optimal portfolio choice with path dependent labor income ...€¦ · optimal portfolio choice with...

Optimal portfolio choice with path dependentlabor income: the infinite horizon case

Fausto Gozzi

joint work with Enrico Biffis and Cecilia Prosdocimi

Milano, April 20, 2017

MotivationBenchmark model without path dependency

Sticky wages

Outline

1 Motivation

2 Benchmark model without path dependency

3 Sticky wages

Fausto Gozzi Optimal portfolio choice with path dependent labor income: the infinite horizon case


Sticky wages

Asset allocation

Merton (1971): wealth of the investor is given by risky andnon risky assets. Optimal for agents to put a constant fractionof their wealth in the risky asset throughout all their life

Starting from the ’90: models which add labor income to thewealth, see e.g. Bodie et al. (’92), Campbell-Viceira (’02),Fahri-Panageas (’07) Dybvig-Liu (’10). The total wealth of anagent is given by her financial wealth and her human capital,i.e. the present value of future labor income. Investors putoptimally a constant fraction of their financial wealth in therisky asset.

Empirical evidence of sticky wages: they respond to shocks inthe market, but with a delay (see Meghir-Pistaferri 2004)



Sticky wages

Our Goal.

Study a model of asset allocation where the wages arepath-dependent to describe more precisely their real behavior.First step of a project in progress.



Sticky wages

Outline

1 Motivation


3 Sticky wages



Sticky wages

The model of Dybvig, P.H. and Liu, H. (2010)

The market is Black & Scholes type:

dS0(t) =rS0(t)dt

dS1(t) =S1(t)µdt + S1(t)σdZ (t),



Sticky wages

dW (t) =[W (t)r + θ(t)(µ− r)− c(t) + δ

(W (t)− B(t)

)]dt

+ (1− R(t))y(t)dt + θ(t)σdZ (t), W (0) = W0

dy(t) =y(t)(µydt + σydZ (t)

), y(0) = y0

W (t) the wealth process, state

y(t) the labor income, state

θ(t) the dollar investment in the risky asset, control

c(t) the consumption, control

B(t) the bequest, control

R(t) := IT≤t and T is the retirement time, control

δ constant rate of mortality



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The death time τδ is modeled as a Poisson arrival time, withhazard rate δ.τδ is independent of the Wiener process (Z (t)).

We should consider as basic filtration the one generated byFτδ ∨ FZ , but we will actually work on τδ > t.

B(t) is the bequest-target we want to leave to the recipient:

for W (t)− B(t) < 0 we have to finance it buyingcontinuously a life insurance with premium δ(B(t)−W (t))

for W (t)− B(t) > 0 then the term δ(B(t)−W (t)) can beseen as a life annuity since it trades wealth in the event ofdeath for a cash inflow while living.



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Goal: maximize over(c(·),B(·), θ(·),T

)E∫ τδ

0e−ρt

((1− R(t))

c(t)1−γ

1− γ+ R(t)

(Kc(t))1−γ

1− γ

)dt

+e−ρτδ

(kB(τδ)

)1−γ

1− γdt

,

K > 1, different marginal utility for unit of c before and after T .The constant k > 0 measures the intensity of preference forleaving a large bequest.The expectation above can be written as J(W0, y0; c,B, θ,T ) :=

E

∫ +∞

0e−(ρ+δ)t

((KR(t)c(t))1−γ

1− γ+ δ

(kB(t)

)1−γ

1− γ

)dt



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The constraints

Problem 1 of Dybvig-Liu 2010

Given retirement time T , no borrowing-without-repaymentconstraint

W (t) ≥ −g(t)y(t),

where

g(t) :=

(1−e−β1(T−t)

β1

)+if β1 6= 0

(T − t)+ if β1 = 0

with

β1 := r + δ − µy +(µ− r)

σσy .



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Meaning of the constraints

Let ξ(t) be the state price density adjusted to condition on living;

ξ(t) := e−(r+δ+ 12

(µ−r)2

σ2 )t− (µ−r)σ

Z(t).

i.e. the solution ofdξ(t) = −ξ(t)(r + δ)dt − ξ(t)κ>dZ (t),ξ(0) = 1.

(1)

Then

g(t)y(t) = E(∫ T

ty(s)ξ(s)ds | Ft

).

ξ(t)−1E( ∫ T

t y(s)ξ(s)ds | Ft

)is the human capital at time t i.e.

the value at t of subsequent labor income.



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Results of Dybvig-Liu 2010

Find an explicit expression for the value function

V (W0, y0) := supadmissible strategies

J(W0, y0; c ,B, θ,T )

and for the optimal strategies (consumption, bequest, portfolio).

Also other problems with different constraints are studied pointingout the financial meaning of the results.



Sticky wages

Outline

1 Motivation


3 Sticky wages



Sticky wages

Empirical evidence of sticky wages: they respond to shocks in themarket, but with a delay (see Meghir-Pistaferri 2004)

We expect that considering sticky wages will change optimal assetallocation and optimal retirement.We start with the model with no retirement, for simplicity.



Sticky wages

The model

dW (t) =[W (t)r + θ(t)(µ− r)− c(t) + δ

(W (t)− B(t)

)]dt

+ y(t)dt + θ(t)σdZ (t), W (0) = W0

dy(t) =

(y(t)µy +

∫ 0

−dα(ξ)y(t + ξ)dξ

)dt + y(t)σydZ (t),

y(0) =y0, y(ξ) = y1(ξ) ∀ξ ∈ [−d , 0).

W (t), y(t), θ(t), c(t), B(t), as before;

α(·) a square integrable (L2) function.



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J1(W0, y0, y1; c ,B, θ) :=

E

∫ +∞

0e−(ρ+δ)t

(c(t)1−γ

1− γ+ δ

(kB(t)

)1−γ

1− γ

)dt

. (2)

Problem

Given T , choose c(·), θ(·), B(·) to maximize (2), with thefollowing no-borrowing-without-repayment constraint

W (t) ≥ −Gy(t)−∫ 0

−dH(ξ)y(t + ξ)dξ,



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The constant G and the function H are

G := (β1 − β∞)−1

H(ξ) :=∫ ξ−d e−(r+δ)(ξ−s)α(s)ds

where

β∞ :=

∫ 0

−de−(r+δ)sα(s)ds

As before

Gy(t) +

∫ 0

−dH(ξ)y(t + ξ)dξ = ξ(t)−1E

(∫ T

ty(s)ξ(s)ds | Ft

)is the human capital at time t i.e. the value at t of subsequentlabor income. This is a nontrivial result, see Biffis et al ’15.Note. Human capital must take into account the past of y . Forα = 0, H = 0 and G coincides with g .



Sticky wages

Stochastic control problem, finite horizon

state equationdx(t) = b

(x(t), c(t)

)dt + σ

(x(t), c(t)

)dZ (t)

x(0) = x0

set of admissible controls (here C is bounded)

U := c : [0,T ]× Ω −→ C | c is Ft-adapted.

objective functional

J(c(·)

):= E

∫ T

0f(t, x(t), c(t)

)dt + h

(x(T )

)Goal : maximize J subject to the state equation over U .



Sticky wages

Dynamic programming

Consider a family of problems, s ∈ [0,T ]

state equation

(∗)

dx(t) = b(x(t), c(t)

)dt + σ

(x(t), c(t)

)dZ (t) t ∈ [s,T ]

x(s) = y(3)

set of admissible controls

U s := c : [s,T ]× Ω −→ C | c is (F st )t≥s -adapted.


J(s, y ; c()

):= E

∫ T

sf(t, x (s,y)(t), c(t)

)dt + h

(x (s,y)(T )

),

where x (s,y)(t) is the solution at time t of (∗).



Sticky wages

Define the value function

V (s, y) := supc(·)∈U s

J(s, y ; c(·)

), for any (s, y) ∈ [0,T ]× R

Dynamic Programming Principle, s ∈ [s,T ]

V (s, y) = supc(·)∈U s

E∫ s

sf(t, x (s,y)(t), c(t)

)dt

+V(s, x (s,y)(s)

)If c(·) |[s,T ] is optimal on [s,T ] with initial value (s, y), then

c(·) |[s,T ] is optimal on [s,T ] with initial value (s, x (s,y)(s)).



Sticky wages

Dynamic programming cont’d

Passing to the limit we get the Hamilton-Jacobi-Bellman (HJB)equation

−vt = H

(t, x , vx , vxx

)for any (s, y) ∈ [0,T ]× R

v(T , y) = h(y) for any y ∈ R

where

H(t, x , p,P

)= sup

c∈Cf (t, x , c) + b(x , c)p +

1

2σ2(x , c)P

If V is C1,2([0,T ]×R

)+ reasonable conditions, then V solves the

HJB equation.



Sticky wages

Dynamic programming cont’d

Verification TheoremLet v a C1,2

([0,T ]× R

)-solution of the HJB and let exist a

function c : [0,T ]× R −→ C such that for any (t, x)

c(t, x) = argmaxf (t, x , c) + b(x , c)vx +1

2σ2(x , c)vxx

and c(·) is admissible, then v is the value function and c() is anoptimal control.



Sticky wages

Infinite horizon

state equation and admissible controls as before


J(s, y ; c(·)

):= E

∫ +∞

se−ρt f

(x (s,y)(t), c(t)

)dt,

value function

V (s, y) := supc(·)∈U s

J(s, y ; c(·)

), for any (s, y) ∈ [0,+∞)× R

we have

V (s, y) = e−ρsV (0, y) = e−ρsV0(y).

Hamilton-Jacobi-Bellman equation for V0

ρv = H(x , vx , vxx

)for any y ∈ R

where

H(x , p,P

)= sup

c∈Cf (x , c) + b(x , c)p +

1

2σ2(x , c)P



Sticky wages

Delay equations as ODEs in infinite dimensional spaces

The state equation of y(·) is a stochastic delay differentialequation. Classical theory works for Markovian state-equation.Consider the Hilbert space

H := R× L2([−d , 0];R

),

with inner product for x = (x0, x1), z = (z0, z1) ∈ H

〈x , z〉H := x0z0 +

∫ 0

−dx1(ξ)z1(ξ)dξ

= x0z0 + 〈x1, z1〉L2



Sticky wages

SetX (t) =

(X0(t),X1(t)

):=(y(t), y(t + ξ)|ξ∈[−d ,0]

),

X (t) is an element of H for all t ∈ [0,+∞). Let X satisfy

dX (t) = AX (t)dt + CX (t)dZ (t), X (0) = (y0, y1) ∈ H

with

A(x0, x1) :=(µyx0 + 〈α(·), x1(·)〉L2 , x ′1(·)

),

C (x0, x1) := (x0σy , 0)

Then the original problem is equivalent to the control problem withstate X in the infinite dimensional space H(see Gozzi-Marinelli ’04).



Sticky wages

Our result

Theorem

The value function V0 is

V0(W , x0, x1) := f γ∞Γ1−γ

1− γ,

where

f∞ := (1 + δk1γ−1)ν,

ν :=γ

ρ+ δ − (1− γ)(r + δ + κ>κ2γ )

> 0.

Γ := W0 + Gx0 + 〈H, x1〉L2 ≥ 0,



Sticky wages

The optimal strategies are

c∗(t) := f −1∞ Γ∗(t)

B∗(t) := k−bf −1∞ Γ∗(t)

θ∗(t) :=(µ− r)Γ∗(t)

γσ2− Gσyy(t)

σ,

where Γ∗(t) := W ∗(t) + GX0(t) + 〈H,X1(t, ·)〉L2 .We have

dΓ∗(t)

Γ∗(t)=[r + δ +

1

γ(µ− r

σ)2

− f −1∞(1 + δk−b

)]dt

+µ− r

γσdZ (t).



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First findings

with no-labor risk (σy = 0), the optimal ratio θ∗

Γ∗ and c∗

Γ∗ areconstant, as in the Merton model

taking α = 0, we recover Dybvig-Liu result

taking α 6= 0, we recover Dybvig-Liu result, but substitutingto the optimal total wealth Λ∗ (financial wealth + humancapital) of Dybvig-Liu the quantityΓ∗(t) = W ∗(t) + G (t)X0 + 〈H(t, ·),X1(t, ·)〉Λ∗ and Γ∗ evolve with the same dynamic



Sticky wages

Sketch of the proof

Guess the value function to be

V (W0, x0, x1) := f γ∞(W0 + Gx0+〉H, x1〈L2)1−γ

1− γ,

putting V in the HJB equation, gives equations for f ,G ,H

solving these equations, we get that f ,G ,H are the constantas in the main Theorem

V is C1,2

Verification Theorem holds and the optimal feedbackstrategies are admissible



Sticky wages

Thank you for your attention



Sticky wages

References

Benzoni, L., Collin-Dufresne, P. and Goldstein, R. (2007)Portfolio Choice over the life-cycle when the stock and labormarkets are cointegrated in The Journal of Finance, 62, 5,pp.2123-2167

Dybvig, P.H. and Liu, H. (2010) Lifetime consumption andinvestment: retirement and constrained borrowing in Journalof Economic Theory, 145, pp. 885-907

Farhi, E. and Panageas, S. (2007), Saving and investing forearly retirement: a theoretical analysis, in Journal of FinancialEconomics 83, pp. 87-121



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Gozzi, F. and Marinelli, C. (2006), Stochastic optimal controlof delay equations arising in advertising models, in StochasticPartial Differential Equations and Application VII. Lect. NotesPure Appl. Math., 245, pp. 133 - 148. Chapman &Hall/CRC, Boca Raton

Meghir, C. and Pistaferri, L. (2004) Income variance dynamicsand heterogeneity, Econometrica, 72, 1, pp. 1-32


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