Perturbative Approaches for Robust Intertemporal Optimal Portfolio Selection F. Trojani and P. Vanini ECAS Course, Lugano, October 7-13, 2001
1 Contents Introduction Merton s Model and Perturbative Solution Approach Preferences for Robustness Perturbative Solutions Some Explicit Computations Conclusions, Further Research
2 Introduction, Motivation Only a few intertemporal optimal portfolio problems can be solved explicitly (cf. Kim and Omberg (1996), Chacko and Viceira (1999)) Existence of closed-form solutions depends on assumptions regarding agent s utility functions, the opportunity set dynamics, intermediate consumption, aversion to model misspecification... Perturbation Theory (PT) is based on approximation methods by which approximate analytical expressions can be achieved Kogan and Uppal (2000): PT is a powerful approximation method for financial optimal decision making also
2.1 Preferences for Robustness (I) Agents have a reference model in mind which describes the approximateprobabilisticfeaturesofsomeunderlyingstatevariablesprocess; AHS (2000), Maenhout (1999), Lei (2001), Trojani and Vanini (2001), Uppal and Wang (2001) Agents believe in the possibility that the benchmark model could be slightly misspecified Model deviations that are particularly different from the reference model are penalized in their impact on the final decision The entity of the penalization is parameterized by a parameter that is interpreted as the strength of a preference for robustness
2.2 Preferences for Robustness (II) Differences through the way by which model deviations are penalized AHS (2000): penalizes deviations proportionally to their relative entropy w.r.t. the reference model Meanhout (1999): penalizes relative entropy, however in a way that is scaled by the current level of indirect utility AHS (1998), Lei (2001), Trojani and Vanini (2001): put a maximal bound on the relative entropy distance of a relevant candidate misspecification The first two approaches produce second order, the third first order risk aversions
3 Merton s Model ³ Z P t, ³ Z X t standard BM in R with covariance ρ Price, state and wealth dynamics db t = r t B t dt, dp t = α t P t dt + σ t P t dzt P dx t = ζ t dt + ξ t dzt X dw t = [w t W t (α t r t )+(r t W t C t )] dt + w t W t σ t dzt P Utility and implied objective function u (C t )= Cγ t 1 γ, V γ (W, X) =E Z 0 e δt u (C t ) dt
3.1 Vector Notation Covariance matrix Σ t of (dx t,dw t ) 0 canbefactorizedasσ t = Λ t Λ 0 t, where Λ t = ξ t 0 ³ 2 12 ρw t W t σ t 1 ρ w t W t σ t µ 0, Orthogonalization Z t = Zt X,ZX t Z P t = ρzt X + ³ 2 12 1 ρ Zt X, gives for the vector valued state variable Y t =(X t,w t ) 0 dy t = µ t dt + Λ t dz t, where µ t =(ζ t,w t W t (α t r t )+(r t W t C t )) 0
3.2 Optimization Problem 1. HJB equation ³ c := W C 0 = sup {u (cw ) δj + A W J + A X J + wwρσξj WX } c,w A W = (r + w(α r) C) W W + 1 2 w2 σ 2 W 2 2 W 2 A X = ζ X + 1 2 ξ2 2 X 2 2. Homogeneous functional form for candidate solution: ³ e g(γ,x) W γ 1 J(W, X) = 1 δ γ
3.3 Perturbative Approach First order expansion J(W, X) = ³ 1 e g(γ,x) W γ 1 1 δ γ γ 0 δ (ln(w )+g 0(X)) g(γ,x) = g 0 (X)+γg 1 (X)+O(γ 2 ) Approximate optimal Policies c(x) = w(x) = = µ 1 δ eγg(γ,x) 1 γ 1 = δ (1 γ(g0 (X) ln (δ))) + O ³ γ 2 Ã 1 α r 1 γ σ 2 + γ g(γ,x)! ρξ X σ Ã 1 α r 1 γ σ 2 + γ g! 0(X) ρξ + O ³ γ 2 X σ
3.4 Remarks g 1 can be neglected in first order analysis g 0 is sufficient to determine the optimal policies up to first order in γ g 0 is obtained from the solution of a log utility agent g 0 (X) =ln(δ) 1+E Z e δt 0 r t + 1 2 Ã αt r t σ t and is typically easier to compute than the function g.! 2 dt
3.5 Summary on Kogan and Uppal (2000) Approach 1. Parameterize the problems under scrutiny and identify a specific parameter value for which the solution is known explicitly 2. Determine a functional form for the solution, such that to first order only the solvable benchmark model enters in the optimality conditions 3. Compute the optimal policies using the functional form of step 2 4. Expand the optimal policies to first order and determine the value function for the explicitly solvable model
4 Introducing Preferences for Robustness Step 1: Define the reference model for asset prices and state dynamics Step 2: Define the candidate model misspecifications Step 3: Define the relevant model misspecifications Step 4: Solve a max-min expected utility problem
4.1 Reference Model ³ Z P t, ³ Z X t standard BM in R with covariance ρ Price, state and wealth dynamics db t = r t B t dt, dp t = α t P t dt + σ t P t dzt P dx t = ζ t dt + ξ t dzt X dw t = [w t W t (α t r t )+(r t W t C t )] dt + w t W t σ t dzt P Orthogonalization Z t = gives µ Z X t,zx t dy t = µ t dt + Λ t dz t, 0, Z P t = ρz X t + ³ 1 ρ 2 12 Z X t
4.2 Model Misspecifications Model contaminations ν =(ν t ) t 0 are modelled as absolutely continuous changes of measure: where Z t = ν t =exp µ Z X t,zx t µ Z t Z t h s dz s 1 0 2 h s 2 ds, 0 0 and for a suitable process (hs )= ³ h X s,h P s 0 By Girsanov Theorem, agents are thus concerned only with misspecifications in the drift of risky assets and state dynamics
4.3 Relative Entropy as a Measure of Model Discrepancy Relative entropy at time t I t (ν) =E (ν t ln (ν t )), Continuous-time relative entropy d dt I t (ν) = 1 2 h0 th t.
4.4 Max Min Expected Utility Problem Optimization problem of a robust agent J (W, X) = R sup C,w inf h E h 0 e δtcγ t 1 γ dt 1 2 h0 h η. A preference for robustness is modelled by a bound η on the rate at which continuous time relative entropy can increase over time Larger η s represent larger preferences for robustness
4.4.1 Worst Case Scenario Infimization w.r.t. h yields the implied worst case model (cf. also AHS (1998) Ã!1 2η 2 h = ξj X + ρwwσj W ³ Γ (w) 2 12, 1 ρ wwσj W where Γ (w) =w 2 W 2 σ 2 J 2 W + ξ2 J 2 X +2wW ρξσj W J X
4.4.2 Single-Agent Optimization Problem HJB equation (c := C W ) 0 = sup c,w ½ u (cw ) δj + A W J + A X J + wwρσξj WX (2ηΓ (w)) 1 ¾ 2 Homogeneous functional form ³ e g(γ,η,x) W γ 1 J(W, X) = 1 δ γ
4.4.3 Robust Optimal Policies Optimal consumption and risky asset allocation c = w = Ã e γg δ! 1 γ 1, 1 1 ³ 1 2η 2 JW 2 Γ J WW 1 (1 γ) α r σ 2 + γ g X µ 2η Γ 1 2 JX ρξ σ. The functional form of c is the same as in the non robust case w is characterized by the solution of an implicit equation through the function Γ (w)
5 Perturbative Optimal Policies First order expansion in γ and η 1 2: w (X) = α r σ 2 + γw 1 (X)+(2η) 1 2 w 2 (X)+O 2 γ, η 1 2 g (X) = g 0 (X)+γg 1 (X)+(2η) 1 2 g 2 (X)+O µγ, 2 η 1 2 µ., Optimal policies, G 0 (X) = ³ α r σ 2 + ³ ξ g 0 X c (X) = δ (1 γ (g 0 (X) ln (δ))), w (X) = 1 Ã 1 γ ³ 2η G 0 1 2! α r σ 2 + 2 ³ +2 α r σ ρξ g 0 X : γ Ã 2η G 0!1 2 g 0 X ρξ σ
5.1 Remark 1 To γ, η 1 2 first order, robustness influences both the myopic and the hedging demand for risky assets µ 1 α r w (X) = µ 1 1 γ 2η 2 σ 2 G 0 (X) {z } MD µ 1 γ 2η 2 G 0 (X) + γ µ 2η 1 2 g 0 X ρξ σ 1 G 0 (X) {z } HD
5.2 Remark 2 The robust risky allocation is the portfolio strategy of an investor with a state dependent effective risk aversion 1 γ µ 2η G 0 (X) 1 2 The state dependent effective risk aversion correction depends on the state X only through the risk factors φ = α r σ and ψ = ξ g 0 X The largest relative portofolio corrections are realized when φ, ψ 0, in a neighborhood of the origin in (φ, ψ)-space Robustness affects optimal portfolios precisely when the standard myopic and intertemporal demands for risky assets are small, that is when risk exposure is low.
6 Some Explicit Computations Version of Kim and Omberg s (1996) model allowing for intermediate consumption db t = rb t dt, dp t = α t P t dt + σp t dzt P, dx t = λ ³ X X t dt + ξdz X t, where r, σ, ξ, λ, X > 0, and α t = r + σx t. In this model the market price of risk α t r σ process is an Ornstein Uhlembeck
6.1 Computation of g 0 Solution g 0 = a 0 + a 1 X + 1 2 a 2X 2, a 0 = ln(δ) 1+ r δ + ξ 2 2δ (δ +2λ) + λx a 1 = (δ + λ)(δ +2λ) > 0, 1 a 2 = δ +2λ > 0. ³ λx 2 δ (δ + λ)(δ +2λ), In this model the two relevant risk factors φ = α r σ, ψ = ξ g 0 X are perfectly correlated
6.1.1 Sketch of the Proof. g 0 (X) = ln(δ) 1+E Since E ³ X 2 t = ln(δ) 1+ X 0 = X " Z Ã e δt 0 Z 0 r + 1 µ αt r dt 2 σ X0 = X e µr δt + 1 2 E ³ Xt 2 X 0 = X dt. 2! = Var(X t X 0 = X)+(E (X t X 0 = X)) 2 # = ξ 2 1 e 2λt 2λ a final integration gives the result. + h e λt ³ X + X ³ e λt 1 i 2,
6.2 Optimal Policies Proposition 6.2 In the given model the following first order optimal policies hold true for a robust agent µ µ c (X) = δ 1 γ a 0 + a 1 X + 1 2 a 2X 2 ln (δ) w (X) = where 1 Ã 1 γ ³ 2η G 0 1 2! X σ + γ Ã 2η G 0!1 2 ρξ (a 1 + a 2 X) σ G 0 = ³ ξ 2 a 2 2 +2ρξa 2 +1 X 2 +2a 1 ξ (ρ + ξa 2 ) X + ξ 2 a 2 1
6.3 Conclusions, Further Research PT provides analytical solutions for consumption/portfolio problems where otherwise only numerical results are available Robustness affects basically only the investment side, by reducing both the myopic and the intertemporal exposure to risky assets CR induces significant portfolio modifications already for low risk exposures Work in progress: Relation between MR and ER in the intertemporal setting, impact of the investment horizon (see also Barberis (1999))