The Poisson log-bilinear Lee Carter model:

Efficient bootstrap in life annuity actuarial analysis

Valeria D’AmatoValeria D’Amato11, Emilia Di Lorenzo, Emilia Di Lorenzo22, Steven Haberman, Steven Haberman33, Maria Russolillo, Maria Russolillo11, , Marilena SibilloMarilena Sibillo11

11Department of Economics and Statistics - University of Salerno, ItalyDepartment of Economics and Statistics - University of Salerno, Italyvia Ponte Don Melillo, campus universitario via Ponte Don Melillo, campus universitario

(e-mail: vdamato@unisa.it, mrussolillo@unisa.it, msibillo@unisa.it)(e-mail: vdamato@unisa.it, mrussolillo@unisa.it, msibillo@unisa.it)

22Department of Mathematics and Statistics -University of Naples Federico II, Italy Department of Mathematics and Statistics -University of Naples Federico II, Italy via Cintia, Complesso Monte S. Angelovia Cintia, Complesso Monte S. Angelo

(e-mail:diloremi@unina.it(e-mail:diloremi@unina.it)

33Faculty of Actuarial Science and Statistics, Cass Business School, City University, Bunhill Row London – UK - e-mail: Faculty of Actuarial Science and Statistics, Cass Business School, City University, Bunhill Row London – UK - e-mail:

s.haberman@city.ac.uks.haberman@city.ac.uk

Agenda

• The Aim• The Funding Ratio• The Poisson Lee Carter model• The Simulation Approach• Numerical Applications

• 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 Aim

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

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

(1) t

tt L

AF

In particular, we have:

where

The Funding Ratio

At N j X jj w t, j

Lt N jY jj w t, j

jtsignvjtw ,

Because the number of deaths is a counting random variable, the Poisson assumption appears to be plausible:

being exposures to the risk of death at age x,

the central mortality rate for age x at time t .

The Poisson Lee Carter Model

xtxtxt ePoissonD txxxt k exp

xte

xt

The parameters are subjected to the following constraints:

The Poisson Lee Carter Model

0t

tk 1x

x

The force of mortality is thus assumed to have the log-bilinear form:

The Poisson Lee Carter Model

txxtx k ,ln

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 so-called 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

XE

Repeated simulation runs, the i-th yielding the

output variable , are performed. The simulation study is terminated when runs have been performed and the estimate of is given by . Because this results in an unbiased estimate of , it follows that its mean square error is equal to its variance. That is

iXn

n

XVarXVarXEMSE 2

nXXn

ii /

1

The Simulation Approach

If we can obtain a different unbiased estimate of having a smaller variance than does we would obtain an improved estimator.

X

The Simulation Approach

In the Stratified Sampling technique, the whole region of interest is split into disjoint subsets, the so-called strata:

ZZK

kk

1

K is the number of the strata

The Simulation Approach

The Variance Reduction Techniques

In the proportional allocation we obtain samples proportional to the size of the stratum which comes out, so that :

where is the size of the population stratum and size of the population

kk

NW

N

kN

K

kKNNN

1

,

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:

E yss Wk y kk1

K

Its variance is given by:

k

K

kkSS yVarWyVar

1

2

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:

The Simulation Approach

SSyVar

SPVarEfficiency

)(

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 Simulation Approach

• 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

The Numerical Applications

Financial scenario:

HJM

TfTf

tdWTtdtTtTtdfM ,0,0

,,,,

• 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

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

Figure 1. Funding Ratio: x=30, Poisson Lee Carter with three different forecasting method, t=5, 20, 35, 40, 45

00,20,40,60,8

11,21,41,61,8

2

5 20 35 40 45

valuation time,t

Funding ratio, insured aged x=30

Iterative Procedure

Standard

Stratified SamplingBootstrap

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

1,44 1,46 1,48 1,50 1,52 1,54 1,56 1,58 1,60

Standard

Iterative

Efficient

Assets, time t=35 - in millions

Figure 2. Asset at valuation time t=35, Poisson Lee Carter with three different forecasting method

The Numerical Applications

€ 0,90 € 0,95 € 1,00 € 1,05 € 1,10

Standard

Iterative

Efficient

Liabilities, time t=35, in millions

Liabilities, time t=35

Figure 3. Liabilities at valuation time t=35, Poisson Lee Carter with three different forecasting method

The Numerical Applications

tK

tKtKt

FEVarDMRM

FEVar]]F[Var[E]F[Var

t

tt

The Numerical Applications

t=5 t=20 t=35 t=40 t=450.2196% 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.

• 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.

Concluding Remarks

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

Further Research

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