As you will recall, under the classical economic model, households purchase insurance to maximise the utility of their consumption. This maximisation largely occurs by smoothing consumption through external shocks.

Information asymmetries such as adverse selection and moral hazard can lead to consumers not being able to insure some risks at an actuarially fair price. These often provide explanations for the failure of a consumer to smooth their consumption. However, explanations for many behaviours requires other tools.

## 16.1 Prospect theory

The classical economic explanation for the purchase of insurance is based on the risk aversion of consumers. 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 negative expected value is due to the insurer’s profit and administrative costs.)

Prospect theory provides an alternative explanation. The purchase of insurance involves a certain loss (the premium) or a gamble involving the possibility of either a large loss or the status quo. As prospect theory has people as risk seeking in the loss domain, we would not expect them to purchase insurance.

However, under prospect theory people also overweight small probabilities. This overweighting of small probabilities can make the purchase of insurance attractive even though it is in the loss domain. This combination of the loss domain but small probabilities is the bottom-right quadrant of the fourfold pattern to risk attitudes generated by prospect theory.

The following numerical example is an illustration.

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.)

Consider an agent who is risk seeking in the domain of losses but weights probability linearly. Their value function is:

$v(x)=\left\{\begin{matrix} x^{0.8} &\textrm{where} &x \geq 0\\ -2(-x)^{0.8} &\textrm{where} &x < 0 \end{matrix}\right.$

Where $$x$$ is the realised outcome relative to the reference point.

Determination of the reference point can be arbitrary. What if you pay insurance every year? Could the reference point then be wealth minus the insurance payment (meaning the insurance payment is in the gain domain)?

Taking the reference point as current wealth, would this agent purchase the insurance?

\begin{align*} V(purchase)&=v(-1,100) \\ &=-(1,100)^{0.8} \\ &=-271.1 \\ \\ V(don't)&=0.999*(0)+0.001*v(-1,000,000) \\ &=0.999*0-0.001*(1,000,000)^{0.8} \\ &=-63.1 \end{align*}

As $$V(purchase)<V(don't)$$, the agent does not purchase insurance. The diminishing feeling of loss leads to them weigh the certain loss of the premium relatively more heavily than the chance of losing the value of their house.

Including loss aversion in the value function does not change the decision as all possible outcomes are in the loss domain.

Would a person who is risk seeking in the domain of losses (i.e. the value function with reflection effect above) and applies the decision weights described below purchase the insurance?

They apply decision weights as per the following table:

 Probability 0.001 0.01 0.1 0.25 0.5 0.75 0.9 0.99 0.999 Weight 0.01 0.05 0.15 0.3 0.5 0.7 0.85 0.95 0.99

\begin{align*} V(purchase)&=v(-1,100) \\ &=-(1,100)^{0.8} \\ &=-271 \\ \\ V(don't)&=\sum_{i=1}^n \pi(p_i)v(x_i) \\ &=\pi(0.999)*v(0)+\pi(0.001)*v(-1,000,000) \\ &=0.99*0-0.01*(1,000,000)^{0.8} \\ &=-631 \end{align*}

Although the diminishing feeling of loss leads to them weigh the certain loss of the premium relatively more heavily than the chance of losing the value of their house, the overweighting of the probability of fire leads them to purchase insurance. Again, if we had included loss aversion it would not have changed the decision as all possible outcomes are in the loss domain.

## 16.2 Life insurance and annuities

At the household level, the standard economic model predicts a household will purchase insurance to protect against the death of household members, particularly those that are the highest earning. This is not, however, the pattern that is observed. Households often insure spouses when they would suffer no decline in living standard were their spouse to die. They also often fail to insure when they would suffer a substantial decline (Bernheim et al. (2003)).

A similar puzzle exists around life annuities. A life annuity is a product that a consumer purchases through payment of a lump sum in return for a stream of income that lasts until they or the beneficiaries in the household dies. Life annuities protect against the risk of living too long and running out of assets. The fact that only 1% of US households over the age of 65 hold life annuities is often called the “annuity puzzle”.

### 16.2.1 Rational explanations

There are some rational explanations for this puzzle. Life annuities are often priced poorly and offer low yields relative to alternative investments. Public pensions already provide protection against longevity risk. There are also arguments that many people have bequest motives, which life annuities cannot satisfy as they only have value while the annuity holder is alive. Finally, annuities are a poor option if there are other uninsurable risks in the future, such as medical costs, which will require access to lump sums rather than an income stream.

All of these are likely factors, although the evidence that people are responsive to prices is weak.

### 16.2.2 Psychological explanations

The link between financial literacy and insurance through annuities is complex and debated. The decision to annuitise is complex, although the decisions required through alternative options such as maintaining assets and determining drawdown requirements each period are possibly more difficult. The result is that in different contexts low financial literacy has been linked to both lower and higher rates of annuitision.

The choice of annuities is sensitive to the frame. When consumers were told about the potential returns from purchasing an annuity (an investment frame), they were far less likely to annuitise than if they were told about the potential future consumption from the annuity (Brown et al. (2016)).

Loss aversion can also make annuities unattractive, as the possibility of an early death might be seen as a potential loss. The future income stream is “lost” in the event of death.

## 16.3 Under-insurance

Households often fail to insure against catastrophic risks to their property, and when they do, they often under-insure against the full extent of the catastrophe. For example, Quantum Market Research (2014) found that 81% of homeowners and renters do not have insurance that enables them to resume the same standard of living in the event of a crisis.

While some of this failure to fully insure is rational, due to the small maximum possible loss, the main explanation for this under-insurance is simply that households underestimate the probability of a large loss. They also do little to understand the extent of the risk.

## 16.4 Low excess

Once households do insure they often over-insure against small losses. They do this by choosing low levels of deductibles, also called “excess”. Excess is the amount the policy holder must contribute in the event of a claim. Excess is designed to reduced moral hazard through sharing risk, and administration costs by reducing the number of claims.

The increased premium required to be paid for a low excess means that those who choose it must be very risk averse; in fact, an implausible level of risk aversion under standard economic models.

Loss aversion is one possible alternative explanation, as the potential for loss, even if small, is strongly felt. The difficulty with this explanation, however, is that the premium itself should also be felt as a loss. Prospect theory also provides another challenge to explaining this phenomena in that people tend to be risk seeking in the domain of losses, making the certain loss of the insurance premium unattractive when they have a chance of going uninsured but not suffering the negative event.

Another element of prospect theory, however, can increase the attractiveness of insurance. This is probability weighting, which can lead to small probability events being given greater weight. This exaggeration of the probability could be sufficient to overcome the risk seeking behaviour in the loss domain.

If you remember the four-fold pattern of risk attitudes generated by Prospect Theory, insurance is a combination of low probability and potentially large loss. In that schema, a person will be on net risk averse and seek to insure.

## 16.5 Junk insurance

People regularly buy insurance with limited value.

The prototypical junk insurance in the Australian market is consumer credit insurance. Consumer credit insurance is sold to consumers to cover them in the event that they cannot meet the minimum payments of a loan due to unemployment, injury or illness, or to pay the balance in the event that they die.

Australian Securities and Investments Commission (2019) found that for consumer credit insurance, only around 19 cents in the dollar was paid out. For insurance associated with credit cards, that payout rate was only 11 cents in the dollar.

Most major consumer credit providers have ceased selling many, if not all, of the forms of consumer credit insurance since ASIC’s report. But this still leaves open the question of why consumers were purchasing this insurance in the first place.

A major issue was understanding the products. Many people were ineligible to ever claim as they were not meeting work requirements such as a working a minimum number of days or having permanency, or having a pre-existing condition excluded by the policy. They simply did not know (and were not told) this.

Another factor is the attention of the customers. They are primarily engaged in obtaining a credit card or loan at the time of purchasing the insurance. The add-on insurance is an immaterial part of the overall purchase, so receives little attention or scrutiny. There is also little opportunity for the consumer to shop around or compare prices.

## 16.6 Evidence of adverse selection and moral hazard

Adverse selection emerges where there is an information asymmetry between the insurer and potential customer about what type of customer is seeking insurance. Only high-risk customers buy coverage, whereas low-risk customers find the pricing unattractive.

The evidence for adverse selection actually occurring is ambiguous (Kunreuther et al. (2013)). In support of the concept, some studies have found that drivers who choose a lower excess tend to be higher risk drivers.

Other evidence provides little support. For example, people with lower life expectancy are not more likely to purchase life insurance than those likely to live longer.

There is even some evidence of an opposing trend, whereby low-risk customers are more likely to seek coverage (Fang et al. (2008)). “Advantageous risk selection” occurs where risk averse people attach a high value to insurance due to their risk aversion, but are also lower risks due to this risk aversion. There is also evidence that higher risks are less capable of making insurance decisions involving comparison of costs and benefits than those who are lower risk, affecting their insurance purchase decisions.

Evidence for moral hazard is more robust, although not always consistent (Kunreuther et al. (2013)). It is also difficult to disentangle moral hazard from adverse selection. Moral hazard has been found in health, medical and automobile insurance markets.