Ruin probability in the presence of risky investments

TL;DR

This paper derives asymptotic bounds and exact asymptotics for the ruin probability of an insurance company investing in a risky asset, depending on the investment parameters and premium rate dynamics.

Contribution

It provides the first precise asymptotic characterization of ruin probabilities under stochastic premium rates and risky investments, including exact bounds and asymptotics.

Findings

Ruin probability decays as a power law with exponent β when β>0.

Exact asymptotics are obtained for exponential premium growth rates.

Ruin probability equals one for all initial capitals when β≤0.

Abstract

We consider an insurance company in the case when the premium rate is a bounded non-negative random function $c_\zs{t}$ and the capital of the insurance company is invested in a risky asset whose price follows a geometric Brownian motion with mean return $a$ and volatility $\sigma>0$. If $\beta:=2a/\sigma^2-1>0$ we find exact the asymptotic upper and lower bounds for the ruin probability $\Psi(u)$ as the initial endowment $u$ tends to infinity, i.e. we show that $C_*u^{-\beta}\le\Psi(u)\le C^*u^{-\beta}$ for sufficiently large $u$. Moreover if $c_\zs{t}=c^*e^{\gamma t}$ with $\gamma\le 0$ we find the exact asymptotics of the ruin probability, namely $\Psi(u)\sim u^{-\beta}$. If $\beta\le 0$, we show that $\Psi(u)=1$ for any $u\ge 0$.