SIEMBRA

In the previous section, the investor’s decision to choose share A was based on the maximisation of expected return. Decision-making, however, involves not only quantitative rules but also qualitative intangibles such as personal attitudes to risk. There are situations where numerical measures, such as an investment’s return or expected monetary value, are not sufficient and could lead to the wrong course of action being taken.

Utility theory, as presented in Von Neumann and Morgenstern’s theory of games, incorpo- rates risk-preference attitudes into the decision-making process by introducing the concept of ‘utility functions’. Every decision-maker is assumed to have a personal utility function which is used to convert quantitative values into non-monetary measures called utilities. A ‘utility’ expands the narrow monetary concept to include individual (or company) preferences to risk and return. The decision-maker then evaluates each alternative on the basis of its utility and identifies the best outcome by maximising expected utility rather than expected value. Because each decision-maker has a unique utility function, there can be a wide variation in the shape of utility curves.

Utility functions for the three most common risk-preference attitudes are shown in Figure 4.2. The risk-averse curve illustrates the concept of ‘diminishing marginal utility’ which states that the more a person has of a particular resource, the less satisfying becomes the next increment. For example, a salary increase of £1000 means less to a person earning £60,000 than to someone earning £15,000. The risk-averse graph shows that utility increases as monetary value increases. However, as monetary value continues to grow, the curve flattens out reflecting the decision-maker’s desire to avoid the higher risks associated with larger monetary values.

The risk-seeking curve illustrates the characteristic of increasing marginal utility whereby utility increases faster than monetary value. This curve reflects the speculative nature of the risk-seeker who is prepared to let utility grow at a faster rate in order to gain some smaller monetary value. The risk-neutral function is a straight line with a constant marginal utility, indicating that the decision-maker is indifferent to risk. Because the risk-indifferent person is concerned with expected monetary value (EMV) rather than risk, utility and monetary values increase proportionately.

Some people can perceive the same situation differently. It is therefore possible to have all three risk-preference attitudes being applied to the same problem. Consider, for example, the reactions of individuals to the situation seen regularly on TV quiz shows. Each participant

Risk-averse (concave curve)

Utility

Risk-neutral (equivalent to EMV)

Risk-seeking (convex curve)

Monetary value

Figure 4.2 Utility functions for risk-preference attitudes.

has to choose between accepting some monetary value being offered by the quiz-master or opening a box which may contain either a star prize or a booby prize. Some contestants will gamble on winning the star prize regardless of the risk involved. On the other hand, if the quiz- master continues to increase the amount on offer, many participants will eventually change their mind and accept the money.

EXAMPLE 4.1 Constructing a utility curve

Joe Bloggs is a contestant on the ‘Open the Box’ quiz show. The star prize is a motor-car worth £16,000 while the booby prize is worth £5. The quiz-master has made an offer of £2000. Joe must now decide what to do. The first step in constructing a utility function is to determine two monetary values representing the worst and best outcomes to the problem. In Joe Bloggs’s situation, the worst and best outcomes are described by the monetary values of £5 (booby prize) and £16,000 (car). A common practice is to assign utility values of 0 to represent the worst outcome and 1 to the best outcome. The associated utility U values are therefore given by the following two equations

U (5)= 0 and U(16,000) = 1

Joe Bloggs has two alternatives, namely (i) accept the quiz-master’s offer of £2000 with certainty, or (ii) gamble on winning the star prize, in which case he also has a 50% chance of receiving the booby prize. The expected utilities for each alternative are given by the following two equations:

First alternative, A1 U(A1)= U(2,000)

Second alternative, A2 U(A2)= 0.5*U(5) + 0.5*U(16,000) = 0.5*0 + 0.5*1 = 0.5

As matters stand, Joe prefers the second alternative which means that U ( A1)< U(A2), i.e., the

utility of £2000 is less than 0.5. However, he waits to see if the quiz-master will increase the offer. When a new offer of £4000 is made, Joe immediately accepts. This decision implies that U (4000)> U(A2), i.e., the utility of £4000 is greater than 0.5. Because Joe has changed his

mind, it can also be deduced that there is some value between £2000 and £4000, at which Joe is ‘indifferent’ to the two alternatives. Indifference implies that both alternatives are equally

acceptable to Joe, i.e., each has a probability of 0.5. In this situation, Joe has adopted a risk- neutral attitude. If it is established that Joe’s ‘indifference value’ is £3000, another utility value can be calculated as follows:

U (3000)= 0.5∗U (5)+ 0.5∗U (16,000) = 0.5∗0+ 0.5∗1= 0.5

Thus a new utility value, U (3000), has been derived from the two known utility values, U (5) and U (16,000). A further point on the utility curve can be found by incorporating U (3000) into a different monetary option. For example, if the booby prize is replaced by £3000 in cash, Joe Bloggs faces a new dilemma. In this situation, he prefers to gamble on winning the star prize unless the quiz-master increases the current offer of £4000. If Joe decides that his indifference value is now £7000, then:

U (7000)= 0.5∗U (3000)+ 0.5∗U (16, 000) = 0.5∗0.5 + 0.5∗1= 0.75

Other utility values may be found in a similar manner. The four points (5, 0), (3000, 0.5), (7000, 0.75) and (16000, 1) are used to construct Joe’s utility curve as shown in Figure 4.3.

Joe Bloggs’s indifference value(s) represents the minimum amount of money that he is prepared to accept for avoiding a risky alternative, i.e., opening the box. This minimum amount of money is called the ‘certainty equivalent’. Because Joe is indifferent between opening or not opening the box, the certainty equivalent can also be interpreted as the maximum amount that a person is willing to forfeit in order to participate in a risky alternative. The term ‘risk premium’ is closely associated with the certainty equivalent. Risk premium is defined as the difference between the risky alternative’s expected monetary value (EMV) and its certainty equivalent.

It should be noted that utility analysis is not easy to implement in practice because of its subjective nature. Finding the exact point of an individual’s indifference can be a complex process. This problem is accentuated when trying to establish a company’s utility function. Managers with different attitudes to risk usually find it very difficult to agree on a common utility function. Furthermore, a person’s perspective can alter over time making any previous utility assessment obsolete.

0 0.25 0.5 0.75 1 1.25 0 4 8 12 16 Monetary value (£'000) Utility

The Exponential Utility Function (EUF)

The concave curve of a risk-averse investor shown in Figure 4.2 can be approximated by an exponential utility function (EUF) as given by the equation

U(x)= 1 − e−x/R where e= 2.71828 the base of natural logarithms

R= the decision-maker’s risk tolerance (R > 0)

The risk-tolerance parameter R controls the shape of the utility function. As R increases the curve flattens out, i.e., the decision-maker becomes less risk-averse. In order to use the exponential utility function, a suitable R-value must be found that best fits the decision-maker’s risk tolerance. One approach is to find a value for X that will persuade the decision-maker to participate in the following gamble: toss a coin to win £X or lose £X/2. The value of X that convinces the decision-maker to accept this gamble can be taken as a reasonable estimate for R, e.g., a person who accepts a gamble to win £200 or lose £100, has an R-value of 200.

EXAMPLE 4.2 Using the EUF to maximise expected utility

AstroReturns stockbrokers has been asked by a long-standing client to invest £30,000. The company has provided details of three stocks X, Y and Z. Stock X is a government bond guaranteeing a fixed return of 8%. The returns from stocks Y and Z depend upon the prevailing market conditions. AstroReturns estimate the probability of a good, average, or poor market as 0.3, 0.6, and 0.1 respectively. The company has calculated relevant stock returns (%) for each market condition as shown in Table 4.1. Since the client wants all her money to be invested in the same stock, AstroReturns must now decide which of the three stocks will maximise her expected utility.

Because the client is well known to AstroReturns, the company can use the exponential utility function. An R-value of 4000 has been assessed as the most reasonable figure for their client’s risk tolerance. The expected utility model of Figure 4.4 has found that stock Z produces a maximum expected utility of 0.5 (see cell H27). If the client’s R-value is reduced to 1500 (cell F11), i.e., she is now more risk-averse, stock X then has the largest utility.