The Poisson log-bilinear Lee Carter
model:
Efficient bootstrap in life annuity
actuarial analysis
Valeria D’Amato1, Emilia Di Lorenzo2, Steven Haberman3, Maria Russolillo1,
Marilena Sibillo1
1Department
of Economics and Statistics - University of Salerno, Italy
via Ponte Don Melillo, campus universitario
(e-mail: vdamato@unisa.it, mrussolillo@unisa.it, msibillo@unisa.it)
2Department
3Faculty
of Mathematics and Statistics -University of Naples Federico II, Italy
via Cintia, Complesso Monte S. Angelo
(e-mail:diloremi@unina.it)
of Actuarial Science and Statistics, Cass Business School, City University, Bunhill Row London – UK - e-mail:
s.haberman@city.ac.uk
Agenda
•
•
•
•
•
The Aim
The Funding Ratio
The Poisson Lee Carter model
The Simulation Approach
Numerical Applications
The Aim
• In the context of the stochastic interest rates,
we investigate the impact of mortality
projection refined methodologies on the
insurance business financial situation.
The Aim
• We identify the actual contribution of the
considered
refined
methodologies
in
constructing the survival probabilities in
actuarial entities that can be immediately
interpreted and used for risk management
and solvency assessment purposes.
The Funding Ratio
Let us consider a portfolio of identical
policies, with benefits due in the case of
survival to persons belonging to an initial
group of c individuals aged x:
- w(t,j) is the value at time t of one monetary
unit due at time j
- Xj and Yj are the stochastic cash flow
respectively of assets coming into the
portfolio and liabilities going out of it
The Funding Ratio
At
Ft 
Lt
(1)
indicating by Ft the funding ratio at time t.
The funding ratio of the portfolio at time t is
expressed by the ratio between At, the
market value of the projected assets and Lt,
the market value of the projected liabilities,
both referred to the considered portfolio
and valued at time t.
The Funding Ratio
In particular, we have:
At   N j X j wt, j 
j
where

Lt   N jY j wt, j 
j
wt , j   v
 signt  j 
The Poisson Lee Carter Model
Because the number of deaths is a counting
random variable, the Poisson assumption
appears to be plausible:
Dxt  Poisson ext  xt 
 xt  exp  x   x kt 
ext being exposures to the risk of death at
age x,  xt the central mortality rate for age x
at time t .
The Poisson Lee Carter Model
The parameters are subjected to the
following constraints:
k
t
t
 0

x
x
1
The Poisson Lee Carter Model
The force of mortality is thus assumed to
have the log-bilinear form:

ln  x , t
 
x
  x kt
The Simulation Approach
Different simulation strategies have been
applied in the log-bilinear Poisson setting:
connected to the Monte Carlo approach and
to the Bootstrap procedures.
The Simulation Approach
We consider the bootstrap simulation
approach allowing for the measurement of
the mortality projections uncertainty.
The Simulation Approach
The bootstrap procedure can be accomplished
in different ways:
• the semi parametric bootstrap from the
Poisson distribution (Brouhns et al. 2005);
• the semi parametric from the multinomial
distribution (Brouhns et al. 2005),
The Simulation Approach
The bootstrap procedure can be accomplished
in different ways:
• the residual bootstrap proposed by Koissi et
al (2006);
• the variant of the latter illustrated in Renshaw
and Haberman (2008).
The Simulation Approach
We resort to the bootstrap simulation
approach:
- the Standard Procedure
- the Stratified Sampling Boostrap.
The Simulation Approach
As regards the Stratified Sampling Bootstrap,
we propose our bootstrap simulation
approach.
In particular, we intend to make efficient the
bootstrap procedure by using a specific
Variance Reducing Technique (VRT), the socalled stratified sampling.
The Simulation Approach
The Stratified Sampling Procedure combines
the stratified sampling technique together
with standard Bootstrap on the Poisson Lee
Carter model.
The Simulation Approach
In a typical scenario for a simulation study,
one is interested in determinig  , a parameter
connected with some stochastic model. To
estimate  , the model is simulated to obtain,
among other things, the output X which is
such that   EX 
The Simulation Approach
Repeated simulation runs, the i-th yielding the
output variable X i , are performed. The
simulation study is terminated when n runs
have been performed
and the estimate of  is
n
given by X   X i / n . Because this results in
i 1
an unbiased estimate of  , it follows that its
mean square error is equal to its variance.
That is
Var  X 
2
MSE  E  X      Var X  
n
The Simulation Approach
If we can obtain a different unbiased estimate
of  having a smaller variance than does
we X would obtain an improved estimator.
The Simulation Approach
In the Stratified Sampling technique, the
whole region of interest is split into disjoint
subsets, the so-called strata:
K
Z
k
Z
k 1
K
is the number of the strata
The Variance Reduction Techniques
In the proportional allocation we obtain
samples proportional to the size of the
stratum which comes out, so that :
Nk
Wk 
N
where N k is the size ofKthe population
stratum and N , N   N K size of
k 1
the population
The Variance Reduction Techniques
Within each stratum, B semi-parametric
bootstrap samples are drawn as in the
efficient algorithm (D’Amato et al. 2009c)
and the sample mean obtained after the
stratified sampling is calculated as:
K
E y ss    W k y k
k1
The Simulation Approach
Its variance is given by:
K
Var  ySS    Wk2Var  yk 
k 1
The Simulation Approach
To quote the efficiency gain of the
Stratified Sampling Bootstrap (SSB) in
respect to the Standard Procedure (SP),
we have resorted to the following index:
Var ( SP )
Efficiency 
Var  y SS 
The Simulation Approach
We can observe the improvement in the
efficiency due to the greater reduction of the
SSB variance.
Apparent small differences in the efficiency
index can lead to significantly accurate
mortality projections.
The Numerical Applications
• The Funding Ratio is calculated in the closed
form presented above in formula (1).
The Numerical Applications
• We consider the case of c=1000 contracts
issued at age x=35 with anticipated premiums
payed during the accumulation period,
extended from the issue time to age 65, and
anticipated instalments R=100 payed from age
65 till the insured is living.
The Numerical Applications
Demographic scenario:
Lee Carter model applied to the Italian male
population death rates (1950-2006)
Iterative Procedure Standard Procedure Stratified Sampling
Bootstrap Procedure
The Numerical Applications
Financial scenario:
HJM
df t , T    t , T dt   t , T dW t ,
f 0, T   f
M
0,T 
The Numerical Applications
• In this framework, for the specific age at issue
x=30, the funding ratio has been calculated at
five different times of valuation
t=5, 20, 35, 40, 45.
The Numerical Applications
Survival Approach
Premium
Standard Procedure
35.83861
Iterative Procedure
36.18761
Stratified Sampling
Bootstrap
37.35852
Table 1. Premium amounts: x=30, Poisson Lee Carter, fixed rate 4%.
The Numerical Applications
Funding Ratio
Valuation Time
Survival
Approach
t=5
t=20
t=35
t=40
t=45
Standard Procedure
1.1347 1.2757
1.4515
1.6330 1.7936
Iterative Procedure
1.2214 1.3993
1.5538
1.6540 1.8182
Stratified Sampling
Bootstrap
1.2330 1.3879
1.5775
1.6646 1.8616
Table 2. Funding Ratio: x=30, Poisson Lee Carter, interest rates HJM, t=5, 20, 35, 40, 45
The Numerical Applications
Funding ratio, insured aged x=30
2
1,8
1,6
1,4
1,2
1
0,8
0,6
0,4
0,2
0
Iterative Procedure
Standard
Stratified Sampling
Bootstrap
5
20
35
40
45
valuation time,t
Figure 1. Funding Ratio: x=30, Poisson Lee Carter with three different forecasting
method, t=5, 20, 35, 40, 45
The Numerical Applications
Survival approach
t=5
t=20
t=35
t=40
t=45
Iterative Procedure
7.64
9.69
7.05
1.29
1.37
Stratified Sampling
Bootstrap
8.66
8.80
8.68
1.94
3.79
Table 3. % change in funding ratio at time t relative to the standard survival assumption
The Numerical Applications
Assets, time t=35 - in millions
Efficient
Iterative
Standard
1,44
1,46
1,48
1,50
1,52
1,54
1,56
1,58
1,60
Figure 2. Asset at valuation time t=35, Poisson Lee Carter with three different
forecasting method
The Numerical Applications
Liabilities, time t=35, in millions
Efficient
Iterative
Liabilities, time t=35
Standard
€ 0,90
€ 0,95
€ 1,00
€ 1,05
€ 1,10
Figure 3. Liabilities at valuation time t=35, Poisson Lee Carter with three different
forecasting method
The Numerical Applications
Var [ Ft ]  E [ VarK t [ Ft ]]  VarK t E Ft 
DMRM  VarK t E Ft 
The Numerical Applications
t=5
t=20
t=35
t=40
t=45
0.2196%
0.280%
0.35%
0.093%
0.0635%
Table 4. Projection risk in the case of probabilities 0.6, 0.3 and 0.1
to choose respectively the Stratified Sampling Bootstrap, the Iterative
Procedure and the Standard Procedure.
Concluding Remarks
• The funding ratio indicates the degree to
which the pension liabilities are covered by
the assets, measuring the relative size of
pension assets compared to pension liabilities.
Concluding Remarks
• If the funding ratio is greater than 100%, then
the pension fund is overfunded, otherwise it is
underfunded or exactly fully funded if it is
respectively less than or equal to 100%.
Concluding Remarks
• It follows that managing the pension funding
ratio constitutes one of the insurance
company’s main goals.
Further Research
Further developments of the research could be
• Combining the Bootstrap on the Poisson LC
with other variance reduction technique, in
order to derive more reliable confidence
intervals for mortality projections.
• Quantifying the impact on actuarial measure
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