Time to ruin, insolvency penalties and dividends in a Markov-modulated multi-risk model with common shocks

We consider a main insurance company with K subcompanies (or lines of busi- ness). The joint evolution of the surpluses of these lines of business is modeled by a Markov-modulated multivariate compound Poisson model with Poisson common shocks, modified by interactions between the lines of business and paiement of divi- dends. We assume that the financial situation of the subcompanies has an impact on the other companies, for example because they have part of their surplus invested in one another. If a line of business is in the red, the others have to pay a penalty, which is traduced by a decrease of the premium received by unit of time, or by a lost of dividends for the shareholders if the other line of business is "doing well". Conversely, a line of business with a high surplus level may increase the premium by unit of time of the others as they receive part of the dividends. In this paper, we focus on a particular line of business, and provide an approximation for expected time to ruin, and the expected amounts of dividends paid to the shareholders, and used to pay penalty due to insolvency of some subcompany. The method is to discretize claim amounts and to approximate the multidimensional surplus process of the subcompanies with a continuous time Markov process with finite state space. A technique of Frostig (2005) and Kella and Whitt (1992) enables us to get approximates, which are shown to converge to the desired values. It is possible to compare the behavior of the main company with and without the other subcompanies, which could provide a tool to help making consortium building decision.multi-risk model; ruin theory; dividends; lines of business; Markovian environment; common shock

Differentiation of some functionals of risk processes.

For general risk processes, the expected time-integrated negative part of the process on a fixed time interval is introduced and studied. Differentiation theorems are stated and proved. They make it possible to derive the expected value of this risk measure, and to link it with the average total time below zero studied by Dos Reis (1993) and the probability of ruin. Differentiation of other functionals of unidimensional and multidimensional risk processes with respect to the initial reserve level are carried out. Applications to ruin theory, and to the determination of the optimal allocation of the global initial reserve which minimizes one of these risk measures, illustrate the variety of application fields and the benefits deriving from an efficient and effective use of such tools.Ruin theory; Sample path properties; Optimal allocation; Multidimensional risk process; Risk measures

On Finite-Time Ruin Probabilities for Classical Risk Models

This paper is concerned with the problem of ruin in the classical compound binomial and compound Poisson risk models. Our primary purpose is to extend to those models an exact formula derived by Picard and Lefèvre (1997) for the probability of (non-)ruin within finite time. First, a standard method based on the ballot theorem and an argument of Seal-type provides an initial (known) formula for that probability. Then, a concept of pseudo-distributions for the cumulated claim amounts, combined with some simple implications of the ballot theorem, leads to the desired formula. Two expressions for the (non-)ruin probability over an infinite horizon are also deduced as corollaries. Finally, an illustration within the framework of Solvency II is briefly presented.ruin probability; finite and infinite horizon; compound binomial model; compound Poisson model; ballot theorem; pseudo-distributions; Solvency II; Value-at-Risk.

Risk aggregation in Solvency II: How to converge the approaches of the internal models and those of the standard formula?

Two approaches may be considered in order to determine the Solvency II economic capital: the use of a standard formula or the use of an internal model (global or partial). However, the results produced by these two methods are rarely similar, since the underlying hypothesis of marginal capital aggregation is not verified by the projection models used by companies. We demonstrate that the standard formula can be considered as a first order approximation of the result of the internal model. We therefore propose an alternative method of aggregation that enables to satisfactorily capture the diversity among the various risks that are considered, and to converge the internal models and the standard formula.

The win-first probability under interest force

In a classical risk model under constant interest force, we study the probability that the surplus of an insurance company reaches an upper barrier before a lower barrier. We define this probability as win-first probability. Borrowing ideas from life-insurance theory, hazard rates of the maximum of the surplus before ruin, regarded as a remaining future lifetime random variable, are studied, and provide an original derivation of the win-first probability. We propose an algorithm to efficiently compute this risk-return indicator and its derivatives in the general case, as well as bounds of these quantities. The efficiency of the proposed algorithm is compared with adaptations of other existing methods, and its interest is illustrated by the computation of the expected amount of dividends paid until ruin in a risk model with a dividend barrier strategy.Ruin probability; hazard rate; upper absorbing barrier; constant interest force; risk-return indicator; win-first probability

In the core of longevity risk: hidden dependence in stochastic mortality models and cut-offs in prices of longevity swaps

In most stochastic mortality models, either one stochastic intensity process (for example a jump-diffusion process) or a collection of independent processes is used to model the stochastic evolution of survival probabilities. We propose and calibrate a new model that takes inter-age correlations into account. The so-called stochastic logit's Deltas model is based on the study of the multivariate time series of the differences of logits of yearly mortality rates. These correlations are important and we illustrate our study on a real-life portfolio. We determine their impact on the price of a longevity swap type reinsurance contract, in which most of the financial risk is taken by a third party. The hypotheses of our model are statistically tested and various measures of risk of the present value of liabilities are found to be significantly smaller in our model than in the case of one common underlying stochastic process.Longevity risk; longevity swap; inter-age correlations; stochastic mortality; multivariate process; logit; Lee-Carter

Finite-Time Ruin Probabilities for Discrete, Possibly Dependent, Claim Severities

This paper is concerned with the compound Poisson risk model and two generalized models with still Poisson claim arrivals. One extension incorporates inhomogeneity in the premium input and in the claim arrival process, while the other takes into account possible dependence between the successive claim amounts. The problem under study for these risk models is the evaluation of the probabilities of (non-)ruin over any horizon of finite length. The main recent methods, exact or approximate, used to compute the ruin probabilities are reviewed and discussed in a unified way. Special attention is then paid to an analysis of the qualitative impact of dependence between claim amounts.compound Poisson model; ruin probability; finite-time horizon; recursive methods; (generalized) Appell polynomials; non-constant premium; non-stationary claim arrivals; interdependent claim amounts; impact of dependence; comonotonic risks; heavy-tailed distributions

Risk aggregation in Solvency II: How to converge the approaches of the internal models and those of the standard formula?

Two approaches may be considered in order to determine the Solvency II economic capital: the use of a standard formula or the use of an internal model (global or partial). However, the results produced by these two methods are rarely similar, since the underlying hypothesis of marginal capital aggregation is not verified by the projection models used by companies. We demonstrate that the standard formula can be considered as a first order approximation of the result of the internal model. We therefore propose an alternative method of aggregation that enables to satisfactorily capture the diversity among the various risks that are considered, and to converge the internal models and the standard formula.

Stationary-excess operator and convex stochastic orders

The present paper aims to point out how the stationary-excess operator and its iterates transform the s-convex stochastic orders and the associated moment spaces. This allows us to propose a new unified method on constructing s-convex extrema for distributions that are known to be t-monotone. Both discrete and continuous cases are investigated. Several extremal distributions under monotonicity conditions are derived. They are illustrated with some applications in insurance.Insurance risks; s-convex stochastic orders; Extremal distributions; t-monotone distributions; Stationary-excess operator; Discrete and continuous versions.

Correlation crises in insurance and finance, and the need for dynamic risk maps in ORSA

We explain why correlation crises may occur in insurance and finance. These phenomena are not taken into account in Solvency II standard formula. We show the importance of taking them into account in internal models or partial internal models. Given the variety of scenarios that could lead to correlation crises and their different potential impacts, we support the idea that ORSA (Own Risk and Solvency Assessment) reports of insurance companies should include dynamic and causal correlation crises analyzes.