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The Poisson log-bilinear Lee Carter model: Efficient bootstrap in life annuity actuarial analysis

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### 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 of Mathematics and Statistics -University of Naples Federico II, Italy

via Cintia, Complesso Monte S. Angelo

(e-mail:diloremi@unina.it)

3Faculty 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

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 Poisson Lee Carter Model

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

The parameters are subjected to the following constraints:

The Poisson Lee Carter Model

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

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

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

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

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

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:

where is the size of the population stratum and sizeof 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:

Its variance is given by:

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:

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

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

Table 1. Premium amounts: x=30, Poisson Lee Carter, fixed rate 4%.

The Numerical Applications

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

Iterative Procedure

1,2

1

Standard

0,8

0,6

0,4

Stratified Sampling

0,2

Bootstrap

0

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

Table 3. % change in funding ratio at time t relative to the standard survival assumption

The Numerical Applications

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

The Numerical Applications

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

The Numerical Applications

Table 4. Projection risk in the case of probabilities 0.6, 0.3 and 0.1 to choose respectivelythe 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

Main References

- Alho, J.M., 2000, Discussion of Lee. North American Actuarial Journal (with discussion) 4 (1), 80–93.
- Brouhns, N., Denuit, M., van Keilegom, 2005, Bootstrapping the Poisson log-bilinear model for mortality forecasting, Scandinavian Actuarial Journal, 3, 212-224.
- Brillinger, D.R., 1986, The natural variability of vital rates and associated statistics, Biometrics 42, 693–734.
- Commission of the European Communities, 2008,Directive of the European Parliament and of the Council on the taking up and pursuit of the business of Insurance and Reinsurance (“Solvency II”), available at: http://ec.europa.eu/internal_market/insurance/docs/solvency/proposal_en.pdf

Main References

- Coppola, M., E. Di Lorenzo, A. Orlando, M. Sibillo, 2007, Measuring demographic uncertainty via actuarial indexes, in Recent Advances in Stochastic Modelling and Data Analysis, CHRISTOS H. SKIADAS, p. 122-129, World Scientific Publishing Co Pte Ltd, ISBN/ISSN: ISBN 978-981-270-968-4
- D’Amato, V., E. Di Lorenzo, M. Russolillo, M. Sibillo, 2009a, The Interplay between Financial and Demographic Risks in a Pension Annuity System, accepted for presentation and publication in the proceedings of ASMDA 2009 Conference, Vilnius (Lithuania)
- D’Amato, V., S. Haberman, M. Russolillo, 2009b, Efficient Bootstrap applied to the Poisson Log-Bilinear Lee Carter Model, Applied Stochastic Models and Data Analysis – ASMDA 2009 Selected Papers, ISBN 978 -9955-28-463-5

Main References

- D’Amato, V., S. Haberman, M. Russolillo, 2009c, The Stratified Sampling Bootstrap: an algorithm for measuring the uncertainty in forecast mortality rates in the Poisson Lee - Carter setting, working paper
- Di Lorenzo, E., M. Sibillo, 2002, Longevity Risk: Measurement and Application Perspectives, Proceedings of 2nd Conference in Actuarial Science and Finance on Samos – Karlovassi (Greece) http://www.stat.ucl.ac.be/Samos2002/proceedSibillo.pdf
- Fornero E., E. Luciano, 2004, Developing an annuity market in Europe, Edward Elgar Cheltenham
- Groupe Consultatif Actuarial Europeen CEA, 2007,Solvency II Glossary available at: http://www.gcactuaries.org/documents/sol2_glossary_final_160307.pdf

Main References

- Hári, N., A. De Waegenaere, B. Melenberg, T.E. Nijman, 2008, Longevity Risk in Portfolios of Pension Annuities, Insurance: Mathematics and Economics 42, 505 - 519
- Heath, D., R. A. Jarrow, A. Morton, 1992, Bond Pricing and the Term Structure of Interest Rates: A New Methodology for Contingent Claims Valuation, Econometrica, vol. 60, issue 1, 77-105
- International Association of Insurance Supervisors,2008,Guidance paper no. 2.2.6 prepared by the Solvency Actuarial Issues Sub-Committee on The Use of Internal Models for Regulatory Capital Purposes, available at:
- http://www.iaisweb.org/__temp/15__Guidance_paper_No__2_2_6_on_the_use_of_Internal_Models_for_regulatory_capital_purposes.pdf
- Koissi, M.C., A.F. Shapiro, G. Hognas, 2006, Evaluating and Extending the Lee–Carter Model for Mortality Forecasting: Bootstrap Confidence Interval. Insurance: Mathematics and Economics 26: 1-20.

Main References

- Lee, R.D., L. R. Carter, 1992, Modelling and Forecasting U.S. Mortality, Journal of the American Statistical Association, 87, 659-671.
- Loladze, I. 2002, Rising atmospheric CO2 and human nutrition: toward globally imbalanced plant stoichiometry? Trends in Ecology & Evolution 17: 457-461
- Melnikov, A., Y. Romaniuk, 2006, Evaluating the Performance of Gompertz, Makeham and Lee-Carter Mortality Models for Risk Management, Insurance: Mathematics and Economics 39: 310-329.
- Olshansky, S. J., D. J. Passaro, 2005, A Potential Decline in Life Expectancy in the United States in the 21st Century, New England Journal of Medicine, March 17, 1138-1145
- Olivieri, A., E. Pitacco, 2003, Solvency requirements for pension annuities, Journal of Pension Economics and Finance 2, 127–154.

Main References

- Pitacco, E., M. Denuit, S. Haberman, A. Olivieri, 2009, Modelling longevity dynamics for pensions and annuity business, Oxford University Press
- Renshaw, A.E., S. Haberman, 2003a, Lee-Carter mortality forecasting: a parallel generalised linear modelling approach for England and Wales mortality projections, Applied Statistics, 52, 119-137.
- Renshaw, A.E., S. Haberman, 2003b, On the forecasting of mortality reduction factors. Insurance: Mathematics and Economics,32, 379-401.
- Renshaw, A.E., S. Haberman, 2003c, Lee-Carter mortality forecasting with age specific Enhancement, Insurance: Mathematics and Economics 33, 255-272.
- Renshaw, A.E., Haberman, S. 2008. On simulation-based approaches to risk measurement in mortality with specific reference to Poisson Lee-Carter modelling. Insurance: Mathematics and Economics 42, 797-816.
- Ross S., (2002). Simulation, Hartcourt Academic Press.

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