Insurance Risk with Variable Number of Policies

In this paper we consider an insurance company selling life insurance policies. New policies are sold at random points in time, and each policy stays active for a random amount of time, during which the policyholder pays premiums continuously at rate r. When the policy expires, the insurance company pays a claim of random size. The aim is to compute the probability of eventual ruin starting with a given number of policies and a given level of insurance reserves. We establish a remarkable result that, if the lifetimes of policies are i.i.d. exponential random variables with rate μ, then the ruin probability is identical to the one in the standard compound Poisson model where the reserves increase at constant rate r and claims occur according to a Poisson process with rate μ. We conclude that the ruin probability does not depend on the initial number of active policies, nor on the arrival process of new policies.