Stochastic mortality and securitization of longevity risk Pierre DEVOLDER ( Université Catholique de Louvain) Belgium

Edinburgh 2005 Devolder2 Purpose of the presentation Suggestions for hedging of longevity risk in annuity market Design of securitization instruments Generalization of Lee Carter approach of mortality to continuous time stochastic mortality models Application to pricing of survival bonds

Edinburgh 2005 Devolder3 Outline 1. Securitization of longevity risk 2. Design of a survival bond 3. From Lee Carter structure of mortality… 4. …To continuous time models of stochastic mortality 5. Valuation of survival bonds 6. Conclusion

Edinburgh 2005 Devolder4 1. Securitization of longevity risk Basic idea of insurance securitization: transfer to financial markets of some special insurance risks Motivation for insurance industry : - hedging of non diversifiable risks - financial capacity of markets Motivation for investors : -risks not correlated with finance

Edinburgh 2005 Devolder5 1. Securitization of longevity risk 2 important examples : CAT derivatives in non life insurance Longevity risk in life insurance - Increasing move from pay as you go systems to funding methods in pension building - Importance of annuity market - Continuous improvement of longevity THE CHALLENGE :

Edinburgh 2005 Devolder6 1. Securitization of longevity risk 1880/901959/ x= x= x= x= Evolution of qx in Belgium ( men) -return of population :

Edinburgh 2005 Devolder7 1. Securitization of longevity risk Hedging context : Initial total lump sum :

Edinburgh 2005 Devolder8 1. Securitization of longevity risk -cash flow to pay at time t : -cash flow financed by the annuity :

Edinburgh 2005 Devolder9 1. Securitization of longevity risk Longevity risk at time t ( « mortality claim » ): Random variable Initial Life table

Edinburgh 2005 Devolder10 1. Securitization of longevity risk Decomposition of the longevity risk : LR Diversifiable part ( number of annuitants) General improvement of mortality Specific improvement of the group

Edinburgh 2005 Devolder11 1. Securitization of longevity risk -Hedging strategy for the insurer/ pension fund : - selling and buying simultaneously coupon bonds: Floating leg: Index-linked bond with floating coupon Fixed leg: Fixed rate bond with coupon SURVIVAL BOND CLASSICAL BOND

Edinburgh 2005 Devolder12 2. Design of a survival bond Classical coupon bond : t=0t=n kkk1+k Survival index-linked bond : t=0 t=n

Edinburgh 2005 Devolder13 2. Design of a survival bond Definition of the floating coupons : Hedging of the longevity risk LR -General principle : the coupon to be paid by the insurer will be adapted following a public index yearly published by supervisory authorities and will incorporate a risk reward through an additive margin Transparency purpose for the financial markets : hedging only of general mortality improvement

Edinburgh 2005 Devolder14 2. Design of a survival bond Form of the floating coupons: The coupon is each year proportionally adapted in relation with the evolution of the index. Initial life table Mortality Index Additive margin

Edinburgh 2005 Devolder15 2. Design of a survival bond Valuation of the 2 legs at time t=0 : Principle of initial at par quotation : Zero coupon bonds structure Mortality risk neutral measure

Edinburgh 2005 Devolder16 2. Design of a survival bond Value of the additive margin of the floating bond : 1° model for the stochastic process I 2° mortality risk neutral measure

Edinburgh 2005 Devolder17 3.From classical Lee Carter structure of mortality…. Classical Lee Carter approach in discrete time: (Denuit / Devolder - IME Congress- Rome- 06/2004 submitted to Journal of risk and Insurance) Probability for an x aged individual at time t to reach age x+1 Time series approach

Edinburgh 2005 Devolder18 3.From classical Lee Carter structure of mortality…. Lee Carter framework : Initial shape of mortality Mortality evolution ARIMA time series

Edinburgh 2005 Devolder19 4….To continuous time models of stochastic mortality Continuous time model for the mortality index :

Edinburgh 2005 Devolder20 4….To continuous time models of stochastic mortality Example of stochastic one factor model 4 requirements for a one factor model : 1° generalization of deterministic and Lee Carter models; 2° …taking into account dramatic improvement in mortality evolution ; 3° …in an affine structure ; 4°… with mean reversion effect and limit table. (+strictly positive process !!!!!!!!!!)

Edinburgh 2005 Devolder21 4….To continuous time models of stochastic mortality Step 1 : static deterministic model : Initial deterministic force of mortality : ( classical life table = initial conditions of stochastic differential equation)

Edinburgh 2005 Devolder22 4….To continuous time models of stochastic mortality Step 2: dynamic deterministic model taking into account dramatic improvement in mortality evolution : (prospective life table )

Edinburgh 2005 Devolder23 4….To continuous time models of stochastic mortality Step 3:stochastic model with noise effect – continuous Lee Carter : z= brownian motion

Edinburgh 2005 Devolder24 4….To continuous time models of stochastic mortality This stochastic process is solution of a stochastic differential equation : Classical model Time evolution Randomness

Edinburgh 2005 Devolder25 4….To continuous time models of stochastic mortality Step 4: affine continuous Lee Carter ( Dahl) : Change in the dimension of the noise

Edinburgh 2005 Devolder26 4….To continuous time models of stochastic mortality Step 5: affine continuous Lee Carter with asymptotic table : We add to the dynamic a mean reversion effect to an asymptotic table Deterministic force of mortality Introduction of a mean reversion term :

Edinburgh 2005 Devolder27 4….To continuous time models of stochastic mortality Mean reversion effect

Edinburgh 2005 Devolder28 4….To continuous time models of stochastic mortality Step 6: affine continuous Lee Carter with asymptotic table and limit table : Introduction of a lower bound on mortality forces: Present life table Biological absolute limit Expected limit

Edinburgh 2005 Devolder29 4….To continuous time models of stochastic mortality …in the historical probability measure…

Edinburgh 2005 Devolder30 4….To continuous time models of stochastic mortality Survival probabilities : In the affine model :

Edinburgh 2005 Devolder31 4….To continuous time models of stochastic mortality Particular case : - initial mortality force : GOMPERTZ law: - constant improvement coefficient : -constant volatility coefficient :

Edinburgh 2005 Devolder32 4….To continuous time models of stochastic mortality Explicit form for A and B :

Edinburgh 2005 Devolder33 5. Valuation of survival bonds Introduction of a market price of risk for mortality : Equivalent martingale measure Q Valuation of the mortality index :

Edinburgh 2005 Devolder34 5. Valuation of survival bonds Affine model in the risk neutral world: Mortality index :

Edinburgh 2005 Devolder35 5. Valuation of survival bonds Valuation of the additive margin : Interpretation : weighted average of mortality margins

Edinburgh 2005 Devolder36 5. Valuation of survival bonds Decomposition of the mortality margin :

Edinburgh 2005 Devolder37 5. Valuation of survival bonds = longevity pure price =market price of longevity risk

Edinburgh 2005 Devolder38 5. Valuation of survival bonds Particular case : GOMPERTZ initial law and constant

Edinburgh 2005 Devolder39 5. Valuation of survival bonds

Edinburgh 2005 Devolder40 6. Conclusions Next steps : Calibration of the mortality models on real data Estimation of the market price of longevity risk Other stochastic mortality models for the valuation model