Residual and Past Entropy in Actuarial Science and Survival Models

Abstract

The best policy for an insurance company is that which lasts for a long period of time and is less uncertain with reference to its claims. In information theory, entropy is a measure of the uncertainty associated with a random variable. It is a descriptive quantity as it belongs to the class of measures of variability, such as the variance and the standard deviation. The purpose of this paper is to investigate the effect of inflation, truncation or censoring from below (use of a deductible) and truncation or censoring from above (use of a policy limit) on the entropy of losses of insurance policies. Losses are differentiated between per-payment and per-loss (franchise deductible). In this context we study the properties of the resulting entropies such as the residual loss entropy and the past loss entropy which are the result of use of a deductible and a policy limit, respectively. Interesting relationships between these entropies are presented. The combined effect of a deductible and a policy limit is also studied. We also investigate residual and past entropies for survival models. Finally, an application is presented involving the well-known Danish data set on fire losses.