Portfolio Choice under Probability Distortion

Portfolio Choice with Life Annuities under

Probability Distortion

Wenyuan Zheng

James G. Bridgeman

Department of Mathematics

University of Connecticut

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

The Need for Behavioral Economics

I

Policyholder’s behaviors affect insurance companies’

operations

I

Behavioral Economics help understand policyholder’s

Probability Distortion

I

People overweight small probabilities and underweight

large probabilities

Weighted Cumulative Probabilities

Distribution Function

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Literature Review

I

Yarri (1965)

I

_{Optimal for an individual without a bequest motive to fully}

annuitize

I

Milevsky and Young (2007)

I

_{Realistically incorporated mortality-contingent payout}

annuities (e.g. DB plan)

I

Wang and Young (2012)

I

_{Commutable life annuities to maximize the lifetime utility}

I

Young and Zariphopoulou (1999)

I

Derive stochastic differential equation for a distorted

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Goal

I

Behavioral Economics

I

_{Probability distortion}

I

Portfolio Choice

I

_{Investment, consumption and annuitization strategy}

I

Continuous-time setting

I

Maximize the lifetime utility

Market

I

A riskless asset

I

A risky asset

I

Commutable life annuities

I

_{A single premium immediate annuity with a surrender}

New Probability Distortion

I

Typical distortion function

I

w

(

p

) =

p

(

p

₊₍

₁

₋

_p

₎

₎

I

Why difficult to apply?

I

_{Hard to derive its stochastic differential equation}

I

We propose a new distortion function

I

_w

(

p

) =

1

−

1

Weibull Distribution

I

Why Weibull distribution for stock price?

I

Explicit hazard function

I

Original SDE

dX

_s

= [

−

X

s

γ

+

γX

s

γ

−

β σ

β

]

ds

+ (

2

X

s

γ

−

β

+

1

σ

β

)

dB

_s

I

Distorted SDE

_dX

_s

= [

−

X

s

γ

+

γX

s

γ

−

β σ

β

+

2

X

s

γ

(

−

1

+

2

δ

1

+

δ

)]

ds

+

(

2

X

s

γ

−

β

+

1

σ

β

)

dB

_s

I

β

: shape parameter

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Model

I

Wealth dynamics

dW

s

= [

r

(

W

s

−

Π

s

)

−

Π

s

γ

+

γ

Π

s

γ

−

β σ

β

+

2

Π

s

γ

₍

₋

1

+

2

δ

1

+

δ

)

−

C

s

+

A

s

]

ds

+ (

2

Π

s

γ

−

β

+

1

σ

β

)

dB

_s

I

Value function

U

(

W

,

A

) =

sup

_π

_,

_c

E

[

R

₀

∞

e

−(

r

+

λ

)

s

u

(

c

s

)

ds

|

W

0

=

W

,

A

0

=

A

]

Model

I

HJB equation

(

r

+

λ

)

U

= (

rW

s

+

A

s

)

U

w

+

max

π

{

[

−

r

Π

s

−

Π

s

γ

+

γ

Π

s

γ

−

β σ

β

+

2

Π

s

γ

(

−

1

+

2

δ

1

+

δ

)]

U

w

+ Π

s

γ

−

β

+

1

σ

β

U

ww

}

+

max

c

(

c

1

−

γ

−

cU

w

)

Numerical Method

I

U

w

(i

,

j

+

1

) =

U

w

(i

,

j) + [W

(

2

)

−

W

(

1

)]

·

U

ww

(i

,

j

+

1

)

Parameters

I

r

=

0

.

04

I

λ

=

0

.

04

I

β

=

1

I

σ

=

5

I

γ

=

2

I

α

=

2

.

5

I

p

=

0

.

2

I

δ

=

2

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Investment

Investment

Investment

Weighted Cumulative Probabilities

Distribution Function

Distorted Distribution Function

Fear of Large Loss

Investment

I

−

X

s

γ

+

γX

s

γ

−

β σ

β

+

2

X

s

γ

(

−

1

+

2

δ

1

+

δ

)

>

0

Stochastic Differential Equation for Weibull Distribution

dx

g(x)+h(x) g(x) g(x)−h(x)

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Consumption

Consumption

Consumption

Utility

Before Distortion (less risk averse)

Outline

Introduction

Motivation

Literature Review

Model

Model Formulation

Theoretical Results

Numerical Results

Investment

Consumption

Annuitization Strategy

Conclusion

Annuitization Strategy

Behavior

Utility

Buy

U(A+4,W-50)

Do nothing

U(A,W)

Annuitization Strategy

Undistorted Case

+

: buy

◦: do nothing

×

: surrender

Annuity Buy/Surrender Behavior (undistorted)

Wealth

Annuitization Strategy

Distorted Case

+

: buy

◦: do nothing

×

: surrender

Annuity Buy/Surrender Behavior (distorted)

Wealth

Annuitization Strategy

I

Need more annuities to against fear

I

_{Stop buying annuity at a higher level}

_{Begin surrendering annuity at a higher level}

I

Different

z

0

: critical ratio of wealth-to-annuity

_{An unique}

_z

₀

_{in Wang and Young (2012)}

I

Behavior pattern

Ann

uity

−

→

Surrender

Surrender

→

Do nothing

Do nothing

Do nothing

→

Buy

Buy

Wealth

−→

Illustration

W=500 A=62

No Distortion

Distortion

Stock

4.97

4.57

Bond

495.00

445.43

Consumption

138.42

145.36

Annuitization

Do nothing

Buy

One year later...

No Distortion

W=468 A=62

Illustration

W=1000 A=74

No Distortion

Distortion

Stock

4.96

4.57

Bond

995.04

995.43

Consumption

149.04

157.72

Annuitization

Surrender

Do nothing

One year later...

No Distortion

W=1030 A=70

Sensitivity Analysis

Weighted Cumulative Probabilities (delta=2)

Distribution Function

Distorted Distribution Function

0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

Weighted Cumulative Probabilities (delta=5)

Distribution Function

Sensitivity Analysis

Annuity Buy/Surrender Behavior (delta=2)

Wealth Annuity 0 200 400 600 800 1000 0 10 20 30 40 50 60 70 80

Annuity Buy/Surrender Behavior (delta=5)

Wealth

Conclusion

I

Probability distortion brings more fear

I

To against fear

I

_{Invest less on risky asset}

_{Consume more}

I

_{Need more annuity (also support more consumption)}

I

Contribution of this work

I

A new distortion function

I

_{Weibull distribution for stock price}

I

_{Annuitization behavior available for each pair of (Wealth,}