PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
4.5 RISK ATTITUDES
EXAMPLE 4.1
(Continued) EUB r2
=0.5×U(0)+0.5×U(0.06)=0+0.5×(6)=3 for asset 2, and E
UB r3
=1.0×U(0.03)=1.0×(3)=3 for asset 3.
The investor with utility equation (4.5) is indifferent among the three assets, because the utilities from all three are the same. Since assets 1, 2, and 3 differ only with respect to their risk, the risk-indifferent investor has an expected utility of three utils from each of the choices.
Investor C’s expected utilities from the three assets are as follows:
E UC
r1
= 2
i=1
piU(ri)=0.5×U(−0.03)+0.5×U(0.09)
=0.5×
100(−0.03)+50(−0.03)2 +0.5
×
100(0.09)+50(0.09)2
=0.5×(−2.055)+0.5×(9.405)=3.225 for asset 1, E
UC r2
=0.5×U(0)+0.5×U(0.06)=0+0.5×(6.18)
=3.09 for asset 2, and E
UC r3
=1.0×U(0.03)=1.0×(3.045)=3.045 for asset 3.
The risk lover with utility equation (4.6) derives the most satisfaction from asset 1, which has the largest variability of return (the most risk).
Summarizing, when the expected returns of all investments are equal, investor A prefers the investment that has the least risk, as computed with equation (4.4). Risk- indifferent investor B is indifferent among the investments as long as their expected returns are the same. And investor C prefers the investment with the largest risk because investor C’s utility function, equation (4.6), represents risk-loving behavior.
This example shows how investment decisions are determined by the shape of investors’ utility functions.
To determine whether marginal utility is rising or falling, the slope of the utility function, as measured by the sign of the second derivative of the utility function, is observed. Decreasing marginal utility exists when the utility function rises at a diminishing rate, or, equivalently, when the second derivative of the utility function is negative, ∂2U(W) /∂W2=U(W) <0. Figure 4.1 illustrates an example of decreasing marginal utility. Diminishing marginal utility of wealth leads to risk-avoiding behavior because, from any point on the utility-of-wealth (or utility- of-returns) curve, a risky investment has a lower expected utility than a risk-free investment with the same expected return. Economics assumes it is always rational to be risk averse.4
Suppose an investor has a utility function with decreasing marginal utility.
Further suppose that this investor is offered a fair game that has a chance of winning
$300 (= ˜z) with probability of 1/2 or losing $300 with probability of 1/2.5 The investor’s initial wealth is $1,000. Would this investor be willing to enter into this fair game? The answer depends on the investor’s attitude toward risk. Figure 4.5 illustrates the amount of utility before and after playing the fair game. If the investor wins this fair game, her wealth increases to $1,300
W+=W0+ ˜z
and the amount of utility is U
W+
. If the investor loses the game, her wealth decreases to $700 W−=W0− ˜z
and the amount of utility is U W−
. Thus, the expected utility after playing the game will be E [U(W)]=12U
W+ +12U
W−
. Whether or not this investor is willing to play this fair game depends on whether the investor’s expected utility is increased or decreased by playing the game. U
W0
indicates the investor’s utility of wealth before playing the game. If E [U(W)]>U
W0
, the investor would be willing to play the game, and if E [U(W)]<U
W0
, the investor would not be willing to play. Figure 4.5 shows the game decreases the investor’s expected utility by U
W0
−E [U(W)]. Therefore, this investor would decline to participate in this fair game. In other words, this investor exhibitsrisk aversion.
A risk averter’s utility function will always be concave (toward the wealth or returns axis), as in Figures 4.1 and 4.5. Risk averters prefer to hold W0 sure cash
$700(W–) U (W–)
U(W+) U(W)
W E[U(W)]
U[E(W)] = U(W0)
$1,000
(W0) $1,300 (W+)
FIGURE 4.5 Risk Averter’s Utility Before and After Playing Fair Game
rather than gamble (assume risk) to increase their wealth aboveW0. If the probability of winning the risky investment is large enough, or if the reward for winning is large enough, a risk averter can be induced to gamble. Stated differently, risk averters may purchase risky investments if they feel the odds are in their favor.
In the previous example, playing the risky game decreases the risk averter’s utility by U
W0
−E [U(W)]. Suppose, the investor’s utility after playing the risky game equals the utility of holding a certainty equivalent (CE) amount of cash, symbolically E [U(W)]=U(CE). Figure 4.6 illustrates the certainty equivalent; it is a certain amount of cash that leaves the investor indifferent between a risky investment and a certain amount of cash, denoted CE. By playing a risky game, this investor loses utility of U
W0
−E [U(W)], or equivalently, loses a certainty equivalent amount of cash equal to E(W)˜ −CE, where E(W)˜ =(1/2)
W0+ ˜z
+(1/2)
W0− ˜z is the expected wealth from the gamble. In order for this investor to avoid this risky game, she would be willing to pay an amount of cash equal to E(W)˜ −C; the amount of this payment is called therisk premium.
Risk premium=Expected wealth−Certainty equivalent or, φ
W0,z˜
=E(W)˜ −CE (4.7)
whereφ W0,z˜
is therisk premium, which is a function of the initial wealthW0and a random payoffz. The risk premium is always positive for risk averters.˜
In summary, the risk-averse investor’s utility function is concave and has the following mathematical characteristics:
U [E(W)]>E[U(W)]⇐⇒U(W) >0, and U(W) <0 (4.8)
W0+z W0−z
U(W)
U(W+)
U(W–) U(W0) E[U(W)]
Risk premium
CE
E(W) W0 W
FIGURE 4.6 Risk Averter’s Utility Function
EXAMPLE 4.2
An investor whose initial wealth is $1,000 is offered an opportunity to play a fair game with two possible outcomes: winning $200 with probability of 1/2 or losing $200 with probability of 1/2. This investor’s utility function equals the natural (or Naperian) logarithm of wealth, U(W)=ln(W). What are the certainty equivalent (CE) and risk premium of this risky game?
Because the logarithmic utility function has the following first- and second- order derivatives with respect to wealth, it satisfies three conditions.
∂U(W)
∂W =U(W)= 1 W >0
∂2U(W)
∂W2 =U(W)= − 1 W2 <0
First, the investor prefers to have more wealth rather than less wealth (i.e., the first derivative is positive). Second, the ln(W)function will always generate risk-averse decisions because the second derivative of U(W)=ln(W) is negative. Third, the investor will require a positive risk premium. The expected utility is
E [U(W)]=0.5×ln(1,200)+0.5×ln(800)=6.8873
Because E [U(W)]=ln(CE), the antilogarithm of E [U(W)] (using base e=2.718. . .) equals the antilog of ln(CE). Therefore, the certainty equivalent and the risk premium of this game are
Certainty equivalent (CE)=e6.8873=$979.80
Risk premium=Expected wealth−Certainty equivalent wealth
=$1,000−$979.80
=$20.20
EXAMPLE 4.3
Assume the facts used in Example 4.2, except that the payoffs will be more widely dispersed. The probability of winning $500 is 1/2 and the probability of losing $500 is 1/2. Because the payoffs are more volatile than the payoffs in Example 4.2, this game is riskier, and as a result, the investor requires a larger risk premium.
The expected utility is E [U(W)]=0.5×ln(1,500)+0.5×ln(500)=6.7639.
Certainty equivalent (CE)=e6.7639=$866.03
Risk premium=$1,000−$866.03=$133.97
EXAMPLE 4.4
Utilize the facts of Example 4.2 again, except that the investor’s initial wealth increases to $2,000. What is the risk premium in this wealthier case?
Depending on the outcome of the game, the investor’s wealth will be either [$2,000+$200=] $2,200 or [$2,000−$200=] $1,800 with equal chance.
The expected utility is
E [U(W)]=0.5×ln(2,200)+0.5×ln(1,800)=7.5959.
Certainty equivalent (CE)=e7.5959=$1,989.97
Risk premium=$2,000−$1,989.97=$10.03
Note that the risk premium decreases as the wealth increases. This indicates that as the investor gets richer, she becomes less afraid of risk. That is, the investor exhibits decreasing risk aversion.
4.5.2 Risk-Loving Behavior
If an investor has the convex utility function illustrated in Figure 4.7, the investor will make riskier decisions about the same fair game. His expected utility from playing, E [U(W)], is greater than without playing the game, U
W0
. In other words, playing the risky game increases the risk lover’s utility by E [U(W)]−U
W0
. This investor expects his utility will be increased by gambling or speculating; this investor is a
U(W)
U(W+)
E[U(game)]
U[E(W)] = U(W0)
E(W) U(W–)
$700(W–)
$1,000 (W0)
$1,300 (W+) CE W Risk premium
FIGURE 4.7 Risk Lover’s Utility Before and After Playing a Fair Game
risk lover. This risk lover’s utility after playing the risky game equals the utility of holding a certain amount of cash, denoted CE in Figure 4.7. The certainty equivalent of this fair game is greater than the expected wealth. Thus, the risk premium is negative:φ
W0,˜z
=E(W)˜ −CE<0.
The utility function for the risk lover can be mathematically summarized by U [E(W)]<E[U(W)]⇐⇒U(W) >0 and U(W) >0 (4.9)
EXAMPLE 4.5
Assume the facts used in Example 4.2 again, except that the investor’s utility function is quadratic; U(W)=W2. What are the certainty equivalent (CE) and risk premium of this risky game?
Because the quadratic utility function has the following first- and second- order derivatives with respect to wealth, it satisfies three conditions.
∂U(W)
∂W =U(W)=2W>0
∂2U(W)
∂W2 =U(W)=2>0
First, the investor prefers to have more wealth than less wealth (i.e., the first derivative is positive). Second, it will generate risk-loving decisions (i.e., the second derivative is positive). Third, the investor will require a negative risk premium. The expected utility is
E [U(W)]=0.5×1,2002+0.5×8002=1,040,000
Because E [U(W)]=U(CE), the square root of E [U(W)] equals the CE.
Therefore, the certainty equivalent and risk premium of this game are Certainty equivalent (CE)=
1,040,000=$1,019.80
Risk premium=Expected wealth−Certainty equivalent
=$1,000−$1,019.80= −$19.80
4.5.3 Risk-Neutral Behavior
If an investor has the linear utility function illustrated in Figure 4.8, the investor would be indifferent about playing a fair game. Her expected utility from playing, E [U(W)], is equal to that from not playing the game, U
W0
. It is said that this investor exhibits risk-neutral behavior. The certainty equivalent of this fair game equals the expected wealth. As illustrated in Figure 4.8, risk-neutral investors require no risk premium:φ
W0,z˜
=E(W)˜ −CE=0.
U(W)
E[U(game)] = U[E(W)] = U(W0)
E(W)
$700(W–)
$1,000 (W0)
$1,300 (W+)
W CE
U(W+)
U(W–)
FIGURE 4.8 Risk Neutral Investor’s Utility Before and After Playing Fair Game
Mathematically, the utility function for risk-neutral investors is
U [E(W)]=E [U(W)]⇐⇒U(W) >0 and U(W)=0 (4.10)
EXAMPLE 4.6
Assume the same facts used in Example 4.2, except that the investor’s utility function is a linear function of wealth: U(W)=W. What are the certainty equivalent (CE) and risk premium of this risky game?
Because the linear utility function has the following first- and second-order derivatives with respect to wealth, it satisfies three conditions.
∂U(W)
∂W =U(W)=1>0
∂2U(W)
∂W2 =U(W)=0
First, the investor prefers to have more wealth than less wealth (i.e., the first derivative is positive). Second, the linear utility function generates risk- neutral decisions (i.e., the second derivative is zero). Third, the investor will require no risk premium. The expected utility is: E [U(W)]=0.5×$1,200+ 0.5×$800=$1,000.
EXAMPLE 4.6
(Continued)Because E [U(W)]=U(CE), the E [U(W)] itself equals the CE. Therefore, the certainty equivalent and risk premium of this game are
Certainty equivalent (CE)=$1,000
Risk premium=Expected wealth−Certainty equivalent
=$1,000−$1,000=0