In the Life Insurance Business Risky Investments are Dangerous

TL;DR

This paper analyzes the risk of ruin in life insurance when reserves are invested in risky assets, providing asymptotic formulas for ruin probabilities and insights into optimal investment strategies.

Contribution

It offers an exact asymptotic analysis of ruin probabilities in life insurance with risky investments, extending previous models to include exponential and general claim distributions.

Findings

Ruin probabilities decrease at a power rate for low volatility

Ruin probabilities approach certainty for high volatility

Quantifies investment share to prevent catastrophic ruin

Abstract

We investigate models of the life annuity insurance when the company invests its reserve into a risky asset with price following 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 unit 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: 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.