The Korean Journal of Applied Statistics (응용통계연구)

The Korean Statistical Society (한국통계학회)
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A Two Factor Model with Mean Reverting Process for Stochastic Mortality

평균회귀확률과정을 이용한 2요인 사망률 모형

  • Lee, Kangsoo (Korea Insurance Development Institute) ;
  • Jho, Jae Hoon (School of International Economics and Business, Yeungnam University)
  • Received : 2015.01.14
  • Accepted : 2015.04.28
  • Published : 2015.06.30
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Abstract

We examine how to model mortality risk using the adaptation of the mean-reverting processes for the two factor model proposed by Cairns et al. (2006b). Mortality improvements have been recently observed in some countries such as United Kingdom; therefore, we assume long-run mortality converges towards a trend at some unknown time and the mean-reverting processes could therefore be an appropriate stochastic model. We estimate the parameters of the two-factor model incorporated with mean-reverting processes by a Metropolis-Hastings algorithm to fit United Kingdom mortality data from 1991 to 2015. We forecast the evolution of the mortality from 2014 to 2040 based on the estimation results in order to evaluate the issue price of a longevity bond of 25 years maturity. As an application, we propose a method to quantify the speed of mortality improvement by the average mean reverting times of the processes.

본 논문은 2요인(two-factor) 사망률 모형에 평균회귀모형(mean reverting process)을 적용하여 2요인의 확률적 변동을 모형화하여 사망률리스크(mortality risk)와 장수리스크(longevity risk)를 분석하였다. 최근 고령사회로 진입한 국가들에서 사망률 개선의 둔화가 관측되고 있는 시점에서 기존의 선형증가 또는 감소의 사망률 개선 모형을 보완함에 그 목적을 두었다. 영국의 1991~2015년 사망률 자료를 이용하여 제시한 모형의 모수를 메트로폴리스 알고리듬을 이용해 추정하였고 추정된 모수 값을 이용하여 다수 시뮬레이션을 통하여 장기간의 미래 사망률 예측값을 계산하였다. 평균회귀 모형의 특성으로 인해 약 60년의 시간이 지난 뒤부터는 사망률 개선이 거의 사라져 사망률이 일정한 값에 근접하였다. 사망률 개선이 둔화되는 현상이 관측되는 특정 집단(국가, 사회)의 경우 2요인 평균회귀 모형은 장기간 사망률 예측방법의 대안으로 간주될 것으로 기대되며, 모형의 응용으로서 평균회귀율의 추정결과로부터 사망률 개선의 속도를 계량화하는 기준을 제시하였다. 끝으로, 2014년~2040 기간의 사망률 예측값을 이용하여 25년 만기 장수채권의 발행가격을 산출하였다.

Keywords

References

  1. Cairns, A. J., Blake, D. and Dowd, K. (2006a). Pricing death: Frameworks for the valuation and securitization of mortality risk, ASTIN Bulletin, 36, 79-120. https://doi.org/10.2143/AST.36.1.2014145
  2. Cairns, A. J., Blake, D. and Dowd, K. (2006b). A two-factor model for stochastic mortality with parameter uncertainty: Theory and calibration, Journal of Risk and Insurance, 73, 687-718. https://doi.org/10.1111/j.1539-6975.2006.00195.x
  3. Cairns, A. J., Blake, D., Dowd, K., Coughlan, D., Epstein, D., Ong, A. and Balevich, I. (2009). A quantitative comparison of stochastic mortality models using data from England & Wales and the United States, North American Actuarial Journal, 13, 1-35. https://doi.org/10.1080/10920277.2009.10597538
  4. Cheon, S. and Lee, H. (2012). Metropolis-Hastings expectation maximization algorithm for incomplete data, The Korean Journal of Applied Statistics, 25, 183-196. https://doi.org/10.5351/KJAS.2012.25.1.183
  5. Cummins, J. D. (2008). CAT bonds and other risk-linked securities: State of the market and recent developments, Risk Management and Insurance Review, 11, 23-47. https://doi.org/10.1111/j.1540-6296.2008.00127.x
  6. Currie, I. D. (2006). Smoothing and forecasting mortality rates with P-splines, Talk given at the Institute of Actuaries, June 2006, Available from http://www.macs.hw.ac.uk/iain/research/talks/London.2007.pdf.
  7. Lee, R. and Carter, L. (1992). Modeling and forecasting US mortality, Journal of the American Statistical Association, 87, 659-671.
  8. Lynch, S. M. (2007). Introduction to Applied Bayesian Statistics and Estimation for Social Scientists, Springer, New York.
  9. Plat, R. (2009). On stochastic mortality modeling, Insurance: Mathematics and Economics, 45, 393-404. https://doi.org/10.1016/j.insmatheco.2009.08.006
  10. Perks, W. (1932). On some experiments in the graduation of mortality statistics, Journal of the Institute of Actuaries, 63, 12-57.
  11. Renshaw, A. and Haberman, S. (2006). A Cohort-based extension to the Lee-Carter model for mortality reduction factors, Insurance: Mathematics and Economics, 38, 255-272.
  12. Sung, J. and Shin, H. (2007). Diagnosis of the current securitization markets hedging mortality-related risks and some notes on their crucial issues, Korean Journal of Insurance, 77, 291-323.
  13. UK office for National Statistics (2013a). Mortality assumptions, 2012-based national population projections, Technical Paper, Available from: http://www.ons.gov.uk/ons/publications/index.html
  14. UK office for National Statistics (2013b). Historic and projected mortality rates (qx) from the 2012-based EW life tables, Technical Paper, Available from: http://www.ons.gov.uk/ons/publications/index.html

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  1. Longevity Bond Pricing by a Cohort-based Stochastic Mortality vol.28, pp.4, 2015, https://doi.org/10.5351/KJAS.2015.28.4.703