PRINCIPLES BOX: DEFINITION OF AN EFFICIENT INVESTMENT
4.8 MEASURING RISK AVERSION
The tools just introduced allow the reader to make studied judgments about logic and rationality of alternative utility functions.
4.8.1 Assumptions
The previous sections suggest the following four assumptions are consistent with rational investment behavior.
Assumption 1: Investors prefer more to less (nonsatiation);∂U/∂W=U(W) >0.
Assumption 2: Investors are risk averse;∂2U/∂W2=U(W) <0.
Assumption 3: Investors exhibit decreasing absolute risk aversion; A(W) <0.
Assumption 4: Investors exhibit constant relative risk aversion; R(W)=0.
Table 4.3 summarizes risk aversion assessments for various utility functions of interest. Subject to the restrictions noted in the table, all these functions meet the first two assumptions about investor behavior; nonsatiation (U(W) >0) and risk aversion (U(W) <0). However, the quadratic utility function displays increasing absolute risk aversion, which violates the third assumption. The exponential utility function also violates this assumption, because it has constant absolute risk aversion.8 Both the logarithmic and power utility functions, on the other hand, are consistent with the assumption of decreasing absolute risk aversion and also have constant relative risk aversion.9These latter two functions are of particular interest.10 4.8.2 Power, Logarithmic, and Quadratic Utility
Comparisons of the quadratic, logarithmic, and power utility functions, reveal several advantages from using the latter two functions. First, the power function exhibits positive marginal utility over all ranges. In contrast, the quadratic func- tion’s marginal utility becomes negative for large wealth levels, an unrealistic and perplexing problem that must be dealt with by appending bounds to the function.11 Another advantage of both the logarithmic and power functions relative to the quadratic is that the investor’s wealth and returns are separable with the logarithmic and power functions, but not with the quadratic.12Separability of wealth and returns means that indifference curves, to be discussed in Section 4.10 in this chapter, and
TABLE4.3AssessingRiskAversionInVariousUtilityFunctions UtilityfunctionRestrictionsU(W)U(W)A(W)R(W) FunctionA(W)FunctionR( 1.Quadratic: U=bW−cW2b>0;c>0; W≤b 2c
b−2cW−2c2c b−2cWpositive2cW b−2cWpositive 2.Exponential: U=−exp(−cW)c>0cexp(−cW)−c2exp(−cW)czerocWpositive 3.Logarithmic: U=lnWNone1 W−1 W21 Wnegative1zero 4.Power:U=W1−γor0<γ<1(1−γ)W−γ−γ(1−γ)W−γ−1γ Wnegativeγzero U=−W−γγ>0γW−γ−1−γ(γ+1)W−γ−21+γ Wnegative1+γzero
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hence optimal portfolio allocations, are independent of the level of initial wealth W0. For example, suppose that an investor has the logarithmic utility of terminal wealth function
U WT
=ln(W)T
=ln W0
1+rp
=lnW0+ln 1+rp
(4.18) The term ln
W0
on the right-hand side of equation (4.18) is a positive constant that can be ignored because a utility function’s preference orderings are invariant under positive linear transformations. Thus, utility analysis can focus on the term involving returns,
1+rp
, without being affected by the level of the investor’s initial wealth.
Consider next the power utility in terminal wealth. As in the case of the logarithmic function, the investor’s initial wealth is separable from the rate of return as shown below:
U WT
=WT1−γ
= W0
1+rp1−γ
=W01−γ
1+rp1−γ
(4.19) The termW01−γ on the right-hand side of equation (4.19) is a positive constant that can be ignored because a utility function’s preference orderings are invariant under positive linear transformations. Thus, utility analysis can focus on the term involving returns,
1+rp1−γ
, without being affected by the level of the investor’s initial wealth. This property of separability has an important consequence for portfolio optimization, which is discussed further in the next section.
The quadratic function does not allow separation of the investor’s initial wealth and returns. A quadratic wealth function implies a quadratic returns function, but the coefficients of the returns function are dependent on the level of the investor’s initial wealth.
4.8.3 Isoelastic Utility Functions
Isoelastic utility functions enjoy a quality calledconstant elasticity of substitution.
The Arrow-Pratt relative risk-aversion measure may be interpreted as a measure of the elasticity of utility with respect to wealth. Those classes of utility functions that have constant relative risk aversion (CRRA) may be called isoelastic utility functions. As an example of isoelastic utility functions, consider the following utility function:
U(W)= 1
1−γW1−γ (4.20)
where 0< γ <1 for risk-averse investors. This function can be viewed as a positive linear transformation of the power utility function because the coefficient 1/(1−γ) can be deleted by such a transformation. This function’s name comes from the fact that the elasticity of utility with respect to terminal wealth is constant
∂U U
∂W W
= ∂U
∂W W
U
=1−γ (4.21)
Isoelastic utility functions allow wealth and returns to be separated so that the investor’s expected utility can be maximized by analyzing investment returns without reference to the investor’s level of wealth—that is, the optimum portfolio is not dependent on the level of the investor’s wealth. This separability property greatly simplifies the analysis of an investor’s utility in a multiperiod context where the investor probably begins every period with a different amount of investable wealth.
Isoelastic utility functions allow the investor to act as if he or she had a short-term (single-period) horizon even though he or she actually has a long-term (multiperiod) horizon. Investors with isoelastic utility functions can simply maximize the same utility of returns function in every period, regardless of the level of their wealth.
Furthermore, this period-by-period utility of returns function is of the same form as the investor’s utility of terminal wealth function.
Among those classes of utility functions that possess positive but diminishing marginal utility, Mossin (1968) has shown that three classes of utility functions are isoelastic. These classes are:13
1. U=W1−γ for 0< γ <1, a positive power function.
2. U= −W−γ forγ >0, a negative power function.
3. U=ln(W)forγ =1, a logarithmic function.
4.8.4 Myopic, but Optimal
An investor who disregards past and future periods in making investment decisions for the current period could be called nearsighted. Instead of calling them nearsighted, financial economists prefer to say they behave myopically. Simply delineating and selecting an optimal single-period efficient portfolio is optimal multiperiod behavior and will maximize multiperiod expected utility under three conditions. These three conditions are:
1. The investor has positive but diminishing utility of terminal wealth, that is, U(W) >0 and U(W) <0.
2. The investor has an isoelastic utility function, which makes wealth and returns separable.
3. The single-period rates of return for securities are distributed according to a multivariate normal distribution and are independent of prior period returns.
These conditions simplify the mathematics sufficiently to prove that expected utility in the multiperiod case can be maximized by acting myopically.14For example, if the investor has a logarithmic utility function, the utility of the terminal wealth is U
WT
=ln WT
, where WT =W0 1+r1
· · · 1+rT
. In this case, for any single time period t between 0 and T, maximizing E
U WT
is equivalent to maximizing E
ln 1+rt
. In other words, a young investor with a logarithmic utility of retirement-age wealth can behave consistently by maximizing the log of end-of-year wealth year after year until reaching retirement. Stated differently, if you have a log utility function, it is not only easy to be an economically nearsighted (myopic) Markowitz portfolio manager who always focuses on rates of return and ignores wealth levels, it is also optimal wealth-maximizing behavior.