folio, the higher its expected return. Just as in the mean-variance world, these results apply only to efficient portfolios. The proper measure of risk for individual securities equivalent to beta in the CAPM is developed later.
One further clarification about efficient portfolios is important. The set of efficient port-folios does not include every portfolio for which there is not another less risky portfolio with the same mean return, The latter set is similar to the set of minimum-variance portfolios in the mean-variance problem, which includes both the mean-variance efficient portfolios and the lower limb of the hyperbola (or lower projection of the capital market line with a riskless asset). These excluded portfolios correspond to solutions to (24a) where marginal utility is not uniformly positive, and investors trade off variance to get lower expected return.
5.10 Verifying The Efficiency of a Given Portfolio
In some contexts it is important to verify whether or not a given portfolio is efficient. In the mean-variance problem this is a simple matter, In all mean-variance efficient portfolios, the mean and variance must satisfy the quadratic relation of Chapter 4, Equation (6), and the portfolio weights must satisfy the linear relations of Chapter 4, Equation (10). In gen-eral, we must either describe a utility function satisfying (24a) or (25) for the portfolio in question or prove that one does not exist.
We examine this problem for the case of discrete outcomes for the returns on the assets.
We denote as separate events or, in the terminology to be adopted later, separate states those realizations which have distinct patterns of returns on the assets. The return on theith asset in thesth state is Zsi. The portfolio in question will have a pattern of returns of
Zs∗ =X
i
w∗iZsi. (5.33)
We order states such thatZ1∗ ≤ Z2∗ ≤ · · · ≤ ZS∗. For a given utility function we also denote vs ≡ u0(Zs∗). Then (24a) can be stated as
XπsZsivs = λ alli. (5.34)
Ifu is a valid utility function, it must have positive marginal utility vs ≥ 0 and decreasing marginal utility v1 ≥ v2 ≥ · · · ≥ vS. Since utility is defined only up to a linear trans-formation, the scale of thev’s is arbitrary. To answer our question, therefore, we may set λ = 1 without loss of generality. Note that the only way in which the portfolio in question can affect the requirements above is through the ordering it imposes on the states. Thus, all portfolios which impose the same ordering on thev’s are efficient if and only if there is a feasible solution to the problem
XπsZsivs= 1, i = 1, . . . , N, (5.35a)
vs≥ vs+1, s = 1, . . . , S − 1. (5.35b)
vS ≥ 0. (5.35c)
If there are no redundant securities, then the rank of the matrix (Zsi), and, therefore, of (πsZsi), is N ≤ S, and (35a) imposes N constraints on the S unknowns. If N = S, then there is a unique solution to (35a), so only one ordering of the states represents efficient portfolios, and all efficient portfolios can be immediately identified as soon as one is found.
This case is the “complete markets” problem discussed in Chapter 8.
IfN < S, then all efficient portfolios need not have the same state ordering of their returns, As an example of this latter case consider a three-state two-asset economy. The matrix(Zsi)0 is
state: a b c
asset
1 0.6 1.2 3.0 2 2.4 1.5 0.6
(5.36)
With three states there is a potential for3! = 6 orderings; however, portfolios of only two assets can achieve at most only four of these. In this case portfolios holding more than 35 in asset 1 will have returns ordered from low to high as(a, b, c). Those with 37 < w1 < 35 will be ordered(b, a, c). Those with 13 < w1 < 37 will be ordered(b, c, a). Finally, those with w1 < 13 will be ordered(c, b, a).
Which of these orderings represent efficient portfolios depends upon the probabilities of the states. With only two assets one of the extreme casesw1 = ±∞ will correspond to the portfolio with the highest expected return, depending on which mean is higher. Investors with utility “close” to risk neutral will choose such portfolios, so in our example one of the orderings(a, b, c) or (c, b, a) will always indicate efficient portfolios. Suppose that the state probabilities are all 13. Thenz¯1 = 1.6 > ¯z2 = 1.5, so w1 = ∞ is the risk-neutral investors’
choice, and the ordering (a, b, c) represents efficient portfolios. This means, as discussed before, that all portfolios with this ordering (i.e., all portfolios withw1 > 35) are efficient.
Other orderings also come from efficient portfolios. For example, the exponential class of utility functionsu(Z) = −e−aZ has the following optimal portfolios:
a 0 .001 .01 .1 1 10 100 ∞
w1∗ ∞ 33.88 3.77 .75 .45 .40 .34 13 (5.37) Plainly, each of the three regions with w1 > 13 is represented. whereas the last is not.
Therefore, portfolios withw1 < 13 are probably not efficient. For now this is only a conjec-ture. As we shall see later, there is no guarantee that a single class of utility functions will generate all efficient portfolios even when there are only two assets. When there are more than two assets, a single class of utility functions almost certainly will not do so.
In general. we require a constructive method of demonstrating efficiency, that is, a method of finding all efficient orderings. The method following is one such.
To find all orderings representing efficient portfolios, we set up the system represented by (35a):
0.2va+ 0.4vb+ 1.0vc = 1.0,
0.8va+ 0.5vb+ 0.2vc = 1.0. (5.38)
The solution set to (38) is va = 21
11vc− 5
11, vb = −38
11vc+ 30
11, (5.39)
which is graphed in Figure 5.6. From the figure it is clear that four orderings of utility represent efficient portfolios, namely,
vb > vc > va, vb > va > vc, va> vb > vc, va> vc > vb. (5.40) Recalling that marginal utility is inversely related to return, we see that the first three orderings in (40) correspond, in reverse order, to the first three orderings under Equation (36). This confirms our finding that all portfolios investing more than 13 in asset 1 are efficient. it also confirms our guess that portfolios withw1 < 13are not efficient. A portfolio
5.10 Verifying The Efficiency of a Given Portfolio 91
withZa < Zc < Zb would be efficient if it were feasible (without changing the economy).
A portfolio withZc < Za< Zbwould be inefficient even if available.
Figure 5.6 Marginal Utilities in Each State Marginal Utility in State c bca bac abc acb
State a
State b State c Marginal
Utilities
It is interesting to compare these results to those obtained by using mean-variance anal-ysis. The variances areσ12 = 1.04, σ22 = 0.54, and σ12 = −0.72. The global minimum-variance portfolio holds
wg1= σ22− σ12
σ21+ σ22− 2σ12 ≈ 0.417. (5.41) Since there are only two assets, the mean-variance efficient frontier includes all portfolios withw1 ≥ wg1. As we just saw, however, the efficient set includes those down tow1 = 13.
If the state probabilities are changed, then different orderings become efficient. For example, if the probabilities are(12,13,16), then the exponential class of utility functions has the following optimal portfolios:
a 0 .001 .01 .1 1 10 100 ∞
w∗1 −∞ -227.24 -22.35 -1.86 .19 .37 .34 13 (5.42) Note that nowz˜2 > ˜z1so a risk-neutral investor would select w1 = −∞. In this case two orderings are represented: (c, b, a) when w1∗ < 13 and (b, c, a) when 13 < w∗1 < 37. Note that not all portfolios in the second region are represented. w∗1 peaks at a value less than 0.38 between values of 8 and 9 for a, but 37 > 0.42. Nevertheless, we know that these other portfolios must also be efficient since they have the same state orderings as those for a = 10 or 100.
Using the constructive method for this case, we obtain that (35a) is 0.3va+ 0.4vb+ 0.5vc = 1.0,
1.2va+ 0.5vb+ 0.1vc = 1.0. (5.43) The solution set is now
va = 7
11vc− 10
33, vb = 30 11 −19
11vc, (5.44)
which is plotted in Figure 5.7. Now only three orderings represent efficient portfolios vb > vc > va, vc > vb > va, vc > va > vb. (5.45)
Of these only the first two are feasible. Together they correspond to all portfolios with w1 < 37.
Figure 5.7 Marginal Utilities in Each State Marginal Utility in State c
Characterizing utility functions for which given portfolios are optimal is now a simple mat-ter (although finding “nice” functions is not). For example, under the first set of probabili-ties, the efficient portfolio holdingw1 = .4 is optimal for vc = 2255(among other values), and from (39)va = 1755,vb = 7455. This portfolio has returnsZa = 1.68, Zb = 1.38, Zc = 1.56, so any utility function possessing u0(1.68) = 1755 < u0(1.56) = 25 < u0(1.38) = 7455 will find it optimal.