## UTILITY THEORY

When discussing the expected value and the standard deviation we noted that decision makers can either be risk seekers, risk averse or risk neutral. Therefore, we cannot be able to tell with certainty whether a decision maker will choose a project with a high expected return and a high standard deviation, or a project with comparatively low expected return and low standard deviation.

Utility theory aims at incorporating the decision maker's preference explicitly into the decision procedure. We assume that a rational decision maker maximises his utility and therefore would accept the investment project which yields maximum utility to him.

Note that utiles is a relative measure of utility. For the risk averse decision maker, the utility for wealth curve is upward-sloping and is convex to the origin. This curve indicates that an investor always prefer a higher return to a lower return, and that each successive identical increment of wealth is worth less to him than the preceding one - in other words, the marginal utility for money is positive but declining.

For a risk seeker, the marginal utility is positive and increasing. For a risk neutral decision maker, the marginal utility is positive but constant. To derive the utility function of an individual, we let him consider a group of lotteries within boundary limits.

Illustration:Derivation of utility functions

Assume that utiles of 0 and 1 are assigned to a pair of wealth representing two extremes (say, Sh 0 and Sh 100,000 respectively). To determine the utility function of a decision maker, we offer him a lottery with 0.5 chance of receiving no money and 0.5 chance of receiving Sh 100,000. Assume he is willing to pay Sh 33,000 for this lottery. (Therefore 0.5 utile = Sh 33,000).

Next, consider a lottery providing a 0.4 chance of receiving Sh 33,000 and a 0.6 chance of receiving Sh 100,000. Assume that the decision maker is willing to buy this lottery at Sh 63,000. The utile value of Sh 63,000 is

U(Sh 63,000) = 0.4 U(Sh 33,000) + 0.6 U(Sh 100,000)

= 0.4 x 0.5 + 0.6 x 1

= 0.8

Assume also a lottery providing a 0.3 chance of receiving Sh 0 and a 0.7 chance of receiving Sh 33,000 is also offered. The decision maker is willing to pay Sh 21,000 for this lottery. The utile value for Sh 21,000 can be computed as follows.

U (Sh 21,000) = (0.3 U(Sh 0) + 0.7 U(Sh 33,000)

= 0.3 x (0) + 0.7(0.5)

= 0.35

Note that other lotteries can be provided to the decision maker until we have enough points to construct his utility function.