Annuity Decisions with Systematic Longevity Risk Ralph Stevens Netspar, CentER, Tilburg University The Netherlands Annuity Decisions with Systematic Longevity Risk 1 / 29
Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Longevity forecast UK Annuity Decisions with Systematic Longevity Risk 2 / 29
Contribution Goal: obtain the optimal annuity decisions; Setting: - Normative approach; - Life Cycle model; Contribution to existing literature: - Allow for systematic longevity risk; - Allow for deferred. : - Systematic longevity risk makes less attractive; - Postponing annuity purchase not optimal; - Utility loss of deferred is small. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Longevity forecast UK Annuity Decisions with Systematic Longevity Risk 3 / 29
Annuity menu Annuity: Product which provides a yearly income stream until death; Consider two types of : - Immediate ; - Deferred. Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Longevity forecast UK Timing decision: - Purchasing at retirement; - Postponing the purchase of. Annuity Decisions with Systematic Longevity Risk 4 / 29
Literature on immediate Yaari (1965): Full annuitization is optimal; Empirical results: few voluntary purchase. Extensive literature with possible explanations: - Actuarially unfairness; - Equity risk premium. 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; Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Longevity forecast UK Blake, Cairns, and Dowd (2003): Optimal annuitization age in the range of 65 to 80. Annuity Decisions with Systematic Longevity Risk 5 / 29
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 Longevity forecast UK Annuity Decisions with Systematic Longevity Risk 6 / 29
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 Longevity forecast UK Annuity Decisions with Systematic Longevity Risk 7 / 29
Longevity forecast UK Contribution Annuity menu Literature on immediate Literature on deferred Outline of presentation Longevity forecast UK Annuity Decisions with Systematic Longevity Risk 8 / 29
Preferences Optimization Annuity Decisions with Systematic Longevity Risk 9 / 29
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.. Preferences Optimization Constraints: - Wealth dynamics; - Short-selling constraint; - Positive wealth level. Annuity Decisions with Systematic Longevity Risk 10 / 29
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 11 / 29
Financial market Longevity uncertainty Annuity Decisions with Systematic Longevity Risk 12 / 29
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, Financial market Longevity uncertainty where µ = r rf + λσ. λ: Equity risk premium; σ: Volatility parameter. -. Annuity Decisions with Systematic Longevity Risk 13 / 29
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) Financial market Longevity uncertainty Forecast processes using random walk with drift; Allow for parameter risk using Jeffreys prior and Bayesian updating. Annuity Decisions with Systematic Longevity Risk 14 / 29
Pricing Price of deferred Price of immediate Annuity Decisions with Systematic Longevity Risk 15 / 29
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 Price of deferred Price of immediate Annuity Decisions with Systematic Longevity Risk 16 / 29
Price of deferred 14 100 90 Price 12 10 8 6 4 2 Risk margin (% of market price) 80 70 60 50 40 30 20 10 Pricing Price of deferred Price of immediate 0 65 70 75 80 85 90 95 Age at first payment (65+d) Deferred are much cheaper; 0 65 70 75 80 85 90 95 Age at first payment (65+d) Loading factor increases with deferral period. Annuity Decisions with Systematic Longevity Risk 17 / 29
Price of immediate Present value of the market price of an annuity 15 10 5 Pricing Price of deferred Price of immediate 0 65 70 75 80 85 90 95 Age at first payment (65+s+d) Generally cheaper to postpone the purchase of ; Future prices of an are currently stochastic. Annuity Decisions with Systematic Longevity Risk 18 / 29
Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Longevity risk Annuity Decisions with Systematic Longevity Risk 19 / 29
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 Longevity risk Annuity Decisions with Systematic Longevity Risk 20 / 29
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 Longevity risk Annuity Decisions with Systematic Longevity Risk 21 / 29
CEC deferred 1.05 1 CEC / CEC fa 0.95 0.9 0.85 0.8 1 3 5 7 9 11 13 15 Deferral period (d) Solid and dashed curves: Pricing using risk-adjusted probabilities; Dashed-dotted curve: Pricing using (7.3%) loading factor; Dotted curve: without systematic longevity risk. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Longevity risk Annuity Decisions with Systematic Longevity Risk 22 / 29
Annuity income Optimal normalized annuity income 1 0.8 0.6 0.4 0.2 0 1 3 5 7 9 11 13 15 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 Longevity risk Annuity Decisions with Systematic Longevity Risk 23 / 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 Longevity risk Annuity Decisions with Systematic Longevity Risk 24 / 29
CEC immediate CEC / CEC fa 1.05 1 0.95 0.9 0.85 0.8 0 5 10 15 20 Postponement period (s) i) Equity risk premium; ii) Mortality credit; iii) Conversion rate risk. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Longevity risk Annuity Decisions with Systematic Longevity Risk 25 / 29
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. Deferral period Quantification utility CEC deferred Annuity income Postponing CEC immediate Systematic longevity risk Longevity risk - prices are stochastic postponing less favorable. Annuity Decisions with Systematic Longevity Risk 26 / 29
Longevity risk Optimal fraction annuitized wealth 1 0.95 0.9 0.85 0.8 0.75 65 70 75 80 85 Age 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 Longevity risk Annuity Decisions with Systematic Longevity Risk 27 / 29
Annuity Decisions with Systematic Longevity Risk 28 / 29
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 29 / 29