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ANTHOLOGY

Sufficient Condition for Buying Full Insurance Against Risk

JONG-SHIN WEI Feng Chia University-Taiww~

It is now well-known that an expected utility-maximizing risk-averse consumer will completely insure himself against the property loss when the expected profit of the insurance policy is zero (which implies that the probability of loss equals the premium rate). Economists prove it by making use of the first-order condition.

This note extends (by a margin of epsilon) the existing result by showing that a sufficient condition for buying full coverage of property loss is any of the following three: (1) the premium is actuarially fair; (2) the premium is superfair and overinsurance is prohibited; or (3) the premium is unfair and underinsurance is prohibited. In what follows, the standard model is first set up and then the result is shown without help from calculus.

Denote W the market value of some property. Let the probability of having a loss, in the amount of L, bep e ]0, 1[. Note that p, W, and L are given positive real numbers. The owner's attitude toward risk is represented by a Von Neumann-Morgenstern utility function U assumed to be both strictly concave and increasing in income. Now consider the insurance option. For the purchase of insurance coverage q > 0, he pays a premium in the amount of 7r .q, where the premium rate 7r e ]0, 1[ is set by the insurance provider. Given ~r, he shall choose q > 0 to maximize his

expected utility denoted by V(q) - ( 1 - p )" U( W - r "q ) + p" U( W - L - 7r" q + q) . Proposition. Under any of the following three conditions, the consumer will completely insure

himself against the loss L. (1) The premium is actuarially fair (i.e., 7r = p due to

( 1 - p ) -Tr -q +p . ( 7r "q - q) = 0 ) . (2) The premium is superfair (or p > 70 and overinsurance is prohibited ( or q < L). (3) The premium is unfair ( orp < ~r) and underinsurance is prohibited (or q > L).

Proof. Let q > 0 be arbitrary. By recalling that U is a concave function of income, one has V(q) < U ( ( 1 - p ) ' ( W - T r ' q ) + p ' ( W - L - T r ' q + q ) ) . Note that the right-hand side of the

inequality above is U( W-Tr "L + (p - 7 r ) - (q-L)) , which is nogreater than U( W- 7r "L), provided

that ( p - T r ) . ( q - L ) < O. Hence, V(L) --- (1 - p ) ' U ( W - T r ' L ) + p ' U ( W - L - ~r'L +L) ;

V ( L ) = U( W - r "L) ; V(L) > v(q) for all q > 0 provided that (p - 7r) "(q - L ) < 0. It has been

shown that under the condition (p - a-) .(q - L) < 0, the optimal insurance plan is to have full coverage.

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