Stochastic mortality and securitization of longevity risk Pierre DEVOLDER ( Université Catholique...

Stochastic mortality and securitization of longevity risk

Pierre DEVOLDER

( Université Catholique de Louvain)Belgium

[email protected]

Edinburgh 2005 Devolder 2

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


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



Basic idea of insurance securitization:transfer to financial markets of some specialinsurance risks

Motivation for insurance industry : - hedging of non diversifiable risks - financial capacity of markets

Motivation for investors : -risks not correlated with finance



2 important examples :

CAT derivatives in non life insurance

Longevity risk in life insurance

- Increasing move from pay as you gosystems to funding methods in pension building- Importance of annuity market- Continuous improvement of longevity

THE CHALLENGE :



1880/90 1959/63 2000

x= 20 0.00688 0.0014 0.001212

x= 45 0.01297 0.00491 0.002866

x= 65 0.04233 0.03474 0.0175

x= 80 0.1500 0.11828 0.0802

Evolution of qx in Belgium ( men) -return of population :



Hedging context :

0ttimeatxagedtstanannuiofcohortinitialLx

yprobabilitsurvivalreferencep:where

vpLaLP

r

tx

1t

rxtxxx

Initial total lump sum :



)processstochastic

yprobabilitsurvivalactualp(

pLCF

p

pxtxt

rxtxt pL*CF

-cash flow to pay at time t :

-cash flow financed by the annuity :



Longevity risk at time t ( « mortality claim » ):

)pp(LLR rxt

pxtxt

Randomvariable

InitialLifetable



Decomposition of the longevity risk :

LR

Diversifiablepart ( number ofannuitants)

Generalimprovement of mortality

Specificimprovementof the group



-Hedging strategy for the insurer/ pension fund : - selling and buying simultaneously coupon bonds:

Floating leg:Index-linked bondwith floating coupon

Fixed leg:Fixed rate bond with coupon

SURVIVAL BOND CLASSICAL BOND



Classical coupon bond :

t=0 t=n

k k k 1+k

Survival index-linked bond :

t=0 t=n

1k 2k 3k nk1



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 rewardthrough an additive margin

Transparency purpose for the financial markets : hedging only of general mortality improvement



Form of the floating coupons:

The coupon is each year proportionally adapted in relation with the evolution of the index.

*k)Ip1(kk trxtt

Initiallife table Mortality

Index

Additivemargin



Valuation of the 2 legs at time t=0 :Principle of initial at par quotation :

n

1ttQ

n

1t

)n,0(P)t,0(P)k(E)n,0(P)t,0(Pk

Zero couponbondsstructure

Mortality riskneutralmeasure



Value of the additive margin of the floating bond :

n

1t

n

1t

rxttQ

)t,0(P

)t,0(P)p)I(E(k*k

1° model for the stochastic process I2° mortality risk neutral measure


3.From classical Lee Carter structure of mortality….

Classical Lee Carter approach in discrete time: (Denuit / Devolder - IME Congress- Rome- 06/2004submitted to Journal of risk and Insurance) )t(px

Probability for an x aged individual at time tto reach age x+1

))t(exp()t(p xx

Time series approach


3.From classical Lee Carter structure of mortality….

)t(exp)t( txxxx

Lee Carter framework :

Initialshapeof mortality

Mortalityevolution

ARIMAtimeseries


4….To continuous time models of stochastic mortality

Continuous time model for the mortality index :

stimeatsxageatforcemortalitystochastic)s(

)ds)s(exp(I

x

t

0xt



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 improvementin mortality evolution ;3° …in an affine structure ;4°… with mean reversion effect and limit table .

(+strictly positive process !!!!!!!!!!)



Step 1 : static deterministic model :

Initial deterministic force of mortality :

)(exp)0( sxsxsx

( classical life table = initial conditionsof stochastic differential equation)



Step 2: dynamic deterministic model taking into account dramatic improvement in mortality evolution :

)s)s(exp()s)s(exp()s( xsxxsxx

(prospective life table )



Step 3:stochastic model with noise effect – continuous Lee Carter :

)du)u(2

1)u(dz)u(exp()s(

:martingaleonentialexpwith

)s()s)s(exp()s(

s

0

2

xsxx

z= brownianmotion



This stochastic process is solution of a stochastic differentialequation :

)s(dz)s()s(ds)s(s)s()s()s(d xxx

sx

sxxx

Classicalmodel

Time evolution

Randomness



Step 4: affine continuous Lee Carter ( Dahl) :

)s(dz)s()s(ds)s(s)s()s()s(d xxx

sx

sxxx

Change in the dimension of the noise



Step 5: affine continuous Lee Carter with asymptotic table :

We add to the dynamic a mean reversion effect to anasymptotic table

sx~ Deterministic force of mortality

Introduction of a mean reversion term :

))s(~(k xsx



)s(dz)s()s(

ds))s(~(kds)s(s)s()s()s(d

x

xsxxx

sx

sxxx

Mean reversioneffect



Step 6: affine continuous Lee Carter with asymptotic table and limit table :

Introduction of a lower bound on mortality forces:

sxsxsx *~

Presentlife table

Biological absolute limit

Expectedlimit



)s(dz*)s()s(

ds))s(~(kds)s(s)s()s()s(d

sxx

xsxxx

sx

sxxx

…in the historical probability measure…



Survival probabilities :

T

ttxPtxtT )ds)s((expE)t(p

In the affine model :

))t()x,T,t(B)x,T,t(Aexp()t(p xtxtT



Particular case :

- initial mortality force : GOMPERTZ law:

sxsx cb

- constant improvement coefficient :

-constant volatility coefficient :



Explicit form for A and B :

22

)tT(

)tT(

2

kcln

:with

2)1e)((

)1e(2)x,T,t(B

T

t

2sx

2sx ds))x,T,s(B*

2

1)x,T,s(B~k()x,T,t(A



Introduction of a market price of risk for mortality :

Equivalent martingale measure Q

t

0x

Q ds))s(,s(h)t(z)t(z

t

0xQtQ )ds)s(exp(E)I(E

Valuation of the mortality index :



Affine model in the risk neutral world:

sxsx

sxsxsxsxx

*

**))s(,s(h

))0()x,t,0(B)x,t,0(Aexp()I(E xQQ

tQ

Mortality index :



Valuation of the additive margin :

n

1t

n

1t

rxtx

QQ

)t,0(P

)t,0(P)p))0()x,t,0(B)x,t,0(A(exp(k*k

Interpretation : weighted average of mortality margins



n

1t

n

1tt

)t,0(P

))t,0(PMM(k*k

Decomposition of the mortality margin :

)2(t

)1(tt MMMMMM



rxtx

)1(t p))0()x,t,0(B)x,t,0(Aexp(MM

= longevity pure price

))0()x,t,0(B)x,t,0(Aexp(

))0()x,t,0(B)x,t,0(Aexp(MM

x

xQQ)2(

t

=market price of longevity risk



Particular case : GOMPERTZ initial law and constant ,,

22

)tT(~

)tT(~

Q

2~~

kcln~:with

~2)1e)(~~(

)1e(2)x,T,t(B



T

t

2sx

2

T

tsxsx

Q

ds)x,T,s(B*2

1

ds)x,T,s(B)*~k*()x,T,t(A


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

Stochastic mortality and securitization of longevity risk Pierre DEVOLDER ( Université Catholique...

Documents

Transcript of Stochastic mortality and securitization of longevity risk Pierre DEVOLDER ( Université Catholique...