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7th International Workshop on Pensions, Insurance and Savings
Modelling Mortality using Multiple
Jorge Miguel Bravo (2009) 1
JORGE MIGUEL BRAVO
University of vora Department of Economics and CIEF/CEFAGE-UE
Paris, May 28-29, 2009
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Presentation Outline
1. Introduction and motivation
2. Modelling mortality and longevity risks
3. Affine-Jump diffusion processes for mortality
Jorge Miguel Bravo (2009) 2
3.1. Mathematical framework
3.2. Multiple stochastic latent factors
4. Revisiting the Gompertz-Makeham law5. Final remarks
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Introduction and motivation
Decline in mortality at all ages
Reforms in social security systems
Increase in contribution rates/retirement age Reduction in pension/salary ratios
Defined Benefit Defined Contribution
Jorge Miguel Bravo (2009) 3
Longevity risk cannot be diversified away Changes in the regulatory framework (Solvency II)
Hedging strategies
Product redesign, natural hedging, stochastic solvency rules New reinsurance treaties
Capital market solutions: longevity bonds, mortality-linked
derivatives,
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Modelling mortality and longevity risks
Longevity-linked products market models for
mortality/longevity risk measurement
Traditional actuarial approach: deterministic mortalityintensity + best estimate interest rates
Discrete-time d namic a roach
Jorge Miguel Bravo (2009) 4
Extrapolative methods
Parametric vs statistical methods (e.g., Poisson-Lee-Carter)
Stochastic mortality modeling Mortality intensity as a stochastic process
Time dependency and uncertainty in future development
Arbitrage-free framework
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Affine-Jump diffusion processes
Main idea
Draw parallel between insurance contracts and credit-sensitive
securities
Analogy between default and insureds death and between
intensity of defaultand mortality intensity
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Mathematical framework Complete filtered probability space
Random lifetime of an individual agedxat time 0 as an
stopping time with random intensity where is the first jump-time of a nonexplosive counting
process recording at any time whether the individual as
died or survived
x
x
N
x
0( )tN
0( )tN
=
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Mathematical framework
Then, for such that we can write
Assuming that N is a Cox process with predictable
intensity , then
0, ,t ( ) ,x t >
( ) ( )t t t t x E N N F t t +
Jorge Miguel Bravo (2009) 6
Model the survival probability by using affine-jump
diffusion processes
( )( )
T
x st
s ds
x t t P T F E e F
+
> =
1
( , ) ( , )m
h
t t t t t
h
dX t X dt t X dW dJ =
= + +
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Multiple Latent Factors Approach
Pioneered by Schrager (2006)
Goal: model the intensity x+t(t) for all ages simultaneously
Assumption
0
1
+
=
= + ( ) ( , ) ( , ) ( )M
x t j j j
t g x t g x t X t
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with M-dimensional factor dynamics
Instantaneous drift, variance-covariance matrix and jump-arrival
intensity are affine functions of the latent factors
Main contribution of paper: inclusion of positive/negative jumps
0 = + + =( ) ( ( )) ( ), ( ) ,Pt tdX t X t dt V dW dJ t X X
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Multiple Latent Factors Approach
Survival probability represented by an exponential affine function
Feyman-Kac representation
+ +
= + = exp ( ) ( ) ( ) ,T t x t x t p A B t T t
{ }21 + + + + ( ) ( )M
t t t t t t i i t i A B X X B B X
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WhereAtand Btare solutions of Riccati ODE
( )0 1 01
1 0
=
=
+ + + = ( , ) ( , ) ( , )m
h h h
t t th
X t B g x t X g x t
2
0 0
1 1
2
1
1 1
11
2
11
2
= =
= =
= +
= +
( , ) ( , )
( , ) ( , )
M mh h
t t t i t ii h
M mh h
t t t i t ii h
A B B t B g x t
B B B t B g x t
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Revisiting Gompertz-Makeham law
Gompertz-Makeham (GM) deterministic mortality law
GM stochastic mortality law
1 1 1 20 1 +
+= + > >, , ,x tx t X X c X X c
+= +
x tt X t X t c
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with factorXj(j=1,2) dynamics
1 2
0
= + + =
=
( ) ( ( )) ( ), ( ) ,P j j j j j j jt j j
P P
t t
dX t a X t dt dW dJ t X X
dW dW dt
0 0 0
> > >, ,
j j j a
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Including jumps
We assumeJ(t) is a compound Poisson process with constant
jump-arrival intensity
1
=
= ( ) , i.i.d.t
N
i ii
J t
0
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1 2
1 1
1 0 2 0
1 2
1 2 1 2 1 2
1 1
0 1
> >, , j j j a
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Final remarks
Longevity-linked products market models for
mortality/longevity risk measurement
Issues for future research
Model calibration
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Model consistency with biological evidence Inclusion of heterogeneity risk classification
Causes of death
Actuarial neutrality of social security systems Market price of longevity risk
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THANK YOU
Jorge Miguel Bravo (2009) 15
JORGE MIGUEL BRAVO
University of vora Department of Economics and CIEF/CEFAGE-UE