Finite-Horizon Ruin Probabilities in a Risk-Switching Sparre Andersen Model

Lesław Gajek,Marcin Rudź

doi:10.1007/s11009-018-9627-2

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.

Highlights

Solvency II changes the insurance industry completely, in the EU and in the USA and Asia
The probability of default should be analyzed by insurers and supervisors when Solvency Capital Requirement (SCR) is to be established, and the model and methods investigated in the present paper can be found useful from this perspective
By the ideas of Gajek (2005), one can verify the following relationship concerning n+1, 1 and the risk operator L: Theorem 1 Let the following assumptions hold for all i, j ∈ S, k ∈ N1 and t, x ∈ R+: (i) The conditional distribution of the random variables Z2, ..., Zk, given I0 = i, I1 = j, T1 = t, X1 = x, is the same as the conditional distribution of the random variables Z1, ..., Zk−1, given I0 = j ; (ii) H ij (t, x) = F ij (x)Gij (t)

Summary

Introduction

Solvency II changes the insurance industry completely, in the EU and in the USA and Asia. 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. The continuous-time risk theory in a Markovian environment is presented e.g. in Asmussen and Albrecher (2010) This approach dates back to Reinhard (1984) and Asmussen (1989), where the detailed references to queuing theory can be found. The conditional probability that τ (u) is not greater than n, given the initial state i, is called the probability of ruin at or before the nth claim Let F ij (respectively Gij ) denote the conditional distribution of X1 (respectively T1), given the initial state i and the state j at the moment A1.

Auxiliary Results

Main Results

Iterating the Risk Operator - Numerical Aspects and Applications

Conclusion