Journal of the Korean Data and Information Science Society

The Korean Data and Information Science Society (한국데이터정보과학회)
권호 표지
DOI QR코드

DOI QR Code

New approximations of the ruin probability in a continuous time surplus process

보험상품 파산확률의 새로운 근사방법

  • Kwon, Cheonga (Department of Statistics, Sookmyung Women's University) ;
  • Choi, Seung Kyoung (Department of Statistics, Sookmyung Women's University) ;
  • Lee, Eui Yong (Department of Statistics, Sookmyung Women's University)
  • 권청아 (숙명여자대학교 통계학과) ;
  • 최승경 (숙명여자대학교 통계학과) ;
  • 이의용 (숙명여자대학교 통계학과)
  • Received : 2013.10.17
  • Accepted : 2013.11.26
  • Published : 2014.01.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study approximations of the ruin probability in a continuous time surplus process. First, we introduce the well-known approximation formulas of the ruin probability such as Cram$\acute{e}$r, Tijms' and De Vylder's methods. We, then, suggest new approximation formulas of two types, which improve the existing approximation formulas. One is Cram$\acute{e}$r and Tijms' type which makes use of the moment generating function of distribution of a claim size and the other is De Vylder's type which makes use of the surplus process with exponential claims. Finally, we compare, by illustrating numerical examples, the newly suggested approximation formulas with the existing approximation formulas of the ruin probability.

논문에서는 보험상품 파산확률의 근사값을 구하는 두 가지 새로운 방법을 제시한다. 첫 번째 방법은 기존의 Cram$\acute{e}$r와 Tijms의 근사방법을 가중평균한 것으로, 초기잉여금 값이 클 때 파산확률에 가까운 Cram$\acute{e}$r 방법과 초기잉여금이 작은 값일 때 파산확률에 가까운 Tijms 방법의 장점을 모두 고려한 방법이다. 두 번째 방법은 De Vylder의 근사식에 Tijms의 아이디어를 이용하여 De Vylder의 근사식을 확장한 방법이다. 또한 두 가지 새로운 방법과 기존의 근사방법 중 어느 것이 더 실제 파산확률에 가까운지 예를 통해 비교해 보았다.

Keywords

References

  1. Beekman, J. (1969). A ruin function approximation. Transactions of the Society of Actuaries, 21, 41-48.
  2. Choi, S. K., Choi, M. H., Lee, H. S. and Lee, E. Y. (2010). New approximations of ruin probability in a risk process. Quality Technology & Quantitative Management, 7, 377-383. https://doi.org/10.1080/16843703.2010.11673239
  3. Cramer, H. (1930). On the mathematical theory of risk. In Harald Cramer Collected Works, Vol. I, Springer, Berlin, 601-678.
  4. De Vylder, F. E. (1978). A practical solution to the problem of ultimate ruin probability. Scandinavian Actuarial Journal, 114-119.
  5. Grandell, J. (1977). A class of approximation of ruin probabilities. Scandinavian Actuarial Journal (Suppl.), 38-52.
  6. Grandell, J. (1991). Aspects of risk theory, Springer, New York.
  7. Grandell. J. (2000). Simple approximation of ruin probabilities. Insurance: Mathematics and Economics, 26, 157-173. https://doi.org/10.1016/S0167-6687(99)00050-5
  8. Hadwiger, H. (1940). Uber die wahrscheinlichkeit des ruins bei einer grossen zahl von geschaften. Arkiv fur Mathematische Wirtschaft-und Sozialforschung, 6, 131-135.
  9. Jung, S. C. (2011). The preference for direct marketing according to the characteristics of policyholders in the life insurance industry. Journal of the Korean Data & Information Science Society, 22, 1137-1143.
  10. Klugman, S. A., Panjer, H. and Willmot, G. E. (2004). Loss models: From data to decision, 2nd Ed., John Wiley & Sons, Hoboken.
  11. Lee, H. S., Choi, S. K. and Lee, E. Y. (2009). An improvement of the approximation of the ruin probability in a risk process. The Korean Journal of Applied Statistics, 22, 937-942. https://doi.org/10.5351/KJAS.2009.22.5.937
  12. Lundberg, O. (1964). On random processes and their application to sickness and accident statistics, 1st Edition, Almqvist & Wiksell, Uppsala.
  13. Song, M. J., Kim, J. W. and Lee, J. Y. (2012). A compound Poisson risk model with variable premium rate. Journal of the Korean Data & Information Science Society, 23, 1289-1297. https://doi.org/10.7465/jkdi.2012.23.6.1289
  14. Tijms, H. (1994). Stochastic models- An algorithmic approach, Wiley, Chichester.
  15. Wikstad, N. (1971). Exemplification of ruin probabilities. Astin Bulletin, 6, 147-152. https://doi.org/10.1017/S0515036100010874
  16. Won, H. J., Choi, S. K. and Lee, E. Y. (2013). Ruin probabilities in a risk process perturbed by diffusion with two types of claims. Journal of the Korean Data & Information Science Society, 24, 1-12. https://doi.org/10.7465/jkdi.2013.24.1.1

Cited by

  1. Stationary analysis of the surplus process in a risk model with investments vol.25, pp.4, 2014, https://doi.org/10.7465/jkdi.2014.25.4.915
  2. The cost efficiency analysis of Korean life insurance companies by data envelopment analysis vol.26, pp.6, 2015, https://doi.org/10.7465/jkdi.2015.26.6.1523
  3. Signed Hellinger measure for directional association vol.27, pp.2, 2016, https://doi.org/10.7465/jkdi.2016.27.2.353
  4. Bounds of PIM-based similarity measures with partially marginal proportion vol.26, pp.4, 2015, https://doi.org/10.7465/jkdi.2015.26.4.857
  5. Generally non-linear regression model containing standardized lift for association number estimation vol.27, pp.3, 2016, https://doi.org/10.7465/jkdi.2016.27.3.629
  6. Defining a cluster market: The case of the Korean Internet portal service market vol.39, pp.11, 2015, https://doi.org/10.1016/j.telpol.2015.08.009
  7. Various types of analysis of warranty returns data vol.26, pp.1, 2015, https://doi.org/10.7465/jkdi.2015.26.1.11
  8. 흥미도 측도 관점에서 상대적 인과 강도의 고찰 vol.28, pp.1, 2014, https://doi.org/10.7465/jkdi.2017.28.1.49
  9. 재무비율을 활용한 포트폴리오 최적화 전략 vol.28, pp.6, 2017, https://doi.org/10.7465/jkdi.2017.28.6.1481