Utility Theory or the Maximization of Economic Happiness

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When academics discuss utility theory (theory of economic satisfaction), the conversation centers on von Neumann and Morgenstern (1944) utility functions. The Fishburn family of LPM utility functions asserts that the investor is risk averse (or risk seeking, depending on the value of a) below the target return and risk neutral above the target return. This utility function is a combination of being very conservative with the risk potential of a portfolio and very aggressive with the upside potential of a portfolio. Fishburn (1977, 121-122) examined a large number of von Neumann and Morgenstern utility functions that have been reported in the investment literature and finds a wide range of a values ranging from less than 1 to a value greater than 4. The a = 2 target semivariance utility function was not commonly found. Given this result, Fishburn concluded that the generalized a - t (LPM) model is superior to the target semivariance because it is more flexible at matching investor utility.

There is a caveat discussed by Fishburn, which is the decreasing marginal utility of wealth. Very simply, an additional dollar of income

figure 13.1 Fxample of Degrees of the lower Partial Moment

COMPANY A

COMPANY B

RETURN PROB.

RETURN PROB.

-5.00 0.20

10.0

0 0.80

20.00 0.80

35.0

0 0.20

Mean return

15.00

15.00

Variance

100.00

100.00

Skewness

-1.50

1.50

LPM a = 0.0

t = 15

0.20

0.80

LPM a = 0.5

t = 15

0.89

1.79

LPM a = 1.0

t = 15

4.00

4.00

LPM a = 1.5

t = 15

17.89

8.94

LPM a = 2.0

t = 15

80.00

20.00

LPM a = 3.0

t = 15

1,600.00

100.00

Note: When a =

1.0 and t is equal to the mean return, then the two LPM values are

equal. If t is set to some other return, then the LPM values will depend on the degree

of skewness in the return distribution.

to a wealthy person provides less economic happiness (utility) than an additional dollar of income to a poor person. Concerning the LPM (a,t) measure, the risk-aversion coefficient, a, is dependent on the amount of the investor's total wealth. If the amount of wealth at risk is very small relative to the investor's total wealth, then the investor can be very aggressive in terms of investing in risky investments (low values of a). If the amount of money at risk is a substantial portion of the investor's total wealth, then the investor will be more risk averse (higher values of a).

Laughhunn, Payne, and Crum (1980) developed an interactive computer program in BASIC that used Fishburn's (1977) methodology for estimating the value of a for an individual. They studied 224 corporate middle managers by giving them a number of small investment projects from which to choose. They found that 71 percent of the managers exhibit risk-seeking behavior (a < 1). Only 9.4 percent of the managers had a values around 2, and only 29 percent of the managers were risk averse (a > 1).

Next, they studied the impact of ruinous loss. Without the threat of ruinous loss, most managers are risk seeking. However, if the investment projects include ruinous losses, there is a statistically significant shift to risk-averse behavior by the managers. With the chance of a ruinous loss, the majority of the corporate managers were risk averse. Therefore, the estimation of the investor's risk coefficient, a, depends on the relationship between the value of the investment portfolio and the investor's total wealth.

To provide investment advice, the use of an appropriate risk measure is imperative. The factors affecting the choice of the risk measure are:

□ Investors perceive risk in terms of below-target returns.

□ Investor risk aversion increases with the magnitude of the probability of ruinous losses.

□ Investors are not static. As the investor's expectations, total wealth, and investment horizon change, the investor's below-target-return risk aversion changes. Investors should be constantly monitored for changes in their level of risk aversion.

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