Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications

Transcription

1 1 Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications Jérôme Detemple Boston University School of Management and CIRANO Marcel Rindisbacher Rotman School of Management, University of Toronto and CIRANO

2 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, 23): 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, 25) 2

3 3 Contribution: New decomposition of optimal portfolio (hedging terms): Formula rests on change of numéraire: use pure discount bonds as units of account Passage to a new probability measure: forward measure (Geman (1989) and Jamshidian (1989)) General context: Ito price processes, general utilities

4 4 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.

5 5 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.

6 6 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

7 7 2 The Model Standard Continuous Time Model: Complete markets and Ito price processes Brownian motion W, d-dimensional Flow of information F t = σ(w s : s [, t]) Finite time period [, T ]. Possibly non-markovian dynamics

8 8 Assets: Price Evolution Risky assets (dividend-paying assets): ds i t S i t = ( r t δ i t) dt + σ i t (θ t dt + dw t ), S i given σt: i volatility coefficients of return process (1 d vector) r t : instantaneous rate of interest δt: i dividend yield θ t : 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

9 9 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: dx t = r t X t dt + π tσ t (θ t dt + dw t ), subject to X = x. Admissibility: π is admissible (π A) if and only if wealth is non-negative: X.

10 1 Asset Allocation Problem: Investor maximizes expected utility of terminal wealth: Utility function: U : R + R max π A E [U(X T )] Strictly increasing, strictly concave and differentiable Inada conditions: lim X U (X) = and lim X U (X) = Includes CRRA U(x) = 1 1 R X1 R where R >. Property: Strictly decreasing marginal utility in (, ) Inverse marginal utility I (y) exists and satisfies U (I (y)) = y Derivative: I (y) = 1/U (I(y))

11 11 3 The Optimal Portfolio Complete Markets: Market price of risk: θ t = (θ 1t,..., θ dt ) State price density: ξ v exp ( v ( rs + 1 ) v ) 2 θ sθ s ds θ sdw s converts state-contingent payoffs into values at date Conditional state price density: ξ t,v exp ( v t ( rs θ sθ s ) ds v t θ sdw s ) = ξv /ξ t

12 12 Optimal Portfolio: Ocone and Karatzas (1991), Detemple, Garcia and Rindisbacher (23) where π t = π m t + π r t + π θ t MV: π m t = E t [ξ t,t Γ T ] (σ t) 1 θ t IRH: π r t = (σ t) 1 E t [ ξ t,t (X T Γ T ) T t ] D t r s ds MPRH: π θ t = (σ t) 1 E t [ ξ t,t (X T Γ T ) T t (dw s + θ s ds) D t θ s ] Optimal terminal wealth X T = I(y ξ T ) Constant y solves x = E [ξ T I(y ξ T )] (static budget constraint) Γ (X) U (X)/U (X): measure of absolute risk tolerance Γ T Γ (X T ): risk tolerance evaluated at optimal terminal wealth D t is Malliavin derivative

13 13 Structure of Hedges: IRH: π r t = (σ t) 1 E t [ ξ t,t (X T Γ T ) T t ] D t r s ds Driven by sensitivities of future IR and MPR to current innovations in W t. Sensitivities measured by Malliavin derivatives D t r s and D t θ s Sensitivities are adjusted by factor ξ t,t (XT Γ T ): depends on preferences, terminal wealth and conditional state prices. Optimal terminal wealth: I(y ξ T ) Date t cost: ξ t,t I(y ξ T ) = ξ t,t I(y ξ t ξ t,t ) Sensitivity to change in conditional SPD ξ t,t (ξ t,t I(y ξ t ξ t,t )) ξ t,t = I(y ξ t ξ t,t ) + y ξ t ξ t,t I (y ξ t ξ t,t ) = X T Γ T Sensitivity of conditional SPD to fluctuations in IR and MPR ξ t,t T t D t r s ds and ξ t,t T t (dw s + θ s ds) D t θ s.

14 14 Constant Relative Risk Aversion (CRRA) π r t X t π m t X t = ρ (σ t) 1 E t [ = 1 R (σ t) 1 θ t ξ ρ T E t[ξ ρ T ] T t ] D t r s ds π θ t X t = ρ (σ t) 1 E t [ ξ ρ T E t[ξ ρ T ] T t (dw s + θ s ds) D t θ s ] ρ = 1 1/R y = ( E [ ξ ρ ] ) R T /x Xt [ = E t ξt,t (y ξ T ) 1/R] Hedging terms are weighted averages of the sensitivities of future interest rates and market prices of risk to the current Brownian innovations.

15 15 4 A New Decomposition of the Optimal Portfolio 4.1 Bond Pricing and Forward Measures Pure Discount Bond Price: B T t = E t [ξ t,t ] Forward T -Measure: (Geman (1989) and Jamshidian (1989)) Random variable: Z t,t ξ t,t E t[ξ t,t ] = ξ t,t B T t Properties: Z t,t > and E t [Z t,t ] = 1. Use Z t,t as density Probability measure: dq T t Equivalent to P = Z t,t dp

16 16 Change of Numéraire: unit of account is T -maturity bond Under Q T t price V (t) of a contingent claim with payoff Y T is [ ] ξ V (t) = E t [ξ t,t Y T ] = E t [ξ t,t ] E t,t t E t[ξ t,t ] Y T = Bt T Et T [Y T ] Et T [ ] E t [Z t,t ] is expectation under Q T t Martingale property: V (t) /Bt T = Et T [Y T ] = E t [Z t,t Y T ]. Density Z t,t is stochastic discount factor: converts future payoffs into current values measured in bond unit of account.

17 17 Characterization (Theorem 2): The forward T -density is given by ( T Z t,t exp t σ Z (s, T ) dw s 1 ) T 2 t σ Z (s, T ) σ Z (s, T ) ds volatility at s [t, T ]: σ Z (s, T ) σ B (s, T ) θ s bond return volatility: σ B (s, T ) D s log B T s Contribution(s): Identify volatility of forward measure Application of Exponential Clark-Haussmann-Ocone formula Market price of risk in the numéraire

18 Portfolio allocation and long term bonds An Alternative Portfolio Decomposition Formula: π t = π m t Mean variance demand: + π b t + π z t π m t = E T t [Γ T ] BT t (σ t) 1 θ t Hedge motivated by fluctuations in price of pure discount bond with matching maturity πt b = (σ t) 1 σ B (t, T ) Et T [XT Γ T ] BT t Hedge motivated by fluctuations in density of forward T -measure π z t = (σ t) 1 E T t [(X T Γ T ) D t log (Z t,t )] B T t.

19 19 Essence of Formula: change of numéraire SPD representation: ξ t,t = B T t Z t,t Optimal terminal wealth: X T = I ( y ξ t B T t Z t,t ) Cost of optimal terminal wealth: B T t Z t,t I ( y ξ t B T t Z t,t ) Hedging portfolio: D t ( B T t Z t,t I ( y ξ t B T t Z t,t )) Chain rule of Malliavin calculus: ( Z t,t I ( y ξ t B T t Z t,t ) + B T t Z t,t I ( y ξ t B T t Z t,t ) y ξ t Z t,t ) Dt B T t ( B T t I ( y ξ t B T t Z t,t ) + B T t Z t,t I ( y ξ t B T t Z t,t ) y ξ t B T t ) Dt Z t,t B T t Z t,t I ( y ξ t B T t Z t,t ) B T t Z t,t D t (y ξ t )

20 2 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 Z t,t (instantaneous and delayed) Source of fluctuations are bond return volatilities and MPRs: σ Z (s, T ) σ B (s, T ) θ s D t σ Z (s, T ) = D t σ B (s, T ) D t θ s.

21 21 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

22 22 Signing the Static Hedge: Bond prices negatively related to IR IR innovation negatively related to equity innovation In one factor (BMP) model σ B > : boost demand for stocks

23 Constant Relative Risk Aversion Hedging Terms are: π z t X t π b t X t = ρ (σ t) 1 E T t = ρ (σ t) 1 σ B (t, T ) B T t [ ] Z ρ 1 t,t Et T [Z ρ 1 t,t ] D t log (Z t,t ) Bt 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

24 24 Literature: special cases of this result analyzed by Bajeux-Besnainou, Jordan and Portait (21): Vasicek short rate and constant market prices of risk. Forward density hedge vanishes Lioui and Poncet (21) and Lioui (25): Diffusion models with power utility. Lioui and Poncet (21): last hedging component in terms of unknown volatility function (PDE). Lioui (25): 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.

25 Illustration: optimal stock-bond mix for CRRA investor Model: T maturity bond is traded Two assets: Stock and investment horizon matching bond σ t = σstock 1t σ1t B σ stock 2t σ B 2t Optimal portfolio weight: static hedging component π b t X t If π z t hedging is very simple: no hedging component for stocks = ρ(σ t) 1 σt B = ρ 1 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

26 26 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: e (t, T ) = σ S t θ 2t +σ2t B (Q(t,T ) 1) σ2t B θ 1t σ1t B θ 2t Q (t, T ) E T t [X T ] /ET t [Γ T ] For HARA utility u(x) = (x A) 1 R /1 R, Q (t, T ; R) + BT h(,t;r) x/a B T! B T /Bt B T t Z t! 1/R 1 A = R 1 + ABT t X t ABT t «B T ρ /Bt with h(, t; R) exp (ρ/r) R t 12 θ s + σ B (s, T ) 2 σ B (s, T ) 2 ds. 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.

27 27 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 1 Bonds to equities ratio Risk aversion parameter R Forward density Z(t) 5 Vasicek interest rate model:r = r =.6, κ r =.5, σ r1 =.2, σ r2 =.15 and market prices of risk are constants θ s =.3 and θ b =.15. The interest rate at t = 5 is r t =.2. Other parameter values are A = 2,, x = 1, and T = 1.

28 28 5 Intermediate Consumption 5.1 The Investor s Preferences Consumption-portfolio Problem: [ ] T max π,c A E u (c t, t) dt + U(X T ) Utility function: u (, ) : R + [, 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 [, T ] Inverse marginal bequest function I (y) exists: U (I (y)) = y

29 Portfolio Representation and Coupon-paying Bonds Decomposition: π t = π m t + π b t + π z t Mean variance demand: ( ) πt m T = t Et v [Γ v] Bt v dv + Et T [Γ T ] BT t (σ t) 1 θ t Hedge motivated by fluctuations in price of coupon-paying bond with matching maturity: π b t = (σ t) 1 T t σ B (t, v) B v t E v t [c v Γ v] dv + (σ t) 1 σ B (t, T ) B T t E T t [X T Γ T ] Hedge motivated by fluctuations in density of forward T -measure: π z t = (σ t) 1 ( T t ) Et v [(c v Γ v) D t log Z t,v ] Bt v dv + (σ t) 1 ( E T t [(X T Γ T ) D t log Z t,t ] B T t )

30 3 Static Hedge πt b : hedge against fluctuations in value of coupon-paying bond Coupon payments C (v) Et v [c v Γ v] at intermediate dates v [, T ) Bullet payment F E T t [XT Γ T ] at terminal date T Coupon payments and face value are time-varying tailored to individual s consumption profile and risk tolerance Bond value Instantaneous volatility B (t, T ; C, F ) T t B v t C (v) dv + B T t F. σ (B (t, T ; C, F )) B (t, T ; C, F ) = T t σ B (t, v) B v t C (v) dv +σ B (t, T ) B T t F Hedge: (σ t) 1 σ (B (t, T ; C, F )) B (t, T ; C, F )

31 31 Forward Density Hedge πt z : Motivation: desire to hedge fluctuations in forward densities Z t,v Static hedge already neutralizes impact of term structure fluctuations on PV of future consumption Given ξ t,v = Bt v Z t,v it remains to hedge fluctuations in risk-adjusted discount factors Z t,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?

32 Constant Relative Risk Aversion Relative risk aversion parameters R u, R U for utility and bequest functions. Portfolio: Mean-variance term π m t = (σ t) 1 ( T t 1 R u E v t [c v] B v t dv + 1 R U E T t [X T ] BT t Hedge motivated by fluctuations in price of coupon-paying bond with matching maturity π b t = (σ t) 1 ( ρ u T t σ B (t, v) B v t E v t [c v] dv + ρ U σ B (t, T ) B T t E T t [X T ] ) Hedge motivated by fluctuations in densities of forward measures ) θ t π z t = ρ u (σ t) 1 T t E v t [c vd t log Z t,v ] B v t dv +ρ U (σ t) 1 E T t [X T D t log Z t,t ] B T t

33 33 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 ρ u and ρ U Knife edge property traditionally associated with power utility function Possibility of positive annuity hedge (R u > 1) combined with negative bequest hedge (R U < 1).

34 34 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.

35 35 Formula shows that optimal behavior naturally induces a demand for certain types of bonds in specific maturity ranges π t = w m t (X t B (t, T ; C, F )) + w b tb (t, T ; C, F ) + π z t w m t = arg max w {w σ t θ t : w σ t σ tw = k}. w b t = arg max w {w σ t σ (B (t, T ; C)) : w σ t σ tw = k} π z t = arg max π {π σ t σ (t, T ) : π σ t σ tπ = k} where σ t,t T t E v t [(c v Γ v) D t log Z t,v ] B v t dv +E T t [(X T Γ T ) D t log Z t,t ] B T t

36 36 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

37 37 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

38 38 Motivation for preferred habitat here is different from Riedel (21) 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

39 6.2 Universal Fund Separation Y : vector of N < d state variables with evolution described by the functional stoachastic differential equation Suppose that B v t = B ( t, v, Y ( ) ) σ Z (t, v) = σ Z ( t, v, Y ( ) ) dy t = µ(y ( ) ) t dt + σ(y ( ) ) t dw t are Fréchet 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 (σ t) 1 σt Y ( ) Y( ) to synthesize the static bond hedge and the forward density hedge. 39

40 4 6.3 Extreme Behavior Assume risk tolerances go to zero: Intermediate utility and bequest functions: (Γ u (z, v), Γ U (z)) (, ) for all z [, + ) and all v [, T ] Relative behaviors: for some constant k [, + ): Γ u (z 1,v) Γ U (z 2 ) k for all z 1, z 2 [, ) and all v [, T ] Γ u (z 1,v 1 ) Γ u (z 2,v 2 ) 1 for all z 1, z 2 [, ) and all v 1, v 2 [, T ]

41 41 Limit Allocations: coupon-paying bond with constant coupon C and face value F given by C = R x x T and F = Bv dv+bt /k R T Bv dvk+bt If k = exclusive preference for pure discount bond, (C, F ) = (, x/b T ) If k ( preference is for a pure coupon bond, (C, F ) = x/ ) T Bv dv, 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..

42 42 Illustration: CARA preferences Γ u and Γ U constant, k Γ u /Γ U. Slope of indifference curves: dx dc = 1 k ( e X c/k ) 1 Γ U Budget Constraint k=1 1 8 X c

43 43 k k= k=.5 Budget Constraint k=1 8 X c

44 44 k Budget Constraint k=1 1 8 X 6 4 k=1 2 k=infinity c

45 45 Special case examined by Wachter (22) 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.

46 Order of Convergence As (Γ u (z, v), Γ U (z)) (, ), the limit portfolios π m t = π z t = π b t = (σ t) T σb (t, v)b v t dvc + σ B (t, T )B T t F have scaled asymptotic errors: ɛ α t (ν) = (Γ ν ( )) 1 (π α t π α t ) with α {m, b, z} and ν {u, U}, [ ] [ɛ m t (U), ɛ m t (u)] (σ t) 1 T θ t t Bt v dv Bt T K [ ɛ b t (U), ɛ b t(u) ] [ (σ t) 1 T t σ B (t, v) Bt v dv σ B (t, T ) Bt T [ [ɛ z t (U), ɛ z t (u)] (σ t) 1 ] T t N t,v Bt v dv N t,t Bt T K where N t,τ is given by K is given by. N t,τ E τ t h R τt σ Z (r, τ) dw r 1 2 K 46 " k 1 R τt σ Z (r, τ) 2 dr (D t log Z t,τ ) i 1 1 k # ] K

47 Term structure models and asset allocation Integration of term structure models and asset allocation models: Forward rate representation of bonds Bt v = exp ( v t f t s ds ) Continuously compounded forward rate: f s t v log (Bv t ) Bond price volatility: σ B (t, v) = D t log B v t = v t D tf s t ds = v t σf (t, s)ds Volatility of forward rate: σ f (t, s) Forward rate dynamics: No arbitrage condition (HJM (1992)): df v t = σ f (t, v) ( dw t + ( θ t σ B (t, v) ) dt ), f v given Dynamics completely determined by forward rate volatility function and initial forward rate curve

48 48 Optimal Portfolio: previous formula with D t log Z t,v = R v t dw s + θ s + R v s σ f (s, u)du ds D t θ s + R v s D t σ f (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.

49 49 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 θ s Pure expectation hypothesis holds under forward measure: f(t, v) = Et v [r v ] Standard version of PEH (f(t, v) = E t [r v ]) fails when Z t,v 1 Density process Z t,v measures deviation from PEH Malliavin derivative D t log Z t,v captures sensitivity of deviation with respect to shocks Dynamic hedge = hedge against deviations from PEH If Z t,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

50 5 Literature: Gaussian models: Merton (1974), Vasicek (1977), Hull and White (199), 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 (21) also falls in this category (one factor Vasicek)

51 51 Numerical Results: Forward measure hedges in one factor CIR model CIR interest rates: dr t = κ r ( r r t ))dt + σ r rdwt ; r = r Parameter values (Durham (JFE, 23)): κ r =.2 r =.497 σ r =.62 r =.6 Market price of risk: θ t = γ r rt Parameter values: γ r =.3/ r such that θ = γ r r =.3 CRRA preferences for terminal wealth

52 52 Mean-variance demand: πt mv /Xt = 1 R (σ t) 1 θ t 1.5 Mean variance portfolio weight Investment horizon Relative risk aversion 1

53 53 Static term structure hedge: π b t /X t = ρ(σ t) 1 σ B (t, T ).4 Static hedge portfolio weight Investment horizon Relative risk aversion 1

54 54 Dynamic forward measure hedge: π z t /X t 2 = ρ `σ t 1 E T t 4 Zρ 1 t,t E T h t Z ρ 1 t,t 3 `D i t log Z t,t 5 Forward measure hedge portfolio weight Investment horizon Relative risk aversion 1

55 55 Total portfolio weight: π t /Xt = πmv t /Xt + πb t /X t + πz t /X t 1.5 Total portfolio weight Investment horizon Relative risk aversion 1

56 56 Changing initial interest rate: Relative risk aversion fixed at R = 4 Mean-variance demand: πt mv /Xt = 1 R (σ t) 1 θ t.6 Mean variance portfolio weight Investment horizon Initial short rate.15.2

57 57 Static term structure hedge: π b t /X t = ρ(σ t) 1 σ B (t, T ).4 Static hedge portfolio weight Investment horizon Initial short rate.15.2

58 58 Dynamic forward measure hedge: π z t /X t 2 = ρ `σ t 1 E T t 4 Zρ 1 t,t E T h t Z ρ 1 t,t 3 `D i t log Z t,t 5 Forward measure hedge portfolio weight Investment horizon Initial short rate.15.2

59 59 Total portfolio weight: π t /Xt = πmv t /Xt + πb t /X t + πz t /X t 1 Total portfolio weight Investment horizon Initial short rate.15.2

60 6 Changing initial interest rate: Investment horizon fixed at T = 15 Mean-variance demand: πt mv /Xt ast = 1 R (σ t) 1 θ t 2.5 Mean variance portfolio weight Relative risk aversion Initial short rate.15.2

61 61 Static term structure hedge: π b t /X t = ρ(σ t) 1 σ B (t, T ).2 Static hedge portfolio weight Relative risk aversion Initial short rate.15.2

62 62 Dynamic forward measure hedge: π z t /X t 2 = ρ `σ t 1 E T t 4 Zρ 1 t,t E T h t Z ρ 1 t,t 3 `D i t log Z t,t 5 Forward measure hedge portfolio weight Relative risk aversion Initial short rate.15.2

63 63 Total portfolio weight: π t /Xt = πmv t /Xt + πb t /X t + πz t /X t 2.5 Total portfolio weight Relative risk aversion Initial short rate.15.2

64 64 Approximate forward density portfolio weight: π fa t /Xt = σ 1 1 t R ET t [N t,t ] Aprooximate forward measure hedge portfolio weight Investment horizon Relative risk aversion 1 Forward measure hedge portfolio weight Investment horizon Relative risk aversion 1 Approximate True

65 65 Approximate forward density portfolio weight: π fa t /Xt = σ 1 1 t R ET t [N t,t ] Aprooximate forward measure hedge portfolio weight Investment horizon Spot rate.15.2 Forward measure hedge portfolio weight Investment horizon Spot rate.15.2 Approximate True

66 66 7 Conclusion Contributions: Asset allocation formula based on change of numéraire 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