# Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window

## Author

Listed:
• Corina Constantinescu

() (Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK)

• Suhang Dai

() (Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK)

• Weihong Ni

() (Institute for Financial and Actuarial Mathematics, Department of Mathematical Sciences, University of Liverpool, Liverpool L69 7ZL, UK)

• Zbigniew Palmowski

() (Mathematical Institute, University of Wrocław, Wroclaw 50-384, Poland)

## Abstract

We analyse the ruin probabilities for a renewal insurance risk process with inter-arrival times depending on the claims that arrive within a fixed (past) time window. This dependence could be explained through a regenerative structure. The main inspiration of the model comes from the bonus-malus (BM) feature of pricing car insurance. We discuss first the asymptotic results of ruin probabilities for different regimes of claim distributions. For numerical results, we recognise an embedded Markov additive process, and via an appropriate change of measure, ruin probabilities could be computed to a closed-form formulae. Additionally, we employ the importance sampling simulations to derive ruin probabilities, which further permit an in-depth analysis of a few concrete cases.

## Suggested Citation

• Corina Constantinescu & Suhang Dai & Weihong Ni & Zbigniew Palmowski, 2016. "Ruin Probabilities with Dependence on the Number of Claims within a Fixed Time Window," Risks, MDPI, Open Access Journal, vol. 4(2), pages 1-23, June.
• Handle: RePEc:gam:jrisks:v:4:y:2016:i:2:p:17-:d:72026
as

File URL: http://www.mdpi.com/2227-9091/4/2/17/pdf

File URL: http://www.mdpi.com/2227-9091/4/2/17/

## References listed on IDEAS

as
1. Albrecher, Hansjörg & Constantinescu, Corina & Loisel, Stephane, 2011. "Explicit ruin formulas for models with dependence among risks," Insurance: Mathematics and Economics, Elsevier, vol. 48(2), pages 265-270, March.
2. Li, Bo & Ni, Weihong & Constantinescu, Corina, 2015. "Risk models with premiums adjusted to claims number," Insurance: Mathematics and Economics, Elsevier, vol. 65(C), pages 94-102.
3. Afonso, Lourdes B. & dos Reis, Alfredo D. Egídio & Waters, Howard R., 2009. "Calculating Continuous Time Ruin Probabilities for a Large Portfolio with Varying Premiums," ASTIN Bulletin: The Journal of the International Actuarial Association, Cambridge University Press, vol. 39(01), pages 117-136, May.
4. Albrecher, Hansjorg & Boxma, Onno J., 2004. "A ruin model with dependence between claim sizes and claim intervals," Insurance: Mathematics and Economics, Elsevier, vol. 35(2), pages 245-254, October.
5. Søren Asmussen, 2014. "Modeling and Performance of Bonus-Malus Systems: Stationarity versus Age-Correction," Risks, MDPI, Open Access Journal, vol. 2(1), pages 1-25, March.
Full references (including those not matched with items on IDEAS)

### Keywords

regenerative risk process; ruin probability; subexponential distribution; Cramér asymptotics; importance sampling; crude Monte Carlo; Markov additive process;

### 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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