Interplay of insurance and financial risks in a discrete-time model with strongly regular variation

Consider an insurance company exposed to a stochastic economic environment that contains two kinds of risk. The first kind is the insurance risk caused by traditional insurance claims, and the second kind is the financial risk resulting from investments. Its wealth process is described in a standard discrete-time model in which, during each period, the insurance risk is quantified as a real-valued random variable $X$ equal to the total amount of claims less premiums, and the financial risk as a positive random variable $Y$ equal to the reciprocal of the stochastic accumulation factor. This risk model builds an efficient platform for investigating the interplay of the two kinds of risk. We focus on the ruin probability and the tail probability of the aggregate risk amount. Assuming that every convex combination of the distributions of $X$ and $Y$ is of strongly regular variation, we derive some precise asymptotic formulas for these probabilities with both finite and infinite time horizons, all in the form of linear combinations of the tail probabilities of $X$ and $Y$. Our treatment is unified in the sense that no dominating relationship between $X$ and $Y$ is required.Comment: Published at http://dx.doi.org/10.3150/14-BEJ625 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotic Ruin Probabilities of the Renewal Model with Constant Interest Force and Regular Variation

Klüppelberg and Stadtmüller (1998, Scand. Actuar. J., no. 1, 49{58) obtained a simple asymptotic formula for the ruin probability of the classical model with constant interest force and regularly varying tailed claims. This short note extends their result to the renewal model. The proof is based on a result of Resnick and Willekens (1991, Comm. Statist. Stochastic Models 7, no. 4, 511{525)

The Finite Time Ruin Probability of the Compound Poisson Model with Constant Interest Force

In this paper we establish a simple asymptotic formula with respect to large initial surplus for thefinite time ruin probability of the compound Poisson model with constant interest force and subexponential claims. The formula is consistent with known results for the ultimate ruin probability and, in particular, it is uniform for all time horizons when the claim size distribution is regularly varying tailed

Finite and Infinite Time Ruin Probabilities in the Presence of Stochastic Returns on Investments

This paper investigates the finite and infinite time ruin probabilities in a discrete time stochastic economic environment. Under the assumption that the insurance risk - the total net loss within one time period - is extended-regularly-varying or rapidly varying tailed, various precise estimates for the ruin probabilities are derived. In particular, some estimates obtained are uniform with respect to the time horizon, hence apply for the case of infinite time ruin

Introducing a Dependence Structure to the Occurences in Studying Precise Large Deviations for the Total Claim Amount

In this paper we study precise large deviations for a compound sum of claims, in which the claims arrive in groups and the claim numbers in the groups may follow a certain negative dependence structure. We try to build a platform both for the classical large deviation theory and for the modern stochastic ordering theory

Asymptotics of Random Contractions

In this paper we discuss the asymptotic behaviour of random contractions $X=RS$, where $R$, with distribution function $F$, is a positive random variable independent of $S\in (0,1)$. Random contractions appear naturally in insurance and finance. Our principal contribution is the derivation of the tail asymptotics of $X$ assuming that $F$ is in the max-domain of attraction of an extreme value distribution and the distribution function of $S$ satisfies a regular variation property. We apply our result to derive the asymptotics of the probability of ruin for a particular discrete-time risk model. Further we quantify in our asymptotic setting the effect of the random scaling on the Conditional Tail Expectations, risk aggregation, and derive the joint asymptotic distribution of linear combinations of random contractions.Comment: 25 page