The section on risk-aversion referred to insurance as a classic illustration of the difference between risk-aversion and risk-neutrality. We saw how risk-averse individuals will always choose to insure valuable assets, since although the probability of a loss may be small, the potential loss of the asset itself would be so large that most people would rather pay small amounts of money as a premium for certain than risk the loss.

On the other hand, insurance companies are risk-neutral, and earn their profits from the fact that the value of the premiums they receive is either greater than or equal to the expected value of the loss.

In this section, we will see how people decide whether or not to insure, and explain what economists mean when they speak of actuarially fair insurance. For the discussion on the insurance purchase decision, a basic knowledge of differential calculus is advised.

Our discussion will assume that apart from his own wealth, an individual making the decision to insure or not also knows for certain the probability of a loss or accident.

Say you, as a risk-averse consumer have initial wealth w, and a von Neumann-Morgenstern utility function u(.). You own a car of value L, and the probability of an accident which would total the car is p (we might imagine p as the current accident rate in the state where you live). If x is the amount of insurance you can purchase, how much should x be?

The answer to this question depends, very simply, on the price of insurance - the premium you'd have to pay. Let's say this price is r, for $1 worth of insurance, so for $x of insurance, you'd be paying $rx as a premium.

For insurance to be actuarially fair, the insurance company should have zero expected profits. We can set up their problem as under:

With probability p, the insurance company must pay $x, while receiving $rx in premiums. With probability (1-p), they pay nothing, and continue to receive $rx in premiums. So their expected profit is:

p(rx - x) + (1-p)rx

If this equals zero, we have: px(r-1) + (1-p)rx = 0

Dividing throughout by x, we get: pr - p + r - pr = 0

i.e. p = r.

So for insurance to be actuarially fair, the premium rate must equal the probability of an accident.

In actual practice, even if the premium does not equal the probability of an accident, it certainly depends on it - which is why different demographic groups pay widely differing automobile insurance premiums. Since single men under the age of 25 have the highest accident risk, they also pay the highest premiums.

As a risk-averse consumer, you would want to choose a value of x so as to maximize expected utility, i.e.

Given actuarially fair insurance, where p = r, you would solve: max pu(w - px - L + x) + (1-p)u(w - px), since in case of an accident, you total wealth would be w, less the loss suffered due to the accident, less the premium paid, and adding the amount received from the insurance company.

Differentiating with respect to x, and setting the result equal to zero, we get the first-order necessary condition as: (1-p)pu'(w - px - L + x) - p(1-p)u'(w - px) = 0,

which gives us: u'(w - px - L + x) = u'(w - px)

Risk-aversion implies u" < 0, so that equality of the marginal utilities of wealth implies equality of the wealth levels, i.e.

w - px - L + x = w - px,

so we must have x = L.

So, given actuarially fair insurance, you would choose to fully insure your car. Since you're risk-averse, you'd aim to equalize your wealth across all circumstances - whether or not you have an accident.

However, if p and r are not equal, we will have x < L; you would under-insure. How much you'd underinsure would depend on the how much greater r was than p.

So far, we've assumed that consumers take their loss probability as given. But that's not always the case - often, the amount of care you take, or don't take, can influence the chance of a loss. This may be something as simple as driving more carefully if you don't have insurance, or not bothering to lock your car door if you do.

The example below assumes a risk-neutral consumer, and a hypothetical cost of loss prevention.

Let c, the cost required to achieve loss probability p be:
c = b(1-p)², where b > (1/2)L

"p" is the choice variable in this case, and a risk-neutral consumer's expected utility function can be written as:

u(p) = p[w - L - b(1-p)²] + (1-p)[w - b(1-p)²]
So u(p) = w - pL - b(1-p)²

We take the first derivative of u with respect to p and set it equal to zero, to find the value of p that maximizes u(p):
u'(p*) = -L + 2b(1-p*) = 0, where p* refers to the optimal value of p.
So p* = 1 - (L/2b)

By full insurance, we refer to full coverage, i.e., the insurance company pays the entire amount of the loss L in case of an accident. In regular parlance, this means a zero deductible. Our risk-neutral consumer can purchase this insurance at a cost of rL, which equals rx (in this case, with full coverage, since x = L), where x is the quantity of insurance purchased.

So now, if full insurance has already been purchased, what loss probability will our consumer choose?

We write the consumer's expected utility function as under:

u(p) = p[w - L + L - rL - b(1-p)²] + (1-p)[w - rL - b(1-p)²]
So u(p) = w - rL - b(1-p)²

As we saw earlier, the value of p that maximizes expected utility, i.e. p⁰ in this case, can be obtained by taking the first derivative of u with respect to p and setting it equal to zero:

u'(p⁰) = 2b(1-p⁰) = 0
So p⁰ = 1, meaning a risk-neutral consumer will not take any care at all to prevent a loss.

In this situation, what should an insurance company do in order to at least break even, i.e. earn zero expected profits?

Well, we know that the optimal premium, r⁰, must satisfy the equation:
r⁰L - p⁰L = 0
So r⁰ = 1, meaning the cost of the policy should equal the amount of the loss!

No, of course not, as one can imagine intuitively even without working through the math. Of course, it's more convincing if we do work through the math.

If full insurance coverage costs L, and the loss probability equals 1, the consumer's expected utility is:

u = p[w - L + L - rL - b(1-p)²] + (1-p)[w - rL - b(1-p)²]
So u = w - L.

But if a consumer does not buy insurance at all, and exercises the implied optimal amount of care, the expected utility is:

u(p*) = w - p*L - b(1-p)²
So u(p*) = w - L[1 - (L/2b)] - b[L/2b]²,
i.e. u(p*) = w - L + (1/4)*(L²/b), which is greater than w - L. The consumer has no incentive to buy insurance.

So why do we observe people buying insurance on a daily basis? Well, first of all, most people are risk-averse. And secondly, most insurance companies do not offer full insurance coverage. Apart from what is detailed above, moral hazard is another very important reason for this - refusing to offer full coverage forces the buyer of insurance to exercise care to avoid a loss.