In the insurance business risky investments are dangerous: the case of negative risk sums

Abstract

We investigate models with negative risk sums when the company invests its reserve into a risky asset whose price follows a geometric Brownian motion. Our main result is an exact asymptotic of the ruin probabilities for the case of exponentially distributed benefits. As in the case of non-life insurance with exponential claims, the ruin probabilities are either decreasing with a rate given by a power function (the case of small volatility) or equal to one identically (the case of large volatility). The result allows us to quantify the share of reserve to invest into such a risky asset to avoid a catastrophic outcome, namely the ruin with probability one. We address also the question of smoothness of the ruin probabilities as a function of the initial reserve for generally distributed jumps.

Appendix: Ergodic theorem for an autoregression with random coefficients

Let $(a_{n},b_{n})_{n\ge1}$ be an i.i.d. sequence of random variables in ${\mathbb {R}}^{2}$ and $x_{0}$ an arbitrary constant. Define the sequence $(x_{n})$ recursively by the formula

$$ x_{n}=a_{n}\,x_{n-1}+b_{n}, \quad n\ge1. $$

Proposition A.1

Assume that there exists $\delta\in(0,1]$ such that

$$ {\mathbf {E}}|a_{1}|^{\delta}< 1, \qquad {\mathbf {E}}|b_{1}|^{\delta}< \infty. $$

Then for any bounded uniformly continuous function $f$,

$$ {\mathbf {P}}\hbox{-}\lim_{N}\frac{1}{N}\sum^{N}_{n=1}f(x_{n})= {\mathbf {E}}f(\zeta), $$

where

$$ \zeta=\sum^{\infty}_{k=1}\,b_{k}\, \prod^{k-1}_{j=1}\, a_{j} \quad\textit{with}\quad \prod^{0}_{j=1}a_{j}=1. $$

The proof is given in [28].