# On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy

## Author

Listed:
• Eric C.K. Cheung

() (Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong)

• Haibo Liu

() (Department of Statistics and Actuarial Science, University of Iowa, Iowa City, IA 52242, USA)

• Jae-Kyung Woo

() (Department of Statistics and Actuarial Science, University of Hong Kong, Pokfulam, Hong Kong)

## Abstract

In the compound Poisson insurance risk model under a dividend barrier strategy, this paper aims to analyze jointly the aggregate discounted claim amounts until ruin and the total discounted dividends until ruin, which represent the insurer’s payments to its policyholders and shareholders, respectively. To this end, we introduce a Gerber–Shiu-type function, which further incorporates the higher moments of these two quantities. This not only unifies the individual study of various ruin-related quantities, but also allows for new measures concerning covariances to be calculated. The integro-differential equation satisfied by the generalized Gerber–Shiu function and the boundary condition are derived. In particular, when the claim severity is distributed as a combination of exponentials, explicit expressions for this Gerber–Shiu function in some special cases are given. Numerical examples involving the covariances between any two of (i) the aggregate discounted claims until ruin, (ii) the discounted dividend payments until ruin and (iii) the time of ruin are presented along with some interpretations.

## Suggested Citation

• Eric C.K. Cheung & Haibo Liu & Jae-Kyung Woo, 2015. "On the Joint Analysis of the Total Discounted Payments to Policyholders and Shareholders: Dividend Barrier Strategy," Risks, MDPI, Open Access Journal, vol. 3(4), pages 1-24, November.
• Handle: RePEc:gam:jrisks:v:3:y:2015:i:4:p:491-514:d:58578
as

File URL: http://www.mdpi.com/2227-9091/3/4/491/pdf

File URL: http://www.mdpi.com/2227-9091/3/4/491/

## References listed on IDEAS

as
1. Feng, Runhuan, 2009. "On the total operating costs up to default in a renewal risk model," Insurance: Mathematics and Economics, Elsevier, vol. 45(2), pages 305-314, October.
2. Cheung, Eric C.K. & Landriault, David, 2010. "A generalized penalty function with the maximum surplus prior to ruin in a MAP risk model," Insurance: Mathematics and Economics, Elsevier, vol. 46(1), pages 127-134, February.
3. Dickson, David C. M. & Hipp, Christian, 2001. "On the time to ruin for Erlang(2) risk processes," Insurance: Mathematics and Economics, Elsevier, vol. 29(3), pages 333-344, December.
4. Cheung, Eric C.K. & Feng, Runhuan, 2013. "A unified analysis of claim costs up to ruin in a Markovian arrival risk model," Insurance: Mathematics and Economics, Elsevier, vol. 53(1), pages 98-109.
5. Taylor, G. C., 1979. "Probability of Ruin under Inflationary Conditions or under Experience Rating," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 10(02), pages 149-162, March.
6. Dickson, David C.M. & Waters, Howard R., 2004. "Some Optimal Dividends Problems," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 34(01), pages 49-74, May.
7. Li, Shuanming & Garrido, Jose, 2004. "On ruin for the Erlang(n) risk process," Insurance: Mathematics and Economics, Elsevier, vol. 34(3), pages 391-408, June.
8. Gerber, Hans U. & Shiu, Elias S.W. & Smith, Nathaniel, 2006. "Maximizing Dividends without Bankruptcy," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 36(01), pages 5-23, May.
Full references (including those not matched with items on IDEAS)

## Citations

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Cited by:

1. repec:eee:apmaco:v:331:y:2018:i:c:p:358-377 is not listed on IDEAS
2. repec:eee:insuma:v:79:y:2018:i:c:p:26-42 is not listed on IDEAS

### Keywords

compound Poisson risk model; dividend barrier strategy; aggregate discounted claims until ruin; discounted dividend payments; joint moments; covariance; Gerber–Shiu function;

### JEL classification:

• C - Mathematical and Quantitative Methods
• G0 - Financial Economics - - General
• G1 - Financial Economics - - General Financial Markets
• G2 - Financial Economics - - Financial Institutions and Services
• G3 - Financial Economics - - Corporate Finance and Governance
• K2 - Law and Economics - - Regulation and Business Law

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