PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
4.10 INDIFFERENCE CURVES
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.
σp,E rp
space such that the investor’s satisfaction (that is, expected utility) is equal throughout its length. Indifference curves are utility isoquants. The slope of an indifference curve depends on the investor’s particular preference for a lower but safer return versus a larger but riskier return.
Figures 4.10A and 4.10B illustrate positively-sloped indifference curves in σp,E
rp
space for inverstors A and B. Indiffernce curves for timid, risk-averse investors are positively sloped, because they require higher expected returns as an inducement to assume larger risks. Let us contrast the behavior of the two risk-averse investors illustrated in Figures 4.10A and 4.10B. Investors A’s indifference curve has lower slope (is flatter) than investor B’s. When an investor increases the level of risks fromσ1toσ2, byσ, that investor requires more expected return to maintain the same level of utility. He or she would ask for an additional amount of the expected return, denotedE(r)[fromE
r1 toE
r2
]. Note that portfoliosP1and P2in Figure 4.10A produce the same amount of utility. For the same amount of risk change (fromσ1toσ2, byσ), investor B would ask for a larger additional amount of expected return,E(r), fromE
r1 toE
r2
, than investor A to maintain the same level of utility. This difference indicates that investor B is more risk averse than investor A and requires a larger risk premium. In other words, indifference curves with steeper slopes are for the more risk-averse investors. The slope of the indifference curves is discussed further in Section 4.2 of this chapter.
Figure 4.11 illustrates a negatively sloped indifference curve that represents the preferences of a risk seeker. When thisrisk loverincreases the level of risk fromσ1
toσ2, byσ, the investor would ask for less expected return to maintain the same level of utility. In other words, when the level of risk is increased, the risk seeker would be willing to give up a certain amount of expected return to maintain the same level of satisfaction. This type of investor enjoys taking risk and would pay money to acquire more risk (that is, would ask for a negative risk premium).
The flat linear indifference curves in Figure 4.12 represent arisk-neutralinvestor.
Even though the level of risk is increased, a risk-neutral investor would not ask for additional expected return to maintain the same level of utility. In other words, any change in the level of risk does not affect the utility of a risk-neutral investor. Only the level of expected return determines the utility. No risk premium is required by risk neutral-investors. As long as the expected returns are the same, the utility is
E (rp)
E (r2)
A
P2 P1 ΔE(r)
Δσ
σP σ1 σ2
E (r1)
FIGURE 4.10A Quadratic Utility Function of Returns
E (r2)′
E (rp)
E (r1)
B
Q2
Q1 ΔE (r)′
Δσ
σP σ1 σ2
FIGURE 4.10B Indifference Curve for a Moderately Risk-Averse Investor
P1 P2
σ2 σp
Δσ ΔE (r)
σ1 E (r2)
E (r1) E (rp)
FIGURE 4.11 Indifference Curve for a Risk Seeker
E (rp)
E (r3) E (r2)
E (r1) P2
U1 U2 U3
P1
σ1 σ2 σp
Δσ
FIGURE 4.12 Indifference Lines for a Risk Neutral Investor
the same. As the expected return increases fromE r1
toE r3
, the utility increases from U1 to U3, regardless of the level of risk in Figure 4.12.
4.10.1 Selecting Investments
The previous paragraphs showed how an investor’s utility function can be used to select a portfolio in terms of risk and expected return statistics; Figure 4.13 illustrates how to do this. The figure is a graph inrisk-return spaceillustrating the seven hypothetical portfolios listed in Table 4.4.
Figure 4.13 shows an indifference map in risk-return space representing the preferences of a risk-averse investor whose utility function is a quadratic function of the form U
rp
=brp−cr2p. Because portfolios G, D, and F are all located on indifference curve U2, the investor obtains equal expected utility from all three, although their expected returns and risks differ considerably.
An infinite number of indifference curves could be drawn for the risk averter depicted in Figure 4.13, but they would all be similar in shape and would all possess the following characteristics:
1. Higher indifference curves represent higher levels of expected utility and thus more investor satisfaction. Symbolically, U5 >U4>U3>U2>U1. This family
TABLE 4.4 Expected Returns and Standard Deviations of Seven Hypothetical Portfolios
Portfolio Expected return (%) Risk (%)
E(rp) σp
A 7 3
B 7 4
C 15 15
D 3 3
E 7 12
F 9 13
G 2 0
U5 > U4 > U3 > U2 > U1 U5U4
U3 U2U1 C
F
E
D B A 9%
15%
E (rp)
7%
3%
3% 4% 12% 13% 15%
G = 2%
σp
FIGURE 4.13 Opportunities and Preferences in Risk-return Space
of indifference curves indicates the investor likes higher expected return and dislikes higher risk.
2. All indifference curves slope upward. This is because the investor requires higher expected returns as an inducement to assume larger risks. Consider, for example, the investor’s indifference between D, F, and G. This is due to the fact that F’s expected return is just enough above the expected return of D to compensate the risk-averse investor for assuming the additional risk incurred in going from D to F. Riskless portfolio G has just enough reduction in risk below the risk of D to compensate the investor for accepting G’s lower rate of return. G is called thecertainty equivalentof portfolios D and F because it involves no risk, yet is equally desirable. That is, the investor finds the riskless 2 percent return (portfolio G) to be equally desirable to 3 percent return with 3 percent risk (portfolio D) or 9 percent return with 13 percent risk (portfolio F).
3. The indifference curves grow steeper at higher levels of risk. This reflects the investor’s diminishing willingness to assume higher and higher levels of risk.
Given the investment opportunities and the investor preferences shown in Figure 4.13, we see that the investor prefers A over any of the other portfolios because A lies on the highest indifference curve. More generally, if we are given several pairs of risk-return values
σp,E rp
, and each pair represents a different portfolio, an investor could plot these risk-return pairs on a graph that also contains that investor’s indifference curves. The portfolio on the highest indifference curve would be optimal (most preferred) because it would provide the investor with the highest level of expected utility.
4.10.2 Risk-Aversion Measures
Figures 4.14, 4.15, and 4.16 illustrate different families of indifference maps for three investors with different risk preferences. Figure 4.14 shows that the slope
Slope decreasing;
A′(W) < 0
E (rp) U(W3)
U(W2)
U(W1) W1 < W2 < W3
σp
FIGURE 4.14 Indifference Curves Exhibiting Decreasing Absolute Risk Aversion
Slope decreasing;
A′(W) > 0
E (rp) U(W3)
U(W2)
U(W1) W1 < W2 < W3
σp
FIGURE 4.15 Indifference Curves Exhibiting Increasing Absolute Risk Aversion
Slope no change;
A′(W) = 0
E (rp) U(W3)
U(W2) U(W1)
W1 < W2 < W3
σp
FIGURE 4.16 Indifference Curves Exhibiting Constant Absolute Risk Aversion
of the indifference curves becomes lower (flatter) as the level of wealth increases from W1, W2 to W3. Because an indifference curve with lower slope indicates less risk aversion, this figure represents decreasing absolute risk aversion. In contrast, Figure 4.15 shows that as the level of wealth increases, the slope of the indifference curves steepens, indicating more risk aversion and increasing absolute risk aversion.
Figure 4.16 shows that although the level of wealth changes, the slopes of the indifference curves do not change. The indifference curves in Figure 4.16 remain parallel with respect to wealth change, representingconstant absolute risk aversion.21