Managing Longevity Risk: Tontines vs. Annuities An Chen, Peter Hieber, Jakob Klein | Lyon, September 2015 | Jakob Klein University of Ulm
Page 2 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Content Table of contents Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion
Page 3 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction Table of contents Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion
Page 4 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction Introduction Annuity providers face systematic mortality risk: ◮ Solvency regulations force insurers to set aside capital ◮ Possible consequences: High annuity/reinsurance premiums, solvency risk when capital requirements are not sufficient, . . . Measures taken: ◮ Risk transfer to other parties (e.g. Swaps) or policyholders
Page 5 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction Objectives ◮ Derive optimal payouts for expected-utility-maximizers ◮ Fairness restrictions ◮ Analyze risks borne by providers ◮ Calculation of risk-adequate loadings ( → Solvency II) ◮ Multiple perspectives
Page 6 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Introduction Tontines: Past and present ◮ Early suggestion by Tonti (17 th century) ◮ Collection of money in the UK ◮ Popular product in the US - now forbidden ◮ Milevsky, Salisbury (2015): Optimal Retirement Tontines
Page 7 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Table of contents Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion
Page 8 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Relevant quantities ◮ Tontine contract: ◮ Provider pays a fixed amount to a group of policyholders ◮ alive policyholders share the payout ◮ Annuity contract: ◮ Provider pays a fixed amount to each alive individual
Page 9 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Contract payoffs At time t > 0 ◮ an individual tontine-policyholder receives nd ( t ) b • ( t ) := 1 { ζ> t } N ( t ) , (1) ◮ an annuitant receives b ◦ ( t ) := 1 { ζ> t } c ( t ) . (2) where ζ is the residual lifetime of the individual and N ( t ) is the number of policyholders at time t .
Page 10 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Value of a tontine ∞ � P • � · p ∗ e − rt b • ( t ) d t � x , d ( · ) , n := E 0 ∞ n − 1 � � n − 1 � x ) n − 1 − k nd ( t ) e − rt t p ∗ � ( t p ∗ x ) k ( 1 − t p ∗ = k + 1 d t x k k = 0 0 ∞ � e − rt � 1 − ( 1 − t p ∗ x ) n � = d ( t ) d t . (3) 0
Page 11 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Value of an annuity ∞ ∞ � � P ◦ � · p ∗ e − rt b ◦ ( t ) d t = e − rt t p ∗ � x , c ( · ) , n := E x c ( t ) d t . (4) 0 0
Page 12 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Expected utility - Policyholder perspective Assume an investor with; Power utility with constant relative risk aversion (CRRA) u ( X ) = X 1 − γ 1 − γ Expected utility of a tontine policyholder: ∞ � � nd ( t ) � U • � · p ∗ 1 { ζ> t } e − rt u � x , d ( · ) , n := E d t N ( t ) 0 ∞ n − 1 � n − 1 � � nd ( t ) � � x ) k + 1 ( 1 − t p ∗ x ) n − 1 − k d t . � e − rt ( t p ∗ = u k k + 1 k = 0 0 (5)
Page 13 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Optimal tontine payout d ∗ ( t ) := max d ( t ) U • � · p ∗ � x , d ( · ) , n , (6) ∞ � e − rt d ( t ) 1 − ( 1 − t p ∗ x ) n � � s . t . d t ≤ 1 . 0 Solution: 1 n − 1 � 1 − γ � � x ) k + 1 ( 1 − t p ∗ γ � n − 1 x ) n − 1 − k n ( t p ∗ � k k + 1 k = 0 d ∗ ( t ) = , (7) 1 − ( 1 − t p ∗ λ ∗ � x ) n �
Page 14 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Optimal annuity payout ∞ � c ∗ ( t ) := max c ( t ) U ◦ � · p ∗ e − rt t p ∗ � x , c ( · ) , n = max x u ( c ( t )) d t , (8) c ( t ) 0 ∞ � e − rt c ( t ) t p ∗ s . t . x d t ≤ 1 . 0 Solution: − 1 ∞ � e − rt t p ∗ c ( t ) = x d t (9) . 0
Page 15 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Mortality assumptions ◮ Gompertz law ◮ Binomial distribution for number of survivors up to time t = ( t p x ) 1 − ǫ , where ǫ is ◮ life tables with mortality shock: t p new x the (random) magnitude of a longevity shock
Page 16 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Risk Margin Calculation of the risk margin (see e.g. Börger (2010)) ◮ Solvency II: Technical Provisions = Best Estimate Liabilities + Risk Margin ◮ In numerical illustrations: Fair Premium = Technical Provision SCR t ◮ Risk Margin = CoC � ( 1 + r ) t t ≥ 0 ◮ Simplifications allowed, e.g. SCR ( t ) = BEL t BEL 0 SCR 0 ◮ CoC = 6% � � � � BEL 1 − CF 1 ◮ SCR = argmin x P − BEL 0 > x ≤ 0 . 005 1 + r
Page 17 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Future losses At time t > 0 the losses generated by a longevity shock can be calculated as � ∞ e − rs � � x ) 1 − ǫ − s p ∗ L ◦ t , · p ∗ x , d ∗ ( · ) ( s p ∗ c ∗ ( s ) d s , � � := (10) x ǫ t � ∞ x ) 1 − ǫ � n − e − rs �� � n � L • t , · p ∗ x , c ∗ ( · ) 1 − ( s p ∗ 1 − ( s p ∗ d ∗ ( s ) d s , � � � := x ) ǫ t (11)
Page 18 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Model setup Switching from tontine to annuity Fix a switching time t ∗ , at time t a policyholder receives: nd ( t ) N ( t ) + 1 { ζ ≥ t ∗ } c , (12) 1 { 0 ≤ ζ< t ∗ } Fair value: t ∗ ∞ � � x ) n ) d ( t ) d t + e − rt ∗ e − r ( t − t ∗ ) t − t ∗ p ∗ e − rt ( 1 − ( 1 − t p ∗ t ∗ p ∗ x + t ∗ c d t = 1 x t ∗ 0 (13)
Page 19 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Table of contents Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion
Page 20 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Tontine vs Annuity: Loss distribution Figure: loss distribution: age 65, r=4%
Page 21 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Tontine vs Annuity:Risk margin Figure: Risk margins: age 65, different longevity shock magnitudes
Page 22 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Annuity: Risk margin Figure: Risk margin: age 65, various portfolio sizes at inception
Page 23 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Tontine: risk margin Figure: Risk margin: age 65, various portfolio sizes at inception
Page 24 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Expected utility - risk loading Figure: Expected utility with risk-based loading: varying interest and shock magnitude
Page 25 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Numerical illustrations Switching times - Solvency Capital Requirement Figure: SCR for deferred payout: age 65
Page 26 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion Table of contents Introduction Model setup Contract specifications Contract value Optimization problem Mortality New product Numerical illustrations Conclusion
Page 27 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion Conclusions ◮ Fairness restrictions ◮ Products with different risk structures: take into account compensation for risk transfer ◮ New products: multiple perspectives have to be analyzed
Page 28 Managing Longevity Risk: Tontines vs. Annuities | An Chen, Peter Hieber, Jakob Klein | Conclusion Outlook/Paper ◮ Mortality models ◮ Detailed proofs ◮ Sensitivity analyses ◮ ...
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