The Ruin Problem in Multiple-Line Insurance. Comment

Abstract

Ruin Problem in Multiple-Line Insurance-A Simplified Model,' by Hofflander and Duvall, which appeared in the Journal of Financial and Quantitative Analysis, (June, 1967) is one of the more significant efforts in applying statistical techniques devise rules for testing the ratio of premiums written policyholders' surplus. The authors state that their purpose is to make use of a generally accepted statistical tool devise and explain a new rule for relating premiums written (i.e., sales) policyholders' surplus, taking the loss reserve and stock portfolio into account.2 They concluded at the end of their paper that under the worst possible conditions a property and liability insurance company could write up $9.10 of premiums for every dollar of policyholders' surplus with only the slightest probability of becoming insolvent. This was a rather astounding conclusion, given the emphasis on the traditional 2:1 rule which has been expounded in industry circles for years. The conclusion may have been so astounding as turn practical insurance men and women off from considering their model. (Just some more ivory tower professors playing with complicated mathematical models which have no applicability the real world.) Unfortunately, Hofflander and Duvall may have unknowingly blunted the importance of their efforts by a rather elementary arithmetic error. Rather than a 9.1:1 premiums policyholders' surplus ratio, it should have been .91:1. Apparently they misplaced the decimal by one place. A ratio of .91:1 is obviously much more realistic and certainly a more palatable result for practitioners in the industry. A brief look at their work will show where they went astray. In constructing their model, Hofflander and Duvall make the convenient assumption that the combined loss and expense ratios of the various lines of insurance are normally distributed. They propose techniques for estimating the mean and variance of the combined loss and expense ratio for a given property and liability insurance company. By making use of these estimates, they illustrate how establish standards of minimum policy-