Optimal Investment, Consumption and Retirement Decision with

Optimal Investment, Consumption and Retirement Decision with Disutility and Borrowing Constraints Yong Hyun Shin Joint Work with Byung Hwa Lim(KAIST)

June 29 – July 3, 2009

Yong Hyun Shin (KIAS)

Workshop on Stochastic Analysis & Finance

June 29 – July 3, 2009

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Contents 1

Historical Remarks Portfolio Selection Borrowing Constraints

2

The Model Main Problem Duality Approaches and Variational Inequality

3

Solutions

4

Examples: CRRA Utility Classes CRRA Utility Class Numerical Results

5

Conclusion



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Historical Remarks

Portfolio Selection

Portfolio Selection (1)

Merton (1969 RESTAT, 1971 JET) Formulate a continuous time consumption/investement problem. (dynamic programming method) Z ∞ max E e−βt u(ct )dt c,π

0

subject to dXt = [rXt + πt (µ − r ) − ct ]dt + πt σdBt .



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Historical Remarks

Portfolio Selection


Karatzas and Wang (2000 SICON) Consider the mixture problem (c, π, τ ). (martingale method) Z τ e−βt u1 (ct )dt + e−βτ u2 (Xτ ) max E c,π,τ

0

subject to dXt = [rXt + πt (µ − r ) − ct ]dt + πt σdBt .



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Historical Remarks

Portfolio Selection


Choi and Shim (2006 MF), Benchmark Model Labor income, disutility and optimal retirement time. (dynamic programming without considering borrowing constraints) Z ∞ max E u(ct ) − l1{0≤t 0 retirement time: a stopping time τ Rt consumption process : ct with 0 cs ds < ∞, a.s. Rt portfolio process : πt with 0 πs2 ds < ∞, a.s. wealth process Xt with an initial wealth X0 = x ≥ 0 dXt = [rXt + πt (µ − r ) − ct + 1{0≤t 0 retirement time: a stopping time τ Rt consumption process : ct with 0 cs ds < ∞, a.s. Rt portfolio process : πt with 0 πs2 ds < ∞, a.s. wealth process Xt with an initial wealth X0 = x ≥ 0 dXt = [rXt + πt (µ − r ) − ct + 1{0≤t 0 Borrowing Constraint Z E Hτ Xτ + t


τ

Hs (cs − )ds Ft ≥ 0, for all 0 ≤ t < τ.


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The Model

Definition 1 (General Utility Function) A function u : (0, ∞) → R is a utility function if it is strictly increasing, strictly concave, continuously differentiable and satisfies u 0 (0+) , lim u 0 (c) = ∞, c↓0

u 0 (∞) , lim u 0 (c) = 0. c↑∞

Labor Income : the agent receives the income continuously Disutility l: disutility comes from labor Retirement Time τ : the immortal agent can choose when to retire



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The Model


u 0 (∞) , lim u 0 (c) = 0. c↑∞




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The Model


u 0 (∞) , lim u 0 (c) = 0. c↑∞




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The Model


u 0 (∞) , lim u 0 (c) = 0. c↑∞




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The Model

Main Problem

Main Problem

Expected Utility Maximization Problem The immortal investor wants to maximize her/his expected utility: Z ∞ J(x; c, π, τ ) , E e−βt u(ct ) − l1{0≤t 0 is the subjective discount rate.



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The Model

Main Problem

Expected Utility Maximization Problem The investor wants to maximize her/his expected utility Z ∞ sup J(x; c, π, τ ) , sup E e−βt u(ct ) − l1{0≤t 0 is an arbitrary constant, I1 (·) is the inverse function of u 0 (·) and λ∗∗ is determined by the algebraic equation −

2n+ (λ∗∗ )n+ −1 θ2 (n+ − n− )

λ∗∗

zI1 (z) − u(I1 (z)) dz z n+ +1 Z ∗∗ 2n− (λ∗∗ )n− −1 λ zI1 (z) − u(I1 (z)) + 2 dz = x. θ (n+ − n− ) yˆ z n− +1

Z

yˆ

Here n+ > 1 and n− < 0 are two roots of the following quadratic equation 1 2 2 1 θ n + β − r − θ2 n − β = 0. 2 2



(1)

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The Model

Main Problem

The Value Function

The Value Function The value function of our problem is given by V (x) =

sup

J(x; c, π, τ )

(c,π,τ )∈A(x)

= sup τ

sup

J(x; c, π, τ )

(c,π)∈Πτ (x)

, sup Vτ (x) τ

where A(x) is the set of an admissible triple (c, π, τ ) and Πτ (x) is the set of τ -fixed consumption-portfolio plan (c, π) for which (c, π, τ ) ∈ A(x)



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The Model

Duality Approaches and Variational Inequality

Duality Approaches (1)

´ (1993)) Individual’s Shadow Prices Problem (He and Pages τ

Z inf e J(λ, Dt ; τ ) , inf E

Dt >0

Dt >0

e−βt

0

n o e λ) , e(ytλ ) + ytλ − l dt + e−βτ U(y u τ

where Dt is the non-negative, decreasing, and progressively measurable process, ytλ = λDt eβt Ht e(y ) , sup {u(c) − cy} = u(I1 (y )) − yI1 (y ) u c

e ) , sup {U(x) − xy} = U(I2 (y )) − yI2 (y ), U(y x

−1

where I1 (·) , u 0 (·) and I2 (·) , U 0 (·) decreasing, strictly convex.


−1

e are strictly e(·) and U(·) . Moreover u


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The Model


Duality Approaches (2) τ

Z J(x; c, π, τ ) = E

e−βt {u(ct ) − l − λDt eβt Ht ct }dt

0

Z τ i Dt Ht ct dt + Dτ Hτ Xτ + e−βτ {U(Xτ ) − λDτ eβτ Hτ Xτ } + λE 0 Z τ Z τ −βt e βt −βτ e ≤E e u (λDt e Ht )dt + e U(λDτ eβτ Hτ ) − e−βt ldt 0 0 Z τ Dt Ht ct dt + Dτ Hτ Xτ + λE 0

Z

τ

βτ e e(λDt eβt Ht )dt + e−βτ U(λD e−βt u Hτ ) − τe 0 Z τ λDt Ht dt + λx +E

τ

Z

≤E

e−βt ldt

0

0

Z

τ

n o e(λDt eβt Ht )dt + λDt eβt Ht − l dt u 0 i −βτ e +e U(λDτ eβτ Hτ ) + λx.

=E


e−βt


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The Model



Z

τ

Dt Ht ct dt + Dτ Hτ Xτ 0 Z τ Z τ =E Dt Ht (ct − )dt + Dτ Hτ Xτ + Dt Ht dt 0 Z0 τ Z τ Z τ =E Dt Ht dt + Ht ct dt − Ht dt + Hτ Xτ 0 0 Z τ Z τ 0 Z τ +E E Hs cs ds + Hτ Xτ − Hs ds Ft dDt t t Z τ 0 Dt Ht dt + x. ≤E

E

0



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The Model



For any fixed τ ∈ S, previous inequalities hold as equality if ct = I1 (λDt eβt Ht ),

Xτ = I2 (λDτ eβτ Hτ ), for all 0 ≤ t ≤ τ,

τ

Z

Z Ht ct dt + Hτ Xτ −

E

τ

Ht dt = x,

0

0

and Z

Z Hs cs ds + Hτ Xτ −

E t


τ

t

τ

Hs ds Ft = 0.


(2)

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The Model


Duality Approaches (5) (Based on Karatzas and Wang (2000)) h i e Dt ; τ ) + λx V (x) = sup Vτ (x) = sup inf J(λ, τ ∈S τ ∈S {λ>0,Dt >0} h i e Dt ; τ ) + λx , = inf sup J(λ, {λ>0,Dt >0} τ ∈S

Proposition (Value Function) Define e (λ) , sup inf e V J(λ, Dt ; τ ) = inf sup e J(λ, Dt ; τ ), τ ∈S Dt >0

Dt >0 τ ∈S

e (λ) exists and is differentiable for λ > 0, then then if V h i e (λ) + λx , V (x) = inf V λ>0

for any x ∈ (0, ∞). Yong Hyun Shin (KIAS)


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The Model


Duality Approaches (5) (Based on Karatzas and Wang (2000)) h i e Dt ; τ ) + λx V (x) = sup Vτ (x) = sup inf J(λ, τ ∈S τ ∈S {λ>0,Dt >0} h i e Dt ; τ ) + λx , = inf sup J(λ, {λ>0,Dt >0} τ ∈S

Proposition (Value Function) Define e (λ) , sup inf e V J(λ, Dt ; τ ) = inf sup e J(λ, Dt ; τ ), τ ∈S Dt >0

Dt >0 τ ∈S

e (λ) exists and is differentiable for λ > 0, then then if V h i e (λ) + λx , V (x) = inf V λ>0

for any x ∈ (0, ∞). Yong Hyun Shin (KIAS)


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The Model


Variational Inequality (1)

e (λ), define To find V yt =y

Z

φ(t, y ) , sup inf E τ >t Dt >0

τ

e t

−βs

−βτ e e(ys ) + ys − l ds + e u U(yτ ) ,

where yt = λDt eβt Ht , y0 = λ > 0. Then dyt dDt = + (β − r )dt − θdBt . yt Dt e (λ). φ(0, λ) = V This optimal stopping problem can be solved by the variational inequality.



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The Model



Suppose Dt has a following differential form dDt = −ψ(t)Dt dt for some ψ(t) ≥ 0. The Bellman equation is given by ∂φ −βt e min Lφ(t, y ) + e {u (y ) + y − l}, − =0 ∂y with the differential operator L=

∂ ∂ 1 ∂2 + (β − r )y + θ2 y 2 2 . ∂t ∂y 2 ∂y

´ (1993)) (He and Pages



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The Model


Variational Inequality (3) Variational Inequality 2.1 Find the free boundary y¯ , yˆ which makes zero wealth level and a function e ·) ∈ C 1 ((0, ∞) × R+ ) ∩ C 2 ((0, ∞) × (R+ \ {y¯ })) satisfying φ(·, e + e−βt {u e(y) + y − l} = 0, y¯ < y ≤ yˆ (1) Lφ e + e−βt {u e(y) + y − l} ≤ 0, 0 < y ≤ y¯ (2) Lφ e e y) > e−βt U(y), y > y¯ (3) φ(t, e e y) = e−βt U(y), (4) φ(t, 0 < y ≤ y¯ , (5)

e ∂φ (t, y) ∂y

≤ 0, 0 < y ≤ yˆ

(6)

e ∂φ (t, y) ∂y

= 0, y ≥ yˆ

for all t > 0, with boundary conditions e ∂φ (t, yˆ ) = 0 and ∂y


e ∂2φ (t, yˆ ) = 0. ∂y 2


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The Model


∃ one-to-one correspondence between y and x.

y 0


y¯

yˆ


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The Model



y y¯

0

e ) e−βt U(y


yˆ

e(y ) + y − l} = 0 Lφe + e−βt {u


e ∂φ ∂y (t, y)

=0

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The Model



y y¯

0

e ) e−βt U(y

∞


yˆ

e(y ) + y − l} = 0 Lφe + e−βt {u

x¯

e ∂φ ∂y (t, y)

=0

x

0


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The Model



Proposition 2 Consider the function  C1 y n+ + C2 y n−   R y l+z(I1 (z)−)−u(I1 (z)) n+    + θ2 (n2y −n ) yˆ dz  z n+ +1  + −  n R y l+z(I1 (z)−)−u(I1 (z))  2y − dz, − θ2 (n −n ) yˆ n +1 v (y) = + z −  R−y zI1 (z)−u(I1 (z)) n+  2y  dz   θ 2 (n+ −n− ) yˆ z n+ +1   n R y zI1 (z)−u(I1 (z))  2y −  dz, − θ2 (n −n ) yˆ n− +1 +

−

z

if y¯ < y ≤ yˆ ,

if 0 < y ≤ y¯ ,

e y) = e−βt v (y ) is a solution to Variational Inequality. And the then φ(t, coefficients C1 , C2 , yˆ and the free boundary value y¯ are determined implicitly. e (λ) = φ(0, e λ) = v (λ) V



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Solutions

Value Function Theorem 3 The value function V (x) is given by  C1 (λ∗ )n+ + C2 (λ∗ )n− + (λ∗ )x    2(λ∗ )n+ R ∗ l+z(I1 (z)−)−u(I1 (z))   + θ2 (n −n ) yˆλ dz z n+ +1 + − V (x) = n 2(λ∗ ) − R λ∗ l+z(I1 (z)−)−u(I1 (z))  − dz,  n +1  θ 2 (n+ −n− ) yˆ  z −  U(x),

if 0 ≤ x < x¯ , if x ≥ x¯

where x¯ = I2 (¯ y ),

where λ∗ is determined from the following algebraic equation ∗ n −1 ∗ n −1 −n+ C1 (λ ) + − n− C2 (λ ) −


−

2n+ (λ∗ )n+ −1 Z λ∗ l + z(I1 (z) − ) − u(I1 (z)) dz θ 2 (n+ − n− ) yˆ z n+ +1

+

n −1 Z ∗ 2n− (λ∗ ) − λ l + z(I1 (z) − ) − u(I1 (z)) dz = x n +1 yˆ θ 2 (n+ − n− ) z −


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Solutions

Optimal Wealth Processes Before Retirement ∗ λ∗ n+ −1 λ∗ n− −1 X (t) = − n+ C1 (yt ) − n− C2 (yt ) −

∗ ∗ 2n+ (ytλ )n+ −1 Z y λ l + z(I1 (z) − ) − u(I1 (z)) t dz yˆ θ 2 (n+ − n− ) z n+ +1

+

∗ n −1 Z λ∗ 2n− (ytλ ) − l + z(I1 (z) − ) − u(I1 (z)) yt dz n +1 2 yˆ θ (n+ − n− ) z −

After Retirement X

∗∗

(t) = −

∗∗ n −1 Z λ∗∗ 2n+ (ytλ ) + zI1 (z) − u(I1 (z)) yt dz yˆ θ 2 (n+ − n− ) z n+ +1

+


∗∗ n −1 Z λ∗∗ ) − zI1 (z) − u(I1 (z)) yt dz n +1 yˆ θ 2 (n+ − n− ) z −

2n− (ytλ


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Solutions

Optimal Policies (1)

Theorem 4 The optimal policies (c ∗ , π ∗ , τ ∗ ) are given by ∗ I1 (ytλ ), if 0 ≤ Xt < x¯ ∗ ∗∗ ct = I1 (ytλ ), if Xt ≥ x¯ , With the optimal wealth process X ∗ (t), the optimal stopping time τ ∗ is determined by τ ∗ = inf {t > 0 | X ∗ (t) ≥ x¯ } .



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Solutions

Optimal Policies (2) Theorem 5 (Continued)

∗ πt =

                                                                  

∗ ∗ n −1 n+ (n+ − 1)C1 (ytλ )n+ −1 + n− (n− − 1)C2 (ytλ ) − ∗ ∗ ∗ λ λ λ l+yt (I1 (yt )−)−u(I1 (yt )) + 2 ∗ θ2 yλ t ∗ ∗ 2n+ (n+ −1)(ytλ )n+ −1 R ytλ l+z(I1 (z)−)−u(I1 (z)) + dz yˆ θ 2 (n+ −n− ) z n+ +1  ∗ n −1 ∗ λ λ − ) 2n (n −1)(yt R yt l+z(I1 (z)−)−u(I1 (z))  − − − dz , 2 n +1 ˆ y  θ (n+ −n− ) z −   y λ∗∗ I (y λ∗∗ )−u(I (y λ∗∗ )) 1 t 1 t 2 t ∗∗ σθ  yλ t ∗∗ ∗∗ n −1 λ + R ytλ ) n+ (n+ −1)(yt zI1 (z)−u(I1 (z)) + dz n+ −n− yˆ z n+ +1  ∗∗ n −1 ∗∗ λ λ − n− (n− −1)(yt ) R yt zI1 (z)−u(I1 (z))  − dz , n− +1 n+ −n− yˆ  z θ σ



if 0 ≤ Xt < x¯

if Xt ≥ x¯ .

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Examples: CRRA Utility Classes

CRRA Utility Class

Examples: CRRA Utility Class

Definition 6 (CRRA Utility Function) A CRRA utility function is defined by 1 1−γ , if γ > 0 and γ 6= 1, 1−γ c u(c) , log c, if γ = 1. Here γ is an investor’s coefficient of relative risk aversion. Merton’s Constant K ,r+


β−r γ−1 2 + θ > 0. γ 2γ 2


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CRRA Utility Class

CRRA Utility Class - Required Functions Expected Utility Maximization Problem (Power-Type) Z

τ

J(x; c, π, τ ) = E

e−βt

0

Z =E 0

τ

e−βt

Z ∞ 1 1 c 1−γ − l dt + c 1−γ dt e−βt 1−γ 1−γ τ 1 1−γ −βτ c − l dt + e U(Xτ ) 1−γ

Required Functions U(x) =

1 1 1−γ K γ 1−γ x

e(y) = u

γ − 1−γ γ 1−γ y

e U(y) =

γ − 1−γ γ K (1−γ) y



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CRRA Utility Class

CRRA Utility Class - Required Functions Expected Utility Maximization Problem (Power-Type) Z

τ

J(x; c, π, τ ) = E

e−βt

0

Z =E 0

τ

e−βt

Z ∞ 1 1 c 1−γ − l dt + c 1−γ dt e−βt 1−γ 1−γ τ 1 1−γ −βτ c − l dt + e U(Xτ ) 1−γ

Required Functions U(x) =

1 1 1−γ K γ 1−γ x

e(y) = u

γ − 1−γ γ 1−γ y

e U(y) =

γ − 1−γ γ K (1−γ) y



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CRRA Utility Class

CRRA Utility Class - Value Function

Proposition (Value Function) From Theorem 3,  1−γ γ  ∗ n ∗ − ∗ n   c1 (λ ) + + c2 (λ ) − + K (1−γ) (λ ) γ V (x) = + x + r (λ∗ ) − βl ,    1 1 x 1−γ , K γ 1−γ

if 0 ≤ x < x¯ if x ≥ x¯

where λ∗ is determined from the algebraic equation −n+ c1 (λ∗ )n+ −1 − n− c2 (λ∗ )n− −1 +

1 ∗ − γ1 (λ ) − = x, for 0 ≤ x < x¯ K r

and the critical wealth level is given by x¯ =


1 ¯ − γ1 . Ky


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CRRA Utility Class

CRRA Utility Class - Optimal Policies Optimal Policies ( ct∗

πt∗ =

        

θ σ

=

∗

1 −γ

(ytλ ) KXt ,

,

if 0 ≤ Xt < x¯ if Xt ≥ x¯ ,

n ∗ n+ (n+ − 1)c1 (ytλ )n+ −1 ∗

+n− (n− − 1)c2 (ytλ )n− −1 +

∗ −1 1 (y λ ) γ Kγ t

θ X, σγ t

,

if 0 ≤ Xt < x¯ if Xt ≥ x¯ ,

τ ∗ = inf {t > 0 | X ∗ (t) ≥ x¯ } , where the optimal wealth process before retirement is given by ∗

∗

X ∗ (t) = −n+ c1 (ytλ )n+ −1 − n− c2 (ytλ )n− −1 +



1 λ∗ − γ1 (y ) − . K t r

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Numerical Results

Numerical Results for a CRRA Utility Function (1) Figure 1: Comparison of amount of wealth invested in the risky asset (β = 0.07, r = 0.01, µ = 0.05, σ = 0.2, γ = 2, = 0.2 and l = 0.5)

120

100

Portfolio

80

60

40

20

-20

0

20

40

60

Wealth Level Yong Hyun Shin (KIAS)


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Numerical Results

Numerical Results for a CRRA Utility Function (2) Figure 2: Comparison of consumption ratio (β = 0.07, r = 0.01, µ = 0.05, σ = 0.2, γ = 2, = 0.2 and l = 0.5)

4 3.5

Consumption

3 2.5 2 1.5 1 0.5

-20

0

20

40

60

Wealth Level Yong Hyun Shin (KIAS)


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Conclusion

Conclusion

We extended the optimal consumption-portfolio selection problem of an infinitely-lived working investor whose wealth is subject to borrowing constraint to the general utility function case We figured out that the critical wealth level with borrowing constraint is lower than the level with no constraint for the CRRA utility case The amount of investing to risky asset with borrowing constraint is lower than the amount with no constraint for the CRRA utility case



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Conclusion

Thank you!



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