Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model
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Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model Lesław Gajek1,2 · Marcin Rud´z1
Received: 13 July 2017 / Revised: 2 March 2018 / Accepted: 12 March 2018 © The Author(s) 2018
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. Keywords Risk operators · Risk-switching models · Ruin probabilities · Mgf’s envelopes · Risk management based on internal models · Solvency II Mathematics Subject Classification (2010) 91B30 · 60J20 · 60J22
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). Marcin Rud´z
[email protected]; marcin [email protected] Lesław Gajek [email protected] 1
Institute of Mathematics Lodz University of Technology, W´olcza´nska 215, 90-924 Ł´od´z, Poland
2
Polish Financial Supervision Authority, Plac Powsta´nc´ow Warszawy 1, 00-950 Warszawa, Poland
Methodol Comput Appl Probab
1 Introduction Solvency II changes the insurance industry completely, not only in the EU but also in the USA and Asia. In accordance with Motive (68) of this EU directive (see Solvency II 2009, p. L 335/7), each insurance company can calculate the Solvency Capital Requirement (SCR) with the help of its own tailor-made partial or full internal model. According to Motive (64) of the directive, SCR should be large enough in order to ensure that ruin occurs no more often than once in every 200 cases over the following 12 months. Thus, the probability of default should be analyzed by insurers and supervisors when SCR is to be established, and the model and methods investigated in the present paper can be found useful from this perspective. To be more concrete, we will show how to calculate finite-horizon ruin probabilities and introduce lower and upper bounds for them in the following risk-switching Sparre Andersen model. Let N denote the set of all positive integers and R - the real line. Set N0 = N ∪ {0}, N1 = N\{1}, R+ = (0, ∞) and R0+ = [0, ∞). All stochastic objects considered in the paper are defined on a probability space (, F , P). Let a random variable Xk denote the amount of the kth claim, T1 - the moment when the first claim appears and Tk - the time between the (k − 1)th claim and the kth on
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