APPENDIX E MODELS OF THE INSOLVENCY PROCESS NOTATION We use the following notation throughout this appendix. Reserves of company available for payment or losses per policy (at time t). Flow of earned premiums per policy. Expenses incurred per policy. Amount of individual loss for an accident occurring at time t. DERIVATION OF RESULTS FOR ONE-PERIOD MODEL: CHEBYSHEV INEQUALITY Chebyshev's theorem on the relationship between the mass in the tails of a distribution and its variance is given by:34 34 See any elementary text in Statistics; for example, Harold Freeman, Introduction to Statistical Inference (Reading, Mass.: Addison-Wesley, 1963), pp. 30 ff. where k is a constant less than unity. For symmetrical distributions, E-1 can be written a We are interested in the probability that at the end of the year the assets of the insurance company are negative; that is, we want to determine: We thus have the fundamental equation for the probability of insolvency: Here k' is a real number less than one-half which depends on the probability distribution. In general, k will not be a constant, although within certain ranges it may be approximately constant.35 * * We can apply the central limit theorem to test the validity of E-5. Where n> 30assume x has finite mean and variance - then the convolution, y = xj. . . +xn, will approach the normal distribution. We can thus assume that x* is normal mean and variance so that +xy as n n→→ In this case, we can use tables on the normal distribution to test equation E-5. Table E-1 below shows the probability of insolvency as predicted by E-5 and as given by normal tables. The table shows the value of I for the normal distribution at different distances from the mean as predicted by E-5, with k = 0.16, and from tables from deviations in large samples, derived from the normal distribution. From Table E-1 it is seen that the Chebyshev inequality overpredicts the probability of insolvency for normal variables by substantial amounts. It is useful to consider the probability of insolvency for normal variables. TABLE E-1 PROBABILITY OF INSOLVENCY FROM EQUATION (5) AND TABLES FOR NORMAL DISTRIBUTION 35 For the normal distribution, k = 0.16 will be approximately correct for 0R+C-m20. |