Asymptotic Estimates for the One-Year Ruin Probability under Risky Investments

Abstract

Abstract:Motivated by the EU Solvency II Directive, we study the one-year ruin probability of an insurer who makes investments and hence faces both insurance and financial risks. Over a time horizon of one year, the insurance risk is quantified as a nonnegative random variable X equal to the aggregate amount of claims, and the financial risk as a d-dimensional random vector Y consisting of stochastic discount factors of the d financial assets invested. To capture both heavy tails and asymptotic dependence of Y in an integrated manner, we assume that Y follows a standard multivariate regular variation (MRV) structure. As main results, we derive exact asymptotic estimates for the one-year ruin probability for the following cases: (i) X and Y are independent with X of Fréchet type; (ii) X and Y are independent with X of Gumbel type; (iii) X and Y jointly possess a standard MRV structure; (iv) X and Y jointly possess a nonstandard MRV structure.

Ultimate Time Survival Probability in Three-Risk Discrete Time Risk Model

“…In such a case, we can use also the upper estimate of ruin probability, which usually decreases with increasing initial capital. The useful estimates for the nonhomogeneous models we can find in [27][28][29][30][31] among others. For instance, results of [28,29] imply that ψ(u) c 1 exp{−c 2 u}, u 0, for all above examples with a positive constants c 1 , and c 2 depending on the numerical characteristics of the random claims X, Y and Z, generating a three-risk discrete time model.…”