stochastic mortality and securitization of longevity risk pierre devolder ( université catholique...
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Stochastic mortality and securitization of longevity risk
Pierre DEVOLDER
( Université Catholique de Louvain)Belgium
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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
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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
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1. Securitization of longevity risk
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
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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 gosystems to funding methods in pension building- Importance of annuity market- Continuous improvement of longevity
THE CHALLENGE :
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1. Securitization of longevity risk
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 :
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1. Securitization of longevity risk
Hedging context :
0ttimeatxagedtstanannuiofcohortinitialLx
yprobabilitsurvivalreferencep:where
vpLaLP
r
tx
1t
rxtxxx
Initial total lump sum :
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1. Securitization of longevity risk
)processstochastic
yprobabilitsurvivalactualp(
pLCF
p
pxtxt
rxtxt pL*CF
-cash flow to pay at time t :
-cash flow financed by the annuity :
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1. Securitization of longevity risk
Longevity risk at time t ( « mortality claim » ):
)pp(LLR rxt
pxtxt
Randomvariable
InitialLifetable
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1. Securitization of longevity risk
Decomposition of the longevity risk :
LR
Diversifiablepart ( number ofannuitants)
Generalimprovement of mortality
Specificimprovementof the group
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1. Securitization of longevity risk
-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
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2. Design of a survival 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
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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 rewardthrough an additive margin
Transparency purpose for the financial markets : hedging only of general mortality improvement
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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.
*k)Ip1(kk trxtt
Initiallife table Mortality
Index
Additivemargin
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2. Design of a survival bond
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
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2. Design of a survival bond
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
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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
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3.From classical Lee Carter structure of mortality….
)t(exp)t( txxxx
Lee Carter framework :
Initialshapeof mortality
Mortalityevolution
ARIMAtimeseries
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4….To continuous time models of stochastic mortality
Continuous time model for the mortality index :
stimeatsxageatforcemortalitystochastic)s(
)ds)s(exp(I
x
t
0xt
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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 improvementin mortality evolution ;3° …in an affine structure ;4°… with mean reversion effect and limit table .
(+strictly positive process !!!!!!!!!!)
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4….To continuous time models of stochastic mortality
Step 1 : static deterministic model :
Initial deterministic force of mortality :
)(exp)0( sxsxsx
( classical life table = initial conditionsof stochastic differential equation)
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4….To continuous time models of stochastic mortality
Step 2: dynamic deterministic model taking into account dramatic improvement in mortality evolution :
)s)s(exp()s)s(exp()s( xsxxsxx
(prospective life table )
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4….To continuous time models of stochastic mortality
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
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4….To continuous time models of stochastic mortality
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
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4….To continuous time models of stochastic mortality
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
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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 anasymptotic table
sx~ Deterministic force of mortality
Introduction of a mean reversion term :
))s(~(k xsx
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4….To continuous time models of stochastic mortality
)s(dz)s()s(
ds))s(~(kds)s(s)s()s()s(d
x
xsxxx
sx
sxxx
Mean reversioneffect
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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:
sxsxsx *~
Presentlife table
Biological absolute limit
Expectedlimit
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4….To continuous time models of stochastic mortality
)s(dz*)s()s(
ds))s(~(kds)s(s)s()s()s(d
sxx
xsxxx
sx
sxxx
…in the historical probability measure…
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4….To continuous time models of stochastic mortality
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
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4….To continuous time models of stochastic mortality
Particular case :
- initial mortality force : GOMPERTZ law:
sxsx cb
- constant improvement coefficient :
-constant volatility coefficient :
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4….To continuous time models of stochastic mortality
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
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5. Valuation of survival bonds
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 :
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5. Valuation of survival bonds
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 :
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5. Valuation of survival bonds
Valuation of the additive margin :
n
1t
n
1t
rxtx
)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
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5. Valuation of survival bonds
n
1t
n
1tt
)t,0(P
))t,0(PMM(k*k
Decomposition of the mortality margin :
)2(t
)1(tt MMMMMM
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5. Valuation of survival bonds
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
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5. Valuation of survival bonds
Particular case : GOMPERTZ initial law and constant ,,
22
)tT(~
)tT(~
Q
2~~
kcln~:with
~2)1e)(~~(
)1e(2)x,T,t(B
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5. Valuation of survival bonds
T
t
2sx
2
T
tsxsx
Q
ds)x,T,s(B*2
1
ds)x,T,s(B)*~k*()x,T,t(A
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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