A note on insurance coverage in incomplete markets.

• Author: Hau, Arthur

1. Introduction

   In his seminal paper, Arrow (1963) shows that if a risk-averse individual is offered insurance against a random loss at an actuarially fair premium rate, she will choose to buy 100% coverage.(1) Recent research, however, indicates that this result does not hold in general when insurance markets become incomplete. Here, market incompleteness means that individuals face some risks that are uninsurable due to the absence of the corresponding insurance markets, possibly as a result of moral hazard or adverse selection.

   By using a simple model with only four states of nature, Doherty and Schlesinger (1983a) showed that full insurance coverage may not be optimal when an individual's wealth consists of not only an insurable random loss (say, medical expenditure) but also an uninsurable idiosyncratic (background) risk (say, income risk). Under their simple model, they show that traditional theory holds only when the insurable and uninsurable risks are statistically independent. Moreover, the correlation between the two risks seems to have a monotonic relation with insurance coverage.(2)

   More recently, Doherty and Schlesinger (1990) studied the effect of the presence of an uninsurable risk of the insurer's default on insurance coverage. They found, surprisingly, that even in their simple model with only three states of nature, in which the default risk and the insurable risk are independent, optimal coinsurance rate can be equal to, greater than, or less than 1 when insurance is fair. Unfortunately, the authors do not provide an intuitive explanation for the apparent difference between the case of an idiosyncratic risk and the case of the risk of insurer's default. So far, very little progress has been made in the theory of insurance demand for the case of default risk.(3)

   Careful inspection of the incomplete insurance market literature reveals the lack of an appropriate definition for "full insurance coverage." In general, when an uninsurable risk exists, a coinsurance rate of 1 on the insurable risk does not eliminate all risk from the individual. Therefore, it is not always meaningful to use that rate as the definition for full insurance and to compare it with an individual's optimal coinsurance rate.(4) As a consequence, before comparing the case of uninsurable background risk and the case of default risk, we need to have a meaningful and workable definition for full-insurance coverage. This paper introduces the concept of "full insurance on average," which is defined as the coinsurance rate at which all risk is eliminated when the uninsurable risk is evaluated at its mean.

   The main purpose of this paper is to check whether full insurance on average is optimal under the case of idiosyncratic risk and the case of default risk. A novelty of this paper is that no specific distributions for the insurable and uninsurable risks are assumed. The only restriction imposed is the "regressibility assumption" employed in the literature on indirect hedging of exchange rate risk (see, e.g., Broll, Wahl, and Zilcha 1995; Broll and Wahl 1996). It turns out that this assumption simplifies the analysis substantially. Moreover, I will show that this assumption is a generalization of the bivariate normal and the perfect correlation assumptions used by Doherty and Schlesinger (1983b) and Schlesinger and Doherty (1985). I will also show that the same results can be reached with the regressibility assumption replaced by the "small-risk assumption," together with a second-order Taylor series approximation.(5)

   This paper is organized as follows. In section 2, a model of insurance demand under uninsurable background risk is analyzed. It is found that under the regressibility assumption, when the background risk and the insurable risk are uncorrelated, full (over-/under-)insurance on average is optimal if and only if insurance is fair (favorable/unfavorable). A negative (positive) correlation between the two risks, however, implies that underinsurance (overinsurance) on average is optimal whenever insurance is actuarially fair or unfavorable (fair or favorable). The former result agrees with that of Eeckhoudt and Kimball (1992), whereas the latter result complements theirs. In section 3, a model of insurer's default is analyzed under the regressibility assumption. In this case, full insurance on average requires the coinsurance rate to be larger than 1. It is found that when the insurable and the default risks are independent, underinsurance on average is optimal if insurance is fair or unfavorable. When the two risks are not independent, the relations between optimal coinsurance rate, loading charges, and the correlation coefficient of the insurable and the default risks are derived. It turns out that knowing the sign of the correlation is insufficient for determining whether full (under-/over-)insurance on average is optimal even when insurance is fair. These results provide an intuitive explanation for the ambiguous results obtained by Doherty and Schlesinger (1990). Section 4 concludes.

2. Additive Idiosyncratic Risk and Insurance Coverage

   The model analyzed in this section is a variation of that of Eeckhoudt and Kimball (1992). An individual with initial wealth w is facing an insurable loss [Mathematical Expression Omitted], which is a random variable with positive realizations, mean [Mathematical Expression Omitted], and standard deviation [[Sigma].sub.2] She can choose a coinsurance rate a to insure against the potential loss by paying a premium. Let k be the constant loading factor of the insurance so that the premium payment equals [Mathematical Expression Omitted]. The insurance is actuarially fair (favorable/unfavorable) if and only if [Lambda] = ([less than]/[greater than])0. In addition to the insurable risk, the individual is also facing an additive uninsurable background risk Y, which is a random variable with mean [Mathematical Expression Omitted] and standard deviation [[Sigma].sub.y]. The background risk reduces (increases) the individual's wealth when its realization is positive (negative).(6) Assume that the joint distribution of [Mathematical Expression Omitted] and [Mathematical Expression Omitted] is known to the insured while the distribution of [Mathematical Expression Omitted] is known to the insurance company. The individual's random wealth level is given by

   [Mathematical Expression Omitted]. (1)

   Her problem is to choose ct to maximize her von Neumann-Morgenstern expected utility

   [Mathematical Expression Omitted], (2)

   where U is assumed to follow standard assumptions with U[prime] [greater than] 0 and U[double prime] [less than] 0. The first-order condition for an optimum is given by

   [Mathematical Expression Omitted], (3)

   where [[Alpha].sup.*] is the optimal coinsurance rate in the presence of the background risk. The second-order condition for a unique maximum is given by

   [Mathematical Expression Omitted]. (4)

   The strict concavity of U guarantees that Equation 4 is satisfied.

   Notice that when [Mathematical Expression Omitted] is absent, I am back to the case of complete insurance market. Denote the optimal coinsurance rate in the absence of 7 by [[Alpha].sub.0]. It is easy to verify from Equation 3 that [[Alpha].sub.0] = ([greater than]/[less than])1 if, and only if k = ([less than]/[greater than])0. In the absence of y, an individual has full insurance coverage if all risk is...