Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution

Transcription

1 Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution Ninna Reitzel Jensen PhD student University of Copenhagen Joint work with Mogens Steffensen (University of Copenhagen) 8th Conference in Actuarial Science & Finance on Samos May 30, 2014 May 30, 2014 Slide 1/18

2 Motivation Preferences Consumption Lifetime Wealth/income Investor Investment Life insurance optimal product design Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 2/18

3 Agenda 1 Optimization problem 2 Challenge & solution 3 Link to recursive utility 4 Hump-shaped consumption (numerics) Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 3/18

4 Mathematical set-up Black-Scholes financial market. Mortality intensities µ og ˆµ. The investor s wealth X is formalized by dx (t) = investment returns {}}{ X (t) ((r + π (t) λ) dt + π (t) σdw (t)) + w (t) dt }{{} c (t) dt }{{} b (t) ˆµ (t) dt }{{}, labor income consumption life insurance premium X (0) = x 0. Power utility from consumption and bequest: u (c) = 1 1 γ c. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 4/18

5 Classical optimization problem (no mortality risk) where Z c,π,b (t, x) = E t,x sup Z c,π,b (t, x) c,π,b n t e δ(s t) 1 1 γ c (s) ds. }{{} utility at time s Solution via Dynamic Programming. Problem: γ accounts for both risk aversion and elasticity of inter-temporal substitution (EIS). Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 5/18

6 Separation of risk aversion and EIS Certainty equivalents: ] u 1 (E t,x [u (c (s))]) = E t,x [c 1 (s). Addition taking into account EIS (= 1 φ ): n t e δ(s t) 1 [ ] 1 φ E t,x c 1 φ (s) ds. Utility not time-additive any more. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 6/18

7 My optimization problem (with mortality risk) Z c,π,b (t, x) = 1 1 γ 1 1 γ n δe δ(s t) t n t ( sup Z c,π,b (t, x), where c,π,b expected utility from consumption { [ }} ]{ Et,x 0 c I (s) ds (s) + [ ds Et,x 0 ε (s) ( X c,π,b (s) + b (s) ) ] dn (s) ds }{{} expected utility from bequest δe δ(s t) e 1 φ s ) 1 φ µ(v) dv t ( m c,π,b (t, s, x) + n c,π,b (t, s, x) ) ds 1 φ. 1 φ ds 1 φ = The special case γ = φ is treated in [Richard, 1975]. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 7/18

8 Agenda 1 Optimization problem 2 Challenge & solution 3 Link to recursive utility 4 Hump-shaped consumption (numerics) Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 8/18

9 Time inconsistency Challenge: non-linearity of Z c,π,b (t, x) = 1 1 γ n t s µ(v) dv δe δ(s t) e 1 φ t ( m c,π,b (t, s, x) + n c,π,b (t, s, x) ) 1 φ ds 1 φ. Consequence: "optimal" control depends on time of solution; Find optimal control (c, π, b ) at time 0 and apply the control up until time t > 0. At time t the control (c, π, b ) is no longer optimal. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 9/18

10 Equilibrium Idea: Equlibrium approach from [Björk & Murgoci, 2010]. Definition: A control (c, π, b ) is an equilibrium control if Z c,π,b (u, y) Z ch,π h,b h (u, y) lim inf 0 h 0 h ( for all (u, y) and c h, π h, b h) with ( c h (t, x), π h (t, x), b h (t, x) ) = { ( c (t, x), π (t, x), b (t, x)), u t < u + h, (c (t, x), π (t, x), b (t, x)), u + h t n. A different kind of optimality. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 10/18

11 Verification theorem Assume that there exists a function U in C 1,2 ([0, n] R) that solves the pseudo-bellman equation c,π,b U t (t, x) = inf U (n, x) = 0, f (t, c, x + b, U (t, x)) x (r + πλ) U x (t, x) ( c ˆµ (t) b + w (t)) U x (t, x) 1 2 σ2 π 2 x 2 U xx (t, x) 1 2 σ2 π 2 x 2 I (t, x), and let (c, π, b ) be the function that realizes the infimum. Then (c, π, b ) is an optimal control, and U = Z c,π,b. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 11/18

12 Agenda 1 Optimization problem 2 Challenge & solution 3 Link to recursive utility 4 Hump-shaped consumption (numerics) Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 12/18

13 Link to recursive utility The function f is given by ( c + ε (t) µ (t) y f (t, c, y, z) = 1 γ 1 φ δz z (1 γ) ( µ (t) + 1 γ ) 1 φ δ z. ) 1 φ Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 13/18

14 Link to recursive utility The function f is given by ( c + ε (t) µ (t) y f (t, c, y, z) = 1 γ 1 φ δz z (1 γ) ( µ (t) + 1 γ ) 1 φ δ z. ) 1 φ In the special case without mortality risk, we get f EZ (c, z) = 1 γ 1 φ δz ( c z (1 γ) ) 1 φ 1. This is the normalized Epstein-Zin aggreagator from recursive utility in continuous time, see e.g. [Duffie & Epstein, 1992]. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 13/18

15 Identity of optimal controls Hence, some sort of equivalence with the recursive utility optimization problem sup V c,π (0) c,π where V c,π is defined via [ n ] V c,π (t) = E f EZ (c (s), V c,π (s)) ds F t t. This equivalence is particularly interesting because... Continuous-time recursive utility optimization with Epstein-Zin preferences is generalized to include mortality risk and life insurance. Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 14/18

16 Agenda 1 Optimization problem 2 Challenge & solution 3 Link to recursive utility 4 Hump-shaped consumption (numerics) Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 15/18

17 Optimal consumption for γ = φ = 2 (Richard) Op#mal consump#on rate (USD) Investor's age (years) delta = 0.03 delta = 0.05 delta = 0.08 delta = 0.10 Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 16/18

18 Optimal consumption for γ = 2 < φ = Op#mal consump#on rate (USD) Investor's age (years) delta = 0.03 delta = 0.05 delta = 0.08 delta = 0.10 Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 17/18

19 References Tomas Björk and Agatha Murgoci. A General Theory of Markovian Time Inconsistent Stochastic Control Problems. Working paper, Darrell Duffie and Larry G. Epstein. Stochastic Differential Utility. Econometrica, 60: , Holger Kraft and Frank Thomas Seifried. Foundations of Continuous-Time Recursive Utility: Differentiability and Normalization of Certainty Equivalents. Mathematics and Financial Economics, 3: , Scott F. Richard. Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics, 2: , Ninna Reitzel Jensen Personal Finance and Life Insurance under Separation of Risk Aversion and Elasticity of Substitution May 30, 2014 Slide 18/18