Dynamic Asset Allocation: a Portfolio Decomposition Formula and Applications

Dynamic Asset Allocation:

a Portfolio Decomposition Formula

and Applications

J´

erˆ

ome Detemple

Boston University School of Management and CIRANO

1 Introduction

I 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)

I Contribution:

• New decomposition of optimal portfolio (hedging terms): → Formula rests on change of num´eraire: use pure discount bonds as

units of account

→ Passage to a new probability measure: forward measure (Geman (1989) and Jamshidian (1989))

• 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

• 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.

I Applications:

• Preferred habitat

• Extreme risk aversion behavior • International asset allocation • Preferences for I-bonds

• Integration of risk management and asset allocation

I Road map:

• Model with utility from terminal wealth • The Ocone-Karatzas formula

• New representation

• Intermediate consumption • Applications

2 The Model

I Standard Continuous Time Model:

• Complete markets and Ito price processes • Brownian motion W, d-dimensional

• Flow of information F_t = σ(W_s : s ∈ [0, t])

• Finite time period [0, T].

I Assets: Price Evolution

• Risky assets (dividend-paying assets):

dS_ti S_ti = rt − δ i t dt + σ_ti (θ_tdt + dW_t) , S₀i given

∗ σ_ti: volatility coefficients of return process (1 × d vector)

∗ r_t: instantaneous rate of interest

∗ δ_ti: 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:

I Investment and Wealth:

• Portfolio policy π: d-dimensional, progressively measurable; integrability conditions

→ amounts invested in assets: π

→ amount in money market: X − π01

• Wealth process:

dX_t = r_tX_tdt + π_t0σ_t (θ_tdt + dW_t) , subject to X₀ = x.

• Admissibility: π is admissible (π ∈ A) if and only if wealth is non-negative: X ≥ 0.

I Asset Allocation Problem:

• Investor maximizes expected utility of terminal wealth:

max_π∈A E [U(X_T )]

• Utility function: U : _R+ → _R

→ Strictly increasing, strictly concave and differentiable

→ Inada conditions: lim_X→∞ U0(X) = 0 and lim_X→0 U0(X) = ∞ • Includes CRRA U(x) = ₁₋1_RX1−R where R > 0.

• Property:

→ Strictly decreasing marginal utility in (0, ∞)

→ Inverse marginal utility I (y) exists and satisfies U0 (I (y)) = y

3 The Optimal Portfolio

I Complete Markets:

• Market price of risk: θ_t = (θ_1t, ..., θ_dt)0

• State price density:

ξ_v ≡ exp − R₀v r_s + 1₂θ_s0θ_sds − R₀v θ_s0 dW_s

→ converts state-contingent payoffs into values at date 0

• Conditional state price density:

I Optimal Portfolio: Ocone and Karatzas (1991), Detemple, Garcia and Rindisbacher (2003) π_t∗ = π_tm + π_tr + π_tθ where MV: π_tm = E_t [ξ_t,TΓ∗_T ] (σ_t0)−1 θ_t IRH: π_tr = − (σ_t0)−1 E_t h ξ_t,T (X_T∗ − Γ_T∗ ) R_tT D_tr_sds i0 MPRH: π_tθ = − (σ_t0)−1 E_t h ξ_t,T (X_T∗ − Γ∗_T ) R_tT (dW_s + θ_sds)0 D_tθ_s i0

• 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) ≡ −U0(X)/U00(X): measure of absolute risk tolerance

I Structure of Hedges:

IRH: π_tr = − (σ_t0)−1 E_t hξ_t,T (X_T∗ − Γ∗_T) R_tT D_tr_sdsi

0

• Driven by sensitivities of future IR and MPR to current innovations in W_t. Sensitivities measured by Malliavin derivatives D_tr_s and D_tθ_s

• Sensitivities are adjusted by factor ξ_t,T (X_T∗ − 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,TI(y∗ξ_tξ_t,T)

• Sensitivity to change in conditional SPD ξ_t,T

∂(ξ_t,TI(y∗ξ_tξ_t,T))

∂ξ_t,T = I(y

∗_ξ

tξt,T ) + y∗ξtξt,TI0(y∗ξtξt,T) = X_T∗ − Γ∗_T

• Sensitivity of conditional SPD to fluctuations in IR and MPR −ξ_t,T R_tT D_tr_sds and − ξ_t,T R_tT (dW_s + θ_sds)0 D_tθ_s.

I Constant Relative Risk Aversion (CRRA) π_tm X_t∗ = 1 R (σ 0 t) −1 θ_t π_tr X_t∗ = −ρ(σ 0 t) −1 E_t ξ_Tρ Et[ξ_Tρ] R T t Dtrsds 0 π_tθ X_t∗ = −ρ(σ 0 t) −1 E_t ξ_Tρ Et[ξ_Tρ ] R T t (dWs + θsds) 0_D tθs 0 • ρ = 1 − 1/R • y∗ = Eξ_Tρ /xR • X_t∗ = E_t ξ_t,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.

4 A New Decomposition of the Optimal Portfolio

4.1 Bond Pricing and Forward Measures

I Pure Discount Bond Price: B_tT = E_t [ξ_t,T ]

I Forward T-Measure: (Geman (1989) and Jamshidian (1989))

• Random variable:

Z_t,T ≡ ξt,T

Et[ξt,T]

= ξt,T

B_tT

• Properties: Z_t,T > 0 and E_t [Z_t,T] = 1. Use Z_t,T as density • Probability measure: dQT_t = Z_t,TdP

I Change of Num´eraire: unit of account is T-maturity bond

• Under QT_t price V (t) of a contingent claim with payoff Y_T is

V (t) = E_t [ξ_t,TY_T] = E_t [ξ_t,T ] E_t ξ_t,T Et[ξt,T] Y_T = B_tTE_tT [Y_T] • E_tT [·] ≡ E_t [Z_t,T ·] is expectation under QT_t • Martingale property: V (t) /B_tT = E_tT [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.

I Characterization (Theorem 2): The forward T-density is given by Z_t,T ≡ exp R T t σ Z _{(s, T)}0 _dW s − 1₂ R T t σ Z _{(s, T)}0 _σZ _{(s, T) ds} • volatility at s ∈ [t, T]: σZ (s, T) ≡ σB (s, T) − θ_s

• bond return volatility: σB (s, T)0 ≡ D_s log B_sT

I Contribution(s):

• Identify volatility of forward measure

• Application of Exponential Clark-Haussmann-Ocone formula • Market price of risk in the num´eraire

4.2 Portfolio allocation and long term bonds

I An Alternative Portfolio Decomposition Formula:

π_t∗ = π_tm + π_tb + π_tz

• Mean variance demand:

π_tm = E_tT [Γ∗_T] B_tT (σ_t0)−1 θ_t

• Hedge motivated by fluctuations in price of pure discount bond with matching maturity

π_tb = (σ_t0)−1 σB (t, T) E_tT [X_T∗ − Γ∗_T ] B_tT

• Hedge motivated by fluctuations in density of forward

T-measure

I Essence of Formula: change of num´eraire • SPD representation: ξ_t,T = B_tT Z_t,T

• Optimal terminal wealth: X_T∗ = I y∗ξ_tB_tTZ_t,T

• Cost of optimal terminal wealth: B_tT Z_t,T I y∗ξ_tB_tT Z_t,T

• Hedging portfolio: D_t B_tTZ_t,TI y∗ξ_tB_tTZ_t,T

• Chain rule of Malliavin calculus:

→ Z_t,TI y∗ξ_tB_tTZ_t,T + B_tT Z_t,T I0 y∗ξ_tB_tTZ_t,Ty∗ξ_tZ_t,TD_tB_tT

→ B_tT I y∗ξ_tB_tT Z_t,T + B_tTZ_t,TI0 y∗ξ_tB_tT Z_t,T y∗ξ_tB_tTD_tZ_t,T

I 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

I Forward Density Hedge:

• Immunizes against fluctuations in forward density Z_t,T

(instantaneous and delayed)

• Source of fluctuations are bond return volatilities and MPRs:

σZ (s, T) ≡ σB (s, T) − θ_s

I 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

I Signing the Static Hedge:

• Bond prices negatively related to IR

• IR innovation negatively related to equity innovation

4.3 Constant Relative Risk Aversion

I Hedging Terms are:

π_tb X_t∗ = ρ(σ 0 t) −1 σB (t, T) B_tT π_tz X_t∗ = ρ(σ 0 t) −1 E_tT Z_t,Tρ−1 E_tT[Z_t,Tρ−1]Dt log (Zt,T) 0 B_tT

I 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

I 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.

I Illustration: optimal stock-bond mix for CRRA investor • Model:

→ T−maturity bond is traded

→ Two assets: Stock and investment horizon matching bond

σ_t =   σ₁stock_t σ₂stock_t σ₁B_t σ₂B_t  

• Optimal portfolio weight: static hedging component

π_tb X_t∗ = ρ(σ 0 t)−1σtB = ρ   0 1  

• If π_tz ≈ 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

I 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_tS θ2t+σ2Bt(Q(t,T)−1) σ₂B_tθ_1t−σ₁B_tθ_2t Q (t, T) ≡ E T t [XT∗] /E T t [Γ∗T ]

– For HARA utility u(x) = (x − A)1−R/1 − R,

∗ Q(t, T;R) ≡ R 0 @1 + BT₀ h(0,t;R) x/A−BT₀ ! BT₀ /Bt₀ BT_{t Zt} !−1/R1 A = R „ 1 + ABTt X_t∗−ABT_t “ B₀T/B₀t ”ρ« ∗ with · h(0, t;R) ≡ exp “ (ρ/R)R₀t “ 1 2kθs + σ B_{(s, T)k}2 _{− kσ}B_{(s, T)k}2”_ds”_.

• In the presence of wealth effect the bonds-to-equities ratio is not necessarily monotone in risk aversion

0 1 2 3 4 5 0 1 2 3 4 0 20 40 60 80 100 Forward density Z(t) Bonds−to−equities ratio: intolerance for shortfall

Risk aversion parameter R

Bonds−to−equities ratio

– Vasicek interest rate model:r₀ = r = 0.06, κ_r = 0.05, σ_r1 = −0.02, σ_r2 = −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.

5 Intermediate Consumption

5.1 The Investor’s Preferences

I Consumption-portfolio Problem:

max_π,c∈A EhR₀T u (c_t, t) dt + U(X_T)i

• 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: u0 (J (y, t) , t) = y for all t ∈ [0, T]

5.2 Portfolio Representation and Coupon-paying Bonds

I Decomposition:

π_t∗ = π_tm + π_tb + π_tz

• Mean variance demand:

π_tm = R T t E v t [Γ∗v] Btvdv + EtT [Γ∗T ] BtT (σ_t0)−1 θ_t

• Hedge motivated by fluctuations in price of coupon-paying bond with matching maturity:

π_tb = (σ_t0)−1 R_tT σB (t, v)B_tvE_tv [c∗_v − Γ∗_v] dv

+ (σ_t0)−1 σB (t, T) B_tT E_tT [X_T∗ − Γ∗_T]

• Hedge motivated by fluctuations in density of forward

T-measure:

π_tz = (σ_t0)−1 R_tT E_tv [(c∗_v − Γ∗_v) D_t log Z_t,v] B_tvdv

0

I Static Hedge π_tb: hedge against fluctuations in value of coupon-paying bond

• Coupon payments C (v) ≡ E_tv [c∗_v − Γ∗_v] at intermediate dates

v ∈ [0, T)

• Bullet payment F ≡ E_tT [X_T∗ − 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

B (t, T; C, F) ≡ R_tT B_tvC (v) dv + B_tTF.

• Instantaneous volatility

σ (B (t, T; C, F)) B (t, T; C, F) = R_tT σB (t, v) B_tvC (v) dv

I Forward Density Hedge π_tz:

• 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 = B_tvZ_t,v it remains to hedge fluctuations in risk-adjusted discount factors Z_t,v, v ∈ [t, T].

I 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)?

5.3 Constant Relative Risk Aversion

I Relative risk aversion parameters R_u, R_U for utility and bequest functions. Portfolio: • Mean-variance term π_tm = (σ_t0)−1 R T t 1 RuE v t [c∗v] Btvdv + _R1_U EtT [XT∗] BtT θ_t

• Hedge motivated by fluctuations in price of coupon-paying bond with matching maturity

π_tb = (σ_t0)−1 ρ_u R_tT σB (t, v) B_tvE_tv [c∗_v] dv + ρ_UσB (t, T) B_tT E_tT [X_T∗]

• Hedge motivated by fluctuations in densities of forward measures

π_tz = ρ_u (σ_t0)−1 R_tT E_tv [c∗_vD_t log Z_t,v]0 B_tvdv

I 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).

6 Applications

6.1 Preferred Habitats and Portfolio Choice

I 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

I Formula shows that optimal behavior naturally induces a demand for certain types of bonds in specific maturity ranges

π_t∗ = w_tm (X_t∗ − B (t, T; C, F)) + w_tbB (t, T; C, F) + π_tz w_tm = arg max_w {w0σ_tθ_t : w0σ_tσ_t0w = k} . w_tb = arg max_w {w0σ_tσ (B (t, T; C)) : w0σ_tσ_t0w0 = k} π_tz = arg max_π {π0σ_tσ_b (t, T) : π0σ_tσ_t0π0 = k} where b σ_t,T0 ≡ R_tT E_tv [(c∗_v − Γ∗_v) D_t log Z_t,v] B_tvdv +E_tT [(X_T∗ − Γ_T∗ ) D_t log Z_t,T ] B_tT

• 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

I 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

I 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

6.2 Universal Fund Separation

• Y : vector of N < d state variables with evolution described by the functional stoachastic differential equation

dY_t = µ(Y_(·))_tdt + σ(Y_(·))_tdW_t

• Suppose that

– B_tv = B t, v, Y_(·)

– σZ(t, v) = σZ t, v, Y_(·)

are Fr´echet 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

6.3 Extreme Behavior

I Assume risk tolerances go to zero:

• Intermediate utility and bequest functions:

(Γ_u(z, v), Γ_U(z)) → (0, 0) for all z ∈ [0, +∞) and all v ∈ [0, T]

• Relative behaviors: for some constant k ∈ [0, +∞):

Γ_u(z₁,v)

Γ_U(z₂) → k for all z1, z2 ∈ [0, ∞) and all v ∈ [0, T] Γu(z1,v1)

I Limit Allocations: coupon-paying bond with constant coupon C and face value F given by

C = RT x 0 B v 0dv+B0T/k and F = R T x 0 B v 0dvk+B0T .

• If k = 0 exclusive preference for pure discount bond,

(C, F) = 0, x/B₀T

• If k → ∞ preference is for a pure coupon bond,

(C, F) = x/ R₀T B₀vdv,0

I 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

I Illustration: CARA preferences Γ_u and Γ_U constant, k ≡ Γ_u/Γ_U. • Slope of indifference curves: −dX_dc = _k1 eX−c/k

1 Γ_U 0 20 40 60 80 100 120 140 X k=1 Budget Constraint

• k → 0 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 c X k=1 Budget Constraint k=0.5 k=0

• k → ∞ 0 20 40 60 80 100 120 0 20 40 60 80 100 120 140 X k=1 Budget Constraint k=10 k=infinity

I 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.

I Order of Convergence

– As (Γ_u(z, v), Γ_U(z)) → (0, 0), the limit portfolios

∗ πm_t = πz_t = 0

∗ πb_t = (σ_t0) R₀T σB(t, v)B_tvdvC + σB(t, T)B_tTF

– have scaled asymptotic errors:

∗ α_t (ν) = (Γ_ν(·))−1 (π_tα − πα_t ) with α ∈ {m, b, z} and ν ∈ {u, U}, [m_t (U), m_t (u)] → (σ_t0)−1 θ_t hR_tT B_tvdv B_tT i K b_t(U), b_t(u) → − (σ_t0)−1 h R T t σ B _{(t, v) B}v t dv σB (t, T) BtT i K [z_t(U), z_t(u)] → − (σ_t0)−1 h R T t Nt,vB v t dv Nt,T BtT i K – where ∗ N_t,τ is given by Nt,τ ≡ E_tτ h“ Rτ t σ Z_{(r, τ)}0_dW r − 1₂ R_tτ kσZ(r, τ)k2dr ” (DtlogZt,τ)0 i

6.4 Term structure models and asset allocation

I Integration of term structure models and asset allocation models:

• Forward rate representation of bonds

B_tv = exp − R_tv f_tsds

→ Continuously compounded forward rate: f_ts ≡ −_∂v∂ log (B_tv)

→ Bond price volatility:

σB(t, v)0 = D_t log B_tv = − R_tv D_tf_tsds = − R_tv σf(t, s)ds

→ Volatility of forward rate: σf(t, s)

• Forward rate dynamics:

→ No arbitrage condition (HJM (1992)):

df_tv = σf(t, v) dW_t + θ_t − σB(t, v) dt, f₀v given → Dynamics completely determined by forward rate volatility

I Optimal Portfolio: previous formula with

D_tlogZ_t,v = R_tv “dW_s + “θ_s + R_sv σf(s, u)du”ds”0“D_tθ_s + R_sv D_tσf(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 σf(s, u)

→ feed into asset allocation formula

I Forward Density Hedge:

• Immunization demand due to fluctuations in future market prices of risk and forward rate volatilities

• Vanishes if deterministic forward rate volatilities σf (s, u) and market prices of risk θ_s

• Pure expectation hypothesis holds under forward measure:

f(t, v) = E_tv[r_v]

→ Standard version of PEH (f(t, v) = E_t[r_v]) fails when Z_t,v 6= 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 σZ deterministic, deviations from PEH are non-predictable and do not need to be hedged

I Literature:

• Gaussian models: Merton (1974), Vasicek (1977), Hull and White (1990), Brace, Gatarek and Musiela (1997)

• Extensively employed in practice

• Forward rate volatilities σf 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)

I Numerical Results: Forward measure hedges in one factor CIR model • CIR interest rates:

dr_t = κ_r(¯r − r_t))dt + σ_r√rdW_t; r₀ = r

→ Parameter values (Durham (JFE, 2003)): · κ_r = 0.002

· r¯ = 0.0497

· σ_r = −0.0062

· r = 0.06

• Market price of risk:

θ_t = γ_r√r_t

→ Parameter values:

· γ_r = 0.3/√r¯ such that θ¯ = γ_r√r¯ = 0.3

• Mean-variance demand: π_tmv/X_t∗ = _R1 (σ_t0)−1θ_t 4 6 8 10 10 20 30 40 0 0.5 1 1.5

• Static term structure hedge: π_tb/X_t∗ = ρ(σ_t0)−1σB(t, T) 0 2 4 6 8 10 0 10 20 30 40 0 0.1 0.2 0.3 0.4

Relative risk aversion Investment horizon

• Dynamic forward measure hedge: πz_t/X_t∗ = ρ`σ0<sub>t</sub>´−1ET<sub>t</sub> 2 4 Zρ−1 t,T ET<sub>t</sub> hZρ−1 t,T i ` D_tlogZ_t,T´0 3 5 6 8 10 20 30 40 0 0.01 0.02 0.03 0.04

• Total portfolio weight: πt/X_t∗ = π_tmv/X_t∗ + π_tb/X_t∗ + πz_t/X_t∗ 0 2 4 6 8 10 0 10 20 30 40 0 0.5 1 1.5

Relative risk aversion Investment horizon

• Changing initial interest rate: Relative risk aversion fixed at R = 4 → Mean-variance demand: π_tmv/X_t∗ = _R1 (σ_t0)−1θ_t 0.15 0.2 20 30 40 0.1 0.2 0.3 0.4 0.5 0.6

→ Static term structure hedge: π_tb/X_t∗ = ρ(σ_t0)−1σB(t, T) 0 0.05 0.1 0.15 0.2 0 10 20 30 40 0 0.1 0.2 0.3 0.4

Initial short rate Investment horizon

• Dynamic forward measure hedge: π_tz/X_t∗ = ρ`σ0<sub>t</sub>´−1ET<sub>t</sub> 2 4 Zρ−1 t,T ET<sub>t</sub> hZρ−1 t,T i ` Dt logZt,T ´0 3 5 0.1 0.15 0.2 20 30 40 0 0.01 0.02 0.03 0.04 0.05

• Total portfolio weight: πt/X_t∗ = π_tmv/X_t∗ + π_tb/X_t∗ + πz_t/X_t∗ 0 0.05 0.1 0.15 0.2 0 10 20 30 40 0 0.2 0.4 0.6 0.8 1

Initial short rate Investment horizon

• Changing initial interest rate: Investment horizon fixed at T = 15

→ Mean-variance demand: π_tmv/X_tast = _R1 (σ_t0)−1θ_t

0.15 0.2 6 8 10 0 0.5 1 1.5 2 2.5

→ Static term structure hedge: π_tb/X_t∗ = ρ(σ_t0)−1σB(t, T) 0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 0 0.05 0.1 0.15 0.2

Initial short rate Relative risk aversion

• Dynamic forward measure hedge: π_tz/X_t∗ = ρ`σ0<sub>t</sub>´−1ET<sub>t</sub> 2 4 Zρ−1 t,T ET<sub>t</sub> hZρ−1 t,T i ` Dt logZt,T ´0 3 5 0.1 0.15 0.2 4 6 8 10 0 0.005 0.01 0.015 0.02 0.025 0.03

• Total portfolio weight: πt/X_t∗ = π_tmv/X_t∗ +πb_t/X_t∗ + π_tz/X_t∗ 0 0.05 0.1 0.15 0.2 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5

Initial short rate Relative risk aversion

• Approximate forward density portfolio weight: π_tf a/X_t∗ = −σ_t−1_R1 E_tT[N_t,T] 0 2 4 6 8 10 0 10 20 30 40 0 0.01 0.02 0.03 0.04 0.05

Relative risk aversion Investment horizon

Aprooximate forward measure hedge portfolio weight

0 2 4 6 8 10 0 10 20 30 40 0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04

Relative risk aversion Investment horizon

Forward measure hedge portfolio weight

• Approximate forward density portfolio weight: π_tf a/X_t∗ = −σ_t−1_R1 E_tT[N_t,T] 0 0.05 0.1 0.15 0.2 0 10 20 30 40 0 0.01 0.02 0.03 0.04 0.05 Spot rate Investment horizon

Aprooximate forward measure hedge portfolio weight

0 0.05 0.1 0.15 0.2 0 10 20 30 40 0 0.01 0.02 0.03 0.04 0.05 Spot rate Investment horizon

Forward measure hedge portfolio weight

7 Conclusion

I Contributions:

• Asset allocation formula based on change of num´eraire

• 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

I Integration of portfolio management and term structure models • Asset allocation in HJM framework