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Dividend Moments in the Dual Risk Model: Exact and Approximate Approaches

  • Eric C.K. Cheung (a1) and Steve Drekic (a2)

Abstract

In the classical compound Poisson risk model, it is assumed that a company (typically an insurance company) receives premium at a constant rate and pays incurred claims until ruin occurs. In contrast, for certain companies (typically those focusing on invention), it might be more appropriate to assume expenses are paid at a fixed rate and occasional random income is earned. In such cases, the surplus process of the company can be modelled as a dual of the classical compound Poisson model, as described in Avanzi et al. (2007). Assuming further that a barrier strategy is applied to such a model (i.e., any overshoot beyond a fixed level caused by an upward jump is paid out as a dividend until ruin occurs), we are able to derive integro-differential equations for the moments of the total discounted dividends as well as the Laplace transform of the time of ruin. These integro-differential equations can be solved explicitly assuming the jump size distribution has a rational Laplace transform. We also propose a discrete-time analogue of the continuous-time dual model and show that the corresponding quantities can be solved for explicitly leaving the discrete jump size distribution arbitrary. While the discrete-time model can be considered as a stand-alone model, it can also serve as an approximation to the continuous-time model. Finally, we consider a generalization of the so-called Dickson-Waters modification in optimal dividends problems by maximizing the difference between the expected value of discounted dividends and the present value of a fixed penalty applied at the time of ruin.

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References

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Avanzi, B., Gerber, H.U. and Shiu, E.S.W. (2007) Optimal dividends in the dual model. Insurance: Mathematics and Economics, 41, 111123.
De Vylder, F. and Goovaerts, M.J. (1988) Recursive calculation of finite-time ruin probabilities. Insurance: Mathematics and Economics, 7, 17.
Dickson, D.C.M. (2005) Insurance Risk and Ruin, Cambridge University Press, Cambridge.
Dickson, D.C.M. and Waters, H.R. (1991) Recursive calculation of survival probabilities. ASTIN Bulletin, 21, 199221.
Dickson, D.C.M. and Waters, H.R. (2004) Some optimal dividends problems. ASTIN Bulletin, 34, 4974.
Gerber, H.U., Lin, X.S. and Yang, H. (2006a) A note on the dividends-penalty identity and the optimal dividend barrier. ASTIN Bulletin, 36, 489503.
Gerber, H.U., Shiu, E.S.W. and Smith, N. (2006b) Maximizing dividends without bankruptcy. ASTIN Bulletin, 36, 523.
Klugman, S.A., Panjer, H.H. and Willmot, G.E. (2004) Loss Models: From Data to Decisions, 2nd edition, Wiley, New York.
Mazza, C. and Rullière, D. (2004) A link between wave governed random motions and ruin processes. Insurance: Mathematics and Economics, 35, 205222.
Seal, H.L. (1969) Stochastic Theory of a Risk Business, Wiley, New York.
Willmot, G.E., Drekic, S. and Cai, J. (2005) Equilibrium compound distributions and stop-loss moments. Scandinavian Actuarial Journal, 624.
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ASTIN Bulletin: The Journal of the IAA
  • ISSN: 0515-0361
  • EISSN: 1783-1350
  • URL: /core/journals/astin-bulletin-journal-of-the-iaa
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