Portfolio Choice under Probability Distortion

Transcription

1 Portfolio Choice with Life Annuities under Probability Distortion Wenyuan Zheng James G. Bridgeman Department of Mathematics University of Connecticut The 49th Actuarial Research Conference, 2014

2 Outline Introduction Motivation Literature Review Formulation Theoretical Results

3 Motivation Literature Review Outline Introduction Motivation Literature Review Formulation Theoretical Results

4 Motivation Literature Review The Need for Behavioral Economics Policyholder s behaviors affect insurance companies operations Behavioral Economics help understand policyholder s behaviors

5 Motivation Literature Review Probability Distortion People overweight small probabilities and underweight large probabilities 1 Weighted Cumulative Probabilities Distorted Distribution Function Distribution Function

6 Motivation Literature Review Outline Introduction Motivation Literature Review Formulation Theoretical Results

7 Motivation Literature Review Literature Review Yarri (1965) Optimal for an individual without a bequest motive to fully annuitize Milevsky and Young (2007) Realistically incorporated mortality-contingent payout annuities (e.g. DB plan) Wang and Young (2012) Commutable life annuities to maximize the lifetime utility Young and Zariphopoulou (1999) Derive stochastic differential equation for a distorted probability via stochastic differential games

8 Formulation Theoretical Results Outline Introduction Motivation Literature Review Formulation Theoretical Results

9 Formulation Theoretical Results Goal Behavioral Economics Probability distortion Portfolio Choice, consumption and annuitization strategy Continuous-time setting Maximize the lifetime utility Commutable life annuity

10 Formulation Theoretical Results Market A riskless asset A risky asset Commutable life annuities A single premium immediate annuity with a surrender option

11 Formulation Theoretical Results New Probability Distortion Typical distortion function p δ w(p) = (p δ +(1 p) δ ) δ 1 Why difficult to apply? Hard to derive its stochastic differential equation We propose a new distortion function w(p) = δ ln(1 p), δ > 1

12 Formulation Theoretical Results Weibull Distribution Why Weibull distribution for stock price? Explicit hazard function Original SDE dxs = [ X γ γ β σ s + γx β s β ]ds + (2X γ β+1 σβ s β ) 1 2 db s Distorted SDE dxs = [ X s γ + γx s γ β σ β β + 2X s γ ( 1 + 2δ (2X s γ β+1 σ β β ) 1 2 db s β: shape parameter σ and γ: scale parameters 1+δ Xs σ β )]ds +

13 Formulation Theoretical Results Outline Introduction Motivation Literature Review Formulation Theoretical Results

14 Formulation Theoretical Results Wealth dynamics dw s = [r(w s Π s ) Π s γ + γπ s γ β σ β β + 2Π s γ ( 1 + 2δ C s + A s ]ds + (2Π s γ β+1 σ β β ) 1 2 db s 1+δ Πs σ Value function U(W, A) = sup πs,c s E[ 0 e (r+λ)s u(c s )ds W 0 = W, A 0 = A] β )

15 Formulation Theoretical Results HJB equation (r + λ)u = (rw s + A s )U w + max πs {[ rπ s Π s γ + γπ s γ β σ β β + 2Π s γ ( 1 + 2δ 1+δ Πs σ γ β+1 σ β )]U w + Π β s β U ww} + max cs ( c1 γ s 1 γ cu w)

16 Numerical Method U w (i, j + 1) = U w (i, j) + [W (2) W (1)] U ww (i, j + 1) U(i, j + 1) = U(i, j) + [W (2) W (1)] U w (i, j + 1)

17 Parameters r = 0.04 λ = 0.04 β = 1 σ = 5 γ = 2 α = 2.5 p = 0.2 δ = 2

18 Outline Introduction Motivation Literature Review Formulation Theoretical Results

19 Strategy undistorted investment distorted investment Annuity Wealth 1000

20 1 Weighted Cumulative Probabilities Distorted Distribution Function Fear of Large Loss Hope of Large Gains Distribution Function

21 X s γ + γx s γ β σ β β + 2X s γ ( 1 + 2δ 1+δ Xs σ β ) > Stochastic Differential Equation for Weibull Distribution dx g(x) h(x) g(x)+h(x) g(x) x

22 Outline Introduction Motivation Literature Review Formulation Theoretical Results

23 Strategy undistorted consumption distorted consumption Annuity 0 0 Wealth

24 30 Utility Function 25 Before Distortion (less risk averse) 20 Utility 15 After Distortion (more risk averse)

25 Outline Introduction Motivation Literature Review Formulation Theoretical Results

26 Behavior Buy Do nothing Surrender U(A+4,W-50) Utility U(A,W) U(A-4,W+40)

27 Undistorted Case 80 Annuity Buy/Surrender Behavior (undistorted) : buy : do nothing : surrender Annuity Wealth

28 Distorted Case 80 Annuity Buy/Surrender Behavior (distorted) : buy : do nothing : surrender Annuity Wealth

29 Need more annuities to against fear Stop buying annuity at a higher level Begin surrendering annuity at a higher level Different z 0 : critical ratio of wealth-to-annuity An unique z0 in Wang and Young (2012) Behavior pattern Annuity Surrender Surrender Do nothing Do nothing Do nothing Buy Buy Wealth

30 Illustration W=500 A=62 No Distortion Distortion Stock Bond Annuitization Do nothing Buy One year later... No Distortion W=468 A=62 Distortion W=422 A=66

31 Illustration W=1000 A=74 No Distortion Distortion Stock Bond Annuitization Surrender Do nothing One year later... No Distortion W=1030 A=70 Distortion W=998 A=74

32 Sensitivity Analysis 1 Weighted Cumulative Probabilities (delta=2) 1 Weighted Cumulative Probabilities (delta=5) Distorted Distribution Function Distorted Distribution Function Distribution Function Distribution Function

33 Sensitivity Analysis 80 Annuity Buy/Surrender Behavior (delta=2) 80 Annuity Buy/Surrender Behavior (delta=5) Annuity 40 Annuity Wealth Wealth

34 Probability distortion brings more fear To against fear Invest less on risky asset Consume more Need more annuity (also support more consumption) Contribution of this work A new distortion function Weibull distribution for stock price Annuitization behavior available for each pair of (Wealth, Annuity)

35 Thank you!