Optimal Insurance Coverage

Abstract

The literature of expected utility theory has treated extensively the problem of optimal portfolio investment, but there is limited treatment of the parallel problem of the optimal protection of assets against casualty or liability loss (Arrow, 1963, 1965). The problem of optimal coverage is formally similar to the problem of optimal inventory stockage under uncertainty. To inventory a product is to insure against sales loss-the larger the inventory, given the distribution of demand, the greater the insurance coverage. If casualty or liability loss (demand) is less than the coverage (inventory level), excessive cost (inventory holding cost) is incurred. If casualty or liability loss (demand) is greater than the coverage (inventory level), one must absorb the cost of the unrecoverable loss (sales loss). These two components of loss must be balanced in determining optimal (inventory) levels. In the analysis to follow, we will use V to denote the given value of an individual's property or assets which are insurable against loss. In deciding how much to buy, an individual must choose A ? 0, the fraction (or multiple) of V which is to be protected against loss. That is, he chooses an amount of or coverage level, A V. We assume, throughout, that the individual can buy as much as he pleases at a fixed price, m > 0 in dollars per dollar of protection, for a given time interval of exposure to risk of loss. His premium for that time interval is then P = mAV if he buys AV dollars of insurance. The analysis will be divided into two sections, the first dealing with against casualty losses of physical property due to fire, wind, storm, vandalism, and so on, in which it is assumed that the loss cannot exceed the value of the property, V. The second section will deal with against liability claims on an individual's tangible and intangible assets, in which it is assumed that the liability claim can exceed the value of assets, but cannot exceed the value of assets plus coverage (1 + A) V.