BAÑOS DE PROCESO

. (7.60)

Example 2 Suppose there is a single factor, and all assets have different factor loadings b_i = b − 2⁻ⁱ. In this case

Q_n= n⁻¹

µ n nb + 1 − 2⁻ⁿ

nb + 1 − 2⁻ⁿ nb²+ (1 − 4⁻ⁿ) /3

¶

. (7.61)

Again Q_∞is given by (60).

Example 3 Suppose there are two factors with loadingsbi andci = 1 − bi+ 2⁻ⁱ. Then

Q_n =



 1 ¯b 1 − ¯b

¯b d¯ ¯b − ¯d 1 − ¯b ¯b − ¯d 1 − 2¯b + ¯d



 + n⁻¹ Xn

1

2⁻ⁱ

à 0 0 1

0 0 b_i

1 bi 2(1 − bi) + 2⁻ⁱ

!

(7.62) If thebiare bounded, then the second matrix vanishes in the limit and the first is singular.

In each of these examples the factor loadings, including a vector of 1’s, are close enough to collinear so that fully diversified portfolios cannot be formed with arbitrary factor re-sponses. Specifically, in Example 1, to form a portfolio withb_p substantially different from b requires nonnegligible investment in some of the first m assets, so any such portfolio cannot be fully diversified.

Exercises for the Reader

1. If Q_∞is singular but B⁰_nB_n/n has a nonsingular limit, show that

a_i ≈ a^o+

K−1X

1

b_ik(a^k− a^o), (7.63)

wherea^kis as defined before anda^ois the expected return on a portfolio with a zero factor loading on the first K − 1 factors. (Note: The omitted factor is not completely arbitrary and might not be the last.)

2. If Q_∞has rankr and B⁰_nB_n/n has a limiting rank of r − 1, show that a_i ≈ R +X

b_ik(a^k− R), (7.64)

where the sum is taken overr − 1 values of k.

3. If both Q_∞and B⁰_nB_n/n have rank r in the limit, show that a_i ≈ a^o+X

b_ik(a^k− a^o), (7.65)

where the sum is taken overr − 1 values of k. The portfolio associated with a^ohasb_ik = 0 for these values ofk.

7.11 Pricing Bounds in A Finite Economy 133

These exercises show two things. First, as seen in Exercise 2, a complete lack of correla-tion among the residuals is not necessary for the APT to hold. Essentially what is required is a sufficient lack of correlation for the law of large numbers to apply. Second, the pricing model can appear in a “Sharpe-Lintner” or “Black” form.

7.11 Pricing Bounds in A Finite Economy

It has been claimed that the APT is not testable because its pricing is meaningful only in an economy with infinite assets. One response to this criticism is to ignore it and test for an exact linear relation among expected returns. In certain respects this is justified.

The CAPM, for example, is based on strong assumptions and yields (objectionably) strong conclusions. The APT, on the other hand, is based on relatively weak assumptions, and it only suggests a “conclusion.” A strong assumption can be made at this point to claim that the conclusion is exact.

A second response is to strengthen the original assumptions, so that the pricing equation becomes exact. We consider this in the next section.

A final response is to derive pricing bounds for each asset. These can then be checked for violations. To show how this can be done, we consider as an example a combination of the CAPM and linear factor models.

Assume that all asset returns are described by the generating model in (1) and that the CAPM pricing result holds. Since σ_im = P

b_ikb_mk + w_ims²_i, expected returns can be written as

a_i = 1 σ²_m

³Xb_ikb_mk+ w_ims²_i

´

(a_m− R)

= X

b_ik µ

b_mka_m− R σ_m²

¶

+ w_ims²_ia_m− R σ_m² .

(7.66) The terms in parentheses can be interpreted as factor premiumsλ_k, so the pricing error is

vi = wims²_ia_m− R

σ_m² . (7.67)

The size of this error is unknown unless the market portfolio is identified; however, it can be bounded. For any value-weighted market proxy0 ≤ wim ≤ wip. Thus, if relative risk aversion is assumed to be less than some bound, the pricing errors are also bounded. For example, if relative risk aversion is less than 20, then a typical stock whose value is about 1/2000 of the NYSE will have a pricing bound of0 < v_i < s²_i/100.

Note that this result answers both the objection that the APT is not testable and that the CAPM is not testable, since the market is not observable. On the other hand, it also assumes both mean-variance maximization and a factor structure.

If we are unwilling to assume the CAPM, the residuals from the APT can still be bounded if other strong assumptions are made. The following theorem of Dybvig is stated without proof.

Theorem 4 Assume that the returns on all assets are given by the linear factor model (1) with ˜f₁, . . . , ˜f_k, ε˜₁, . . . , ˜ε_n all mutually independent. The residuals are bounded below by -100%. The portfolio w is the optimal holding of some investor with a von Neumann-Morgenstern utility funcdon which is strictly increasing, concave, has bounded absolute risk aversion−u⁰⁰(Z)/u⁰(Z) < A, and has u⁰⁰⁰(Z) ≥ 0. Then for each asset

a_i = R +X

b_ikλ_k+ v_i, |v_i| ≤ w_is²_iAe^wiA. (7.68) The assumptions on the utility function and the assumption that the ε˜_i are bounded assure that small contributions to returns by the ε˜_i cannot have large utility effects. The assumption that the residuals and factors are independent guarantees that the residual risk is uncorrelated with the utility of returns on the factors. It has the same role here as in Equation (72) of Chapter 6.

7.12 Exact Pricing in the Linear Model

Thus far we have shown that the APT holds with a zero mean square error. That is, “most”

assets are priced with negligible error. Under what conditions can we assert that all assets are priced with negligible error, or, equivalently, that the total error converges to zero?

Sufficient conditions for this stronger result are that at least one investor chooses to hold a fully diversified portfolio, and that each asset’s idiosyncratic risk be a fair game with respect to all the factors

E(˜ε|f₁, f₂, . . . , f_K) = 0. (7.69) In the limit a fully diversified portfolio has no idiosyncratic risk, so exact pricing follows, as shown for a finite number of assets in Chapter 6.

The question remaining is, What requirements are needed to assure that some investor holds a fully diversified portfolio? It seems natural to view idiosyncratic risk as nonmarket risk which is not priced and conclude that all investors would hold fully diversified port-folios. However, this parallel is deceptive, and the reasoning is circular. In some cases a significant portion of the total risk of the economy comes from the idiosyncratic risk of a few assets. In these circumstances it is priced, and the APT fails - at least for these assets.

As a simple example of a linear factor economy in which not all assets are priced exactly by the arbitrage relation, consider the set of assets with returns

˜

z₁ = a₁+ ˜ε₁, z˜_i = a + ˜ε_i, i = 2, . . . , ∞. (7.70) The ε_i, i > 1, are independent and identically distributed. This is a zero factor economy.

Ifa = R, assets 2, 3, . . . are priced exactly right. However, if a₁ 6= R, then asset 1 is not priced correctly. Can such an economy exist in equilibrium? The answer is yes.

Suppose that the assets are scale-free investment opportunities. By symmetry, investors will purchase assets 2, 3, . . . equally. In the limit this portfolio must duplicate the riskless asset.

Assume that ε˜₁ is normally distributed with mean zero and variance s², and that all investors have exponential utility u(Z) = e^−δZ. From Equation (53) in Chapter 4, the optimal portfolio is the fraction w in asset 1 and 1 − w in the riskless asset (the equally weighted portfolio of2, 3, . . .), where w is

w = a₁ − R

δs² . (7.71)

Thus it is perfectly consistent to find a₁ 6= R, and in this case investors do not hold fully diversified portfolios. (It would be possible to construct a one-factor model fittinga and a₁ exactly by having the factor affect only asset 1. However, in this case the factor loadings on the factor are not sufficiently different. That is, Q_∞. is singular.)

7.12 Exact Pricing in the Linear Model 135

The following theorem, which is stated without proof, gives sufficient conditions under which the arbitrage model holds exactly in the limit.

Theorem 5 If (i) the returns on the risky assets are given by theK factor linear generating model in (1) with E(˜ε1|f1, . . . , fK) = 0, (ii) the market proportion supply of each asset is negligible, (iii) the loadings on each factor are spread “evenly” among many securities (i.e., Q_nis non-singular in the limit), (iv) no investor takes an unboundedly large position in any security, and (v) marginal utility is bounded above zero, then the linear pricing relation in (26) holds, and the pricing errors converge to zero in the sense that

n→∞lim Xn

1

v_i² = 0. (7.72)

Assumptions (ii) and (iii) together describe a “fully diversified economy.” Condition (ii) assures that no asset is a significant proportion of the market portfolio. Condition (iii) serves two purposes. First, many of the coefficientsb_ik (i = 1, 2, . . .) must be nonzero for each k. That is, each ˜f_k affects a nonnegligible proportion of the asset returns. (Note that this does not say that only a finite number are unaffected. For example, ˜f₁could influence only the odd-numbered assets.) Second, there can be no severe multicollinearity in the columns of the factor loading matrix. That is, each factor must make an identifiable independent contribution to the risk of the economy. In a fully diversified economy, all investors can simultaneously hold fully diversified portfolios with the factor effects distributed among them in any fashion.

Assumptions (iv) and (v) guarantee that each investor’s portfolio is an “interior” opti-mum; (iv) assures that risk aversion does not vanish so that variance is always disliked; (v) assures that expected return is always valued.

In this theorem λ₀ = R, provided that the riskless asset exists. Each λ_k is (asymp-totically) exactly equal to the expected return on an asset with b_ik = 1 and b_ij = 0 for j 6= k.

Equilibrium Models with Complete