Abstract
After implementation of Solvency II, insurance companies can use internal risk models. In this paper, we show how to calculate finite-horizon ruin probabilities and prove for them new upper and lower bounds in a risk-switching Sparre Andersen model. Due to its flexibility, the model can be helpful for calculating some regulatory capital requirements. The model generalizes several discrete time- as well as continuous time risk models. A Markov chain is used as a ‘switch’ changing the amount and/or respective wait time distributions of claims while the insurer can adapt the premiums in response. The envelopes of generalized moment generating functions are applied to bound insurer’s ruin probabilities.
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References
Asmussen S (1989) Risk theory in a Markovian environment. Scand Actuar J 1989(2):66–100
Asmussen S (2003) Applied probability and queues, 2nd edn. Springer, New York
Asmussen S, Albrecher H (2010) Ruin probabilities, 2nd edn. World Scientific, Singapore
Chen X, Xiao T, Yang X (2014) A Markov-modulated jump-diffusion risk model with randomized observation periods and threshold dividend strategy. Insurance: Mathematics and Economics 54:76–83
Feng R, Shimizu Y (2014) Potential measures for spectrally negative Markov additive processes with applications in ruin theory. Insurance: Mathematics and Economics 59:11–26
Frühwirth-Schnatter S (2006) Finite mixture and Markov switching models. Springer, New York
Gajek L (2005) On the deficit distribution when ruin occurs - discrete time model. Insurance: Mathematics and Economics 36:13–24
Gajek L, Rudź M (2013) Sharp approximations of ruin probabilities in the discrete time models. Scand Actuar J 2013(5):352–382
Gajek L, Rudź M (2017) A generalization of Gerber’s inequality for ruin probabilities in risk-switching models. Statistics and Probability Letters 129:236–240
Gajek L, Rudź M (2018) Banach Contraction Principle and ruin probabilities in regime-switching models. Insurance: Mathematics and Economics 80:45–53
Guillou A, Loisel S, Stupfler G (2013) Estimation of the parameters of a Markov-modulated loss process in insurance. Insurance: Mathematics and Economics 53:388–404
Landriault D, Li B, Li S (2015) Analysis of a drawdown-based regime-switching Lévy insurance model. Insurance: Mathematics and Economics 60:98–107
Reinhard JM (1984) On a class of semi-Markov risk models obtained as classical risk models in a Markovian environment. Astin Bulletin 16:23–43
Solvency II (2009) Directive 2009/138/EC of the European Parliament and of the Council of 25 November 2009 on the taking-up and pursuit of the business of Insurance and Reinsurance (Solvency II). Official Journal of the European Union L 335/1, L 335/2, L 335/155
Taylor GC (1976) Use of differential and integral inequalities to bound ruin and queuing probabilities. Scand Actuar J 1976(4):197–208
Wang G, Wang G, Yang H (2016) On a multi-dimensional risk model with regime switching. Insurance: Mathematics and Economics 68:73–83
Xu L, Zhang L, Yao D (2017) Optimal investment and reinsurance for an insurer under Markov-modulated financial market. Insurance: Mathematics and Economics 74:7–19
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The authors thank the reviewers and the editors for helpful comments.
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Lesław Gajek is Advisor to the Chairman of the Polish Financial Supervision Authority. This article has been performed in a private capacity and the opinions expressed in it should not be attributed to the PFSA.
Research supported by the National Science Centre, Poland (2014/13/B/HS4/03222).
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Gajek, L., Rudź, M. Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model. Methodol Comput Appl Probab 22, 1493–1506 (2020). https://doi.org/10.1007/s11009-018-9627-2
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DOI: https://doi.org/10.1007/s11009-018-9627-2
Keywords
- Risk operators
- Risk-switching models
- Ruin probabilities
- Mgf’s envelopes
- Risk management based on internal models
- Solvency II