Slides 4 Matthieu Gomez Fall 2017
Portfolio Problem
Optimization A typical optimization problem has the form { + } J(x t ) = maxe t e ρ(τ t) h(u τ )dτ u t t s.t. dx = µ(x, u)dt + σ(x, u)dz t Bellman s principle of optimality: HJB equation Heuristic derivation 0 = max u t { h(u t )dt + E t [dj(x t )] ρj t dt } J t = max u t {h(u t ) t + e ρ t E[J t+ t ]} J t = max u t {h(u t ) t + (1 ρ t)e[j t+ t ]} 0 = max{h(u t ) t + E[J t+ t J t ] ρj t t} u t 0 = max u t { } h(u t )dt + E t [dj(x t )] ρj t dt 1
Investment Opportunities. Suppose the return of an investment is given by dr t = µ R (s)dt + σ R (s)dz t ds t = µ s(s)dt + σ s(s)dz t risky asset state variable In all generality, dz t is a vector of shocks of size n z, ds t is a vector of state variables of length n s, and R t is a vector of returns of size n R. We take n s = 1 to simplify It corresponds to the following system and dr 1t = µ R1 dt + σ R11 dz 1 + + σ R1m dz m. dr nt = µ Rn dt + σ Rn1 dz 1 + + σ Rnm dz m ds t = µ sdt + σ Rn1 dz 1 + + σ Rnm dz m Complete markets iff σ R is a matrix of full row rank (i.e. has rank n z) 2
Investor Problem The investor problem is { + } J(W, s) = max E t e ρ(τ t) U(C)dτ c,α t dw s.t. W = (rdt + α (dr rdt) Cdt W ) W 0 > 0 W t > 0 C t > 0 3
HJB Using Ito s lemma: E[dJ(W, s)] = J W EdW + J seds + 1 2 J WWσ(W)σ(W) W 2 dt + J Ws Wσ(W)σ(s) dt + 1 2 Jssσ(s)σ(s) dt Plugging that into the Bellman equation 0 = max C,α {U(C, t) + J W ( r + α (µ R (s) r(s)) C W ) W + J sµ(s) 1 2 J WWα 2 σ R (s)σ R (s) W 2 + J Ws Wασ R (s)σ s(s) + 1 } 2 Jssσs(s)σs(s) ρj (1) 4
HJB The FOCs are: [C] :U c = J w (2) [α] :WJ W (µ R (s) r(s)) + W 2 J WW ασ R (s)σ R (s) + WJ Ws σ R (s)σ s(s) = 0 (3) 5
HJB FOC gives C = U 1 C (J W) α = J W µ R (s) r(s) WJ WW σ R (s)σ R (s) J WS σ R (s)σ S (s) WJ WW σ R (s)σ R (s) }{{}}{{} myopic demand int. hedging demand The intertemporal hedging demand is nonzero as long as: σ S > 0: Investment opportunities are time-variant. σ R σ S 0: Shocks to state variables are correlated with instantaneous returns on the risky asset. J WS 0: Shocks to state variables affect marginal utility. Last step: plug solutions for C and α in Equation (8) to obtain a PDE of J in term of W and S 6
CRRA utilities. From now on, assume CRRA utilities U(C) = C1 γ 1 γ In this case, guess J(W, s) = (Wξ(s))1 γ 1 γ ξ can be seen as a networth multiplier The FOCs are: [C] : C = ξ(s) 1 1 γ W (4) [α] : α = 1 µ R (s) r(s) γ σ R (s)σ R (s) + 1 γ ξ (s) γ ξ }{{} myopic demand σ R (s)σ S (s) σ R (s)σ R (s) } {{ } int. hedging demand Plugging FOC into HJB, we obtain an ODE for ξ that only depends on s In particular, W does not appear in the ODE = validates our guess (5) 7
CAPM The FOC for α is α = 1 µ R (s) r(s) γ σ R (s)σ R (s) + 1 γ σ R (s)σ ξ (s) γ σ R (s)σ R (s) }{{}}{{} myopic demand int. hedging demand Denote µ ξ and σ ξ the geometric drift and volatility of ξ t, i.e. dξ t ξ t = µ ξt dt + σ ξt dz t Inverting the FOC to get µ R r, we obtain the CAPM: µ R r = γσ R σ W (1 γ)σ R (s)σ ξ (s) The FOC for consumption gives σ C = (1 1 γ )σ ξ + σ W Therefore µ R r = γσ R σ C 8
SDF in Continuous Time
SDF in Discrete Time The pricing equation + P t = E t m t,t+τ D τ t+1 m t,t+τ = m t,t+1... m t+τ 1,t+τ That is P t = E[m t,t+1 (D t+1 + P t+1 )] The instantaneous return is R t+1 = D t+1 + P t+1 P t Pricing equation can be written in term of returns: E[m t,t+1 R t+1 ] = 1 9
SDF in Continuous Time The pricing equation Taking the derivative, we obtain P t = E t + t Λ τ Λ t D τ dτ Dividing by Λ t P t and applying Ito s Lemma we obtain E[d(Λ t P t )] = Λ t D t dt (6) E[ dλ t Λ t ] + E[ dp t P t ] + σ Pt σ Λt = D t P t dt (7) where σ Pt (resp σ Λt ) denotes the geometric volatility of P t (resp Λ t ) The instantaneous return is defined as Plugging into Equation (7), we obtain dr t = D tdt + dp t P t E[dR t ] = E t [ dλ t Λ t ] σ Λ σ R dt 10
SDF in Continuous Time Applying this equation with instantaneous risk free rate r t we obtain r t = E t [ dλ t Λ t ] Denote market price of risk κ defined as E[dR t ] = r t dt + κ t σ Rt dt Or in other words κ t = σ Λt That is, dλ t Λ t = r t dt κ t dz t 11
Obtaining Directy the Euler Equation We can express the SDF in term of consumption process, similarly to discrete time derivation Start from HJB equation for an investor optimization ( 0 = max {U(C, t) + J W E[dR t ] C C W Deriving HJB wrt C, we obtain ) W + J sµ(s) 1 2 J WWσR 2 (s)w2 + J Ws Wσ R (s)σ s(s) + 1 } 2 Jssσ2 s (s) ρj (8) Deriving HJB wrt W, we obtain U (C) = J W 0 = E[dJ W ] + J W E[dR t ] + E[dJ W dr t ] ρj W (9) We conclude that Λ t = e ρt U (C t ) is the SDF since it verifies E[dR t ] = E[ dλ t Λ t ] E[ dλ t Λ t R t ] 12
Alternative Derivation Using SDF Remember that with CRRA utility, the SDF is Λ t = e ρt C γ t Denote dc t C t = µ C dt + σ C dz t By Ito dλ t Λ t = ρdt γµ C dt + (1 + γ)γ σc 2 2 dt Therefore the Euler equations are κ = γσ C r = ρ + γµ C (1 + γ)γ σc 2 2 13
Aside: Euler equations in term of law of motion of log consumption As an aside, we now derive formula for r and κ using the drift and volatility of log consumption Given the process for consumption dc t C t = µ C dt + σ C dz t The process for log-consumption is d ln C t = (µ C σ2 C 2 )dt + σ CdZ t Therefore, we can write the Euler equations as κ = γσ[d ln C t ] r = ρ + γe[d ln C t ] γ2 2 σ[d ln C t] 2 14