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If somebody is risk averse then they are willing to pay a risk premium in order to avoid the risk – usually in order for someone else to bear the risk in the form of insurance.

Consider the case where a person has a utility function of $U=10W^{0.3}$ where U signifies utility and W signifies wealth in £. The more wealth they have, the more utility they have so the happier they are.

Say they start with wealth of £2000. They own a machine that has a 10% chance of breaking down in a particular time period. If it breaks down it will cost £800 to repair it.

There are two possible outcomes here, a good outcome where the machine doesn’t break down and the person has wealth of £2000, or a bad outcome where the machine breaks down and they have £1200. We can use the utility function to calculate the amount of utility in the respective states:
Good outcome: U = 97.793

The machine has a 10% chance of breaking down, so that means a 90% chance of us having the good outcome and 10% chance of the bad outcome. We can use this to calculate the expected utility of this whole situation:

EU = (0.9 x 97.793) + (0.1 x 83.899) = 96.404

Now if we went back into the original utility function to find what wealth corresponds to a utility level of 96.404, we get $96.404=10W^{0.3} \Rightarrow W = 1906.838$.

In other words the expected utility of this situation is equivalent to having wealth of £1906.84. The difference between this and the starting wealth of £2000, £93.16, is the maximum risk premium the machine owner is willing to pay to remove the risk. If an insurer offers to sell insurance that will fully cover the costs of machine repair if it breaks down, then the machine owner is willing to pay up to £93.16 in order to buy that insurance.

Say the insurer offers to provide full insurance for a cost of £90. Then the machine owner has a choice – either he takes the insurance, which means he has guaranteed wealth of £1910, and a corresponding guaranteed utility of 96.452, or he doesn’t take insurance, which will mean he either has £2000 (and utility of 97.793) if it doesn’t break down, or £1200 (and utility of 83.899) if it does. In terms of expected utility, the expected utility from taking insurance is 96.452, and the expected utility from not taking the insurance is 96.404. As the expected utility from taking the insurance is higher than the expected utility from not taking insurance, then it is rational to take the insurance. If the insurance had cost more than £93.16, then the expected utility from taking the insurance would have been lower than the expected utility from not taking the insurance, so it would have been rational not to take out the insurance.

Risk aversion and utility

We can think of risk aversion in the context of utility. The most intuitive way is to consider wealth. Most of us would probably have a utility function that increases with diminishing returns as wealth increases. That is, the more wealth we have, the more utility we get (the happier we are), but the increases in our utility are greatest when we have less wealth. If you have wealth of nothing, and somebody offers you £10000, you are probably going to get a great deal of additional happiness from that £10000. However if you are already a multi-millionaire, then an extra £10000 might increase your happiness but it won’t have the same effect as it would if you had nothing.

A risk averse person will have a utility function relative to wealth, that looks similar to this:

A risk neutral person will have a utility function that looks like this:

A risk loving person will have a utility function that looks like this:

The shapes of the utility functions reflect the extent to which utility increases with wealth. The one of most interest to us is the risk averse one, because people are more likely to be risk-averse than anything else. This brings us to the concept of risk premium and insurance.

Uncertainty and risk

Often the choices we make lead to uncertain outcomes. When we know or can estimate the likelihood of each possible outcome, we can classify the risk of the outcome occurring. When we have know way of knowing or estimating the likelihood, we are just left with uncertainty.

When we know all the possible outcomes that could occur, then the probabilities of these outcomes will sum to 1.

We use the estimated value and variance to illustrate the expected payoff and amount of risk involved with a project, if we are able to assess value to the different outcomes concerned. This is easiest when we can assess it in financial terms.

Where there are two outcomes, A and B, the estimated value is $EV = Probability(A) Value (A) + Probability(B) Value(B)$. This shows us the expected payoff.

The variance is $Var = Probability(A) [Value (A) - EV]^2 + Probability(B) [Value (B) - EV]^2$. This shows us the degree of risk, the higher the variance, the more the risk.

For instance lets say you are offered a bet, and the choice is either to take the bet or not take the bet. The bet costs £100 to make, and involves tossing a coin. If the coin comes up heads, you lose your £100. If it comes up tails, you get £250.

The estimated value of taking the bet is $0.5(-100) + 0.5(250) = 75$
The variance of taking the bet is $0.5(-100-75)^2 + 0.5(250-75)^2 = 30625$

The estimated value of not taking the bet is $1(0) = 0$
The variance of taking the bet is $1(0-0)^2 = 0$

So here we have a higher estimated value for taking the bet than not taking the bet. That means your expected payoff is better for taking the bet. But there is a large variance involved with taking the best, that means you are taking a lot of risk in taking the bet.

Whether or not you take the bet depends on your aversion to risk.

Someone that is risk-neutral does not care either way about risk, and will accept the bet if it has a higher expected value than not accepting the bet, so here a risk-neutral person would accept the best.

Someone that is risk-averse does not like risk, and will not accept a ‘fair bet’ (a bet with an expected value of 0) or a bet with a higher expected value, if there is risk involved. So here a risk-averse person would not accept the bet. Most people are by nature risk averse to some extent.

At the other end of the spectrum some people are risk-loving and will actively seek risk, so they would accept a fair bet or even some unfair bets, if there was risk involved.