ETAPAS DE PREPARACIÓN DE LA SUPERFICIE

It might seem reasonable to expect that the market portfolio would always be an efficient combination. After all, in equilibrium, the market portfolio is just the wealth-weighted average of each investor’s optimal portfolio, and all of these are efficient. Such reasoning would be correct if it were known that the set of efficient portfolios were convex. Unfor-tunately, such is not the case. The efficient set is not necessarily convex, as the following example proves. Suppose that three assets are available to investors and that they have the following pattern of returns:

Asset: 1 2 3 Portfolio: (¹₂, 0,¹₂) (¹₄,¹₂,¹₄) (−.5, −.4, 1.9)

State: a 1.50 2.05 1.80 1.65 1.85 1.85

b 1.00 1.30 1.20 1.10 1.20 1.26

c 1.40 0.95 1.20 1.30 1.125 1.20

d 1.30 1.25 1.20 1.25 1.25 1.13

(6.1) Any portfolio of these three assets may be held. The patterns of returns on three portfolios which we shall need are also given. The probability of each of the outcome states is ¹₄.

Consider two different investors possessing utility functions with the following proper-ties:

u⁰₁(1.65) = 1 < u⁰₁(1.30) = 40 < u⁰₁(1.25) = 59 < u⁰₁(1.10) = 68, (6.2a) u⁰₂(2.05) = 21 < u⁰₂(1.30) = 28 < u⁰₂(1.25) = 39 < u⁰₂(0.95) = 40. (6.2b) Many utility functions satisfying (2a) or (2b) are possible because in both cases marginal utility meets the only two requirements - that it be positive and decreasing.

In the absence of a riskless asset, the criterion for an optimal portfolio for investork, from Equation (26a) in Chapter 5

E£

u⁰_k(Z^k)˜zi

¤ ≡X

s

πsu⁰_k(Z_s^k)Zsi = λk fori = 1, 2, 3. (6.3)

The first investor will optimally choose to hold the portfolio(.5, 0, .5) since

1

4(1.50 × 1 + 1.00 × 68 + 1.40 × 40 + 1.30 × 59) = 50.55,

1

4(2.05 × 1 + 1.30 × 68 + 0.95 × 40 + 1.25 × 59) = 50.55,

1

4(1.80 × 1 + 1.20 × 68 + 1.20 × 40 + 1.20 × 59) = 50.55

(6.4)

Similarly, the second investor will choose the portfolio(0, 1, 0) since

1

4(1.50 × 21 + 1.00 × 28 + 1.40 × 40 + 1.30 × 39) = 41.55,

1

4(2.05 × 21 + 1.30 × 28 + 0.95 × 40 + 1.25 × 39) = 41.55,

1

4(1.80 × 21 + 1.20 × 28 + 1.20 × 40 + 1.20 × 39) = 41.55.

(6.5)

We now pose the question: Is the portfolio (.25, .5, .25), which is an equal mixture of these two investors’ efficient portfolios, efficient as well? If it is not, then we have demonstrated that the efficient set is not convex. Furthermore, since these two investors could be alone in the market, we would have also proved that the market portfolio is not necessarily efficient.

The answer to the question posed is no. We could prove this by examining permitted orderings as in Chapter 5; however, we first choose the simpler and more direct method of finding a better portfolio. The portfolio(.25, .5, .25) is not efficient because it has a lower expected utility than the portfolio(−.5, −.4, 1.9) to any investor who prefers more to less.

This is easily demonstrated by comparing the expected utility of these two portfolios:

1

4[u(1.85) + u(1.20) + u(1.125) + u(1.25)], (6.6a)

1

4[u(1.85) + u(1.26) + u(1.20) + u(1.13)]. (6.6b) For any increasingu, the quantity in (6a) is smaller because its first two terms are equal to the first and third terms of (6b) and its third and fourth terms are smaller than the fourth and second terms in (6b).

Figure 6.1 demonstrates explicitly the nonconvexity of the efficient set. Each point (w₁, w₂) represents a feasible portfolio. Furthermore, since w₃ is pegged by the budget constraint, all portfolios are represented. Shown in the figure are the 18 possible portfolio types, as indicated by their ordered (from high to low) state returns. (The six unlisted permutations are not possible to achieve with feasible portfolios.)

The regions indicated by capital letters correspond to efficient portfolios. As already stated, the portfolio (0, 1, 0) is optimal for the utility function with the properties in (2b).

Its returns are ordered ABDC from high to low. The portfolio (.5, 0, .5) is also efficient, and it lies inACDB.

6.1 Inefficiency of The Market Portfolio: An Example 101

Figure 6.1 Example of Non-Convexity of Efficient Set

The portfolio (−.5, −.4, 1.9) is in the region ABCD and therefore must be efficient if the figure is correct. We do know that it dominates the portfolio(.25, .5, .25), but this is insufficient to prove that it is efficient. To do this, we must find an appropriate utility function. One such function would be that characterized by

u⁰(1.85) = .107 < u⁰(1.26) = .195 < u⁰(1.20) = .234 < u⁰(1.13) = .244. (6.7) Marginal utility is positive and decreasing, and the first-order condition of optimality (Equa-tion (26a) of Chapter 5) is satisfied for each asset:

1

4(1.5O × .107 + 1.00 × .195 + 1.4 × .234 + 1.3 × .244) = .250,

1

4(2.O5 × .107 + 1.30 × .195 + 0.95 × .234 + 1.25 × .244) = .250,

1

4(1.8O × .107 + 1.20 × .195 + 1.20 × .234 + 1.20 × .244) = .250.

(6.8)

Similarly, the regionACBD is supported by the marginal utilities .079, .234, .224, .257 (among other values).

Many portfolios in the regions adbc and adcb are convex portfolio combinations of efficient portfolios lying in the regionsABDC and ACDB. All of them are inefficient, however, just as(.25, .5, .25) in the region adbc is.

This example shows that nonconvexity of the efficient set does not depend on risk aver-sion, since nowhere was it assumed that u was concave. Nonconvexity does, however, require strict monotonicity, a strict preference of more to less. Without strict monotonicity

a constant utility function would be permitted and would be indifferent among all portfolios, so the efficient set would be the entire space of portfolios.

Although we worked in the context of a perfect market and with an unrestricted state space tableau, these assumptions were not crucial. The specific counterexample does sat-isfy the two often-imposed conditions of limited liability, Zsi ≥ 0, and no bankruptcy, PZ_siw^∗_i > 0. It does not meet the no-short-sales constraint; however, it could be suitably modified to do so.

The foregoing example is of minimal complexity. With just two assets the feasible set can be described by a single parameter (for example, w₁ or, ifz¯₁ 6= ¯z₂, ¯Z_p). Using mean-variance analysis, which is always valid for the case of two assets with different means, we have seen in Chapter 4 that the efficient set includes all portfolios satisfying ¯Z_p ≥ ¯Z_p^∗ for some ¯Z_p^∗. These portfolios are those withw₁ ≥ w^∗₁ (ifz¯₁ > ¯z₂) orw₁ ≤ w^∗₁ (ifz¯₁ < ¯z₂).

A set of this type is obviously convex, Therefore, at least three assets are required for the efficient set not to be convex. Furthermore, as we shall see in Chapter 8, there must be fewer assets than states for the market portfolio to be inefficient.