Optimal decumulation into annuity after retirement: a stochastic control approach
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1 Optimal decumulation into annuity after retirement: a stochastic control approach Nicolas Langrené, Thomas Sneddon, Geo rey Lee, Zili Zhu 2 nd Congress on Actuarial Science and Quantitative Finance, Cartagena, 18 June
2 Motivation Defined Contribution plans ) Longevity risk borne by retirees! Solution: Lifelong annuity? Unpopular:. money locked away. fear of immediate death. bequest motives IDEA: Gradual decumulation into annuity! Optimal decumulation into annuity after retirement: a stochastic control approach 2 / 16
3 Retiree situation B Initial wealth W 0 at retirement time t 0 B Consumption C t = c I t, I t =inflation B Consumption covered by: 1. Portfolio of investments n assets, prices S t,weightsx t 2. Inflation-linked Annuity price A t,cash-flowsc t Optimal decumulation into annuity after retirement: a stochastic control approach 3 / 16
4 How much annuity to buy? Percentage 0 apple a t apple 1: a t = 0% ) Consumption fully covered by withdrawals from investment portfolio a t = 100% ) Consumption fully covered by annuity Wealth dynamics W ti = W ti a ti A ti (1 a ti ) C ti t i W ti+1 = W ti apple. x ti + x ti. (S ti+1 S ti ) Optimal decumulation into annuity after retirement: a stochastic control approach 4 / 16 Dynamic annuity purchase. a ti = a ti a ti, x ti = x ti x ti
5 Objective function Maximize expected final utility V 0 = max E [ U (W T)] (a ti,x ti ), 0applet i applet T = (random) time of retiree death Dynamic programming principle V ti = max (a ti,x ti ) E [V t i+1 F ti ] P (T t i+1 T t i ) F ti = (S t,i t,w t,a t,x t ) 0appletappleti (assuming T? F t, t apple T ) Optimal decumulation into annuity after retirement: a stochastic control approach 5 / 16 Stochastic control problem
![Least-squares Monte Carlo (1/2) Parametric approximation of conditional expectation Ê [V ti+1 X ti ] := ˆt i.p(x ti ) ˆti := arg inf 2R B 1 M MX m=1.](/docs-images/90/103263483/images/6-1.jpg "p X m t i V m t i+1 2 Optimal decumulation into annuity after retirement: a stochastic control approach 6 / 16 E h i V ti+1 X ti = arg infuti 2L 2 ( (X ti ),P) E")
6 Least-squares Monte Carlo (1/2) Parametric approximation of conditional expectation Ê [V ti+1 X ti ] := ˆt i.p(x ti ) ˆti := arg inf 2R B 1 M MX m=1.p X m t i V m t i+1 2 Optimal decumulation into annuity after retirement: a stochastic control approach 6 / 16 E h i V ti+1 X ti = arg infuti 2L 2 ( (X ti ),P) E 2 U ti V ti+1
7 Least-squares Monte Carlo (2/2) Optimal decumulation into annuity after retirement: a stochastic control approach 7 / 16 Backward loop
8 LSMC with endogenous state variable The dynamics of W depends on the control =(a, x) W ti = W ti a ti A ti (1 a ti ) C ti t i W ti+1 = W ti apple. x ti + x ti. S ti Control randomization approach. Forward loop with dummy random control := (ã, x): W ti = W ti ã ti A ti (1 ã ti ) C ti t i W ti+1 = W ti apple. x ti + x ti. S ti. Backward loop with state vector X ti := ( W ti,s ti, ti ) Approximate t i ( X ti, ti ):=E V (t i+1, X ti+1 ) Xti, ti Maximize 7! ti ( X m t i, ) for each path m =1,...,M Optimal decumulation into annuity after retirement: a stochastic control approach 8 / 16 Based on theory of Constrained BSDEs
9 Numerical results: parameters B Exponential utility U(W )=exp( 0.5 W ) B C 0 = c =A$42,254 per year B A 0 = C 0 /5% = A$0.84M, thena t decreases B Three starting balances W 0 :A$0.5M,A$0.8M,A$1.0M B Two annuity purchase strategies: 1. Buy full coverage a t =100%when first a ordable 2. Optimal dynamic purchase Optimal decumulation into annuity after retirement: a stochastic control approach 9 / 16
10 Asset dynamics: Extended Wilkie model Price Inflation Salary inflation Australian equity Dividends Australian equity Dividend yield Long-term Interest rate Australian equity Total return Short-term Interest rate Australian bonds Total return International equity Total return International bonds Total return Optimal decumulation into annuity after retirement: a stochastic control approach 10 / 16 Cascade of Autoregressive dynamics
11 Strategy comparison (1/3) Optimal decumulation into annuity after retirement: a stochastic control approach 11 / 16 Initial wealth 0.5 million
12 Strategy comparison (2/3) Optimal decumulation into annuity after retirement: a stochastic control approach 12 / 16 Initial wealth 0.8 million
13 Strategy comparison (3/3) Optimal decumulation into annuity after retirement: a stochastic control approach 13 / 16 Initial wealth 1.0 million
14 Extension: new objective function Optimal decumulation into annuity after retirement: a stochastic control approach 14 / 16 Enclosed Linear utility: always better than classical utilities
15 Conclusion So far ) Dynamic decumulation into annuity after retirement ) Regression Monte Carlo with endogenous wealth Further improvements ) Dynamic mortality model ) Targeted utility function ) Additional info/variables: house prices, health,... ) Dynamic decision visualization Application ) For retirees, advisors, or insurance companies (package) Optimal decumulation into annuity after retirement: a stochastic control approach 15 / 16 And extensions
16 CSIRO Data61 Real Options & Financial Risk Nicolas Langrené t e nicolas.langrene@csiro.au w Data61 website 2 nd Congress on Actuarial Science and Quantitative Finance, Cartagena, 18 June
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