0}, B (e ) = {(ht , gt )|et ≥ 0 for all t ≥ 0}. The method of analyzing the problems in a setting where randomness is modelled by Brownian motion is presented by Fleming and Souganidis (1989). In the following section, we solve the value function with an Isaacs’ equation which is derived from (heuristic) dynamic programming arguments. 3. The solution The value function V in Problem 2.1 under suitable conditions for c, π , g and h, can have a recursive form of
V (wt , et ) =
max min Eth c,π h,g
t
τ
f¯(cu , V (wu , eu ))du + V (wτ , eτ ) ,
for 0 ≤ t < τ .
(3)
The relationship stems from the dynamic programming principle described in Fleming and Souganidis (1989) and a similar proof is also applicable. We obtain the following Isaacs’ equation using standard arguments in online Appendix A.1 Proposition 3.1. The value function V(w, e) in Problem 2.1 is the solution of the following Isaacs’ equation:
0 = max min (w(r + π (μ − r + σ h )) − c )Vw + (δ e − c,π
h,g
h2 )Ve 2 ρ
(4)
1 1 δ (cρ − (αV ) α ) + (π wσ )2Vww + g2Vee + π wσ gVew + , ρ 2 2 ρ (αV ) α −1 where Vw =
∂V ∂ 2V ∂V ∂ 2V ∂ 2V ,V = ,V = ,V = and Vew = . ∂ w ww ∂ w2 e ∂ e ee ∂ e2 ∂ e∂ w
The following properties of the value function are essential for our further analysis. The proof of Proposition 3.2 is presented in online Appendix B. Proposition 3.2. The value function defined in Problem 2.1 is increasing and concave with respect to w, and decreasing and convex with respect to e. From the first order conditions (FOCs) with respect to h and g, we can determine the optimal values h∗ and g∗ as
h∗ =
Vw ( π wσ ) Ve
and g∗ = −
Vew ( π wσ ) . Vee
Substituting these optimal values into the Isaacs’ Eq. (4), we get
0 = max c,π
δ (cρ − (αV ) αρ ) ρ (αV )
ρ α −1
+ (w(r + π (μ − r )) − c )Vw + δ eVe +
1 V2 V2 (π wσ )2 w + Vww − ew 2 Ve Vee
.
The FOCs with respect to c and π also provide the optimal consumption c∗ and risky investment proportion π ∗ :
c∗ =
1 V (αV ) αρ −1 ρ −1 V (μ − r ) w w2
. and π ∗ = − 2 δ wσ 2 Vw + V − Vew
Ve
1
ww
Vee
Our online Appendix is available in Jang’s homepage. (http://bgjang.postech.ac.kr/online_appendix)
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5
Therefore, the value function V(w, e) must satisfy
0=−
ρ ρ −α 1 δα 1−ρ (μ − r )2Vw2 2
. V+ (αV ) α (ρ −1) δ 1−ρ Vwρ −1 + wrVw + δ eVe − 2 ρ ρ 2σ 2 VVwe + Vww − VVew ee
(5)
We conjecture that the solution of the above equation for the value function has the following intuitive form. Theorem 3.3. The solution V(w, e) of (5) can be written as
1
V (w, e ) =
α
f ( e )wα ,
(6)
where f( · ) is a function satisfying the equation
1 ρ δ (1 − ρ )δ 1−ρ δ e f ( e ) 1 μ − r 2 0= r− + f (e ) α (ρ −1) + − ρ ρ α f (e ) 2 σ α
where α
f (0 ) = δ ρ and
f
1 f (e ) f (e )
−
f ( e )2 f (e ) f (e )
+ (α − 1 )
,
(7)
1 μ − r 2
α (ρρ−1) ρ δ − rρ + 1−ρ 2 (α − 1 ) σ
( μ − r )2
2δσ 2
α
= δρ
1
α (ρρ−1) δ − rρ . 1−ρ
(8)
The derivation of the boundary conditions is represented in online Appendix C. The value function V in (6) is a generalized version of that in the classical Merton problem. We prove the existence of the solution for boundary problem (7) in online Appendix D. Our proof is mainly based on the work of Franco and O’Regan (2003). We solve Eq. (7) using the following algorithm. First, we discretize the domain of the function f( · ) with Chebyshev’s nodes which are scaled by the length of +1 the domain, {ek = (1 − cos((k − 1 )/(2n )π ))(μ − r )2 /(2δσ 2 )}nk=1 . We also discretize ordinary differential equation (7) with forward difference. 4. Implications In this section, we investigate how the investor’s ambiguity aversion, which is measured by the level of continuation entropy, affects her optimal behaviors. The parameter values used are μ = 9.32%, r = δ = 5.75%, and σ = 17.95% . We utilize the annual rates on T-bills computed by rolling over twelve 1-month bills during each year as the risk-free rate r. The stock return and volatility, μ and σ , are those of the U.S. large stocks, a capitalization-weighted portfolio composed of the stocks in the S&P 500 index. The data periods are from 1968 to 2009. The subjective discount rate δ is chosen to be equal to the risk-free rate r as Kung and Schmid (2015), Liu and Miao (2015), and Croce et al. (2012) do.2 The relationship of (6) tells us that f(e)/α can be interpreted as the unit-wealth indirect utility. It is quite intuitive that the unit-wealth indirect utility decreases as the initial continuation entropy increases since the investor should waive a higher level of return (or wealth) in exchange for a more robust portfolio. Fig. 1 together with the relationship of the value function (6) shows the fact that the unit-wealth indirect utility decreases as initial entropy increases, i.e., the investor is less satisfactory as her model confidence decreases. For further analysis, let’s rewrite the optimal consumption and investment strategies in terms of f: ρ
c∗ (e, w ) = and
π ∗ (e ) = −
f (e ) α (ρ −1)
δ
1 ρ −1
w
μ−r · σ 2 α f (e ) − f (e )
(9)
1 f ( e )2 f (e ) f (e )
+ (α − 1 )
.
Not surprisingly, both the optimal consumption-to-wealth ratio c∗/w and the proportion of optimal risky investment π ∗ in our model are functions with respect to initial entropy e. 4.1. Optimal consumption Fig. 2a presents the optimal consumption-to-wealth ratios as functions of initial entropy level e with various levels of RRA, γ = 2, 5, 10, and 20.3 As Merton’s result does, the consumption-to-wealth decreases in response to the increase in the 2 We present additional numerical results with higher subjective discount rates in our online Appendix E. (The asset pricing model of Chen et al. (2012) derives a reasonable risk-free rate with a higher subjective discount rate.) They are similar to the result of this article. 3 As a base parameter for EIS, we choose ψ = 0.3 because (Vissing-Jorgensen, 2002) asserts that the estimate of the EIS is around 0.3-0.4 for stockholders.
Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 1. The unit-wealth indirect utility f(e)/α as a function of initial entropy e when α = −4(γ = 5 ) and ρ = −2.33(ψ = 0.3 ). The other parameters are μ = 9.32%, r = δ = 5.75%, and σ = 17.95%.
level of RRA. Although the consumption-to-wealth in Fig. 2a is a decreasing function with respect to initial entropy level, it can be an increasing function when it comes with the EIS greater than one, ψ > 1. Detail explanation for this observation will be covered at the next paragraph. In any case, the figure supports that the investor’s optimal consumption behavior might drastically change according to her model confidence level. A more interesting observation is that the consumptionto-wealth ratio converges to a certain level as initial entropy e approaches to the maximum value, (μ − r )2 /(2δσ 2 ). The investor’s robust optimal consumption behavior is not affected by her risk preference in case with the maximal entropy level, e = (μ − r )2 /(2δσ 2 ), because she does not invest in the risky asset at all. (Fig. 3) From Proposition 3.2 and Eqs. (6) and (9), we can easily derive
⎧ ∂ c∗ (e, w ) ⎨ < 0, ∂e ∗ ⎩ ∂ c (e, w ) > 0, ∂e
for 0 < ψ < 1, or ρ < 0, for ψ > 1, or 0 < ρ < 1.
This is reasonable since an increase in initial entropy level e has two different kinds of impact on the investor’s consumption behavior c∗(e, w), called the substitution effect and the income effect. As the investor becomes less confident in the reference model, she would like to immediately consume her wealth when she can (substitution effect). In contrast, a higher initial entropy makes her feel the wealth is less valuable, which leads to a decrease in current consumption (income effect). If the coefficient ψ of EIS is less than one, the investor reduces today’s consumption as model confidence increases, since the income effect dominates the substitution effect. If ψ is sufficiently big, the substitution effect dominates the income effect, so the consumption rate rises as the initial entropy is elevated. Fig. 2b shows the optimal consumption-to-wealth ratios as functions of initial entropy level e with various levels of EIS, ψ = 0.1, 0.3, 0.8, 1.15, and 3.0. The optimal consumption-to-wealth ratio is a decreasing and convex function when ψ < 1, while it is increasing and concave when ψ > 1. This is consistent to our analytical result in the previous paragraph. Also, the optimal consumption-to-wealth ratios in Fig. 2b with different levels of EIS converge to a certain level as initial entropy level approaches to its maximum, (μ − r )2 /(2δσ 2 ). The optimal consumption-to-wealth ratio with the maximal entropy level, −r ) c ∗ ( (μ , w) 2δσ 2 2
w
=
1 ( δ − r ρ ), 1−ρ
(10)
is calculated from the second boundary condition (8) and the semi-analytic formula (9) for the optimal consumption. When the risk free rate r and the subjective discount rate δ is assumed to be same, the optimal consumption-to-wealth ratio with the maximal entropy level is exactly the level of risk-free rate r, and is independent of EIS level.4
4 Figure E.2 in online Appendix E shows that, when subjective discount rate differs from risk-free rate, the optimal consumption-to-wealth ratio with the maximal entropy level depends on the level of EIS.
Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 2. The optimal consumption-to-wealth ratio c∗/w as a function of initial entropy e.
4.2. Optimal asset demands Using the optimal investment ratio πM = (μ − r )/((1 − α )σ 2 ) of the classical (Merton, 1971) problem, then
πM α f (e ) { f ( e )}2
=1+ − . ∗ π (e ) 1 − α f (e ) f (e ) f (e )
This relationship and the properties of f lead to the fact that π M /π ∗ (e) > 1 for all positive γ = 1 (or equivalently all α = 0) and all e. This implies that the investor in our model takes smaller positions in the stock market compared with the case where she does not care about model misspecification of stock returns. This fact might provide us with an (partial) explanation for the equity premium puzzle: concerns about model misspecification of stock returns might hamper investor’s stock investment. Fig. 3 shows a drastic reduction of the investor’s stockholdings according to the decrease in her confidence in the reference model.5 This result was also obtained by Maenhout (2004), but our model is faithfully based on the classical robust optimization without modification of the relative entropy. Another noticeable result is associated with the relationship between the investor’s EIS level and her optimal asset demands. Optimal investment behavior of the investor in our model could vary across a range of EIS coefficients even under
5
As well, the optimal investment proportion decreases with RRA level for any levels of initial entropy. (Fig. 3a)
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Fig. 3. The optimal investment proportions π ∗ as a function of initial entropy e.
a constant investment opportunity. (Fig. 3b) The optimal risky investment slightly changes when the investor is moderately concerned about model misspecification. This can be intuitively explained as Bhamra and Uppal (2006) do. The investor chooses the worst-case scenario h∗ which is continuously adjusted by the market risk Bt , a random variable. When the investor makes the wealth-maximization decision, she considers μ − h∗ , a random variable, as the expected rate of return on the risky asset. This makes the decision maker’s worst-case investment opportunity set stochastic, so the portfolio rate also depends on the level of EIS. Based on the same rationale, the following observation can be also explained: the optimal investment ratio is not affected by the level of EIS when the initial entropy is the minimum (e = 0) or the maximum (e = (μ − r )2 /(2δσ 2 )). In each case, the expected rate of return on the risky asset is considered to be a constant, μ or 0, respectively. Therefore, the investor’s asset demand is independent of the level of EIS as Svensson (1989) asserts. Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 4. h∗ as a function of initial entropy e when γ = 5 and ψ = 0.3. The parameters are μ = 9.32%, r = 5.75%, δ = 10.54% and σ = 17.95%.
4.3. Behaviors for the worst-case scenario In our model, the worst-case scenario is determined by control variables h and g, which are associated with the drift and volatility of the continuation entropy et . This implies that the investor establishes her worst outlook of the stock market by controlling the first and the second moments of the continuation entropy, or, equivalently, her confidence in the reference model. In optimum, they are described as
Vw f (e ) ( π ∗ wσ ) = α π ∗ ( e ) σ and Ve f (e ) Vew f (e ) g∗ = − (π ∗ wσ ) = −α π ∗ (e )σ . Vee f (e )
h∗ =
Not surprisingly, h∗ and g∗ are independent of the investor’s initial wealth level w. She does not care about her wealth level when considering the worst-case scenario. Fig. 4 represents a decreasing and concave function h∗ (e). As we see in the figure, the quantity h∗ is negative for all e > 0, so the expected return of the stock (μ + h∗ (e )σ ) should always be less than the return μ in the reference model. In particular, it is equal to the risk-free rate r when the initial continuation entropy e goes to the maximum value, in other words, the absolute value of h∗ becomes the Sharpe ratio (μ − r )/σ of the stock if the investor does not rely on the reference model at all. For this case, the optimal investment strategy is obviously to invest all of her wealth in the bond (thus, π ∗ = 0), because the risky stock does not give any risk premium to the market. Fig. 5 also has two implications. First, the sensitivity g∗ (e) of the investor’s confidence level with respect to the market risk dBt decreases with initial entropy level e. This is related to the decrease in the optimal investment rate π ∗ (e) in Fig. 3. As initial entropy level e grows up, the investor intentionally reduces her exposure to the market risk dBt . Consequently, the impact of the market risk on the optimal behaviors, including g∗ (e), decreases. Second, the investor conducts more robust, or conservative, decision making when she meets positive news dBt > 0 whereas she makes decisions more aggressively when she meets negative news dBt < 0. Such an implication is based on the following observation: the instantaneous correlation g∗ (e) between the change of the continuation entropy det and the return rate of the risky asset dSt /St is positive for all e. This observation can be intuitively explained as follows. When the return rate of the risky asset is unexpectedly high, i.e. dBt > 0, the wealth of the investor also increases unexpectedly. As a result, the marginal utility of the investor decreases, so she does not need to hold the risky portfolio that much any more. Rather, the investor may tend to make her investment more robust in order to protect her elevated wealth level. On the other hand, when the investor face an unexpected loss from a negative market shock dBt < 0, she may want to invest more in the risky asset in order to compensate for her loss with the high return rate of the asset. 5. Conclusion We generalize the problem of Hansen et al. (2006) with the Epstein-Zin type utility and a new state variable, continuation entropy, which measures the investor’s confidence level on the reference model. Applying the dynamic programming approach, we find that the value function can be expressed as a generalized version of that of Merton (1971). Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 5. g∗ as a function of initial entropy e when γ = 5 and ψ = 0.3. The parameters are μ = 9.32%, r = 5.75%, δ = 10.54% and σ = 17.95%.
Our numerical analysis explicitly shows many interesting results related to the robust optimal consumption and investment problem of Hansen et al. (2006). The robust optimal behaviors of the investor have different values according to different levels of her confidence in the reference model. We find that the investor’s optimal consumption rate can increase even though her concern about model misspecification increases when her EIS level is high enough. We also show that the investor’s investment proportion in a risky asset might drastically decrease when she significantly distrusts her reference model, which can be an (partial) explanation for the equity premium puzzle, as Maenhout (2004) does. More interestingly, our results show that the investor’s optimal asset demands could be dependent on the coefficient of EIS as well as on the coefficient of risk aversion since she makes consumption and investment decisions as if she is under a stochastic investment opportunity. This numerical result implies that our model can be a compromise between the arguments of Svensson (1989) and Bhamra and Uppal (2006). This research work seems to be important in that our model can be used in any asset pricing or asset allocation models which incorporate a time-varying level of model uncertainty, or the continuation entropy, which is not artificially modified. Acknowledgements We would like to thank Hyeng Keun Koo and Gyoocheol Shim for helpful comments. A preliminary version of the paper was presented at the 2009 APK conference, the 2009 SSS conference, the 2010 World Congress of the BFS, the 2011 KFA conference, the 2011 KDA conference, and the 4th KOFES conference. This research was supported by Global PH.D Fellowship Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2011-0 0 08628). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2013R1A2A2A03068890). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (2014S1A3A2036037, NRF-2014S1A5A8018920, and NRF-2013S1A5A8024023). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.frl.2016.05.012. References Anderson, E., Hansen, L., Sargent, T., 2003. A quartet of semigroups for model specification, robustness, prices of risk, and model detection. J. Eur. Econ. Assoc. 1 (1), 68–123. Anderson, E.W., Ghysels, E., Juergens, J.L., 2009. The impact of risk and uncertainty on expected returns. J. Financ. Econ. 94 (2), 233–263. Barillas, F., Hansen, L.P., Sargent, T.J., 2009. Doubts or variability? J. Econ. Theor. 144 (6), 2388–2418. Ben-Tal, A., Nemirovski, A., 1998. Robust convex optimization. Math. Oper. Res. 23 (4), 769–805. Bhamra, H., Uppal, R., 2006. The role of risk aversion and intertemporal substitution in dynamic consumption-portfolio choice with recursive utility. J. Econ. Dyn. Control 30 (6), 967–991. Branger, N., Larsen, L., Munk, C., 2012. Robust portfolio choice with ambiguity and learning about return predictability. J. Bank. Finance 37, 1397–1411. Chen, H., Joslin, S., Tran, N.-K., 2012. Rare disasters and risk sharing with heterogeneous beliefs. Rev. Financ. Stud. 25 (7), 2189–2224. Croce, M.M., Nguyen, T.T., Schmid, L., 2012. The market price of fiscal uncertainty. J. Monet. Econ. 59 (5), 401–416. Easley, D., O’Hara, M., 2010. Microstructure and ambiguity. J. Finance 65 (5), 1817–1846. Easley, D., O’Hara, M., Yang, L., 2014. Opaque trading, disclosure, and asset prices: implications for hedge fund regulation. Rev. Financ. Stud. 27 (4), 1190–1237.
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Ellsberg, D., 1961. Risk, ambiguity, and the savage axioms. Quart. J. Econ. 75, 643–669. Epstein, L., Schneider, M., 2008. Ambiguity, information quality, and asset pricing. J. Finance 63 (1), 197–228. Epstein, L.G., Schneider, M., 2003. Recursive multiple-priors. J. Econ. Theory 113 (1), 1–31. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57, 937–969. Fleming, W.H., Souganidis, P.E., 1989. On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38. Franco, D., O’Regan, D., 2003. Existence of solutions to second order problems with nonlinear boundary conditions. In: Proceeding of the Fourth International Conference on Dynamical Systems and Differential Equations, Discrete and Continuous Dynamical Systems, pp. 273–280. Garlappi, L., Uppal, R., Wang, T., 2007. Portfolio selection with parameter and model uncertainty: a multi-prior approach. Rev. Financ. Stud. 20 (1), 41–81. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. J. Math. Econ. 18 (2), 141–153. Glasserman, P., Xu, X., 2013. Robust portfolio control with stochastic factor dynamics. Oper. Res. 61 (4), 874–893. Hansen, L., Sargent, T., 2001. Robust control and model uncertainty. Am. Econ. Rev. 91 (2), 60–66. Hansen, L., Sargent, T., 2011. Robustness and ambiguity in continuous time. J. Econ. Theory 146 (3), 1195–1223. Hansen, L., Sargent, T., Turmuhambetova, G., Williams, N., 2006. Robust control and model misspecification. J. Econ. Theory 128 (1), 45–90. Illeditsch, P.K., 2011. Ambiguous information, portfolio inertia, and excess volatility. J. Finance 66 (6), 2213–2247. Kung, H., Schmid, L., 2015. Innovation, growth, and asset prices. J. Finance 70 (3), 1001–1037. Liu, H., Miao, J., 2015. Growth uncertainty, generalized disappointment aversion and production-based asset pricing. J. Monet. Econ. 69, 70–89. Liu, J., Pan, J., Wang, T., 2005. An equilibrium model of rare-event premia and its implication for option smirks. Rev. Financ. Stud. 18 (1), 131–164. Maenhout, P., 2004. Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17 (4), 951–983. Merton, R., 1971. Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413. Skiadas, C., 2003. Robust control and recursive utility. Finance Stochast. 7 (4), 475–489. Svensson, L., 1989. Portfolio choice with non-expected utility in continuous time. Econ. Lett. 30 (4), 313–317. Uppal, R., Wang, T., 2003. Model misspecification and underdiversification. J. Finance 58 (6), 2465–2486. Vissing-Jorgensen, A., 2002. Limited Asset Market Participation and the Elasticity of Intertemporal Substitution. Technical Report. National Bureau of Economic Research.
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Finance Research Letters 0 0 0 (2016) 1–11
Contents lists available at ScienceDirect
Finance Research Letters journal homepage: www.elsevier.com/locate/frl
Robust consumption and portfolio rules with time-varying model confidence Bong-Gyu Jang a, Seungkyu Lee a,∗, Byung Hwa Lim b a b
Department of Industrial and Management Engineering, POSTECH, India Graduate School of Financial Engineering, University of Suwon, South Korean
a r t i c l e
i n f o
Article history: Received 11 January 2016 Revised 24 March 2016 Accepted 15 May 2016 Available online xxx JEL codes: C61 G11 G12
a b s t r a c t This paper investigates robust optimal consumption and portfolio rules for an Epstein-Zin type investor who is concerned about model misspecification. We propose a semi-explicit solution for the generalized problem of [Hansen, L., Sargent, T., 2001. Robust control and model uncertainty. The American Economic Review 91 (2), 60–66.]. Numerical results show that the optimal behaviors change dramatically according to the investor’s confidence level on the estimated model and that the elasticity of intertemporal substitution in consumption can affect investment ratio. In addition, we show how the investor decides her optimal behaviors for the worst-case scenario. © 2016 Elsevier Inc. All rights reserved.
Keywords: Model misspecification Robust control Portfolio selection Stochastic differential utility Elasticity of intertemporal substitution Consumption
1. Introduction This paper presents a generalized form of the robust optimal consumption and investment problem of Hansen and Sargent (2001) and provides a semi-explicit solution to this problem. We generalize the problem of Hansen et al. (2006) with the recursive utility of Epstein and Zin (1989), so our problem and numerical algorithm can be applied to all problems which need to disentangle the elasticity of intertemporal substitution (EIS) effect from the risk aversion effect. In addition, our problem provides how the Epstein-Zin type investors in the problem of Hansen and Sargent (2001) and Hansen et al. (2006) choose their worst-case scenario of financial markets. We are interested in investigating the effect of ambiguity aversion in consumption and investment behaviors. Our representative investor is assumed to choose a worst-case probability measure because she is concerned about model misspecification (estimation error in the expected return of a risky asset), or because she is ambiguity-averse. The investor uses optimal consumption and investment strategies which can be financed even under the worst-case probability measure. The effect of ambiguity aversion in consumption and portfolio choices has been widely studied. The seminal paper of Gilboa and Schmeidler (1989) proposes an atemporal max-min expected utility to explain the (Ellsberg, 1961) paradox.
∗
Corresponding author. E-mail addresses: [email protected] (B.-G. Jang), [email protected] (S. Lee), [email protected] (B.H. Lim).
http://dx.doi.org/10.1016/j.frl.2016.05.012 1544-6123/© 2016 Elsevier Inc. All rights reserved.
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Following this seminal paper, Hansen, Sargent, and their colleagues connect the max-min expected utility problem with the risk sensitivity problem by means of a penalty term (or the relative entropy). (See Hansen and Sargent, 2001; Hansen et al., 2006; Anderson et al., 2003; Barillas et al., 2009, and Hansen and Sargent, 2011.) These studies commonly consider a robust optimal consumption and investment problem with a family of alternative probability measures constrained via relative entropy. Although robust optimization techniques and algorithms have been extensively developed in static or discrete-time problems (Ben-Tal and Nemirovski, 1998; Glasserman and Xu, 2013), the advance in continuous-time robust optimization has been relatively negligible. This is because the derived differential equations are heavily nonlinear so they are hard to solve even by a numerical method. In order to cope with this difficulty, many researchers (Liu et al., 2005; Branger et al., 2012; Anderson et al., 2009) have adopted the artificially modified robust optimization model of Maenhout (2004) or the recursive multiple-priors utility of Epstein and Schneider (Epstein and Schneider, 2003; Easley and O’Hara, 2010; Easley et al., 2014; Epstein and Schneider, 2008; Garlappi et al., 2007; Illeditsch, 2011; Uppal and Wang, 2003). Using those models, they explained several stylized facts in finance with ambiguity aversion. Although ambiguity aversion has become an interesting research subject in finance, no one has provided an exact solution for a robust optimal consumption and investment problem of Hansen and Sargent (2001) with Epstein-Zin type, or even CRRA type, utilities. We consider a continuous-time problem with one risk-free asset and one risky asset. The problem has an investor who does not fully trust the reference probability measure. For the whole life span of the investor, she faces the problem of selecting optimal consumption and investment strategies with concern about the estimated model. The main objective of our study is to formulate a problem with a time-varying model confidence level, which is measured by a continuation entropy, and to present a simple and efficient numerical algorithm. Our problem is directly related to the traditional robust optimization. We show numerical results of our problem with real market data. Our numerical results are interesting because we can simultaneously investigate the EIS effect and the ambiguity effect. There are two interesting observations related to the EIS in consumption. First, depending on the EIS level of the investor, the optimal consumption can monotonically increase or decrease in the investor’s confidence level. Second, the optimal asset demands could vary across a range of EIS even under a constant investment opportunity. Furthermore, we present how the investor determines her worst-case scenario: the adjusted market price of the risk due to the model misspecification and the sensitivity of the investor’s confidence level to market risk. 2. The problem We consider a financial market with a riskless asset St0 (or a bond) and a risky asset St (or a stock). The prices of the two assets are evolved by the following equations:
dSt0 = rSt0 dt,
dSt = μSt dt + σ St dBt , where r is the risk-free interest rate, μ is the expected rate of return of the stock, and σ is the volatility of the stock. Let ct ≥ 0 and π t be the consumption level and the stock-to-wealth ratio of an investor, respectively. The investor’s wealth process is evolved by
dwt = {(r + πt (μ − r ))wt − ct }dt + wt πt σ dBt ,
with w0 = w.
(1)
The standard Brownian motion {Bt } is defined on a probability space (, P, F ). We call the probability measure P the reference measure (or reference model) and assume it is obtained from historical market data. Now, consider an equivalent probability measure Qh as a perturbed measure (or perturbed model), under which the process B˜t satisfying
dB˜t = dBt − ht dt for all t ≥ 0, is a standard Brownian motion for a given Ft -adapted process ht . We assume that ht lies on the interval [−(μ − r )/σ , 0] for all t ≥ 0, which implies that the absolute value, |ht |, of the negative premium occurring from model misspecification cannot exceed the stock market risk premium, (μ − r )/σ . This is quite natural because an investor who is most pessimistic about the current stock market condition will not participate in the market, because she thinks there might be no risk premium in the stock market. It is well known that the exponential martingale
zt := exp 0
t
hu d Bu −
t 0
2
hu du , for all t ≥ 0, 2
is the Radon-Nikodym derivative of the perturbed probability measure Qth with respect to the reference probability measure Pt or, equivalently, is dQth /dPt , where Pt (or Qth ) is the restriction of a given probability measure P (or Qh , respectively) on Ft . Hansen and Sargent (2001) quantify the distance between the perturbed measure Qh and the reference measure P by Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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introducing the relative entropy
R(Q ) := E δ
∞
h
dQh dQth exp(−δt ) t log dt dPt dPt
0
= Eh δ
∞
exp(−δt ) log zt dt = E
0
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∞
h 0
2
ht exp(−δt ) dt , 2
Eh [·]
where E[·] and are the expectations with respect to P and Qh , respectively. This relationship and the lower bound of ht lead us to an upper bound of the relative entropy,
( μ − r )2 . 2δσ 2
R ( Qh ) ≤
With our notation, we can restate the constraint robust control problem of Hansen and Sargent (2001): to find the value function of an investor with a utility function of UHS ,
v(w, η ) =
min Eh
max
(c,π )∈A(w ) Qh ∈Q(η )
∞ 0
exp(−δt )U HS (ct )dt ,
subject to Eq. (1), where
A(w ) = {(ct , πt )|wt ≥ 0
for all t ≥ 0},
and
Q(η ) = {Q ∈ Q|R(Q ) ≤ η}. h
h
Here, Q(η ) is the set of all perturbed measures and η is interpreted as the maximum magnitude of the admitted pertur∗ bation of the stock return distribution from the reference model. Obviously, a perturbed measure Qh ∈ Q(η ) representing ∗ the worst-case scenario exists such that R(Qh ) = η. The investor permits a few models close to the reference model as ∗ candidates of the worst-case scenario if R(Qh ) (or η) is small, whereas she takes lots of perturbed models into considera∗ ∗ h tion if R(Q ) (or η) is large enough. Therefore, we can think of R(Qh ) as the investor’s zero-time confidence level on the ∗ h reference model. As extreme cases, R(Q ) = 0 reflects the case where the investor fully trusts the reference model, and ∗ R(Qh ) = (μ − r )2 /(2δσ 2 ) can be thought of as the case where she does not trust the reference model at all. We reformulate the above problem by employing a new state variable et , continuation entropy, which measures the investor’s confidence level at every time t. Although the motivation for adopting the continuation entropy is similar to the work of Hansen and Sargent (2011), the derivation on the continuation entropy dynamics is based on the definition of the relative entropy, as in Hansen et al. (2006) and Skiadas (2003). We consider the initial continuation entropy e0 to be the ∗ maximum level of the admitted confidence level, e0 := R(Qh ) = η, and derive the dynamics of the continuation entropy as ∗ follows. Skiadas (2003) extend the 0-time relative entropy to t-time relative entropy Rth , ∗
Rth = Eh
∗
∞
t
exp(−δt )
2
ht∗ dt Ft . 2
Skiadas (2003) explicitly shows that the dynamics of the relative entropy is given by ∗
dRth =
∗
δ Rth −
2
ht∗ 2
dt + d (martingale under Qh ).
Therefore, we assume that, as time unfolds, the continuation entropy et grows at the subject discount rate δ and is consumed with the extent of the t-time consumption of the relative entropy ht2 /2. The relative entropy can be consumed at any time by an investor who is anxious about model misspecification. The drift of the continuation entropy, therefore, should be δ et − ht 2 /2. We represent the dynamics of the continuation entropy with the following form:
det =
δ et −
2
ht 2
dt + gt dB˜t
with e0 = e.
(2)
The volatility gt of the continuation entropy et can be interpreted as the sensitivity of the investor’s confidence in the reference model with respect to market shocks and we take gt as a control variable. This implies that we allow the investor’s sensitivity of confidence in the reference model to fluctuate so that she chooses the worst-case market scenario keeping the possibility of such fluctuation in mind. Hansen et al. (2006) also state that the process gt should be a control variable for allocating the continuation entropy across various future states because the minimizer should choose gt to allocate entropy over the next moment. Following (Skiadas, 2003), we do not impose any restriction on gt . In fact, the presence of this additional control variable gt leads to exactly the same Hamilton-Jacobi-Isaacs’ (or Isaacs’) equation as Hansen et al. (2006) do. Note again that the derivation of the continuation entropy is initiated from the original idea of the relative entropy. Because the range of the relative entropy is from 0 to (μ − r )2 /(2δσ 2 ), we only consider the control variables ht and gt which let the continuation entropy et stay in the same range, [0, (μ − r )2 /(2δσ 2 )]. Furthermore, because we extend the investor’s 0-time confidence level (the relative entropy) to the investor’s on-time confidence level (the continuation Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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entropy), we can take the market data-fitted level of the relative entropy as the initial value of the continuation entropy and also use the detection error probabilities technique developed by Anderson et al. (2003) to calibrate the initial value of the continuation entropy. The investor has a recursive type utility preference. We take the normalized stochastic differential utility (SDU) of Epstein and Zin (1989). The formal representation of this recursive utility is
Ut = Et
∞
0
δ csρ − (αUs )ρ /α ds . ρ (αUs )(ρ /α )−1
Here, δ is the subjective discount rate, α = 1 − γ (0 = α ≤ 1) for the coefficient of relative risk aversion (RRA) γ , and ρ = 1 − 1/ψ (0 = ρ ≤ 1) for the coefficient of EIS ψ . Now we can state our problem thoroughly. Problem 2.1. Find the value function
V (w, e ) := max
min Eh
c,π ∈A(w ) h,g∈B (e )
∞ 0
δ ctρ − (αV (wt , et ))ρ /α dt , ρ (αV (wt , et ))(ρ /α )−1
subject to Eqs. (1) and (2), where
A(w ) = {(ct , πt )|wt ≥ 0 for all t ≥ 0}, B (e ) = {(ht , gt )|et ≥ 0 for all t ≥ 0}. The method of analyzing the problems in a setting where randomness is modelled by Brownian motion is presented by Fleming and Souganidis (1989). In the following section, we solve the value function with an Isaacs’ equation which is derived from (heuristic) dynamic programming arguments. 3. The solution The value function V in Problem 2.1 under suitable conditions for c, π , g and h, can have a recursive form of
V (wt , et ) =
max min Eth c,π h,g
t
τ
f¯(cu , V (wu , eu ))du + V (wτ , eτ ) ,
for 0 ≤ t < τ .
(3)
The relationship stems from the dynamic programming principle described in Fleming and Souganidis (1989) and a similar proof is also applicable. We obtain the following Isaacs’ equation using standard arguments in online Appendix A.1 Proposition 3.1. The value function V(w, e) in Problem 2.1 is the solution of the following Isaacs’ equation:
0 = max min (w(r + π (μ − r + σ h )) − c )Vw + (δ e − c,π
h,g
h2 )Ve 2 ρ
(4)
1 1 δ (cρ − (αV ) α ) + (π wσ )2Vww + g2Vee + π wσ gVew + , ρ 2 2 ρ (αV ) α −1 where Vw =
∂V ∂ 2V ∂V ∂ 2V ∂ 2V ,V = ,V = ,V = and Vew = . ∂ w ww ∂ w2 e ∂ e ee ∂ e2 ∂ e∂ w
The following properties of the value function are essential for our further analysis. The proof of Proposition 3.2 is presented in online Appendix B. Proposition 3.2. The value function defined in Problem 2.1 is increasing and concave with respect to w, and decreasing and convex with respect to e. From the first order conditions (FOCs) with respect to h and g, we can determine the optimal values h∗ and g∗ as
h∗ =
Vw ( π wσ ) Ve
and g∗ = −
Vew ( π wσ ) . Vee
Substituting these optimal values into the Isaacs’ Eq. (4), we get
0 = max c,π
δ (cρ − (αV ) αρ ) ρ (αV )
ρ α −1
+ (w(r + π (μ − r )) − c )Vw + δ eVe +
1 V2 V2 (π wσ )2 w + Vww − ew 2 Ve Vee
.
The FOCs with respect to c and π also provide the optimal consumption c∗ and risky investment proportion π ∗ :
c∗ =
1 V (αV ) αρ −1 ρ −1 V (μ − r ) w w2
. and π ∗ = − 2 δ wσ 2 Vw + V − Vew
Ve
1
ww
Vee
Our online Appendix is available in Jang’s homepage. (http://bgjang.postech.ac.kr/online_appendix)
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5
Therefore, the value function V(w, e) must satisfy
0=−
ρ ρ −α 1 δα 1−ρ (μ − r )2Vw2 2
. V+ (αV ) α (ρ −1) δ 1−ρ Vwρ −1 + wrVw + δ eVe − 2 ρ ρ 2σ 2 VVwe + Vww − VVew ee
(5)
We conjecture that the solution of the above equation for the value function has the following intuitive form. Theorem 3.3. The solution V(w, e) of (5) can be written as
1
V (w, e ) =
α
f ( e )wα ,
(6)
where f( · ) is a function satisfying the equation
1 ρ δ (1 − ρ )δ 1−ρ δ e f ( e ) 1 μ − r 2 0= r− + f (e ) α (ρ −1) + − ρ ρ α f (e ) 2 σ α
where α
f (0 ) = δ ρ and
f
1 f (e ) f (e )
−
f ( e )2 f (e ) f (e )
+ (α − 1 )
,
(7)
1 μ − r 2
α (ρρ−1) ρ δ − rρ + 1−ρ 2 (α − 1 ) σ
( μ − r )2
2δσ 2
α
= δρ
1
α (ρρ−1) δ − rρ . 1−ρ
(8)
The derivation of the boundary conditions is represented in online Appendix C. The value function V in (6) is a generalized version of that in the classical Merton problem. We prove the existence of the solution for boundary problem (7) in online Appendix D. Our proof is mainly based on the work of Franco and O’Regan (2003). We solve Eq. (7) using the following algorithm. First, we discretize the domain of the function f( · ) with Chebyshev’s nodes which are scaled by the length of +1 the domain, {ek = (1 − cos((k − 1 )/(2n )π ))(μ − r )2 /(2δσ 2 )}nk=1 . We also discretize ordinary differential equation (7) with forward difference. 4. Implications In this section, we investigate how the investor’s ambiguity aversion, which is measured by the level of continuation entropy, affects her optimal behaviors. The parameter values used are μ = 9.32%, r = δ = 5.75%, and σ = 17.95% . We utilize the annual rates on T-bills computed by rolling over twelve 1-month bills during each year as the risk-free rate r. The stock return and volatility, μ and σ , are those of the U.S. large stocks, a capitalization-weighted portfolio composed of the stocks in the S&P 500 index. The data periods are from 1968 to 2009. The subjective discount rate δ is chosen to be equal to the risk-free rate r as Kung and Schmid (2015), Liu and Miao (2015), and Croce et al. (2012) do.2 The relationship of (6) tells us that f(e)/α can be interpreted as the unit-wealth indirect utility. It is quite intuitive that the unit-wealth indirect utility decreases as the initial continuation entropy increases since the investor should waive a higher level of return (or wealth) in exchange for a more robust portfolio. Fig. 1 together with the relationship of the value function (6) shows the fact that the unit-wealth indirect utility decreases as initial entropy increases, i.e., the investor is less satisfactory as her model confidence decreases. For further analysis, let’s rewrite the optimal consumption and investment strategies in terms of f: ρ
c∗ (e, w ) = and
π ∗ (e ) = −
f (e ) α (ρ −1)
δ
1 ρ −1
w
μ−r · σ 2 α f (e ) − f (e )
(9)
1 f ( e )2 f (e ) f (e )
+ (α − 1 )
.
Not surprisingly, both the optimal consumption-to-wealth ratio c∗/w and the proportion of optimal risky investment π ∗ in our model are functions with respect to initial entropy e. 4.1. Optimal consumption Fig. 2a presents the optimal consumption-to-wealth ratios as functions of initial entropy level e with various levels of RRA, γ = 2, 5, 10, and 20.3 As Merton’s result does, the consumption-to-wealth decreases in response to the increase in the 2 We present additional numerical results with higher subjective discount rates in our online Appendix E. (The asset pricing model of Chen et al. (2012) derives a reasonable risk-free rate with a higher subjective discount rate.) They are similar to the result of this article. 3 As a base parameter for EIS, we choose ψ = 0.3 because (Vissing-Jorgensen, 2002) asserts that the estimate of the EIS is around 0.3-0.4 for stockholders.
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Fig. 1. The unit-wealth indirect utility f(e)/α as a function of initial entropy e when α = −4(γ = 5 ) and ρ = −2.33(ψ = 0.3 ). The other parameters are μ = 9.32%, r = δ = 5.75%, and σ = 17.95%.
level of RRA. Although the consumption-to-wealth in Fig. 2a is a decreasing function with respect to initial entropy level, it can be an increasing function when it comes with the EIS greater than one, ψ > 1. Detail explanation for this observation will be covered at the next paragraph. In any case, the figure supports that the investor’s optimal consumption behavior might drastically change according to her model confidence level. A more interesting observation is that the consumptionto-wealth ratio converges to a certain level as initial entropy e approaches to the maximum value, (μ − r )2 /(2δσ 2 ). The investor’s robust optimal consumption behavior is not affected by her risk preference in case with the maximal entropy level, e = (μ − r )2 /(2δσ 2 ), because she does not invest in the risky asset at all. (Fig. 3) From Proposition 3.2 and Eqs. (6) and (9), we can easily derive
⎧ ∂ c∗ (e, w ) ⎨ < 0, ∂e ∗ ⎩ ∂ c (e, w ) > 0, ∂e
for 0 < ψ < 1, or ρ < 0, for ψ > 1, or 0 < ρ < 1.
This is reasonable since an increase in initial entropy level e has two different kinds of impact on the investor’s consumption behavior c∗(e, w), called the substitution effect and the income effect. As the investor becomes less confident in the reference model, she would like to immediately consume her wealth when she can (substitution effect). In contrast, a higher initial entropy makes her feel the wealth is less valuable, which leads to a decrease in current consumption (income effect). If the coefficient ψ of EIS is less than one, the investor reduces today’s consumption as model confidence increases, since the income effect dominates the substitution effect. If ψ is sufficiently big, the substitution effect dominates the income effect, so the consumption rate rises as the initial entropy is elevated. Fig. 2b shows the optimal consumption-to-wealth ratios as functions of initial entropy level e with various levels of EIS, ψ = 0.1, 0.3, 0.8, 1.15, and 3.0. The optimal consumption-to-wealth ratio is a decreasing and convex function when ψ < 1, while it is increasing and concave when ψ > 1. This is consistent to our analytical result in the previous paragraph. Also, the optimal consumption-to-wealth ratios in Fig. 2b with different levels of EIS converge to a certain level as initial entropy level approaches to its maximum, (μ − r )2 /(2δσ 2 ). The optimal consumption-to-wealth ratio with the maximal entropy level, −r ) c ∗ ( (μ , w) 2δσ 2 2
w
=
1 ( δ − r ρ ), 1−ρ
(10)
is calculated from the second boundary condition (8) and the semi-analytic formula (9) for the optimal consumption. When the risk free rate r and the subjective discount rate δ is assumed to be same, the optimal consumption-to-wealth ratio with the maximal entropy level is exactly the level of risk-free rate r, and is independent of EIS level.4
4 Figure E.2 in online Appendix E shows that, when subjective discount rate differs from risk-free rate, the optimal consumption-to-wealth ratio with the maximal entropy level depends on the level of EIS.
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Fig. 2. The optimal consumption-to-wealth ratio c∗/w as a function of initial entropy e.
4.2. Optimal asset demands Using the optimal investment ratio πM = (μ − r )/((1 − α )σ 2 ) of the classical (Merton, 1971) problem, then
πM α f (e ) { f ( e )}2
=1+ − . ∗ π (e ) 1 − α f (e ) f (e ) f (e )
This relationship and the properties of f lead to the fact that π M /π ∗ (e) > 1 for all positive γ = 1 (or equivalently all α = 0) and all e. This implies that the investor in our model takes smaller positions in the stock market compared with the case where she does not care about model misspecification of stock returns. This fact might provide us with an (partial) explanation for the equity premium puzzle: concerns about model misspecification of stock returns might hamper investor’s stock investment. Fig. 3 shows a drastic reduction of the investor’s stockholdings according to the decrease in her confidence in the reference model.5 This result was also obtained by Maenhout (2004), but our model is faithfully based on the classical robust optimization without modification of the relative entropy. Another noticeable result is associated with the relationship between the investor’s EIS level and her optimal asset demands. Optimal investment behavior of the investor in our model could vary across a range of EIS coefficients even under
5
As well, the optimal investment proportion decreases with RRA level for any levels of initial entropy. (Fig. 3a)
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Fig. 3. The optimal investment proportions π ∗ as a function of initial entropy e.
a constant investment opportunity. (Fig. 3b) The optimal risky investment slightly changes when the investor is moderately concerned about model misspecification. This can be intuitively explained as Bhamra and Uppal (2006) do. The investor chooses the worst-case scenario h∗ which is continuously adjusted by the market risk Bt , a random variable. When the investor makes the wealth-maximization decision, she considers μ − h∗ , a random variable, as the expected rate of return on the risky asset. This makes the decision maker’s worst-case investment opportunity set stochastic, so the portfolio rate also depends on the level of EIS. Based on the same rationale, the following observation can be also explained: the optimal investment ratio is not affected by the level of EIS when the initial entropy is the minimum (e = 0) or the maximum (e = (μ − r )2 /(2δσ 2 )). In each case, the expected rate of return on the risky asset is considered to be a constant, μ or 0, respectively. Therefore, the investor’s asset demand is independent of the level of EIS as Svensson (1989) asserts. Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 4. h∗ as a function of initial entropy e when γ = 5 and ψ = 0.3. The parameters are μ = 9.32%, r = 5.75%, δ = 10.54% and σ = 17.95%.
4.3. Behaviors for the worst-case scenario In our model, the worst-case scenario is determined by control variables h and g, which are associated with the drift and volatility of the continuation entropy et . This implies that the investor establishes her worst outlook of the stock market by controlling the first and the second moments of the continuation entropy, or, equivalently, her confidence in the reference model. In optimum, they are described as
Vw f (e ) ( π ∗ wσ ) = α π ∗ ( e ) σ and Ve f (e ) Vew f (e ) g∗ = − (π ∗ wσ ) = −α π ∗ (e )σ . Vee f (e )
h∗ =
Not surprisingly, h∗ and g∗ are independent of the investor’s initial wealth level w. She does not care about her wealth level when considering the worst-case scenario. Fig. 4 represents a decreasing and concave function h∗ (e). As we see in the figure, the quantity h∗ is negative for all e > 0, so the expected return of the stock (μ + h∗ (e )σ ) should always be less than the return μ in the reference model. In particular, it is equal to the risk-free rate r when the initial continuation entropy e goes to the maximum value, in other words, the absolute value of h∗ becomes the Sharpe ratio (μ − r )/σ of the stock if the investor does not rely on the reference model at all. For this case, the optimal investment strategy is obviously to invest all of her wealth in the bond (thus, π ∗ = 0), because the risky stock does not give any risk premium to the market. Fig. 5 also has two implications. First, the sensitivity g∗ (e) of the investor’s confidence level with respect to the market risk dBt decreases with initial entropy level e. This is related to the decrease in the optimal investment rate π ∗ (e) in Fig. 3. As initial entropy level e grows up, the investor intentionally reduces her exposure to the market risk dBt . Consequently, the impact of the market risk on the optimal behaviors, including g∗ (e), decreases. Second, the investor conducts more robust, or conservative, decision making when she meets positive news dBt > 0 whereas she makes decisions more aggressively when she meets negative news dBt < 0. Such an implication is based on the following observation: the instantaneous correlation g∗ (e) between the change of the continuation entropy det and the return rate of the risky asset dSt /St is positive for all e. This observation can be intuitively explained as follows. When the return rate of the risky asset is unexpectedly high, i.e. dBt > 0, the wealth of the investor also increases unexpectedly. As a result, the marginal utility of the investor decreases, so she does not need to hold the risky portfolio that much any more. Rather, the investor may tend to make her investment more robust in order to protect her elevated wealth level. On the other hand, when the investor face an unexpected loss from a negative market shock dBt < 0, she may want to invest more in the risky asset in order to compensate for her loss with the high return rate of the asset. 5. Conclusion We generalize the problem of Hansen et al. (2006) with the Epstein-Zin type utility and a new state variable, continuation entropy, which measures the investor’s confidence level on the reference model. Applying the dynamic programming approach, we find that the value function can be expressed as a generalized version of that of Merton (1971). Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
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Fig. 5. g∗ as a function of initial entropy e when γ = 5 and ψ = 0.3. The parameters are μ = 9.32%, r = 5.75%, δ = 10.54% and σ = 17.95%.
Our numerical analysis explicitly shows many interesting results related to the robust optimal consumption and investment problem of Hansen et al. (2006). The robust optimal behaviors of the investor have different values according to different levels of her confidence in the reference model. We find that the investor’s optimal consumption rate can increase even though her concern about model misspecification increases when her EIS level is high enough. We also show that the investor’s investment proportion in a risky asset might drastically decrease when she significantly distrusts her reference model, which can be an (partial) explanation for the equity premium puzzle, as Maenhout (2004) does. More interestingly, our results show that the investor’s optimal asset demands could be dependent on the coefficient of EIS as well as on the coefficient of risk aversion since she makes consumption and investment decisions as if she is under a stochastic investment opportunity. This numerical result implies that our model can be a compromise between the arguments of Svensson (1989) and Bhamra and Uppal (2006). This research work seems to be important in that our model can be used in any asset pricing or asset allocation models which incorporate a time-varying level of model uncertainty, or the continuation entropy, which is not artificially modified. Acknowledgements We would like to thank Hyeng Keun Koo and Gyoocheol Shim for helpful comments. A preliminary version of the paper was presented at the 2009 APK conference, the 2009 SSS conference, the 2010 World Congress of the BFS, the 2011 KFA conference, the 2011 KDA conference, and the 4th KOFES conference. This research was supported by Global PH.D Fellowship Program through the National Research Foundation of Korea(NRF) funded by the Ministry of Education (2011-0 0 08628). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIP) (No. 2013R1A2A2A03068890). This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government (2014S1A3A2036037, NRF-2014S1A5A8018920, and NRF-2013S1A5A8024023). Supplementary material Supplementary material associated with this article can be found, in the online version, at doi: 10.1016/j.frl.2016.05.012. References Anderson, E., Hansen, L., Sargent, T., 2003. A quartet of semigroups for model specification, robustness, prices of risk, and model detection. J. Eur. Econ. Assoc. 1 (1), 68–123. Anderson, E.W., Ghysels, E., Juergens, J.L., 2009. The impact of risk and uncertainty on expected returns. J. Financ. Econ. 94 (2), 233–263. Barillas, F., Hansen, L.P., Sargent, T.J., 2009. Doubts or variability? J. Econ. Theor. 144 (6), 2388–2418. Ben-Tal, A., Nemirovski, A., 1998. Robust convex optimization. Math. Oper. Res. 23 (4), 769–805. Bhamra, H., Uppal, R., 2006. The role of risk aversion and intertemporal substitution in dynamic consumption-portfolio choice with recursive utility. J. Econ. Dyn. Control 30 (6), 967–991. Branger, N., Larsen, L., Munk, C., 2012. Robust portfolio choice with ambiguity and learning about return predictability. J. Bank. Finance 37, 1397–1411. Chen, H., Joslin, S., Tran, N.-K., 2012. Rare disasters and risk sharing with heterogeneous beliefs. Rev. Financ. Stud. 25 (7), 2189–2224. Croce, M.M., Nguyen, T.T., Schmid, L., 2012. The market price of fiscal uncertainty. J. Monet. Econ. 59 (5), 401–416. Easley, D., O’Hara, M., 2010. Microstructure and ambiguity. J. Finance 65 (5), 1817–1846. Easley, D., O’Hara, M., Yang, L., 2014. Opaque trading, disclosure, and asset prices: implications for hedge fund regulation. Rev. Financ. Stud. 27 (4), 1190–1237.
Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012
JID: FRL
ARTICLE IN PRESS B.-G. Jang et al. / Finance Research Letters 000 (2016) 1–11
[m3Gsc;May 24, 2016;7:43] 11
Ellsberg, D., 1961. Risk, ambiguity, and the savage axioms. Quart. J. Econ. 75, 643–669. Epstein, L., Schneider, M., 2008. Ambiguity, information quality, and asset pricing. J. Finance 63 (1), 197–228. Epstein, L.G., Schneider, M., 2003. Recursive multiple-priors. J. Econ. Theory 113 (1), 1–31. Epstein, L.G., Zin, S.E., 1989. Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57, 937–969. Fleming, W.H., Souganidis, P.E., 1989. On the existence of value functions of two-player, zero-sum stochastic differential games. Indiana Univ. Math. J. 38. Franco, D., O’Regan, D., 2003. Existence of solutions to second order problems with nonlinear boundary conditions. In: Proceeding of the Fourth International Conference on Dynamical Systems and Differential Equations, Discrete and Continuous Dynamical Systems, pp. 273–280. Garlappi, L., Uppal, R., Wang, T., 2007. Portfolio selection with parameter and model uncertainty: a multi-prior approach. Rev. Financ. Stud. 20 (1), 41–81. Gilboa, I., Schmeidler, D., 1989. Maxmin expected utility with non-unique prior. J. Math. Econ. 18 (2), 141–153. Glasserman, P., Xu, X., 2013. Robust portfolio control with stochastic factor dynamics. Oper. Res. 61 (4), 874–893. Hansen, L., Sargent, T., 2001. Robust control and model uncertainty. Am. Econ. Rev. 91 (2), 60–66. Hansen, L., Sargent, T., 2011. Robustness and ambiguity in continuous time. J. Econ. Theory 146 (3), 1195–1223. Hansen, L., Sargent, T., Turmuhambetova, G., Williams, N., 2006. Robust control and model misspecification. J. Econ. Theory 128 (1), 45–90. Illeditsch, P.K., 2011. Ambiguous information, portfolio inertia, and excess volatility. J. Finance 66 (6), 2213–2247. Kung, H., Schmid, L., 2015. Innovation, growth, and asset prices. J. Finance 70 (3), 1001–1037. Liu, H., Miao, J., 2015. Growth uncertainty, generalized disappointment aversion and production-based asset pricing. J. Monet. Econ. 69, 70–89. Liu, J., Pan, J., Wang, T., 2005. An equilibrium model of rare-event premia and its implication for option smirks. Rev. Financ. Stud. 18 (1), 131–164. Maenhout, P., 2004. Robust portfolio rules and asset pricing. Rev. Financ. Stud. 17 (4), 951–983. Merton, R., 1971. Optimum consumption and portfolio rules in a continuous-time model. J. Econ. Theory 3, 373–413. Skiadas, C., 2003. Robust control and recursive utility. Finance Stochast. 7 (4), 475–489. Svensson, L., 1989. Portfolio choice with non-expected utility in continuous time. Econ. Lett. 30 (4), 313–317. Uppal, R., Wang, T., 2003. Model misspecification and underdiversification. J. Finance 58 (6), 2465–2486. Vissing-Jorgensen, A., 2002. Limited Asset Market Participation and the Elasticity of Intertemporal Substitution. Technical Report. National Bureau of Economic Research.
Please cite this article as: B.-G. Jang et al., Robust consumption and portfolio rules with time-varying model confidence, Finance Research Letters (2016), http://dx.doi.org/10.1016/j.frl.2016.05.012