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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
[email protected]
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
Jorge Miguel Bravo (2009) 5
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
Jorge Miguel Bravo (2009) 7
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
Jorge Miguel Bravo (2009) 8
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
Jorge Miguel Bravo (2009) 9
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
Jorge Miguel Bravo (2009) 10
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
Jorge Miguel Bravo (2009) 14
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
[email protected]
