THE CONCEPT OF UTILITY

3.1 THE CONCEPT OF UTILITY

When the expected monetary value (EMV) is used as the sole criterion to reach

a decision under uncertainty, it can lead to results we might not have intended. Outstanding examples to this effect are noted by observing people gambling in a casino or acquiring insurance. For example, in Monte Carlo, Atlantic City or Las Vegas, we might see people gambling (investing!) their wealth on ventures (such as putting $100 on number 8 in roulette), knowing that these ventures have a negative expected return. To explain such an ‘irrational behaviour’, we may argue that not all people value money evenly. Alternatively we may rationalize that the prospect of winning 36 ∗ 100 = $3600 in a second at the whim of the roulette is worth taking the risk. After all, someone will win, so it might as well be me! Both an attitude towards money and the willingness to take risks, originating in a person’s initial wealth, emotional state and the pleasure to be evoked in some way by such risk, are reasons that may justify a departure from the Bayes EMV criterion. If all people were ‘straight’ expected payoff decision-makers, then there would be no national lotteries and no football or basketball betting. Even the mafia might

be much smaller! People do not always use straight expected payoffs to reach decisions, however. The subjective valuation of money and people’s attitudes towards risk and gambling provide the basic elements that characterize gambling and the utility of money associated with such gambling. Utility theory seeks to represent how such subjective valuation of wealth and attitude towards risk can

be quantified so that it may provide a rational foundation for decision-making under uncertainty. Just as in Las Vegas we might derive ‘pleasure from gambling’, we may be also concerned by the loss of our wealth, even if it can happen with an extremely small probability. To protect ourselves from large losses, we often turn to insurance. Do we insure our house against fire? Do we insure our belongings against theft? Should we insure our exports against currency fluctuations or against default pay- ment by foreign buyers? Do we invest in foreign lands without seeking insurance against national takeovers? And so on. In these situations and in order to avoid large losses, we willingly pay money to an insurance firm – the premium needed to buy such insurance. In other words, we transfer our risk to the insurer who in

Risk and Financial Management: Mathematical and Computational Methods. C. Tapiero C 2004 John Wiley & Sons, Ltd ISBN: 0-470-84908-8

40 EXPECTED UTILITY

R R : ( −π )

Figure 3.1

A lottery.

turn makes money by collecting the premium. Of course, how much premium to pay for how much risk insured underscores our ability to sustain a great loss and our attitude towards risk. Thus, just as our gambler was willing to pay a small amount of money to earn a very large one (albeit with a very small probability), we may be willing to pay a small amount (the premium) to prevent and protect ourselves from having to face a large loss, even if it occurs with a very small probability. In both cases, the Bayes expected payoff (EMV) criterion breaks down, for otherwise there would be no casinos and no insurance firms. Yet, they are here and provide an important service to society. Due to the importance of utility theory to economics and finance, providing a normative framework for decision-making under uncertainty and risk management, we shall outline its ba- sic principles. Subsequently, we shall see how the concepts of expected utility have been used importantly in financial analysis and financial decision-making.

3.1.1 Lotteries and utility functions

Lotteries consist of the following: we are asked to pay a price π (say it is $5) for the right to participate in a lottery and earn, potentially, another amount, R, called the reward (which is say $1 000 000), with some probability, p. If we do not win the lottery, the loss is π . If we win, the payoff is R. This lottery is represented graphically in Figure 3.1 where all cash expenditures are noted. Lotteries of this sort appear in many instances. A speculator buys a stock expecting to make a profit (in probability) or losing his investment. Speculators are varied, however, owning various lotteries and possessing varied preferences for these lotteries. It is the exchange between speculators and investors that create a ‘financial market’ which, once understood, can provide an understanding and a valuation of lotteries pricing.

If we use an EMV criterion for valuing the lottery, as seen in the previous chapter, then the value of the lottery would be:

Expected value of lottery = p(R − π) − (1 − p)π = pR − π < 0 By participating in the lottery, we will be losing money in an expected sense.

In other words, if we had ‘an infinite amount of money’ and were to play the lottery forever, then in the long run we would lose $(π − pR)! Such odds for

lotteries are not uncommon, and yet, however irrational they may seem at first, many people play such lotteries. For example, people who value the prospect of ‘winning big’ even with a small probability much more than the prospect of

41 ‘losing small’ even with a large probability, buy lottery tickets. This uneven

THE CONCEPT OF UTILITY

valuation of money means that we may not be able to compare two sums of money easily. People are different in many ways, not least in their preferences for outcomes that are uncertain. An understanding of human motivations and decision making is thus needed to reconcile observed behaviour in a predictable and theoretical framework. This is in essence what expected utility theory is attempting to do. Explicitly, it seeks to define a scale that values money by some function, called the utility function U (.), whose simple expectation provides the scale for comparing alternative financial and uncertain prospects. The larger the expected utility, the ‘better it is’.

More precisely, the function U (.) is a transformation of the value of money that makes lotteries of various sums comparable. Namely, the two sums (R − π) and

(−π), can be transformed into U (R − π) and U (−π), and then the lottery would be,

r r Make U (R − π) with a probability p. Make (lose) U (−π) with a probability 1 − p.

while its expected value, which tells how valuable it is compared to other lotteries, is:

Expected utility = EU = pU (R − π) + (1 − p)U (−π) This means that:

r r If EU = 0, we are indifferent whether we participate in the lottery or not. If EU > 0, we are better off participating in the lottery. r If EU < 0, we are worse off participating in the lottery.

Thus, participation in a lottery is measured by its expected utility. Further, the price $π we will be willing to pay – the premium, for the prospect of winning $R with probability p – is the price that renders the expected utility null, or EU = 0, found by the solution to

EU = 0 = pU (R − π) + (1 − p)U (−π)

which can be solved for π when the utility function is specified. By the same token, expected utility can be used by an investor to compare various lotteries, various cash flows and payments, noting that the value of each has an expected utility, known for certain and used to scale the uncertain prospects. The ‘expected utility’ approach to decision-making under uncertainty is thus extremely useful, providing a rational approach ‘eliminating the uncertainty from decision-making’ and bringing it back to a problem under certainty, which we can solve explicitly and numerically. But there remains the nagging question: how can we obtain such utility functions? And how justified are we in using them? Von Neumann and Morgenstern, two outstanding mathematicians and economists, concluded in the late 1940s, that for expected utility to be justified as a scaling function for uncertain prospects the following holds:

42 EXPECTED UTILITY

(1) The higher the utility the more desirable the outcome. This makes it possible to look for the best decision by seeking the decision that makes the expected utility largest.

(2) If we have three possibilities (such as potential investment alternatives), then if possibility ‘1’ is ‘better’ than ‘2’ and ‘2’ is better than ‘3’, then necessarily ‘1’ is better than ‘3’. This is also called the transitivity axiom.

(3) If we are indifferent between two outcomes or potential acts, then necessarily the expected utilities will be the same.

These three assumptions, underlie the rational framework for decision making under uncertainty that expected utility theory provides.