Insurance is a method by which an individual or entity can protect themselves against financial loss.

There are many different types of insurance: home and contents, health, life, auto, credit, and income protection, among others. In Australia, these are regulated in three main categories: general insurance, which includes the major forms of property insurance; life insurance, which includes income protection and permanent disability; and health insurance.

There are other ways of insuring, such as private risk pooling and annuities. We will touch on these later in the book.

There are two equivalent ways to think about why people purchase insurance under the classical economic model:

• People use insurance to smooth consumption across different states of the world, maximising their expected utility of consumption. If they were to suffer a major loss when they are not insured, this could result in a sharp change in consumption. As you would have noticed, smoothing is a common theme across consumer savings, borrowing, investing and insurance behaviour.

• Risk averse consumers are willing to buy insurance with a negative expected value as the consumer prefers the certainty of the premium payment to the risk of suffering an uninsured loss.

The equivalence between the two comes from the diminishing marginal utility of consumption. People smooth consumption over time as extra consumption in one period delivers less utility than spreading consumption evenly. Diminishing marginal utility also leads to risk aversion.

Insurance is typically provided as a financial product by an insurer. The insured person or entity buys an insurance policy from the insurer. The insured pay a premium that entitles them to a promise from insurer to be compensated in the event of a loss that is covered by the insurance policy. The insurer collects premiums from the policy holders to cover the losses of those who experience a loss.

Insurance benefits both the insured and the insurer. As the insured are risk averse, they are willing to pay a premium that exceeds their expected loss (the size of the loss multiplied by its probability). Insurers pool risks by insuring many people or entities. If the loss by each individual is statistically independent of the others, by the law of large numbers the average loss experienced by the insurer will be close or equal to the expected loss. The amount that the insured are willing to pay to avoid the risk thus becomes the insurer’s return on their investment.

Insurance is only feasible in the presence of risk or uncertainty. If the insured knew they definitely would not incur the loss, they would not purchase insurance. If insurer knew the insured would definitely incur the loss, they would not insure them.

## 4.1 A numerical example

The following example illustrates why a risk neutral agent (or expected value maximiser) will not purchase insurance, but a risk averse agent might.

An agent is considering insurance against bushfire for its $1,000,000 house. The house has a 1 in 1000 chance of burning down. An insurer is willing to offer full coverage for$1100. (Note: $1000 is the actuarially fair price, the additional$100 might represent profit or administrative costs.)

Would an expected value maximiser or risk neutral person purchase the insurance?

$E[\text{purchase}]=-\text{premium}=-\1,100$

The expected value of purchasing insurance is the guaranteed loss of the premium.

\begin{align*} E[\text{don't}]&=P_{\text{burn}}*-value_{\text{house}} \\ &=-0.001*1000000 \\ &=-\1000 \end{align*}

The expected value of purchasing insurance is $100 less than the expected value of risking the house burning down. A risk neutral agent (who maximises expected value) would not purchase this insurance. Would a risk averse agent purchase the insurance? Suppose they have a logarithmic utility function ($$U(x)=ln(x)$$) and they have$10,000 in cash in addition to their house, giving them wealth ($$W$$) of 1,010,000. \begin{align*} E[U(\text{purchase})]&=ln(W-premium) \\ &=ln(1,008,900) \\ &=13.8244 \end{align*} \begin{align*} E[U(\text{don't})]&=0.999*ln(W)+0.001*ln(W-value_{\text{house}}) \\ &=0.999*ln(1,010,000)+0.001*ln(10,000) \\ &=13.8208 \end{align*} The expected utility of purchasing insurance is greater than the expected utility from not purchasing insurance. This agent will insure against the fire despite it being actuarially unfair. ## 4.2 Adverse selection A problem emerges when the insured and insurer have different information. Suppose there is a population comprising two types of person, high risk and low risk. These two types are found in equal proportions across the population. The high risk people have a 30% chance of experiencing a loss each year, while the low risk have a 10% probability of a loss. In either case, if they experience a loss event, the loss will be100. Since there are equal numbers of each type, the expected loss of a random person in the population is $20. What if an insurer offered to insure anyone who wants insurance for$20? If no-one knew which type was which, this insurance would be attractive to both low and high-risk types and the insurer’s expected losses would equal the premiums it collects.

But what if the people in the population know which type they are, but the insurer doesn’t? Unless they are extremely risk averse, a $20 insurance premium is unattractive to the low risk types, who have an expected loss of only$10. They don’t buy insurance. Only the high-risk types get insured, getting a great bargain of a $20 premium to insure against their expected loss of$30. The insurer would then suffer a loss, unless it boosted premiums to \$30.

This phenomena where only the high-risk types buy coverage, called adverse selection, was highlighted in a classic paper by Rothschild and Stiglitz (1976). The problem can be pervasive. How does an insurer set life premiums for smokers and non-smokers if it can’t differentiate the two? Or good and bad drivers?

## 4.3 Moral hazard

Whereas adverse selection involves an information asymmetry about type, moral hazard emerges when the asymmetry involves information about the insured’s intention to take on risk. (Sometimes the distinction is described as hidden information in the first case, and hidden action in the second).

Moral hazard is the idea that when someone is insured, they may take on greater risks because they know that they will not pay the costs. The insurer will. If their behaviour is not observable or contractable, there are constraints as to what the insurer can do about this.

Moral hazard might be seen in risky driving, not wearing a seatbelt, taking less care on a black diamond ski run, or failing to prepare properly for the bushfire season.