Finite-horizon general insolvency risk measures in a regime-switching Sparre Andersen model

Insolvency risk measures play important role in the theory and practice of risk management. In this paper, we provide a numerical procedure to compute vectors of their exact values and prove for them new upper and/or lower bounds which are shown to be attainable. More precisely, we investigate a general insolvency risk measure for a regime-switching Sparre Andersen model in which the distributions of claims and/or wait times are driven by a Markov chain. The measure is defined as an arbitrary increasing function of the conditional expected harm of the deficit at ruin, given the initial state of the Markov chain. A vector-valued operator L, generated by the regime-switching process, is introduced and investigated. We show a close connection between the iterations of L and the risk measure in a finite horizon. The approach assumed in the paper enables to treat in a unified way several discrete and continuous time risk models as well as a variety of important vector-valued insolvency risk measures.

Deficit distributions at ruin in a regime-switching Sparre Andersen model

In this paper, we investigate deficit distributions at ruin in a regime-switching Sparre Andersen model. A Markov chain is assumed to switch the amount and/or respective wait time distributions of claims while the insurer can adjust the premiums in response. Special attention is paid to an operator {\mathbf{L}} generated by the risk process. We show that the deficit distributions at ruin during n periods, given the state of the Markov chain at time zero, form a vector of functions, which is the n-th iteration of {\mathbf{L}} on the vector of functions being identically equal to zero. Moreover, in the case of infinite horizon, the deficit distributions at ruin are shown to be a fixed point of {\mathbf{L}} . Upper bounds for the vector of deficit distributions at ruin are also proven.

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

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.