Annuity Decisions with Systematic Longevity Risk. Ralph Stevens

Transcription

1 Annuity Decisions with Systematic Longevity Risk Ralph Stevens Netspar, CentER, Tilburg University The Netherlands Annuity Decisions with Systematic Longevity Risk 1 / 34

2 Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 2 / 34

3 Contribution Goal: obtain the optimal annuity decisions; Setting: - Normative approach; - Life Cycle model; What should one do with their money at 65 (and retired) to live the most happy life? Contribution to existing literature: - Allow for systematic longevity risk; - Allow for deferred. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 3 / 34

4 Many interesting findings; w.r.t. optimal annuity choice: - Postponing annuity purchase not optimal; - Optimal deferral period is short; - Utility loss of deferred is small. w.r.t. systematic longevity risk: - Systematic longevity risk makes less attractive; - Systematic longevity risk postponing annuity purchase not optimal; - For the utility of deferred the effect of systematic longevity risk is small. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 4 / 34

5 Annuity menu Annuity: Product which provides a yearly income stream until death; Consider two types of : - Immediate ; - Deferred. Timing decision: - Purchasing at retirement; - Postponing the purchase of. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 5 / 34

6 Literature on immediate Yaari (1965): Full annuitization is optimal; Empirical results: few voluntary purchase. Extensive literature with possible explanations: - Actuarially unfairness; - Equity risk premium. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Milevsky (1998): postpone until mortality credit = equity risk premium; Milevsky and Young (2002): real option to annuitize is still valuable until the mid-70s or mid-80s; Blake, Cairns, and Dowd (2003): Optimal annuitization age in the range of 65 to 80. Annuity Decisions with Systematic Longevity Risk 6 / 34

7 Literature on deferred Idea: Milevsky (2005, NAAJ): Buy with long deferral period at retirement; Deferred : - Cheap; - Provide real longevity insurance; Research on deferral period: - Evidence that it may not reduce lifetime ruin probability (Bayaraktar and Young, 2009, NAAJ); - Deferred annuity (20 years) compared with immediate is utility increasing when loading factor is high enough (Horneff and Maurer, 2008). Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 7 / 34

8 Outline of presentation Systematic longevity risk Model: - Life Cycle Model - - : optimal annuity choices: - Deferred - Immediate Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Annuity Decisions with Systematic Longevity Risk 8 / 34

9 Longevity forecast UK Survivor fan chart Annuity Decisions with Systematic Longevity Risk 9 / 34

10 Longevity forecast UK Longevity forecast UK Survivor fan chart Annuity Decisions with Systematic Longevity Risk 10 / 34

11 Survivor fan chart Nominal payment Longevity forecast UK Survivor fan chart Time (years) Blue bars: 90% CI; Green bars: 75% CI; Black line: Median. Annuity Decisions with Systematic Longevity Risk 11 / 34

12 Preferences Optimization Annuity Decisions with Systematic Longevity Risk 12 / 34

13 Preferences CRRA intertemporally separable expected lifetime utility function: E t Control variables: τ 0 τp x,t β τ C1 γ τ 1 γ - Annuity; - Consumption; - Fraction liquid wealth in risky asset. Constraints: - Wealth dynamics; - Short-selling constraint; - Positive wealth level.. Preferences Optimization Annuity Decisions with Systematic Longevity Risk 13 / 34

14 Optimization Dynamic optimization problem. Without annuity decision: see, among others: Brandt, Goyal, Clara, and Stroud (2005, RFS) & Koijen, Nijman, and Werker (2009, RFS). Maximize utility conditional on: - Timing of purchasing of ; - Type of annuity (deferral period). Preferences Optimization Assumptions: - Purchase only once ; - Only one type (deferral period) of. Compare utility in the different sub-optima. Annuity Decisions with Systematic Longevity Risk 14 / 34

15 Financial market Longevity uncertainty Annuity Decisions with Systematic Longevity Risk 15 / 34

16 Financial market Sources of risk: - Investment risk; - Idiosyncratic (non-systematic) longevity risk; - Systematic longevity risk. Financial market consists of - Riskfree asset: Yearly time-independent return r rf. - Risky asset: Stock price S t follows a Brownian motion with drift: ds t = µs t dt + σs t dz t, where µ = r rf + λσ. λ: Equity risk premium; σ: Volatility parameter. -. Financial market Longevity uncertainty Annuity Decisions with Systematic Longevity Risk 16 / 34

17 Longevity uncertainty Cairns-Blake-Dowd-model is given by: ( ) qx,t log = k (1) t + x k (2) t + ǫ x,t, 1 q x,t where q x,t is the time-t mortality probability for a x-year old. - k (1) t : General level of mortality (generally decreasing over time) - k (2) t : General level of increase in mortality by age (generally increasing over time) Forecast processes using random walk with drift; Allow for parameter risk using Jeffreys prior and Bayesian updating. Financial market Longevity uncertainty Annuity Decisions with Systematic Longevity Risk 17 / 34

18 Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Annuity Decisions with Systematic Longevity Risk 18 / 34

19 Pricing Mitchell et al. (1999, AER): loading factor 7.3% for immediate annuity. Explanation: transaction cost and profit. Actuarially unfairness due to: - Price of systematic longevity risk. When: - Efficient market; - No arbitrage. The time-t value of an annuity with a deferral period of d years: V t (A (d) x,t ) = ( ) 1 τ E Q t [ τ p x,t ] 1 + r rf. τ d Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Annuity Decisions with Systematic Longevity Risk 19 / 34

20 Pricing longevity risk No observable market price for systematic longevity risk; Indications that there is a price: - In six longevity risk transactions in UK; - Solvency II requires market price of longevity risk ( 7% of actuarial fair price). We use three pricing methods: - Loading factor of 7.3% (conservative for deferred ); - Market price calibrated using announced longevity bond (λ = 0.175); - Market price with λ U N IF(0, 0.175). Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Annuity Decisions with Systematic Longevity Risk 20 / 34

21 Price of deferred Price Market price Actuarial fair price Risk premium Age at first payment (65+d) (a) Current price of deferred Deferred are much cheaper; Risk margin (% of market price) Age at first payment (65+d) (b) Risk margin for systematic longevity risk Loading factor increases with deferral period. Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Annuity Decisions with Systematic Longevity Risk 21 / 34

22 Price of immediate Present value of the market price of an annuity Price deferred annuity Median present value of immediate annuity 90% CI present value of immediate annuity Age at first payment (65+s+d) Generally cheaper to postpone the purchase of ; Future prices of an are currently stochastic. Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Annuity Decisions with Systematic Longevity Risk 22 / 34

23 Mortality credit Why are attractive? Advantage : mortality credit (MC); - Pooling effect; - Reallocation of contributions to the survivors. MC: the gain from not postponing the purchase of an annuity; Traditional: without systematic longevity risk: - Independent of deferral period; - Deterministic: 1+rrf 1p x,t ( 1 + r rf). Pricing Pricing longevity risk Price of deferred Price of immediate Mortality credit Allow for systematic longevity risk - Stochastic; - Depends on the deferral period. Annuity Decisions with Systematic Longevity Risk 23 / 34

24 Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 24 / 34

25 Deferral period Advantage longer deferral period: - Lower longevity risk premium; - Cheaper more liquid wealth when MC is low; Allows for more capital gains from equity risk premium; Disadvantage longer deferral period: - Lower gains from mortality credit; - Fewer periods with income guarantee; More uncertainty in consumption level; Less consumption smoothing. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 25 / 34

26 Quantification utility The utility is quantified by: CEC: Certainty Equivalent Consumption: How much deterministic yearly consumption do I need in order to be equally well off as in the optimal strategy; The CEC is normalized by: The CEC in case of fully immediate annuitization (CEC fa ); Hence, CEC/CEC fa relative utility gain by optimal choices instead of currently fully annuitizing. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 26 / 34

27 CEC deferred CEC / CEC fa Risk neutral pricing, calibrated λ 0.85 Risk neutral pricing, stochastic λ Loading factor No systematic longevity risk Deferral period (d) Optimal deferral period is short. Utility gain of deferred is small. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 27 / 34

28 Annuity income Optimal normalized annuity income Risk neutral pricing, calibrated λ 0.2 Risk neutral pricing, stochastic λ Loading factor No systematic longevity risk Deferral period (d) First: increasing function of d; Then: decreasing function of d. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 28 / 34

29 Postponing Postponing the annuity decision allows for consumption smoothing; It also allows for capital gains from equity risk premium; Postponing less longevity risk conditional on realized survival probabilities; Reduces longevity risk premium. Future annuity prices are stochastic. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 29 / 34

30 CEC immediate CEC / CEC fa Postponement period (s) i) Equity risk premium; ii) Mortality credit; iii) Conversion rate risk. Risk neutral pricing, calibrated λ Risk neutral pricing, stochastic λ Loading factor No systematic longevity risk Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 30 / 34

31 Systematic longevity risk Effect systematic longevity risk on consumption level: - Survival probabilities low more consumption; - Survival probabilities high less consumption; Effect including systematic longevity risk on annuity choice: - Immediate becomes less attractive postponing more favorable; Fraction wealth with systematic longevity risk: 84%; without systematic longevity risk: 90%; Due to: uncertainty in the value of. - prices are stochastic postponing less favorable. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 31 / 34

32 Optimal fraction of annuitized wealth (a s * (1)) Median % CI 80% CI 90% CI Age (65+s) Systematic longevity risk large confidence intervals of optimal fraction annuitized wealth. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Annuity Decisions with Systematic Longevity Risk 32 / 34

33 Annuity Decisions with Systematic Longevity Risk 33 / 34

34 There exists systematic longevity risk; Systematic longevity risk has a non-zero market price; Systematic longevity risk affects annuity decision; - provide lower utility; Postponing annuity purchase: - Systematic longevity risk uncertain prices; - Postponing not utility increasing; Type of annuity: - Deferred are preferable; - Optimal deferral period is short. Annuity Decisions with Systematic Longevity Risk 34 / 34