2003: Applications of Simulation Models in Finance and Insurance

We describe a number of applications of simulation methods to practical problems in finance and insurance. The first entails the simulation of a two-stage model of a propertycasualty insurance operation. The second application simulates the operation of an insurance regime for home equity conversion mortgages (also known as reverse mortgages). The third is an application of simulation in the context of Value at Risk, a widely-used measure for assessing the performance of portfolios of assets and/or liabilities. We conclude with an application of simulation in the testing of the efficient market hypothesis of the U.S. stock market. 1 FREQUENCY-SEVERITY INSURANCE MODEL 1.1 Formulating the Problem In property and casualty insurance, as well as in health insurance, the actuary is often asked to predict the amount of insured losses during the next period of observation, such as a calendar year. In doing so, the actuary frequently has the results observed for a number of prior periods. Then if is a random variable representing the amount of aggregate claims during the i i S th policy year (or, equivalently, the i period of observation), the problem may be considered to be the estimation of the quantity ] S , , S , S | s Pr[S m 2 1 1 m 1 m ... + + ≤ where . This is the conditional probability of the incurred losses during period 0 s 1 m ≥ + 1 m + , given the results of the first m periods. Such a probability distribution is usually called a predictive distribution. One way of approaching this problem is to first determine the distribution of the frequency of loss (i.e., the number of insurance claims) and then to determine the severity or amount of each individual claim. We assume that given a parameter θ , the random variables S are independent and identically distributed with conditional probability density function p. We use f to denote the density function of 1 m S , , 2 S , 1 + ... θ . Thus, using Bayes’ Theorem, we can write the conditional density function of S , given S as 1 m+ 1 m 2 S , , S , ...