Dynamic Asset Allocation a Portfolio Decomposition Formula

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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