Main Article Content
Abstract
Joint-life insurance pays a sum insured when the first death occurs. This insurance has a case based on the order of exit from the cohort, namely joint life and last survivor. The former means that one of the insured leaves the cohort, while the latter means the last member of the insured has left his or her cohort. For some reasons of simplicity, the insurance premium is usually calculated with the assumption that the husband and wife are mutually independent. However, this assumption is considered unrealistic. Couples are open to the same risks, hence explaining joint survival model should involve dependence structures between the distribution of spouse mortality. In line with this, to understand the dependence structure of multiple random variables, the approach used is Copula. In this context, Copula relates the marginal distribution function of these variables to the joint life distribution. One of the advantages from Copula is that the random variables do not have to come from the same distribution, hence Copula is considered good enough to explain the dependence of the mortality rate between husband and wife. This study aimed to develop a joint survival model for calculating joint life insurance premiums using the concept of Archimedean Copula to discover the minimum premium value by conducting the following steps: first, identifying the marginal distributions of mortality for genders using Indonesian Mortality Table IV (TMI/
Tabel Mortalitas Indonesia IV); second, Archimedean copula function-based constructing survival models that captures the relationship between these variables; third, setting dependency parameter θ; fourth, calculating the joint life premium using Archimedean copula based survival modeled for each correlation dependency level; and carrying out optimization to find the minimum premium value. This can be achieved by formulating the problem as an optimization problem, considering an objective function that yields the lowest premium till satisfying the financial requirements of the insurance company.
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References
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References
Bowers, N. L., Gerber, H. U., Hickman, J. C., Jones, D. A. and Nesbitt, C. J., Actuarial Mathematics, Second Edition, Illinois, USA: The Society of Actuaries, 1997.
Deresa, N. W., van-Keilegom, I. and Antonio, K., “Copula-based inference for bivariate survival data with left truncation and dependent censoring”, Insurance: Mathematics and Economics, 107 (2022), 1-21.
Denuit, M., Dhaene, J., Le Bailly de Tilleghem, C. and Teghem, S., “Measuring the Impact of a Dependence Among Insured Lifelengths”, Belgian Actuarial Bulletin, 01:1 (2001), 18-39.
Dufresne, F., Hashorva, E., Ratovomirija, G. and Toukourou, Y., “On age difference in joint lifetime modelling with life insurance annuity applications”, Annals of Actuarial Science, 12:2 (2018), 350-371.
Parkes, C. M., Benjamin, B. and Fitzgerald, R. G., “Broken Heart: A Statistical Study of Increased Mortality among Widowers”, British Medical Journal, 01 (1969), 740-743.
Ward, A. W. M., “Mortality of bereavement”, British Medical Journal, 01 (1976), 700-702.
Hippisley-Cox, J., Coupland, C., Pringle, M., Crown, N. and Hammersley, V., “Married couples’ risk of same disease: cross sectional study”, British Medical Journal, 325 (2002), 1-5.
Frees, E. W. and Valdez, E. A., “Understanding Relationships Using Copulas”, North American Actuarial Journal, 02:1 (1998), 1-25.
Zhou, R. and Ji, M., “Modelling mortality dependence: An application of dynamic vine copula”, Insurance: Mathematics and Economics, 99 (2021), 241-255.
Zhu, W., Tan, K. S. and Wang, C. W., “Modeling Multi-Country Longevity Risk with Mortality Dependence: A Levy Subordinated Hierarchical Archimedean Copulas (LSHAC) Approach”, Journal of Risk and Insurance, 84:1 (2017), 477-493.
Dickson, D. C. M., Hardy, M. R. and Waters, H. R., Actuarial Mathematics for Life Contingent Risks, New York: Cambridge University Press, 2009.
Nelsen, R. B., An Introduction to Copulas, New York: Springer, 2006.
Carriere, J. F., “Dependent Decrement Theory”, Transactions of Society of Actuaries, 46 (1994), 45-74.
Hull, J. C., Options, Futures, and Other Derivatives, Eighth Edition, Boston: Prentice Hall, 2012.
AAJI, Tabel Mortalitas Indonesia (TMI) IV, Asosiasi Asuransi Jiwa Indonesia, Jakarta, 2019.
AAJI, Tabel Morbiditas Indonesia: Penyakit Kritis, Asosiasi Asuransi Jiwa Indonesia, Jakarta, 2022.
BPS, ”Statistik Indonesia dalam Infografis Tahun 2022”, Badan Pusat Statistik (BPS) Indonesia 2022. [Online]. Available: https://www.bps.go.id/publication/. [Accessed 28 07 2023].