Transcript Stochastic mortality and securitization of longevity risk

Stochastic mortality and
securitization of longevity risk
Pierre DEVOLDER
( Université Catholique de Louvain)
Belgium
[email protected]
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 special
insurance 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
THE CHALLENGE :
- Increasing move from pay as you go
systems to funding methods in pension building
- Importance of annuity market
- Continuous improvement of longevity
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1. Securitization of longevity risk
Evolution of qx in Belgium ( men) -return of population :
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
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1. Securitization of longevity risk
Hedging context :
Lx  initial cohort of annui tan ts aged x at time t  0
Initial total lump sum :
P  Lx a x  Lx
 x

t 1
r
t
p
v
t x
where : p r  reference survival probabilit y
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1. Securitization of longevity risk
-cash flow to pay at time t :
CFt  L x t p px
( p p  actual survival probability
 stochastic process)
-cash flow financed by the annuity :
CFt *  L x t p rx
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1. Securitization of longevity risk
Longevity risk at time t ( « mortality claim » ):
LR t  L x ( t p px  t p rx )
Random
variable
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Initial
Life
table
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1. Securitization of longevity risk
Decomposition of the longevity risk :
LR
Diversifiable
part
( number of
annuitants)
General
improvement
of mortality
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Specific
improvement
of 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 bond
with floating coupon
SURVIVAL BOND
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Fixed leg:
Fixed rate bond
with coupon
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
k1 k 2 k 3
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1 k n
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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 reward
through 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 t  k (1 t p rx  I t )  k *
Initial
life table
Mortality
Index
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Additive
margin
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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
n
 k P(0, t )  P(0, n)   E
t 1
t 1
Zero coupon
bonds
structure
Q
(k t ) P(0, t )  P(0, n)
Mortality risk
neutral
measure
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2. Design of a survival bond
Value of the additive margin of the floating bond :
n
k*  k
r
(
E
(
I
)

p
 Q t t x ) P(0, t )
t 1
n
 P(0, t )
t 1
1° model for the stochastic process I
2° 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- 06/2004
submitted to Journal of risk and Insurance)
px (t )  Probability for an x aged individual at time t
to reach age x+1
px (t )  exp(x (t ))
Time series approach
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3.From classical Lee Carter
structure of mortality….
Lee Carter framework :
x (t)  exp ( x  x t  x t )
Initial
shape
of mortality
Mortality
evolution
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ARIMA
time
series
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4….To continuous time models
of stochastic mortality
Continuous time model for the mortality index :
t
I t  exp(    x (s) ds )
0
 x (s)  stochastic mortality force at age x  s at time s
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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 improvement
in 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 :
 x s   x s (0)  exp ( x s )
( classical life table = initial conditions
of 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 :
 x (s)  exp(xs  x (s) s)   xs exp(x (s) s)
(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 :
 x (s)   x s exp( x (s) s)  (s)
with   exp onential martingale:
s
1 2
 (s)  exp(  ( u )dz ( u )   ( u ) du )
2
0
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z=
brownian
motion
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4….To continuous time models
of stochastic mortality
This stochastic process is solution of a stochastic differential
equation :
 
 
x s
d x (s)   x (s) 
  x (s)  s x (s) ds   (s)  x (s) dz (s)
  x s

Classical
model
Time
evolution
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Randomness
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4….To continuous time models
of stochastic mortality
Step 4: affine continuous Lee Carter ( Dahl) :
 


d x (s)   x (s)  x s   x (s)  s x (s) ds   (s)  x (s) dz (s)
  x s

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 an
asymptotic table
~  Deterministic force of mortality

x s
Introduction of a mean reversion term :
~   (s))
k(
x s
x
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4….To continuous time models
of stochastic mortality
 


~   (s)) ds
d x (s)   x (s)  x s   x (s)  s x (s) ds  k(
x s
x
  x s

  (s)  x (s) dz(s)
Mean reversion
effect
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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:
~  *
x s  
x s
x s
Present
life table
Expected
limit
Biological
absolute limit
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4….To continuous time models
of stochastic mortality
 


~   (s)) ds
d x (s)   x (s)  x s  x (s)  sx (s) ds  k (
x s
x
  x s

  (s)  x (s)   *x s dz(s)
…in the historical probability measure…
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4….To continuous time models
of stochastic mortality
Survival probabilities :
T
Tt
px t ( t )  E P (exp    x (s) ds t )
t
In the affine model :
Tt
pxt (t)  exp( A(t, T, x)  B(t, T, x) x (t ))
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4….To continuous time models
of stochastic mortality
Particular case :
- initial mortality force : GOMPERTZ law:
x s  b cx s
- constant improvement coefficient :
-constant volatility coefficient :
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

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4….To continuous time models
of stochastic mortality
Explicit form for A and B :
2(e  ( T  t )  1)
B( t , T, x ) 
(  )(e  ( T  t )  1)  2
with :
  ln c    k
   2  2 2
T
1 2
~
A(t, T, x)   (k x s B(s, T, x)    *x s B2 (s, T, x)) ds
2
t
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5. Valuation of survival bonds
Introduction of a market price of risk for mortality :
Equivalent martingale measure Q
t
z Q ( t )  z( t )   h(s,  x (s)) ds
0
Valuation of the mortality index :
t
E Q ( I t )  E Q ( exp   x (s) ds )
0
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5. Valuation of survival bonds
Affine model in the risk neutral world:
h (s,  x (s))   x s  x s   *x s
 *x s

 x s   *x s
Mortality index :
EQ (I t )  exp( AQ (0, t, x)  BQ (0, t, x) x (0))
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5. Valuation of survival bonds
Valuation of the additive margin :
n
k*  k
Q
Q
r
(exp(
A
(
0
,
t
,
x
)

B
(
0
,
t
,
x
)

(
0
))

p

x
t x ) P (0, t )
t 1
n
 P(0, t )
t 1
Interpretation : weighted average of mortality margins
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5. Valuation of survival bonds
n
k*  k
 ( MM
t 1
t
P(0, t ))
n
 P(0, t )
t 1
Decomposition of the mortality margin :
MMt  MM(t1)  MM(t 2)
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5. Valuation of survival bonds
MM(t1)  exp(A(0, t, x)  B(0, t, x)x (0))t prx
= longevity pure price
MM (t 2 )  exp(A Q (0, t , x )  BQ (0, t , x ) x (0))
 exp(A (0, t , x )  B(0, t , x ) x (0))
=market price of longevity risk
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5. Valuation of survival bonds
Particular case : GOMPERTZ initial law and constant , , 
~
 (Tt )
2( e
 1)
B ( t , T, x )  ~ ~ ~ ( T  t )
(   )(e
 1)  2~

with :
~
  ln c    k  
~
 ~
 2  2 2
Q
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5. Valuation of survival bonds
T
~   * )B(s, T, x ) ds
A Q ( t , T, x )   (  * k
x s
x s
t
T
1 2
    *x s B2 (s, T, x ) ds
t 2
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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
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