Dependent Risks and Ruin Probabilities in Insurance

Classical risk process models in insurance rely on independency. However, especially when modeling natural events, this assumption is very restrictive. This paper proposes a new approach to introducing dependency structures between events into the model and investigates its effects on a crucial parameter for insurance companies, the probability of ruin. Explicit formulas, numerical simulations and sensitivity results for dependence are established for different dependency models of first-order markovian type indicating that for various scenarios dependency considerably increases the probability of ruin

On optimal dividend strategies in insurance with a random time horizon

For the classical compound Poisson surplus process of an insurance portfolio we investigate the problem of how to optimally pay out dividends to shareholders if the criterion is to maximize the expected discounted dividend payments until the time of ruin or a random time horizon, whichever is smaller. We explicitly solve this problem for an exponential time horizon and exponential claim sizes. Furthermore, we study the case of an Erlang(2) time horizon by introducing an external state process and derive the solution under the assumption that the external state process is observable. The results are illustrated by numerical examples

Power identities for Lévy risk models under taxation and capital injections

In this paper we study a spectrally negative Lévy process which is refracted at its running maximum and at the same time reflected from below at a certain level. Such a process can for instance be used to model an insurance surplus process subject to tax payments according to a loss-carry-forward scheme together with the flow of minimal capital injections required to keep the surplus process non-negative. We characterize the first passage time over an arbitrary level and the cumulative amount of injected capital up to this time by their joint Laplace transform, and show that it satisfies a simple power relation to the case without refraction, generalizing results by Albrecher and Hipp (2007) and Albrecher, Renaud and Zhou (2008). It turns out that this identity can also be extended to a certain type of refraction from below. The net present value of tax collected before the cumulative injected capital exceeds a certain amount is determined, and a numerical illustration is provided

From ruin to bankruptcy for compound Poisson surplus processes

In classical risk theory, the infinite-time ruin probability of a surplus process Ct is calculated as the probability of the process becoming negative at some point in time. In this paper, we consider a relaxation of the ruin concept to the concept of bankruptcy, according to which one has a positive surplus-dependent probability to continue despite temporary negative surplus. We study the resulting bankruptcy probability for the compound Poisson risk model with exponential claim sizes for different bankruptcy rate functions, deriving analytical results, upper and lower bounds as well as an efficient simulation method. Numerical examples are given and the results are compared with the classical ruin probabilities. Finally, it is illustrated how the analysis can be extended to study the discounted penalty function under this relaxed ruin criterion

Dividends and the Time of Ruin under Barrier Strategies with a Capital-Exchange Agreement

We consider a capital-exchange agreement, where two insurers recapitalize each other in certain situations with funds they would otherwise use for dividend payments. We derive equations characterizing the expected time of ruin and the expected value of the respective discounted dividends until ruin, if dividends are paid according to a barrier strategy. In a Monte Carlo simulation study we illustrate the potential advantages of this type of collaboration

Static hedging of Asian options under Lévy models: the comonotonicity approach.

In this paper we present a simple static super-hedging strategy for the payoff of an arithmetic Asian option in terms of a portfolio of European options. Moreover, it is shown that the obtained hedge is optimal in some sense. The strategy is based on stop-loss transforms and is applicable under general stock price models. We focus on some popular Lévy models. Numerical illustrations of the hedging performance are given for various Lévy models calibrated to market data of the S&P 500.Comonotonicity; Data; Hedging; Market; Model; Models; Optimal; Options; Performance; Portfolio; Strategy;

On ruin probability and aggregate claim representations for Pareto claim size distributions

We generalize an integral representation for the ruin probability in a Cramer-Lundberg risk model with shifted (or also called US-)Pareto claim sizes, obtained by Ramsay (2003), to classical Pareto(a) claim size distributions with arbitrary real values a > 1 and derive its asymptotic expansion. Furthermore an integral representation for the tail of compound sums of Pareto-distributed claims is obtained and numerical illustrations of its performance in comparison to other aggregate claim approximations are provided