Optimal Investment, Consumption and Retirement Decision with - - PowerPoint PPT Presentation

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 1 / 36

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 2 / 36

Historical Remarks Portfolio Selection

Portfolio Selection (1)

Merton (1969 RESTAT, 1971 JET) Formulate a continuous time consumption/investement problem. (dynamic programming method) max

c,π E

∞ e−βtu(ct)dt

• subject to

dXt = [rXt + πt(µ − r) − ct]dt + πtσdBt.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 3 / 36

Historical Remarks Portfolio Selection

Portfolio Selection (2)

Karatzas and Wang (2000 SICON) Consider the mixture problem (c, π, τ). (martingale method) max

c,π,τ E

τ e−βtu1(ct)dt + e−βτu2(Xτ)

• subject to

dXt = [rXt + πt(µ − r) − ct]dt + πtσdBt.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 4 / 36

Historical Remarks Portfolio Selection

Portfolio Selection (3)

Choi and Shim (2006 MF), Benchmark Model Labor income, disutility and optimal retirement time. (dynamic programming without considering borrowing constraints) max

c,π,τ E

∞

• u(ct) − l1{0≤t<τ}
• dt
• subject to

dXt = [rXt + πt(µ − r) − ct + ǫ1{0≤t<τ}]dt + πtσdBt.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 5 / 36

Historical Remarks Borrowing Constraints

Borrowing Constraints

Borrowing Constraint (Liquidity Constraint) The economic agent has limited opportunities to borrow against future labor income and cannot totally insure the risk of income fluctuation. So the borrowing constraint restricts the agent’s choice in a non-trivial way. Literatures He and Pag´ es (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (2005 WP), Farhi and Panageas (2007 JFE), Choi, Shim, and Shin (2008 MF), Saha (2008 WP)

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 6 / 36

Historical Remarks Borrowing Constraints

Borrowing Constraints

Borrowing Constraint (Liquidity Constraint) The economic agent has limited opportunities to borrow against future labor income and cannot totally insure the risk of income fluctuation. So the borrowing constraint restricts the agent’s choice in a non-trivial way. Literatures He and Pag´ es (1993 JET), Duffie, Fleming, Soner, and Zariphopoulou (1997 JEDC), Koo (1998 MF), El Karoui and Jeanblanc (1998 FS), Dybvig and Liu (2005 WP), Farhi and Panageas (2007 JFE), Choi, Shim, and Shin (2008 MF), Saha (2008 WP)

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 6 / 36

The Model

The Basic Financial Market Setup (1)

riskless asset S0(·): dS0(t)

S0(t) = rdt

risky asset St: dSt

St = µdt + σdBt

r, µ, σ: constants Bt is a standard Brownian motion on a probability space (Ω, F, P)

market-price-of-risk θ µ−r

σ

discount process ζt exp{−rt} exponential martingale Zt exp

• −θBt − 1

2θ2t

• pricing kernel(or state-price-density) Ht ζtZt

equivalent martingale measure

• PT(A) E[ZT1A], for any fixed T ∈ [0, ∞) and for A ∈ FT
• BT

t Bt + θt : a standard Brownian motion under the new measure

PT by Girsanov theorem ∃ P on F∞, which agrees with PT on FT for any T ∈ [0, ∞). Furthermore

• Bt is a standard Brownian motion under

P.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 7 / 36

The Model

The Basic Financial Market Setup (2)

labor wage: ǫ > 0, disutility: l > 0 retirement time: a stopping time τ consumption process : ct with t

0 csds < ∞, a.s.

portfolio process : πt with t

0 π2 sds < ∞, a.s.

wealth process Xt with an initial wealth X0 = x ≥ 0 dXt = [rXt + πt(µ − r) − ct + ǫ1{0≤t<τ}]dt + πtσdBt = [rXt − ct + ǫ1{0≤t<τ}]dt + πtσd Bt budget constraint E

• HτXτ +

τ Hscsds − τ Hsǫds

• ≤ x, for all τ

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 8 / 36

The Model

The Basic Financial Market Setup (2)

labor wage: ǫ > 0, disutility: l > 0 retirement time: a stopping time τ consumption process : ct with t

0 csds < ∞, a.s.

portfolio process : πt with t

0 π2 sds < ∞, a.s.

wealth process Xt with an initial wealth X0 = x ≥ 0 dXt = [rXt + πt(µ − r) − ct + ǫ1{0≤t<τ}]dt + πtσdBt = [rXt − ct + ǫ1{0≤t<τ}]dt + πtσd Bt budget constraint E

• HτXτ +

τ Hscsds − τ Hsǫds

• ≤ x, for all τ

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 8 / 36

The Model

Borrowing Constraints

The borrowing constraint means that the investor cannot borrow against her/his future labor income. So the wealth of the investor should always be

• nonnegative. i.e. Xt ≥ 0, ∀t > 0

Borrowing Constraint E

• HτXτ +

τ

t

Hs(cs − ǫ)ds

• Ft
• ≥ 0, for all 0 ≤ t < τ.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 9 / 36

The Model

Borrowing Constraints

The borrowing constraint means that the investor cannot borrow against her/his future labor income. So the wealth of the investor should always be

• nonnegative. i.e. Xt ≥ 0, ∀t > 0

Borrowing Constraint E

• HτXτ +

τ

t

Hs(cs − ǫ)ds

• Ft
• ≥ 0, for all 0 ≤ t < τ.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 9 / 36

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+) lim

c↓0 u′(c) = ∞,

u′(∞) lim

c↑∞ u′(c) = 0.

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36

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+) lim

c↓0 u′(c) = ∞,

u′(∞) lim

c↑∞ u′(c) = 0.

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36

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+) lim

c↓0 u′(c) = ∞,

u′(∞) lim

c↑∞ u′(c) = 0.

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36

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+) lim

c↓0 u′(c) = ∞,

u′(∞) lim

c↑∞ u′(c) = 0.

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 10 / 36

The Model Main Problem

Main Problem

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

• dt
• i.e.,

V(x) = sup

(c,π,τ)∈A(x)

J(x; c, π, τ) subject to the budget constraint and the borrowing constraint, where A(x) is the set of an admissible triple (c, π, τ) and β > 0 is the subjective discount rate.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 11 / 36

The Model Main Problem

Expected Utility Maximization Problem

The investor wants to maximize her/his expected utility sup

(c,π,τ)∈A(x)

J(x; c, π, τ) sup

(c,π,τ)∈A(x)

E ∞ e−βt u(ct) − l1{0≤t<τ}

• dt
• =

sup

(c,π,τ)∈A(x)

E τ e−βt (u(ct) − l) dt + e−βτU(Xτ)

• ,

subject to the budget constraint E τ Hscsds + HτXτ − τ Hsǫds

• ≤ x,

and the borrowing constraint, E

• HτXτ +

τ

t

Hs(cs − ǫ)ds

• Ft
• ≥ 0.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 12 / 36

The Model Main Problem

Lemma 2 (Value Function After Retirement) The value function U(·) is given by

U(x) = 2(λ∗∗)n+ θ2(n+ − n−) λ∗∗

ˆ y

zI1(z) − u(I1(z)) zn++1 dz − 2(λ∗∗)n− θ2(n+ − n−) λ∗∗

ˆ y

zI1(z) − u(I1(z)) zn−+1 dz + (λ∗∗)x,

where ˆ y > 0 is an arbitrary constant, I1(·) is the inverse function of u′(·) and λ∗∗ is determined by the algebraic equation

− 2n+(λ∗∗)n+−1 θ2(n+ − n−) λ∗∗

ˆ y

zI1(z) − u(I1(z)) zn++1 dz + 2n−(λ∗∗)n−−1 θ2(n+ − n−) λ∗∗

ˆ y

zI1(z) − u(I1(z)) zn−+1 dz = x. Here n+ > 1 and n− < 0 are two roots of the following quadratic equation 1 2 θ2n2 +

• β − r − 1

2 θ2

• n − β = 0.

(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

(c,π,τ)∈A(x)

J(x; c, π, τ) = sup

τ

sup

(c,π)∈Πτ (x)

J(x; c, π, τ) sup

τ

Vτ(x) 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)

Individual’s Shadow Prices Problem (He and Pag´ es (1993))

inf

Dt >0

• J(λ, Dt; τ) inf

Dt >0 E

τ e−βt

• u(yλ

t ) + yλ t ǫ − l

• dt + e−βτ

U(yλ

τ )

• ,

where Dt is the non-negative, decreasing, and progressively measurable process, y λ

t = λDteβtHt

• u(y) sup

c

{u(c) − cy} = u(I1(y)) − yI1(y)

• U(y) sup

x

{U(x) − xy} = U(I2(y)) − yI2(y), where I1(·) u′(·)−1 and I2(·) U′(·)−1. Moreover u(·) and U(·) are strictly decreasing, strictly convex.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 15 / 36

The Model Duality Approaches and Variational Inequality

Duality Approaches (2)

J(x; c, π, τ) = E τ e−βt{u(ct) − l − λDteβtHtct}dt + e−βτ{U(Xτ) − λDτeβτHτXτ}

• + λE

τ DtHtctdt + DτHτXτ

• ≤ E

τ e−βt u(λDteβtHt)dt + e−βτ U(λDτeβτHτ) − τ e−βtldt

• + λE

τ DtHtctdt + DτHτXτ

• ≤ E

τ e−βt u(λDteβtHt)dt + e−βτ U(λDτeβτHτ) − τ e−βtldt

• + E

τ λDtHtǫdt

• + λx

= E τ e−βt

• u(λDteβtHt)dt + λDteβtHtǫ − l
• dt

+ e−βτ U(λDτeβτHτ)

• + λx.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 16 / 36

The Model Duality Approaches and Variational Inequality

Duality Approaches (3)

E τ DtHtctdt + DτHτXτ

• = E

τ DtHt(ct − ǫ)dt + DτHτXτ + τ DtHtǫdt

• = E

τ DtHtǫdt + τ Htctdt − τ Htǫdt + HτXτ

• + E

τ E τ

t

Hscsds + HτXτ − τ

t

Hsǫds

• Ft
• dDt
• ≤ E

τ DtHtǫdt

• + x.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 17 / 36

The Model Duality Approaches and Variational Inequality

Duality Approaches (4)

For any fixed τ ∈ S, previous inequalities hold as equality if ct = I1(λDteβtHt), Xτ = I2(λDτeβτHτ), for all 0 ≤ t ≤ τ, E τ Htctdt + HτXτ − τ Htǫdt

• = x,

and E τ

t

Hscsds + HτXτ − τ

t

Hsǫds

• Ft
• = 0.

(2)

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 18 / 36

The Model Duality Approaches and Variational Inequality

Duality Approaches (5)

(Based on Karatzas and Wang (2000)) V(x) = sup

τ∈S

Vτ(x) = sup

τ∈S

inf

{λ>0,Dt>0}

• J(λ, Dt; τ) + λx
• =

inf

{λ>0,Dt>0} sup τ∈S

• J(λ, Dt; τ) + λx
• ,

Proposition (Value Function) Define

• V(λ) sup

τ∈S

inf

Dt >0

• J(λ, Dt; τ) = inf

Dt >0 sup τ∈S

• J(λ, Dt; τ),

then if V(λ) exists and is differentiable for λ > 0, then V(x) = inf

λ>0

• V(λ) + λx
• ,

for any x ∈ (0, ∞).

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 19 / 36

The Model Duality Approaches and Variational Inequality

Duality Approaches (5)

(Based on Karatzas and Wang (2000)) V(x) = sup

τ∈S

Vτ(x) = sup

τ∈S

inf

{λ>0,Dt>0}

• J(λ, Dt; τ) + λx
• =

inf

{λ>0,Dt>0} sup τ∈S

• J(λ, Dt; τ) + λx
• ,

Proposition (Value Function) Define

• V(λ) sup

τ∈S

inf

Dt >0

• J(λ, Dt; τ) = inf

Dt >0 sup τ∈S

• J(λ, Dt; τ),

then if V(λ) exists and is differentiable for λ > 0, then V(x) = inf

λ>0

• V(λ) + λx
• ,

for any x ∈ (0, ∞).

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 19 / 36

The Model Duality Approaches and Variational Inequality

Variational Inequality (1)

To find V(λ), define φ(t, y) sup

τ>t

inf

Dt>0 Eyt=y

τ

t

e−βs u(ys) + ǫys − l

• ds + e−βτ

U(yτ)

• ,

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 20 / 36

The Model Duality Approaches and Variational Inequality

Variational Inequality (2)

Suppose Dt has a following differential form dDt = −ψ(t)Dtdt for some ψ(t) ≥ 0. The Bellman equation is given by min

• Lφ(t, y) + e−βt{

u(y) + ǫy − l}, −∂φ ∂y

• = 0

with the differential operator L = ∂ ∂t + (β − r)y ∂ ∂y + 1 2θ2y2 ∂2 ∂y2 . (He and Pag´ es (1993))

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 21 / 36

The Model Duality Approaches and Variational Inequality

Variational Inequality (3)

Variational Inequality 2.1 Find the free boundary ¯ y, ˆ y which makes zero wealth level and a function

• φ(·, ·) ∈ C1((0, ∞) × R+) ∩ C2((0, ∞) × (R+ \ {¯

y})) satisfying (1) L φ + e−βt{ u(y) + ǫy − l} = 0, ¯ y < y ≤ ˆ y (2) L φ + e−βt{ u(y) + ǫy − l} ≤ 0, 0 < y ≤ ¯ y (3) φ(t, y) > e−βt U(y), y > ¯ y (4) φ(t, y) = e−βt U(y), 0 < y ≤ ¯ y, (5)

∂ φ ∂y (t, y) ≤ 0, 0 < y ≤ ˆ

y (6)

∂ φ ∂y (t, y) = 0, y ≥ ˆ

y for all t > 0, with boundary conditions ∂ φ ∂y (t, ˆ y) = 0 and ∂2 φ ∂y2 (t, ˆ y) = 0.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 22 / 36

The Model Duality Approaches and Variational Inequality

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 23 / 36

The Model Duality Approaches and Variational Inequality

∃ one-to-one correspondence between y and x. y ¯ y ˆ y e−βt U(y) L φ + e−βt{ u(y) + ǫy − l} = 0

∂ φ ∂y (t, y) = 0

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 23 / 36

The Model Duality Approaches and Variational Inequality

∃ one-to-one correspondence between y and x. y ¯ y ˆ y e−βt U(y) L φ + e−βt{ u(y) + ǫy − l} = 0

∂ φ ∂y (t, y) = 0

x ∞ ¯ x

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 23 / 36

The Model Duality Approaches and Variational Inequality

Variational Inequality (4)

Proposition 2 Consider the function

v(y) =                    C1yn+ + C2yn− +

2yn+ θ2(n+−n−)

y

ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn++1

dz −

2yn− θ2(n+−n−)

y

ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn−+1

dz, if ¯ y < y ≤ ˆ y,

2yn+ θ2(n+−n−)

y

ˆ y zI1(z)−u(I1(z)) zn++1

dz −

2yn− θ2(n+−n−)

y

ˆ y zI1(z)−u(I1(z)) zn−+1

dz, if 0 < y ≤ ¯ y,

then φ(t, y) = e−βtv(y) is a solution to Variational Inequality. And the coefficients C1, C2, ˆ y and the free boundary value ¯ y are determined implicitly.

• V(λ) =

φ(0, λ) = v(λ)

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 24 / 36

Solutions

Value Function

Theorem 3 The value function V(x) is given by

V(x) =            C1(λ∗)n+ + C2(λ∗)n− + (λ∗)x +

2(λ∗)n+ θ2(n+−n−)

λ∗

ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn++1

dz −

2(λ∗)n− θ2(n+−n−)

λ∗

ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn−+1

dz, if 0 ≤ x < ¯ x, U(x), if x ≥ ¯ x where

¯ x = I2(¯ y),

where λ∗ is determined from the following algebraic equation

−n+C1(λ∗)n+−1 − n−C2(λ∗)n−−1 − 2n+(λ∗)n+−1 θ2(n+ − n−) λ∗ ˆ y l + z(I1(z) − ǫ) − u(I1(z)) zn++1 dz + 2n−(λ∗)n−−1 θ2(n+ − n−) λ∗ ˆ y l + z(I1(z) − ǫ) − u(I1(z)) zn−+1 dz = x Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 25 / 36

Solutions

Optimal Wealth Processes

Before Retirement

X∗(t) = − n+C1(yλ∗ t )n+−1 − n−C2(yλ∗ t )n−−1 − 2n+(yλ∗ t )n+−1 θ2(n+ − n−) yλ∗ t ˆ y l + z(I1(z) − ǫ) − u(I1(z)) zn++1 dz + 2n−(yλ∗ t )n−−1 θ2(n+ − n−) yλ∗ t ˆ y l + z(I1(z) − ǫ) − u(I1(z)) zn−+1 dz

After Retirement

X∗∗(t) = − 2n+(yλ∗∗ t )n+−1 θ2(n+ − n−) yλ∗∗ t ˆ y zI1(z) − u(I1(z)) zn++1 dz + 2n−(yλ∗∗ t )n−−1 θ2(n+ − n−) yλ∗∗ t ˆ y zI1(z) − u(I1(z)) zn−+1 dz Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 26 / 36

Solutions

Optimal Policies (1)

Theorem 4 The optimal policies (c∗, π∗, τ ∗) are given by c∗

t =

• I1(yλ∗

t

), if 0 ≤ Xt < ¯ x I1(yλ∗∗

t

), 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} .

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 27 / 36

Solutions

Optimal Policies (2)

Theorem 5 (Continued)

π∗ t =                                                                    θ σ

• n+(n+ − 1)C1(yλ∗

t )n+−1 + n−(n− − 1)C2(yλ∗ t )n−−1 + 2 θ2 l+yλ∗ t (I1(yλ∗ t )−ǫ)−u(I1(yλ∗ t )) yλ∗ t + 2n+(n+−1)(yλ∗ t )n+−1 θ2(n+−n−) yλ∗ t ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn++1 dz − 2n−(n−−1)(yλ∗ t )n−−1 θ2(n+−n−) yλ∗ t ˆ y l+z(I1(z)−ǫ)−u(I1(z)) zn−+1 dz    , if 0 ≤ Xt < ¯ x 2 σθ    yλ∗∗ t I1(yλ∗∗ t )−u(I1(yλ∗∗ t )) yλ∗∗ t + n+(n+−1)(yλ∗∗ t )n+−1 n+−n− yλ∗∗ t ˆ y zI1(z)−u(I1(z)) zn++1 dz − n−(n−−1)(yλ∗∗ t )n−−1 n+−n− yλ∗∗ t ˆ y zI1(z)−u(I1(z)) zn−+1 dz    , if Xt ≥ ¯ x. Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 28 / 36

Examples: CRRA Utility Classes CRRA Utility Class

Examples: CRRA Utility Class

Definition 6 (CRRA Utility Function) A CRRA utility function is defined by u(c)

• 1

1−γ c1−γ,

if γ > 0 and γ = 1, log c, if γ = 1. Here γ is an investor’s coefficient of relative risk aversion. Merton’s Constant K r + β − r γ + γ − 1 2γ2 θ2 > 0.

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 29 / 36

Examples: CRRA Utility Classes CRRA Utility Class

CRRA Utility Class - Required Functions

Expected Utility Maximization Problem (Power-Type)

J(x; c, π, τ) = E τ e−βt

• 1

1 − γ c1−γ − l

• dt +

∞

τ

e−βt 1 1 − γ c1−γdt

• = E

τ e−βt

• 1

1 − γ c1−γ − l

• dt + e−βτU(Xτ)
• Required Functions

U(x) =

1 K γ 1 1−γ x1−γ

• u(y) =

γ 1−γ y− 1−γ

γ

• U(y) =

γ K(1−γ)y− 1−γ

γ Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 30 / 36

Examples: CRRA Utility Classes CRRA Utility Class

CRRA Utility Class - Required Functions

Expected Utility Maximization Problem (Power-Type)

J(x; c, π, τ) = E τ e−βt

• 1

1 − γ c1−γ − l

• dt +

∞

τ

e−βt 1 1 − γ c1−γdt

• = E

τ e−βt

• 1

1 − γ c1−γ − l

• dt + e−βτU(Xτ)
• Required Functions

U(x) =

1 K γ 1 1−γ x1−γ

• u(y) =

γ 1−γ y− 1−γ

γ

• U(y) =

γ K(1−γ)y− 1−γ

γ Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 30 / 36

Examples: CRRA Utility Classes CRRA Utility Class

CRRA Utility Class - Value Function

Proposition (Value Function)

From Theorem 3, V(x) =        c1(λ∗)n+ + c2(λ∗)n− +

γ K(1−γ) (λ∗)− 1−γ

γ

+

• x + ǫ

r

• (λ∗) − l

β ,

if 0 ≤ x < ¯ x

1 K γ 1 1−γ x1−γ,

if x ≥ ¯ x

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

γ − ǫ

r = x, for 0 ≤ x < ¯ x and the critical wealth level is given by ¯ x = 1

K ¯

y− 1

γ . Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 31 / 36

Examples: CRRA Utility Classes CRRA Utility Class

CRRA Utility Class - Optimal Policies

Optimal Policies

c∗

t =

• (yλ∗

t

)− 1

γ ,

if 0 ≤ Xt < ¯ x KXt, if Xt ≥ ¯ x, π∗

t =

        

θ σ

• n+(n+ − 1)c1(yλ∗

t

)n+−1 +n−(n− − 1)c2(yλ∗

t

)n−−1 +

1 Kγ (yλ∗ t

)− 1

γ

• ,

if 0 ≤ Xt < ¯ x

θ σγ Xt,

if Xt ≥ ¯ x, τ ∗ = inf {t > 0 | X ∗(t) ≥ ¯ x} , where the optimal wealth process before retirement is given by X ∗(t) = −n+c1(yλ∗

t

)n+−1 − n−c2(yλ∗

t

)n−−1 + 1 K (yλ∗

t

)− 1

γ − ǫ

r .

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 32 / 36

Examples: CRRA Utility Classes 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, ǫ = 0.2 and l = 0.5)

• 20

20 40 60

Wealth Level

20 40 60 80 100 120

Portfolio

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 33 / 36

Examples: CRRA Utility Classes 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, ǫ = 0.2 and l = 0.5)

• 20

20 40 60

Wealth Level

0.5 1 1.5 2 2.5 3 3.5 4

Consumption

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 34 / 36

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

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 35 / 36

Conclusion

Thank you!

Yong Hyun Shin (KIAS) Workshop on Stochastic Analysis & Finance June 29 – July 3, 2009 36 / 36