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Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications eo J´rˆme Detemple Boston University School of Management and CIRANO Marcel Rindisbacher Rotman School of Management, University of Toronto and CIRANO 1 1 Introduction Dynamic consumption-portfolio choice: • Merton (1971): optimal portfolio includes intertemporal hedging terms in addition to mean-variance component (diﬀusion) • Breeden (1979): hedging performed by holding funds giving best protection agst ﬂuctuations in state variable (diﬀusion) • Ocone and Karatzas (1991): representation of hedging terms using Malliavin derivatives (Ito, complete markets) → Interest rate hedge → Market price of risk hedge • Detemple, Garcia and Rindisbacher (DGR JF, 2003): practical implementation of model (diﬀusion, complete markets) → Based on Monte Carlo Simulation → Flexible method: arbitrary # assets and state variables, non-linear dynamics, arbitrary utility functions → Extends to incomplete/frictional markets (DR MF, 2005) 2 Contribution: • New decomposition of optimal portfolio (hedging terms): → Formula rests on change of num´raire: use pure discount bonds as e units of account → Passage to a new probability measure: forward measure (Geman (1989) and Jamshidian (1989)) → General context: Ito price processes, general utilities 3 • New economic insights about structure of hedges: → Utility from terminal wealth: hedge · ﬂuctuations in instantaneaous price of long term bond with maturity date matching investment horizon · ﬂuctuations in future bond return volatilities and future market prices of risk (forward density) · ﬁrst hedge has a static ﬂavor (static hedge) → Utility from terminal wealth and intermediate consumption · static hedge is a coupon-paying bond, with variable coupon payments tailored to consumption needs → Risk aversion properties: · if risk aversion approaches one both hedges vanish: myopia · if risk aversion becomes large mean-variance term and second hedge vanish: holds just long term bonds · if risk aversion vanishes all terms are of ﬁrst order in risk tolerance. → Non-Markovian N + 2 fund separation theorem. 4 • Technical contribution: → Exponential version of Clark-Haussmann-Ocone formula → Identiﬁes volatilities of exponential martingale in terms of Malliavin derivatives → Malliavin derivatives of functional SDEs → Explicit solution of a Backward Volterra Integral Equation (BVIE) involving Malliavin derivatives. 5 Applications: • Preferred habitat • Extreme risk aversion behavior • International asset allocation • Preferences for I-bonds • Integration of risk management and asset allocation Road map: • Model with utility from terminal wealth • The Ocone-Karatzas formula • New representation • Intermediate consumption • Applications • Conclusions 6 2 The Model Standard Continuous Time Model: • Complete markets and Ito price processes • Brownian motion W , d-dimensional • Flow of information Ft = σ(Ws : s ∈ [0, t]) • Finite time period [0, T ]. • Possibly non-Markovian dynamics 7 Assets: Price Evolution • Risky assets (dividend-paying assets): dSti i i i = rt − δt dt + σt (θt dt + dWt ) , S0 given Sti ∗ i σt : volatility coeﬃcients of return process (1 × d vector) ∗ rt : instantaneous rate of interest ∗ i δt : dividend yield ∗ θt : market prices of risk associated with W (d × 1 vector) ∗ (r, δ, σ, θ): progressively measurable processes; standard integrability conditions • Riskless asset: ∗ pays interest at rate r 8 Investment and Wealth: • Portfolio policy π: d-dimensional, progressively measurable; integrability conditions → amounts invested in assets: π → amount in money market: X − π 1 • Wealth process: dXt = rt Xt dt + πt σt (θt dt + dWt ) , subject to X0 = x. • Admissibility: π is admissible (π ∈ A) if and only if wealth is non-negative: X ≥ 0. 9 Asset Allocation Problem: • Investor maximizes expected utility of terminal wealth: maxπ∈A E [U (XT )] • Utility function: U : R+ → R → Strictly increasing, strictly concave and diﬀerentiable → Inada conditions: limX→∞ U (X) = 0 and limX→0 U (X) = ∞ 1 • Includes CRRA U (x) = 1−R X 1−R where R > 0. • Property: → Strictly decreasing marginal utility in (0, ∞) → Inverse marginal utility I (y) exists and satisﬁes U (I (y)) = y → Derivative: I (y) = 1/U (I(y)) 10 3 The Optimal Portfolio Complete Markets: • Market price of risk: θt = (θ1t , ..., θdt ) • State price density: v v ξv ≡ exp − 0 rs + 1 θs θs ds − 2 0 θs dWs → converts state-contingent payoﬀs into values at date 0 • Conditional state price density: v v ξt,v ≡ exp − t rs + 1 θs θs ds − 2 t θs dWs = ξv /ξt 11 Optimal Portfolio: Ocone and Karatzas (1991), Detemple, Garcia and Rindisbacher (2003) ∗ m r θ π t = πt + πt + π t where MV: πt = Et [ξt,T Γ∗ ] (σt )−1 θt m T πt = − (σt )−1 Et ξt,T (XT − Γ∗ ) r ∗ T IRH: T t Dt rs ds πt = − (σt )−1 Et ξt,T (XT − Γ∗ ) θ ∗ T MPRH: T t (dWs + θs ds) Dt θs ∗ • Optimal terminal wealth XT = I(y ∗ ξT ) • Constant y ∗ solves x = E [ξT I(y ∗ ξT )] (static budget constraint) • Γ (X) ≡ −U (X)/U (X): measure of absolute risk tolerance ∗ • Γ∗ ≡ Γ (XT ): risk tolerance evaluated at optimal terminal wealth T • Dt is Malliavin derivative 12 Structure of Hedges: πt = − (σt )−1 Et ξt,T (XT − Γ∗ ) r ∗ T IRH: T t Dt rs ds • Driven by sensitivities of future IR and MPR to current innovations in Wt . Sensitivities measured by Malliavin derivatives Dt rs and Dt θs ∗ • Sensitivities are adjusted by factor ξt,T (XT − Γ∗ ): depends on T preferences, terminal wealth and conditional state prices. • Optimal terminal wealth: I(y ∗ ξT ) • Date t cost: ξt,T I(y ∗ ξT ) = ξt,T I(y ∗ ξt ξt,T ) • Sensitivity to change in conditional SPD ξt,T ∂ (ξt,T I(y ∗ ξt ξt,T )) ∂ξt,T ∗ = I(y ∗ ξt ξt,T ) + y ∗ ξt ξt,T I (y ∗ ξt ξt,T ) = XT − Γ∗ T • Sensitivity of conditional SPD to ﬂuctuations in IR and MPR T T −ξt,T t Dt rs ds and − ξt,T t (dWs + θs ds) Dt θs . 13 Constant Relative Risk Aversion (CRRA) m πt Xt ∗ = 1 R (σt )−1 θt r ρ πt −1 ξT T ∗ Xt = −ρ (σt ) Et ρ Et [ξT ] t Dt rs ds θ ρ πt −1 ξT T Xt∗ = −ρ (σt ) Et ρ Et [ξT ] t (dWs + θs ds) Dt θs • ρ = 1 − 1/R ρ R • y∗ = E ξT /x ∗ • Xt = Et ξt,T (y ∗ ξT )−1/R • Hedging terms are weighted averages of the sensitivities of future interest rates and market prices of risk to the current Brownian innovations. 14 4 A New Decomposition of the Optimal Portfolio 4.1 Bond Pricing and Forward Measures T Pure Discount Bond Price: Bt = Et [ξt,T ] Forward T -Measure: (Geman (1989) and Jamshidian (1989)) • Random variable: ξt,T ξt,T Zt,T ≡ Et [ξt,T ] = Bt T • Properties: Zt,T > 0 and Et [Zt,T ] = 1. Use Zt,T as density • Probability measure: dQT = Zt,T dP t → Equivalent to P 15 e Change of Num´raire: unit of account is T -maturity bond • Under QT price V (t) of a contingent claim with payoﬀ YT is t ξt,T T T V (t) = Et [ξt,T YT ] = Et [ξt,T ] Et Y Et [ξt,T ] T = Bt Et [YT ] T • Et [·] ≡ Et [Zt,T ·] is expectation under QT t T T • Martingale property: V (t) /Bt = Et [YT ] = Et [Zt,T YT ]. • Density Zt,T is stochastic discount factor: converts future payoﬀs into current values measured in bond unit of account. 16 Characterization (Theorem 2): The forward T -density is given by T 1 T Zt,T ≡ exp t σ Z (s, T ) dWs − 2 t σ Z (s, T ) σ Z (s, T ) ds • volatility at s ∈ [t, T ]: σ Z (s, T ) ≡ σ B (s, T ) − θs T • bond return volatility: σ B (s, T ) ≡ Ds log Bs Contribution(s): • Identify volatility of forward measure • Application of Exponential Clark-Haussmann-Ocone formula • Market price of risk in the num´raire e 17 4.2 Portfolio allocation and long term bonds An Alternative Portfolio Decomposition Formula: ∗ m b z π t = πt + πt + π t • Mean variance demand: πt = Et [Γ∗ ] Bt (σt )−1 θt m T T T • Hedge motivated by ﬂuctuations in price of pure discount bond with matching maturity πt = (σt )−1 σ B (t, T ) Et [XT − Γ∗ ] Bt b T ∗ T T • Hedge motivated by ﬂuctuations in density of forward T -measure πt = (σt )−1 Et [(XT − Γ∗ ) Dt log (Zt,T )] Bt . z T ∗ T T 18 e Essence of Formula: change of num´raire T • SPD representation: ξt,T = Bt Zt,T ∗ • Optimal terminal wealth: XT = I y ∗ ξt Bt Zt,T T • Cost of optimal terminal wealth: Bt Zt,T I y ∗ ξt Bt Zt,T T T • Hedging portfolio: Dt Bt Zt,T I y ∗ ξt Bt Zt,T T T • Chain rule of Malliavin calculus: → Zt,T I y ∗ ξt Bt Zt,T + Bt Zt,T I y ∗ ξt Bt Zt,T y ∗ ξt Zt,T Dt Bt T T T T → Bt I y ∗ ξt Bt Zt,T + Bt Zt,T I y ∗ ξt Bt Zt,T y ∗ ξt Bt Dt Zt,T T T T T T → Bt Zt,T I y ∗ ξt Bt Zt,T Bt Zt,T Dt (y ∗ ξt ) T T T 19 Long Term Bond Hedge: • Immunizes against instantaneous ﬂuctuations in return of long term bond with matching maturity date • Corresponds to portfolio that maximizes the correlation with long term bond return • This portfolio is a synthetic asset or maturity matching bond itself, if exists Forward Density Hedge: • Immunizes against ﬂuctuations in forward density Zt,T (instantaneous and delayed) • Source of ﬂuctuations are bond return volatilities and MPRs: σ Z (s, T ) ≡ σ B (s, T ) − θs • Dt σ Z (s, T ) = Dt σ B (s, T ) − Dt θs . 20 Remarks: • Generality of decomposition is remarkable: → Interest rate’s response to Brownian innovations has disappeared → Replaced by bond volatilities and MPRs → Surprising because inﬁnite dim. Ito processes: · Model for prices is not diﬀusion · Current bond prices are not suﬃcient statistics for IR evolution • Formula in spirit of immunization strategies sometimes advocated by practitioners → First term is static hedge: hedge against current ﬂuctuations in LT bond price → To ﬁrst approximation optimal portfolio has mean-variance term + static hedge → Additional hedge ﬁne tunes allocation: captures ﬂuctuations in future quantities → Static hedge is preference independent 21 Signing the Static Hedge: • Bond prices negatively related to IR • IR innovation negatively related to equity innovation • In one factor (BMP) model σ B > 0: boost demand for stocks 22 4.3 Constant Relative Risk Aversion Hedging Terms are: b πt ∗ Xt = ρ (σt )−1 σ B (t, T ) Bt T ρ−1 z πt −1 T Zt,T T Xt∗ = ρ (σt ) Et T ρ−1 Dt log (Zt,T ) Bt [ Et Zt,T ] Highlights knife-edge property of log utility (Breeden (1979)) • Logarithmic investor displays myopia (hedging demands vanish) • More (less) risk averse investors will hold (short) portfolio synthesizing long term bond • More (less) risk averse investors will hold (short) portfolio that hedges forward density → portfolio is individual-speciﬁc: depends on risk aversion of utility function 23 Literature: special cases of this result analyzed by • Bajeux-Besnainou, Jordan and Portait (2001): Vasicek short rate and constant market prices of risk. Forward density hedge vanishes • Lioui and Poncet (2001) and Lioui (2005): → Diﬀusion models with power utility. → Lioui and Poncet (2001): last hedging component in terms of unknown volatility function (PDE). → Lioui (2005): aﬃne model with mean-reverting IR and MPR processes. Forward density hedge is proportional to vector of volatilities of MPR with proportionality factor linear in MPRs. 24 Illustration: optimal stock-bond mix for CRRA investor • Model: → T −maturity bond is traded → Two assets: Stock and investment horizon matching bond stock σ stock σ1t 2t σt = B σ1t σ2tB • Optimal portfolio weight: static hedging component b πt 0 −1 σ B = ρ Xt = ρ(σt ) ∗ t 1 z • If πt ≈ 0 hedging is very simple: → no hedging component for stocks → hedging component does not depend on investment horizon → hedging portfolio only depends on relative risk aversion coeﬃcient → no need to estimate: if risk aversion is R = 4, then static hedging component for bonds is 0.75. 25 Illustration: Asset Allocation Puzzle – Asset Allocation Puzzle (see Canner, Mankiw and Weil (1997)): investment advisors typically recommend an increase in the bonds-to-equities ratio for more conservative investors while mean-variance portfolio theory predicts that a constant ratio is optimal. – Bonds-to-equities ratio in Gaussian terms structure models: B e (t, T ) = S θ2t +σ2t (Q(t,T )−1) σt σB θ −σB θ ∗ Q (t, T ) ≡ Et [XT ] /Et [Γ∗ ] T T T 2t 1t 1t 2t – For HARA utility u(x) = (x − A)1−R /1 − R, 0 ! !−1/R 1 T T t „ T ” « B0 h(0,t;R) B0 /B0 ABt t ρ “ T ∗ Q (t, T ; R) ≡ R @1 + T T A=R 1+ ∗ T B0 /B0 x/A−B0 Bt Zt Xt −ABt ∗ with “ Rt “ ” ” · h(0, t; R) ≡ exp (ρ/R) 0 1 θs + σ B (s, T ) 2 − σ B (s, T ) 2 ds . 2 – Bonds-to-equities ratio risk tolerance, at a given time t, if and only if Q (t, T ) is a monotone function of risk tolerance. 26 • In the presence of wealth eﬀect the bonds-to-equities ratio is not necessarily monotone in risk aversion Bonds−to−equities ratio: intolerance for shortfall 100 80 Bonds−to−equities ratio 60 40 20 0 4 3 5 4 2 3 1 2 1 Risk aversion parameter R 0 0 Forward density Z(t) – Vasicek interest rate model:r0 = r = 0.06, κr = 0.05, σr1 = −0.02, σr2 = −0.015 and market prices of risk are constants θs = 0.3 and θb = 0.15. The interest rate at t = 5 is rt = 0.02. Other parameter values are A = 200, 000, x = 100, 000 and T = 10. 27 5 Intermediate Consumption 5.1 The Investor’s Preferences Consumption-portfolio Problem: T maxπ,c∈A E 0 u (ct , t) dt + U (XT ) • Utility function: u (·, ·) : R+ × [0, T ] → R and bequest function: U : R+ → R satisfy standard assumptions • Maximization over set of admissible portfolio policies π, c ∈ A • Inverse marginal utility function J (y, t) exists: u (J (y, t) , t) = y for all t ∈ [0, T ] • Inverse marginal bequest function I (y) exists: U (I (y)) = y 28 5.2 Portfolio Representation and Coupon-paying Bonds Decomposition: ∗ m b z πt = π t + π t + π t • Mean variance demand: Et [Γ∗ ] Bt dv + Et [Γ∗ ] Bt (σt )−1 θt m T v v T T πt = t v T • Hedge motivated by ﬂuctuations in price of coupon-paying bond with matching maturity: (σt )−1 T B b πt = t σ (t, v) Bt Et [c∗ − Γ∗ ] dv v v v v + (σt )−1 σ B (t, T ) Bt Et [XT − Γ∗ ] T T ∗ T • Hedge motivated by ﬂuctuations in density of forward T -measure: (σt )−1 T z πt = t Et [(c∗ − Γ∗ ) Dt log Zt,v ] Bt dv v v v v + (σt )−1 Et [(XT − Γ∗ ) Dt log Zt,T ] Bt T ∗ T T 29 b Static Hedge πt : hedge against ﬂuctuations in value of coupon-paying bond • Coupon payments C (v) ≡ Et [c∗ − Γ∗ ] at intermediate dates v v v v ∈ [0, T ) ∗ • Bullet payment F ≡ Et [XT − Γ∗ ] at terminal date T T T • Coupon payments and face value are → time-varying → tailored to individual’s consumption proﬁle and risk tolerance • Bond value T v T B (t, T ; C, F ) ≡ t Bt C (v) dv + Bt F. • Instantaneous volatility T σ (B (t, T ; C, F )) B (t, T ; C, F ) = t v σ B (t, v) Bt C (v) dv T +σ B (t, T ) Bt F • Hedge: (σt )−1 σ (B (t, T ; C, F )) B (t, T ; C, F ) 30 z Forward Density Hedge πt : • Motivation: desire to hedge ﬂuctuations in forward densities Zt,v • Static hedge already neutralizes impact of term structure ﬂuctuations on PV of future consumption v • Given ξt,v = Bt Zt,v it remains to hedge ﬂuctuations in risk-adjusted discount factors Zt,v , v ∈ [t, T ]. Optimal Portfolio Composition: • To ﬁrst approximation optimal portfolio has mean-variance term + long term coupon bond hedge • Under what conditions is this approximation exact (i.e. last term vanishes)? • If last term does not vanish what is its size? 31 5.3 Constant Relative Risk Aversion Relative risk aversion parameters Ru , RU for utility and bequest functions. Portfolio: • Mean-variance term πt = (σt )−1 T 1 m t Ru v Et [c∗ ] Bt dv + v v 1 RU T Et ∗ T [XT ] Bt θt • Hedge motivated by ﬂuctuations in price of coupon-paying bond with matching maturity πt = (σt )−1 ρu T b t ∗ σ B (t, v) Bt Et [c∗ ] dv + ρU σ B (t, T ) Bt Et [XT ] v v v T T • Hedge motivated by ﬂuctuations in densities of forward measures ρu (σt )−1 T z πt = t Et [c∗ Dt log Zt,v ] Bt dv v v v −1 T ∗ T +ρU (σt ) Et [XT Dt log Zt,T ] Bt 32 Static Hedge has two parts: • Pure coupon bond (annuity) with coupon given by optimal consumption • Bullet payment given by optimal terminal wealth • Two parts are weighted by risk aversion factors ρu and ρU • Knife edge property traditionally associated with power utility function • Possibility of positive annuity hedge (Ru > 1) combined with negative bequest hedge (RU < 1). 33 6 Applications 6.1 Preferred Habitats and Portfolio Choice Preferred Habitat Theory Modigliani and Sutch (1966): • Individuals exhibit preference for securities with maturities matching their investment horizon • Investor who cares about terminal wealth should invest in bonds with matching maturity • Existence of group of investors with common investment horizon might lead to increase in demand for bonds in this maturity range • Implies increase in bond prices and decrease in yields. Explains hump-shaped yield curves with decreasing proﬁle at long maturities. 34 Formula shows that optimal behavior naturally induces a demand for certain types of bonds in speciﬁc maturity ranges ∗ m ∗ b z πt = wt (Xt − B (t, T ; C, F )) + wt B (t, T ; C, F ) + πt m wt = arg maxw {w σt θt : w σt σt w = k} . b wt = arg maxw {w σt σ (B (t, T ; C)) : w σt σt w = k} z πt = arg maxπ {π σt σ (t, T ) : π σt σt π = k} where T σt,T ≡ t Et [(c∗ − Γ∗ ) Dt log Zt,v ] Bt dv v v v v ∗ +Et [(XT − Γ∗ ) Dt log Zt,T ] Bt T T T 35 • Any individual has preferred bond habitat: → Optimal portfolio includes long term bond with maturity date matching the investor’s horizon → Preferred instrument is coupon-paying bond with payments tailored to consumption proﬁle of investor • Complemented by mean-variance eﬃcient portfolio to constitute the static component of allocation • Under general market conditions the static policy is ﬁne-tuned by dynamic hedge → When bond return volatilities and market prices of risk are deterministic, dynamic hedge vanishes 36 Equilibrium Implications • Existence of natural preferred habitats in certain segments of ﬁxed income market • Existence of equilibrium eﬀects on prices and premia in these habitats depends on characteristics of investors’ population • With suﬃcient homogeneity → Strong demand for structured ﬁxed income products might emerge → Prompt ﬁnancial institutions to oﬀer tailored products appealing to those segments → Yields to maturity would then naturally reﬂect this habitat-motivated demand 37 Motivation for preferred habitat here is diﬀerent from Riedel (2001) • In his model habitat preferences are driven by structure of subjective discount rates placing emphasis on speciﬁc future dates • In our setting preference for long term bonds emerges from the structure of the hedging terms • Optimal hedging combines static hedge (long term bond) with dynamic hedge motivated by ﬂuctuations in forward measure volatilities 38 6.2 Universal Fund Separation • Y : vector of N < d state variables with evolution described by the functional stoachastic diﬀerential equation dYt = µ(Y(·) )t dt + σ(Y(·) )t dWt • Suppose that v – Bt = B t, v, Y(·) – σ Z (t, v) = σ Z t, v, Y(·) e are Fr´chet diﬀerentiable functionals of Y(·) . •: Universal N + 1-fund separation holds: portfolio demands can be synthesized by investing in N + 2 (preference free) mutual funds: 1. riskless asset 2. the mean-variance eﬃcient portfolio 3. N portfolios (σt )−1 σt Y(·) Y to synthesize the static bond hedge and the forward density hedge. 39 6.3 Extreme Behavior Assume risk tolerances go to zero: • Intermediate utility and bequest functions: (Γu (z, v), ΓU (z)) → (0, 0) for all z ∈ [0, +∞) and all v ∈ [0, T ] • Relative behaviors: for some constant k ∈ [0, +∞): Γu (z1 ,v) ΓU (z2 ) →k for all z1 , z2 ∈ [0, ∞) and all v ∈ [0, T ] Γu (z1 ,v1 ) Γu (z2 ,v2 ) →1 for all z1 , z2 ∈ [0, ∞) and all v1 , v2 ∈ [0, T ] 40 Limit Allocations: coupon-paying bond with constant coupon C and face value F given by x x C= RT v T and F = RT v T . 0 B0 dv+B0 /k 0 B0 dvk+B0 • If k = 0 exclusive preference for pure discount bond, (C, F ) = 0, x/B0T • If k → ∞ preference is for a pure coupon bond, T v (C, F ) = x/ 0 B0 dv, 0 Limit Behavior: • Governed by relation between utility functions at diﬀerent dates • As risk tolerances vanish, preference for certainty: coupon-paying bond with bullet payment • Least extreme of the extreme behaviors drives the habitat: → Given a preference for riskless instruments: individuals puts more weight on maturities where risk tolerance is greater → Exhibits a time preference in the limit. 41 Illustration: CARA preferences Γu and ΓU constant, k ≡ Γu /ΓU . 1 • Slope of indiﬀerence curves: − dX dc = 1 k eX−c/k ΓU 140 Budget Constraint 120 k=1 100 80 X 60 40 20 0 0 20 40 60 80 100 120 c 42 • k→0 140 k=0.5 Budget Constraint 120 k=1 k=0 100 80 X 60 40 20 0 0 20 40 60 80 100 120 c 43 • k→∞ 140 Budget Constraint 120 k=1 100 80 X 60 k=10 40 20 k=infinity 0 0 20 40 60 80 100 120 c 44 Special case examined by Wachter (2002) • Arbitrary utility functions over terminal wealth and markets with general coeﬃcients • Documents emergence of preferred habitat when relative risk aversion goes to inﬁnity → Pure discount bond with unit face value and matching maturity • Our analysis shows that preferred habitat for an extreme consumer may take diﬀerent forms depending on nature of behavior → Pure discount bonds, pure annuities or coupon-paying bonds with bullet payments at maturity can emerge in limit. 45 Order of Convergence – As (Γu (z, v), ΓU (z)) → (0, 0), the limit portfolios ∗ πm = πz = 0 t t T v T ∗ π b = (σt ) 0 σ B (t, v)Bt dvC + σ B (t, T )Bt F t – have scaled asymptotic errors: ∗ α (ν) t = (Γν (·))−1 (πt − π α ) with α ∈ {m, b, z} and ν ∈ {u, U }, α t (σt )−1 θt m (U ), m (u)] T v T [ t t → t Bt dv Bt K − (σt )−1 b (U ), b (u) T B v t t → t σ (t, v) Bt dv T σ B (t, T ) Bt K − (σt )−1 t Nt,v Bt dv T [ z (U ), t z (u)] t → v T Nt,T Bt K – where ∗ Nt,τ is given by h“R ” i τ τ Nt,τ ≡ Et t 1 Rτ σ Z (r, τ ) dWr − 2 t σ Z (r, τ ) 2 dr (Dt log Zt,τ ) ∗ K is given by " # k 1 K≡ 1 1 k . 46 6.4 Term structure models and asset allocation Integration of term structure models and asset allocation models: • Forward rate representation of bonds v v s Bt = exp − t ft ds ∂ → Continuously compounded forward rate: fts ≡ − ∂v log (Bt ) v → Bond price volatility: v v v σ B (t, v) = Dt log Bt = − t Dt fts ds = − t σ f (t, s)ds → Volatility of forward rate: σ f (t, s) • Forward rate dynamics: → No arbitrage condition (HJM (1992)): dftv = σ f (t, v) dWt + θt − σ B (t, v) dt , v f0 given → Dynamics completely determined by forward rate volatility function and initial forward rate curve 47 Optimal Portfolio: previous formula with Rv “ “ Rv f ” ” “ Rv f ” Dt log Zt,v = t dWs + θs + s σ (s, u)du ds Dt θs + s Dt σ (s, u)du) • Forward density hedge in terms of forward rate volatilities • Useful for ﬁnancial institution using a speciﬁc HJM model to price/hedge ﬁxed income instruments and their derivatives • Implied forward rates inferred from term structure model and observed prices → estimate volatility function σ f (s, u) → feed into asset allocation formula • Simple integration of ﬁxed income management and asset allocation. 48 Forward Density Hedge: • Immunization demand due to ﬂuctuations in future market prices of risk and forward rate volatilities • Vanishes if deterministic forward rate volatilities σ f (s, u) and market prices of risk θs • Pure expectation hypothesis holds under forward measure: v f (t, v) = Et [rv ] → Standard version of PEH (f (t, v) = Et [rv ]) fails when Zt,v = 1 → Density process Zt,v measures deviation from PEH → Malliavin derivative Dt log Zt,v captures sensitivity of deviation with respect to shocks → Dynamic hedge = hedge against deviations from PEH → If Zt,v = 1 PEH holds under the original beliefs and hedging becomes irrelevant → If σ Z deterministic, deviations from PEH are non-predictable and do not need to be hedged 49 Literature: • Gaussian models: Merton (1974), Vasicek (1977), Hull and White (1990), Brace, Gatarek and Musiela (1997) • Extensively employed in practice • Forward rate volatilities σ f are insensitive to shocks. If MPR also deterministic no need to hedge • Bajeux-Besnainou, Jordan and Portait (2001) also falls in this category (one factor Vasicek) 50 Numerical Results: Forward measure hedges in one factor CIR model • CIR interest rates: √ drt = κr (¯ − rt ))dt + σr rdWt ; r r0 = r → Parameter values (Durham (JFE, 2003)): · κr = 0.002 · ¯ r = 0.0497 · σr = −0.0062 · r = 0.06 • Market price of risk: √ θt = γr rt → Parameter values: √ √ ¯ = γr r = 0.3 · γr = 0.3/ r such that θ ¯ ¯ • CRRA preferences for terminal wealth 51 mv ∗ • Mean-variance demand: πt /Xt = 1 (σt )−1 θt R 1.5 Mean−variance portfolio weight 1 0.5 0 40 30 10 8 20 6 10 4 2 Investment horizon 0 0 Relative risk aversion 52 ∗ • Static term structure hedge: πt /Xt = ρ(σt )−1 σ B (t, T ) b 0.4 Static hedge portfolio weight 0.3 0.2 0.1 0 40 30 10 8 20 6 10 4 2 Investment horizon 0 0 Relative risk aversion 53 2 3 ρ−1 Z • Dynamic forward measure hedge: z ∗ πt /Xt = ` ´−1 T ρ σt Et 4 ET Z ht,T ρ−1 i ` Dt log Zt,T ´ 5 t t,T Forward measure hedge portfolio weight 0.04 0.03 0.02 0.01 0 40 30 10 8 20 6 10 4 2 Investment horizon 0 0 Relative risk aversion 54 • Total portfolio weight: ∗ mv ∗ b ∗ z ∗ πt /Xt = πt /Xt + πt /Xt + πt /Xt 1.5 Total portfolio weight 1 0.5 0 40 30 10 8 20 6 10 4 2 Investment horizon 0 0 Relative risk aversion 55 • Changing initial interest rate: Relative risk aversion ﬁxed at R = 4 mv ∗ → Mean-variance demand: πt /Xt = 1 (σt )−1 θt R 0.6 Mean−variance portfolio weight 0.5 0.4 0.3 0.2 0.1 40 30 0.2 20 0.15 0.1 10 0.05 Investment horizon 0 0 Initial short rate 56 ∗ → Static term structure hedge: πt /Xt = ρ(σt )−1 σ B (t, T ) b 0.4 Static hedge portfolio weight 0.3 0.2 0.1 0 40 30 0.2 20 0.15 0.1 10 0.05 Investment horizon 0 0 Initial short rate 57 • Dynamic forward measure 2 3 ρ−1 Z hedge: z ∗ πt /Xt = ` ´−1 T ρ σt Et 4 ET Z ht,T ρ−1 i ` Dt log Zt,T ´ 5 t t,T Forward measure hedge portfolio weight 0.05 0.04 0.03 0.02 0.01 0 40 30 0.2 20 0.15 0.1 10 0.05 Investment horizon 0 0 Initial short rate 58 • Total portfolio weight: ∗ mv ∗ b ∗ z ∗ πt /Xt = πt /Xt + πt /Xt + πt /Xt 1 0.8 Total portfolio weight 0.6 0.4 0.2 0 40 30 0.2 20 0.15 0.1 10 0.05 Investment horizon 0 0 Initial short rate 59 • Changing initial interest rate: Investment horizon ﬁxed at T = 15 mv ast → Mean-variance demand: πt /Xt = 1 R (σt )−1 θt 2.5 Mean−variance portfolio weight 2 1.5 1 0.5 0 10 8 0.2 6 0.15 4 0.1 2 0.05 Relative risk aversion 0 0 Initial short rate 60 ∗ → Static term structure hedge: πt /Xt = ρ(σt )−1 σ B (t, T ) b 0.2 Static hedge portfolio weight 0.15 0.1 0.05 0 10 8 0.2 6 0.15 4 0.1 2 0.05 Relative risk aversion 0 0 Initial short rate 61 • Dynamic forward measure 2 3 ρ−1 Z hedge: z ∗ πt /Xt = ` ´−1 T ρ σt Et 4 ET Z ht,T ρ−1 i ` Dt log Zt,T ´ 5 t t,T Forward measure hedge portfolio weight 0.03 0.025 0.02 0.015 0.01 0.005 0 10 8 0.2 6 0.15 4 0.1 2 0.05 Relative risk aversion 0 0 Initial short rate 62 • Total portfolio weight: ∗ mv ∗ b ∗ z ∗ πt /Xt = πt /Xt + πt /Xt + πt /Xt 2.5 2 Total portfolio weight 1.5 1 0.5 0 10 8 0.2 6 0.15 4 0.1 2 0.05 Relative risk aversion 0 0 Initial short rate 63 • Approximate forward density portfolio weight: fa ∗ −1 1 T πt /Xt = −σt R Et [Nt,T ] Aprooximate forward measure hedge portfolio weight 0.04 Forward measure hedge portfolio weight 0.05 0.035 0.04 0.03 0.025 0.03 0.02 0.02 0.015 0.01 0.01 0.005 0 0 40 40 30 10 8 30 10 20 6 8 20 10 4 6 2 10 4 Investment horizon 0 0 2 Relative risk aversion 0 0 Investment horizon Relative risk aversion Approximate True 64 • Approximate forward density portfolio weight: fa ∗ −1 1 T πt /Xt = −σt R Et [Nt,T ] Aprooximate forward measure hedge portfolio weight Forward measure hedge portfolio weight 0.05 0.05 0.04 0.04 0.03 0.03 0.02 0.02 0.01 0.01 0 0 40 40 30 0.2 30 0.2 20 0.15 20 0.15 0.1 0.1 10 10 0.05 0.05 Investment horizon 0 0 Investment horizon 0 0 Spot rate Spot rate Approximate True 65 7 Conclusion Contributions: • Asset allocation formula based on change of num´raire e • Highlights role of consumption-speciﬁc coupon bonds as instruments to hedge ﬂuctuations in opportunity set • Formula has multiple applications: preferred habitat, extreme behavior, international asset allocation, demand for I-bonds • Exponential Clark-Haussmann-Ocone formula • Malliavin derivatives of functional SDEs • Solution of linear BVIE Integration of portfolio management and term structure models • Asset allocation in HJM framework • Other applications Universal N + 2 fund separation result 66

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dynamic asset allocation, asset allocation, portfolio choice, optimal portfolio, asset management, stock returns, interest rate, risk aversion, labor income, human capital, asset allocation models, university of british columbia, strategic asset allocation, optimal asset allocation, princeton university

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posted: | 5/26/2010 |

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