PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
4.9 PORTFOLIO ANALYSIS
4.9.1 Quadratic Utility Functions
A quadratic utility function implies certain things about its owner’s behavior.
Consider the following quadratic utility function of the terminal wealth U
WT
=βWT−γW2T (4.27)
where β and γ are positive constants.17 By substituting W0 1+rp
for WT in equation (4.27), the utility can be expressed as a quadratic function inrp:
U rp
=a+brp−crp2 (4.28)
wherea=βW0−γW02,b=βW0−2γW02, andc=γW02. Thus, utility functions that are quadratic inWT are also quadratic inrp.
When the utility function is quadratic, the expected utility is exactly a function of the mean and variance of the wealth. Taking the expectation of equation (4.27) yields
E U
WT
=E
βWT−γWT2
=βE WT
−γE WT2
=βE WT
−γ E
WT2
+σW2
(4.29)
because E WT2
= E
WT2
+σW2.18 Therefore, portfolio analysis based on the quadratic utility function is identical to the standard mean-variance analysis.19
The marginal utility of returns is the change in utility resulting from a tiny change in return. For the quadratic utility function in equation (4.30), the marginal utility of returns is
∂U(r)
∂r =b−2cr>0 (4.30)
The marginal utility from additional returns is positive forr<b/2c. Atr=b/2c, marginal utility is zero and the utility curve in Figure 4.9 peaks. For returns above b/2c, the investor receives negative marginal utility—that is, positive returns greater than r=b/2c are distasteful. This violates the first assumption in Section 4.8.1
U(r)
b/2c r
FIGURE 4.9 Quadratic Utility Function of Returns
(nonsatiation). To avoid the unrealistic portion of the utility curve where additional gains yield negative marginal utility (that is, the dashed portion of Figure 4.9), the analysis must be restricted to returns below a limit. This constraint is r<b/2c. In other words, we must assume that a quadratic utility of returns function has an upper bound.
4.9.2 Using Quadratic Approximations to Delineate Max[E(Utility)] Portfolios
Most investors are not gratified to hear their investment advisor tell them they should maximize their expected utility. This is because utility theory is abstract.
Most investors prefer to discuss dollars or returns or something that is, as a minimum, sufficiently tangible to be measured. Most investors are not interested in utility theory.
Unlike most investors, financial economists like utility theory. They find risk- averse, risk-neutral, and risk-loving behavior interesting. They analyze linear, power, logarithmic, quadratic, and other utility functions and compare and contrast the risk-aversion characteristics of each function, as shown in this chapter’s Table 4.3.
Utility functions that deliver positive but diminishing marginal utility of wealth are held in esteem because they rationalize complicated human behaviors like wealth maximization, risk aversion, and diversification. An unfortunate disadvantage of utility analysis is that delineating Max[E(U{r})] portfolios provides a slow and cumbersome route to selecting efficient portfolios. Markowitz (1959, 282–289) and Markowitz (1987, 59–68) show that a quadratic function of a portfolio’s expected return and standard deviation can provide a fast and easy way to delineate efficient portfolios.
Markowitz (1959) suggests the quadratic utility function U
rp
=a+brp−crp2 (4.28)
E U
rp
=a+bE rp
−c E
rp2
+σr2
(4.28a)
=a+bE rp
−c E
rp2
−cσr2 (4.28b)
sinceE r2p
= E
rp2
+σr2. Markowitz proposes this utility function because (1) as shown in equation (4.28b), it is an exact function of the investment’s expected return and standard deviation; (2) Markowitz (1959, 282–289), Young and Trent (1969), Levy and Markowitz (1979), Kroll, Levy, and Markowitz (1984), Markowitz (1987, 59–68), and others have shown that a custom-fit quadratic function of a portfolio’s expected return and standard deviation can provide a very good shortcut for finding utility-maximizing portfolios for several different types of utility functions;20 and (3) a quadratic function of an investment’s expected return and standard deviation can be combined with mean-standard deviation portfolio analysis to select optimal investment opportunities in risk and return space. This mean-variance approximation technique quickly and efficiently selects portfolios that are almost identical to those selected by the slower and more cumbersome process of computing Max[E(U{r})]
portfolios.
As Levy and Markowitz (1979) first noted, if investors with unidentified utility functions choose off the mean-variance efficient frontier, they can approximately
maximize their expected utility without ever knowing what their utility function is; in fact, without anyone even mentioning the ‘‘expected utility maxim’’ to them.
Finally, in spite of the fact it is possible to find utility-maximizing portfolios without a utility function, this in no way diminishes the insights obtained by analyzing traditional utility functions.
4.9.3 Normally Distributed Returns
Using the expected utility hypothesis causes the investors to maximize their expected utility, even in the presence of uncertainty. If the number of outcomes for a given portfoliopis discrete, its expected utility is
E U
Wp
=k i=1U
WTi pi
WTi
(4.31) where k is the number of possible outcomes, WTi is the level of terminal wealth corresponding to outcomeifor portfoliop, andpi
WTi
is the probability it occurs.
In the more general case where the outcomes are continuously distributed, expected utility was shown to be calculated as
E Up
= +∞
−∞
U WT
p WT
dWT (4.32)
wherep WT
is the probability density function for terminal wealth associated with a given portfoliop. Thus, for any utility function, E
U Wp
depends onp WT
. If the probability density function is normally distributed, it will be of the form
p WT
= 1
√2πσWT
exp
−
WT−E WT2
2σW2T
(4.33) In comparing any set of portfolios that have normally distributed returns, meaning that terminal wealth is normally distributed as in equation (4.33), their corresponding levels of E
U Wp
will differ only to the extent thatp WT
differs between alternative portfolios. In this situation, examination of equation (4.33) indicates thatp
WT
will differ between alternative portfolios if and only if either E
WT
or σWT differs between the portfolios. Accordingly, if terminal wealth is normally distributed (or, equivalently, if portfolio returns are normally distributed), then portfolios can be evaluated by using the magnitudes ofE
WT
andσWT. Thus, mean-variance analysis is exact not only when utility functions are quadratic, but also when returns are normally distributed.