INSURANCE, RISK MANAGEMENT AND EXPECTED UTILITY

3.3 INSURANCE, RISK MANAGEMENT AND EXPECTED UTILITY

How much would it be worth paying for car insurance (assuming that there is such a choice)? This simple question highlights an essential insurance problem. If we are fully insured and the premium is $π , then the expected utility is, for sure, U (w − π) where w is our initial wealth. If we-self-insure for a risk whose

probability distribution is p( ˜ X ), then using the expected utility theory paradigm, we should be willing to pay a premium π as long as U (w − π) > EU (w − ˜X).

In fact the largest premium we would be willing to pay solves the equation above, or

π ∗ =w−U − 1 (EU (w − ˜ X ))

Thus, if the utility function is known, we can find out the premium π ∗ above which we would choose to self-insure.

Problem

For an exponential, HARA and logarithmic utility function, what is the maximal premium an individual will be willing to pay for insurance?

3.3.1 Insurance and premium payments

Insurance risk is not reduced but is transferred from an individual to an insurance firm that extracts a payment in return called the premium and profits from it by investing the premium and by risk reducing aggregation. In other words, it is the difference in risk attitudes of the insurer and the insured, as well as the price insured, and insurers are willing to pay for that to create an opportunity for the insurance business.

INSURANCE , RISK MANAGEMENT AND EXPECTED U TILITY

49 Say that ˜ X is a risk to insure (a random variable) whose density function is

F (˜ X ). Insurance firms, typically, seek some rule to calculate the premium they ought to charge policyholders. In other words, they seek a ‘rule’ ϒ such that a premium can be calculated by:

P = ϒ(F( ˜X))

Although there are alternative ways to construct this rule, the more prominent ones are based on the application of the expected utility paradigm and traditionally based on a factor loading the mean risk insured. The expected utility approach seeks a ‘fair’ premium P which increases the firm expected utility, or:

U (W ) ≤ EU (W + P − ˜X)

where W is the insurance firm’s capital. The loading factor approach seeks, how- ever, to determine a loading parameter λ providing the premium to apply to the insured and calculated by P/n = (1 + δ)E( ˜x), where ˜x denotes the individual risk in a pool of n insured, i.e. ˜x = ˜X/n and P/n is an individual premium share. For the insured, whose utility function is u(.) and whose initial wealth is w, the expected utility of insurance ought to be greater than the expected utility of self- insurance. As a result, a premium P is feasible if the expected utilities of both the insurer and the insured are larger with insurance, or:

u (w − P/n) ≥ Eu(w − ˜x i ), ˜ X= ˜x i ; U (W ) ≤ EU (W + P − ˜X)

i =1

Note that in this notation, the individual risk is written as ˜x i which is assumed to be identically and independently distributed for all members of the insurance pool. Of course, since an insurance firm issues many policies, assumed independent, it will profit from risk aggregation. However, if risks are correlated, the variance of

X ˜ will be much greater, prohibiting in some cases the insurance firm’s ability or willingness to insure (as is the case in natural disaster, agricultural and weather related insurance).

Insurance ‘problems’ arise when it is necessary to resolve the existing dispari- ties between the insured and the insurer, which involves preferences and insurance terms that are specific to both the individual and the firm. These lead to extremely rich topics for study, including the important effects of moral hazard, adverse selection resulting from information asymmetry which will be studied subse- quently, risk correlation, rare events with substantive damages, insurance against human-inspired terrorists acts etc.

Risk sharing, risk transfer, reinsurance and other techniques of risk management are often used to spread risk and reduce its economic cost. For example, let ˜x

be the insured risk; the general form of reinsurance schemes associated with an insurer (I), an insured (i) and a reinsurer (r) and consisting in sharing risk can be

50 EXPECTED UTILITY

written as follows:

Insured: R i ˜x ≤ a

0 ˜x ≤ a Insurer: R I ( ˜x |a, b, c, q ) = q ( ˜x − a) a < ˜x ≤ b

c b> ˜x

Reinsurer: R ˜x ≤ b

r ( ˜x |b, c, q ) = q (b − ˜x) − c ˜x ≥ b Here, if a risk materializes and it is smaller than ‘a’, then no payment is made

by the insurance firm while the insured will be self-insured up to this amount. When the risk is between the lower level ‘a’ and the upper one ‘b’, then only a proportion q is paid where 1−q is a co-participation rate assumed by the insured. Finally, when the risk is larger than ‘b’, then only c is paid by the insurer while the remaining part ˜x − c is paid by a reinsurer. In particular for a proportional risk scheme we have R( ˜x) = q ˜x while for an excess-loss reinsurance scheme we have:

˜x ≤ a

I ( ˜x |a ) = ˜x − a ˜x > a

where a is a deductible specified by the insurance contract. A reinsurance scheme is thus economically viable if the increase in utility is larger than the premium

P r to be paid to the reinsurer by the insurance firm. In other words, for utility functions u I (.), u i (.), u r (.) for the individual, the insurance and the reinsurance firms with premium payments: P i , P I , P r , the following conditions must be held:

u i (w − R i ( ˜x |a, q ) − P i ) ≥ Eu i (w − ˜x) (the individual condition) u I (W ) ≤ Eu I (W + P i −P I −R I ( ˜x |a, b, c, q )) (the insurance firm condition) u r (W r ) ≤ Eu r (W r +P I −R r ( ˜x |b, c, q )) (the reinsurer condition)

Other rules for premium calculation have also been suggested in the insurance literature. For example, some say that in insurance ‘you get what you give’. In this sense, the premium payments collected from an insured should equal what

he has claimed plus some small amounts to cover administrative expenses. These issues are in general much more complex because the insurer benefits from risk aggregation over the many policies he insures, a concept that is equivalent to portfolio risk diversification. In other words, if the insurance firm is large enough it might be justified in using a small (risk-free) discount rate in valuing its cash flows, compared to an individual insured, sensitive with the uncertain losses associated with the risk insured. For this reason, the determination of the loading rate is often

a questionable parameter in premium determination. Recent research has greatly improved the determination of insurance premiums by indexing insurance risk to market risk and using derivative markets (such as options) to value insurance contracts (and thereby the cost of insurance or premium).

CRITIQUES OF EXPECTED UTILITY THEORY