Optimal portfolio choice with path dependent
labor income: the infinite horizon case
Fausto Gozzi
joint work with Enrico Biffis and Cecilia Prosdocimi
Milano, April 20, 2017
Motivation
Benchmark 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: th
Motivation
Benchmark model without path dependency
Sticky wages
Asset allocation
Merton (1971): wealth of the investor is given by risky and
non risky assets. Optimal for agents to put a constant fraction
of their wealth in the risky asset throughout all their life
Starting from the ’90: models which add labor income to the
wealth, see e.g. Bodie et al. (’92), Campbell-Viceira (’02),
Fahri-Panageas (’07) Dybvig-Liu (’10). The total wealth of an
agent is given by her financial wealth and her human capital,
i.e. the present value of future labor income. Investors put
optimally a constant fraction of their financial wealth in the
risky asset.
Empirical evidence of sticky wages: they respond to shocks in
the market, but with a delay (see Meghir-Pistaferri 2004)
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Our Goal.
Study a model of asset allocation where the wages are
path-dependent to describe more precisely their real behavior.
First step of a project in progress.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark 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: th
Motivation
Benchmark model without path dependency
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),
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
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) µy dt + σy dZ (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) := I{T ≤t} and T is the retirement time, control
δ constant rate of mortality
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
The death time τδ is modeled as a Poisson arrival time, with
hazard rate δ.
τδ is independent of the Wiener process (Z (t)).
We should consider as basic filtration the one generated by
F τδ ∨ F Z , 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 buying
continuously 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 be
seen as a life annuity since it trades wealth in the event of
death for a cash inflow while living.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Goal: maximize over c(·), B(·), θ(·), T
Z τδ
c(t)1−γ
(Kc(t))1−γ
−ρt
E
e
(1 − R(t))
+ R(t)
dt
1−γ
1−γ
0
)
1−γ
−ρτδ kB(τδ )
dt ,
+e
1−γ
K > 1, different marginal utility for unit of c before and after T .
The constant k > 0 measures the intensity of preference for
leaving a large bequest.
The expectation above can be written as J(W0 , y0 ; c, B, θ, T ) :=
(Z
1−γ ! )
+∞
R(t) c(t))1−γ
kB(t)
(K
E
e −(ρ+δ)t
+δ
dt
1−γ
1−γ
0
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
The constraints
Problem 1 of Dybvig-Liu 2010
Given retirement time T , no borrowing-without-repayment
constraint
W (t) ≥ −g (t)y (t),
where
(
g (t) :=
1−e −β1 (T −t) +
β1
(T − t)+
if β1 6= 0
if β1 = 0
with
β1 := r + δ − µy +
Fausto Gozzi
(µ − r )
σy .
σ
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Meaning of the constraints
Let ξ(t) be the state price density adjusted to condition on living;
2
(µ−r )
1 (µ−r )
)t− σ Z (t)
σ2
ξ(t) := e −(r +δ+ 2
.
i.e. the solution of
dξ(t) = −ξ(t)(r + δ)dt − ξ(t)κ> dZ (t),
ξ(0) = 1.
Then
Z
g (t)y (t) = E
T
(1)
y (s)ξ(s)ds | Ft .
t
R
T
ξ(t)−1 E t y (s)ξ(s)ds | Ft is the human capital at time t i.e.
the value at t of subsequent labor income.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Results of Dybvig-Liu 2010
Find an explicit expression for the value function
V (W0 , y0 ) :=
sup
J(W0 , y0 ; c, B, θ, T )
admissible strategies
and for the optimal strategies (consumption, bequest, portfolio).
Also other problems with different constraints are studied pointing
out the financial meaning of the results.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark 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: th
Motivation
Benchmark model without path dependency
Sticky wages
Empirical evidence of sticky wages: they respond to shocks in the
market, but with a delay (see Meghir-Pistaferri 2004)
We expect that considering sticky wages will change optimal asset
allocation and optimal retirement.
We start with the model with no retirement, for simplicity.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
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
Z 0
dy (t) = y (t)µy +
α(ξ)y (t + ξ)dξ dt + y (t)σy dZ (t),
−d
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.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
J1 (W0 , y0 , y1 ; c, B, θ) :=
(Z
+∞
E
0
e −(ρ+δ)t
1−γ ! )
kB(t)
c(t)1−γ
+δ
dt .
1−γ
1−γ
(2)
Problem
Given T , choose c(·), θ(·), B(·) to maximize (2), with the
following no-borrowing-without-repayment constraint
Z 0
W (t) ≥ −Gy (t) −
H(ξ)y (t + ξ)dξ,
−d
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
The constant G and the function H are
(β1 − β∞ )−1
G :=
Rξ
H(ξ) :=
−d
e −(r +δ)(ξ−s) α(s)ds
where
Z
0
β∞ :=
e −(r +δ)s α(s)ds
−d
As before
Z
0
Gy (t) +
−1
H(ξ)y (t + ξ)dξ = ξ(t)
−d
E
Z
T
y (s)ξ(s)ds | Ft
t
is the human capital at time t i.e. the value at t of subsequent
labor income. This is a nontrivial result, see Biffis et al ’15.
Note. Human capital must take into account the past of y . For
α = 0, H = 0 and G coincides with g .
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Stochastic control problem, finite horizon
state equation
dx(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
Z
J c(·) := E
T
f t, x(t), c(t) dt + h x(T )
0
Goal : maximize J subject to the state equation over U.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
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)
(∗)
x(s) = y
set of admissible controls
t ∈ [s, T
U s := {c : [s, T ] × Ω −→ C | c is (Fts )t≥s -adapted}.
objective functional
Z
J s, y ; c() := E
T
f t, x (s,y ) (t), c(t) dt + h x (s,y ) (T ) ,
s
where
x (s,y ) (t)
is the solution at time t of (∗).
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Define the value function
V (s, y ) := sup J s, y ; c(·) , for any (s, y ) ∈ [0, T ] × R
c(·)∈U s
Dynamic Programming Principle, ŝ ∈ [s, T ]
Z
ŝ
V (s, y ) = sup E
f t, x (s,y ) (t), c(t) dt
c(·)∈U s
s
+V ŝ, x (s,y ) (ŝ)
If c̄(·) |[s,T ] is optimal on [s, T ] with initial value (s, y ), then
c̄(·) |[ŝ,T ] is optimal on [ŝ, T ] with initial value (ŝ, x (s,y ) (ŝ)).
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
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
1
H t, x, p, P = sup {f (t, x, c) + b(x, c)p + σ 2 (x, c)P}
2
c∈C
If V is C 1,2 [0, T ] × R + reasonable conditions, then V solves the
HJB equation.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Dynamic programming cont’d
Verification Theorem Let v a C 1,2 [0, T ] × R -solution of the HJB and let exist a
function c̄ : [0, T ] × R −→ C such that for any (t, x)
1
c̄(t, x) = argmax{f (t, x, c̄) + b(x, c̄)vx + σ 2 (x, c̄)vxx }
2
and c̄(·) is admissible, then v is the value function and c̄() is an
optimal control.
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Infinite horizon
state equation and admissible controls as before
objective functional
Z
+∞ −ρt
J s, y ; c(·) := E
e f x (s,y ) (t), c(t) dt ,
s
value function
V (s, y ) := sup J s, y ; c(·) , for any (s, y ) ∈ [0, +∞) × R
c(·)∈U s
we have
V (s, y ) = e −ρs V (0, y ) = e −ρs V0 (y ).
Hamilton-Jacobi-Bellman
equation for V0
ρv = H x, vx , vxx for any y ∈ R
where
1
H x, p, P = sup {f (x, c) + b(x, c)p + σ 2 (x, c)P}
2
c∈C
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Delay equations as ODEs in infinite dimensional spaces
The state equation of y (·) is a stochastic delay differential
equation. 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
Z
0
hx, ziH := x0 z0 +
x1 (ξ)z1 (ξ)dξ
−d
= x0 z0 + hx1 , z1 iL2
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Set
X (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 ) := µy x0 + hα(·), x1 (·)iL2 , x10 (·) ,
C (x0 , x1 ) := (x0 σy , 0)
Then the original problem is equivalent to the control problem with
state X in the infinite dimensional space H
(see Gozzi-Marinelli ’04).
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Our result
Theorem
The value function V0 is
γ
V0 (W , x0 , x1 ) := f∞
Γ1−γ
,
1−γ
where
1
f∞ := (1 + δk γ
ν :=
−1
)ν,
γ
ρ + δ − (1 − γ)(r + δ +
κ> κ
2γ )
> 0.
Γ := W0 + Gx0 + hH, x1 iL2 ≥ 0,
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
The optimal strategies are
−1 ∗
c ∗ (t) := f∞
Γ (t)
−1 ∗
B ∗ (t) := k −b f∞
Γ (t)
(µ − r )Γ∗ (t) G σy y (t)
−
,
θ∗ (t) :=
γσ 2
σ
where Γ∗ (t) := W ∗ (t) + GX0 (t) + hH, X1 (t, ·)iL2 .
We have
dΓ∗ (t) h
1 µ−r 2
= r +δ+ (
)
∗
Γ (t)
γ σ
i
−1
− f∞
1 + δk −b dt
+
µ−r
dZ (t).
γσ
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
First findings
with no-labor risk (σy = 0), the optimal ratio
constant, as in the Merton model
θ∗
Γ∗
and
c∗
Γ∗
are
taking α = 0, we recover Dybvig-Liu result
taking α 6= 0, we recover Dybvig-Liu result, but substituting
to the optimal total wealth Λ∗ (financial wealth + human
capital) of Dybvig-Liu the quantity
Γ∗ (t) = W ∗ (t) + G (t)X0 + hH(t, ·), X1 (t, ·)i
Λ∗ and Γ∗ evolve with the same dynamic
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Sketch of the proof
Guess the value function to be
γ
V (W0 , x0 , x1 ) := f∞
(W0 + Gx0 +iH, x1 hL2 )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 constant
as in the main Theorem
V is C 1,2
Verification Theorem holds and the optimal feedback
strategies are admissible
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Thank you for your attention
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
References
Benzoni, L., Collin-Dufresne, P. and Goldstein, R. (2007)
Portfolio Choice over the life-cycle when the stock and labor
markets are cointegrated in The Journal of Finance, 62, 5,
pp.2123-2167
Dybvig, P.H. and Liu, H. (2010) Lifetime consumption and
investment: retirement and constrained borrowing in Journal
of Economic Theory, 145, pp. 885-907
Farhi, E. and Panageas, S. (2007), Saving and investing for
early retirement: a theoretical analysis, in Journal of Financial
Economics 83, pp. 87-121
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th
Motivation
Benchmark model without path dependency
Sticky wages
Gozzi, F. and Marinelli, C. (2006), Stochastic optimal control
of delay equations arising in advertising models, in Stochastic
Partial Differential Equations and Application VII. Lect. Notes
Pure Appl. Math., 245, pp. 133 - 148. Chapman &
Hall/CRC, Boca Raton
Meghir, C. and Pistaferri, L. (2004) Income variance dynamics
and heterogeneity, Econometrica, 72, 1, pp. 1-32
Fausto Gozzi
Optimal portfolio choice with path dependent labor income: th