What is a Utility Function and How is it Used?
A utility function represents the 'worth,' 'happiness' or 'satisfaction associated with goods, services, events, outcomes, levels of wealth, etc. It can be used to rank outcomes, to aggregate 'happiness across individuals and to value games of chance.
You own a valuable work of art; you are going to put it up for auction. You don't know how much you will make but the auctioneer has estimated the chances of achieving certain amounts. Someone then offers you a guaranteed amount provided you withdraw the painting from the auction. Should you take the offer or take your chances? Utility theory can help you make that decision.
The idea is not often used in practice in finance but is common in the literature, especially economics literature. The utility function allows the ranking of the otherwise incomparable, and is used to explain people s actions; rational people are supposed to act so as to increase their utility.
When a meaningful numerical value is used to represent utility this is called cardinal utility. One can then talk about one thing having three times the utility of another, and one can compare utility from person to person. If the ordering of utility is all that matters (so that one is only concerned with ranking of preferences, not the numerical value) then this is called ordinal utility.
If we denote a utility function by U(W)where W is the 'wealth,' then one would expect utility functions to have certain commonsense properties. In the following a prime () denotes differentiation with respect to W.
• The function U(WW) can vary among investors, each will have a different attitude to risk for example.
• U(W) > 0: more is preferred to less. If it is a strict inequality then satiation is not possible, the investor will always prefer more than he has. This slope measures the marginal improvement in utility with changes in wealth.
• Usually U"(W) < 0: the utility function is strictly concave. Since this is the rate of change of the marginal 'happiness,' it gets harder and harder to increase happiness as wealth increases. An investor with a concave utility function is said to be risk averse. This property is often referred to as the law of diminishing returns.
The final point in the above leads to definitions for measurement of risk aversion. The absolute risk aversion function is defined as
The relative risk aversion function is defined as
Utility functions are often used to analyse random events. Suppose a monetary amount is associated with the number of spots on a rolled dice. You could calculate the expected winnings as the average of all of the six amounts. But what if the amounts were $1, $2, $3, $4, $5 and $6,000,000? Would the average, $1,000,002.5, be meaningful? Would you be willing to pay $1,000,000 to enter this as a bet? After all, you expect to make a profit. A more sensible way of valuing this game might be to look at the utility of each of the six outcomes, and then average the utility. This leads on to the idea of certainty equivalent wealth.
When the wealth is random, and all outcomes can be assigned a probability, one can ask what amount of certain wealth has the same utility as the expected utility of the unknown outcomes. Simply solve
The quantity of wealth Wc that solves this equation is called the certainty equivalent wealth. One is therefore indifferent between the average of the utilities of the random outcomes and the guaranteed amount Wc. As an example, consider the above dice-rolling game, supposing our utility function is U(W) = —^e~nW. With r = 1 we find that the certainty equivalent is $2.34. So we would pay this amount or less to play
Figure 2.6: Certainty equivalent as a function of the risk-aversion parameter for example in the text.
the game. Figure 2.6 shows a plot of the certainty equivalent for this example as a function of the risk-aversion parameter n. Observe how this decreases the greater the risk aversion.
References and Further Reading
Ingersoll, JE Jr 1987 Theory of Financial Decision Making. Rowman& Littlefield