7.1 Linear Factor Models
Throughout this chapter we assume that asset returns are described by the linear factor model
˜
zi = ai+ XK k=1
bikf˜k+ ˜εi,
E(˜εi) = E( ˜fk) = E(˜εiε˜j) = E(˜εif˜k) = E( ˜fkf˜m) = 0, E(˜ε2i) ≡ s2i < S2,
E( ˜fk2) = 1,
(7.1)
unless otherwise noted. The ˜fk are the factors, the bik are the factor loadings, and the ε˜i represent the residual or idiosyncratic risks. The factor loadings are bounded: |bij| < b.
This model can also be written conveniently in vector notation as
˜z = a + B˜f + ε, (7.2a)
E(˜ε) = 0, (7.2b)
E(˜f) = 0, (7.2c)
E(˜f˜f0) = I, (7.2d)
E(˜ε˜f0) = 0, (7.2e)
E(˜ε˜ε0) = D, a diagonal matrix. (7.2f) Assumptions (2b)-(2e) are innocuous; (2c), (2d) can be met simply by subtracting any factor means and then orthogonalizing and rescaling the parameters. This does not change any of the stochastic properties of the set of factors. Conditions (2b), (2e) are always possible through the appropriate choices for a and B. Only the assumptions of uncorrelated residuals cannot be guaranteed. Relaxation of (2f) will be considered later.
7.2 Single-Factor, Residual-Risk-Free Models
Before studying the model in (2) in detail, it is useful to examine economies with no id-iosyncratic (asset-specific) risk. Although these are very special cases, much of the intuition developed is useful in general.
7.3 Multifactor Models 121
Suppose that asset returns are related by a single-factor, residual-riskfree linear model
˜
zi = ai+ bif ,˜ i = 1, . . . , n. (7.3) From the basic theorem of arbitrage pricing, any two assets with the same value ofb must have the same expected return a. To price assets with different values of b, consider a portfolio of two assets, withbi 6= bj,bi 6= 0, bj 6= 0, formed by investing a fraction w in asseti and 1 − w in asset j. The return on this portfolio is
Z = w(a˜ i− aj) + aj+ [w(bi− bj) + bj] ˜f . (7.4) If we choosew∗ = bj(bj − bi), then ˜Z∗ = bj(ai − aj)/(bj − bi) + aj, which is certain.
Therefore, to avoid arbitrage, we must haveZ∗ = R. This condition can also be expressed
as aj− R
where λ is some constant known as the factor risk premium. From (6) it is clear that λ is equal to the excess expected return on any asset with b = 1. Even if no riskless asset naturally exists, a similar relation must still obtain since a riskless asset can be constructed from any two risky assets. All such constructed riskless assets must have the same return.
Thus, when no explicit riskless asset exists,
¯
zi = λ0+ biλ1 (7.7)
for all assets described by (3).
7.3 Multifactor Models
For multifactor models a similar analysis can also be performed, provided that the factor loadings are “sufficiently different.” The case of two factors is outlined next.
Suppose that asset returns are generated by the two-factor linear model
˜
zi = ai+ bif˜1+ cif˜2 (7.8) with the vectors b, c, and 1 not collinear. Choose a portfolio of three securities. The return on this portfolio is
wici = 0, then the portfolio is riskless; so if arbitrage opportuni-ties are absent,P
The matrix in (10) must be singular. Because the last two rows are not collinear we have
¯
zi− R ≡ ai− R = λ1bi + λ2ci for alli. (7.11) Again, if the riskless asset does not exist, we replaceR by λ0.
7.4 Interpretation of the Factor Risk Premiums
The only remaining issue is to identifyλ1andλ2. If we choose a portfolio withP
wibi = 1 and P
wici = 0, then we see that its excess expected return Z1− R is λ1. Similarly,λ2 is the excess expected return on a portfolio with b = 0 and c = 1. So λ1 and λ2 can be thought of as the excess return per unit risk associated with thef1 andf2 factors, and (11) may be written
¯
zi = R + bi( ¯Z1− R) + ci( ¯Z2− R). (7.12) The relation in (11) or (12) can also be rewritten in terms of the expected excess return on any portfolios whose two risk components are in different proportions. For example, choose the market portfolio and any other for which bP/cP 6= bM/cM. Solve (11) forλ1 andλ2 in terms of the expected returns on these two portfolios. Then for any securityi,
¯
zi = R + b0i( ¯ZM − R) + c0i( ¯ZP − R), where
b0i ≡ bicP − cibP bMcP − cMbP
, c0i ≡ cibM − bicM bMcP − cMbP
. (7.13)
7.5 Factor Models with “Unavoidable” Risk
Up to now we have been assuming implicitly that the factor loadings are “sufficiently dif-ferent.” We outline now a case in which this is not true. Consider the two-factor model just discussed and assume thatci = c − bi(exceptb0 = c0 = 0). The previous derivation fails because we cannot form a portfolio of risky assets with b = c = 0, as we assumed. This point is emphasized if we rewrite the factor model (8) as
˜
zi = ai+ bi( ˜f1− ˜f2) + c ˜f2 = ai+ bif˜10 + ˜f20. (7.14) (Note that the factors are no longer orthogonal or normalized, but this need not concern us.) Under this restructuring the loadings on ˜f20 are obviously not “sufficiently different”, and we will be unable to form a portfolio which has no ˜f20 risk. Any portfolio formed will have the return
Z = a + b ˜˜ f10 + ˜f20. (7.15) The thing to note from (15) is that any two portfolios with the same value of b are perfectly correlated; therefore, by the basic theorem of arbitrage pricing, they must have the same expected return.
To price securities with differentb’s we introduce ao, which is defined to be the expected return for a portfolio or security withb = 0. From the paragraph above, all such securities will have the same expected return, so ao is well defined. Now consider forming two portfolios from assetsi, j, and o (with bo = 0) as follows:
wi+= wi∗+ δ1, w+j = wj∗+ δ2 w+o = w∗o − δ1− δ2. (7.16) The two portfolios have the properties that
a+ = a∗+ δ1(ai− ao) + δ2(aj − ao), b+ = b∗+ δ1bi+ δ2bj.
(7.17)
7.6 Asymptotic Arbitrage 123
Suppose we chooseδ1bi + δ2bj = 0. Then b+ = b∗, and, as above,a+= a∗. Therefore 0 = δ1(ai− ao) + δ2(aj − ao) = δ1
·
ai− ao− bi
bj(aj − ao)
¸
. (7.18)
Equation (18) must be true for any choice ofδ1. Thus, ai− ao
bi = aj − ao
bj = λ. (7.19)
The linear pricing relation isai = ao+ λbior
¯
zi ≡ ai = ¯Z0+ bi( ¯Z1− ¯Z0), (7.20) where ¯Z0 = ao is the expected return on any portfolio withb = 0, and ¯Z1 is the expected return on a portfolio with b = 1. Note that the riskless return does not appear in this formula at all, unlike Equation (12). In that case riskless portfolios could be constructed by using only the risky assets. Such is not true here, so the absence of arbitrage opportunities alone cannot reveal any relation between the riskless rate and the expected returns on the risky assets. In particular, ¯Z0 need not be equal toR. In general, ¯Z0 > R to compensate investors for bearing ˜f20-type risk. ¯Z1 can be greater than or less than ¯Z0.
Using the same steps as Equations (44)-(49) in Chapter 6, Equation (20) can be reinter-preted as the “two-factor” form of the CAPM. Note. however, that here a riskless asset may be available and there need be no restrictions on borrowing.
We can reintroduce both factor loadings into the no arbitrage condition by selectingλ0, λ1, andλ2such that
ai = ao+ λbi = λ0+ λ1bi+ λ2ci. (7.21) This relation remains valid for all assets if and only if the choices of factor risk premiums satisfy
ao = λ0+ cλ2, λ = λ1− λ2. (7.22) Since there are three variables and only two constraints in (22), this second relation is not unique, and any one of the three factor risk premiums may be arbitrarily specified. For example, if we takeλ0 = R, then the resulting no arbitrage expression is Equation (11).
Thus, the previous “invalid” analysis does come to a valid conclusion, but only one of the many possible.
7.6 Asymptotic Arbitrage
We now examine the general linear factor model described in (1) or (2). Note the similarity in form between this model and the separating distribution model of Chapter 6. One im-portant distinction is that in the latter we assumed a CAPM-like equilibrium or derived it directly from the fair-game assumption on the residuals. Here we are interested in proving this equilibrium, and we have introduced varying intercepts and weakened the fair-game assumption to a lack of correlation with the indices.
We no longer assume that zero residual-risk portfolios are available. Instead we assume that the residuals are mutually uncorrelated and that an infinite number of assets are avail-able. These two assumptions guarantee that investors can form diversified portfolios with near zero residual risks. These portfolios play the role of the idiosyncratic risk-free port-folios of Chapter 6 and are discussed at length later. The assumed existence of an infinite
number of assets is not, of course, true to life and means that the APT is only an asymptotic result for large economies.
One final item which should be noted is that the definition of arbitrage to be used here is also a limiting one. An asymptotic arbitrage opportunity is said to exist if there exists a sequence of arbitrage portfoliosωn,n = 2, . . . , satisfying
XN 1
ωin= 0, (7.23a)
XN 1
ωinz¯i ≥ δ > 0, (7.23b)
Xn 1
Xn 1
ωniωjnσij → 0. (7.23c)
(The notation here is deliberately vague. It is not required that the limit of the variance be zero. What is required is that for some infinite subsequence the limit be zero. However, if the n + 1 asset economy has the same parameters for the first n assets, ˜zinand σnij, as the n asset economy, then the existence of a subsequence arbitrage opportunity will imply the existence of a sequence arbitrage opportunity where the limit of the variance is zero).
This notion of arbitrage is an obvious extension to the definition of a riskless arbitrage opportunity in Chapter 3. As seen there, the scale of an arbitrage opportunity is arbitrary, so that an infinite profit can be earned by holding an unbounded position in the arbitrage portfolio. With an asymptotic arbitrage opportunity the question of scaling becomes a little trickier because the risk only vanishes in the limit. If ωn is a sequence defining an asymptotic arbitrage opportunity, then γωn is as well for any γ > 0. However, γ cannot become unbounded in an arbitrary fashion if (23c) is to remain valid. Clearly, it must depend on n. One construction that will work is to set γn = (ω0Σω)−1/4. Then
ˆ
ωn≡ γnωnis an asymptotic arbitrage opportunity with a limiting infinite profit Xωˆni = γnX
ωin= 0, alln, (7.24a)
Xωˆniz˜i = γnX
ωniz˜i ≥ δγn→ ∞, (7.24b) X Xωˆniωˆjnσij = γn2X X
ωniωnjσij =hX X
ωniωnjσij i1/2
→ 0. (7.24c) A natural question to ask at this point is, What does an asymptotic arbitrage opportunity achieve for an investor? An ordinary arbitrage opportunity when practiced on unlimited scale guarantees infinite wealth and infinite certainty equivalent of return; that is,u( ˜Zn) → u(∞). Chebyshev’s inequality when applied to (24) shows that an asymptotic arbitrage opportunity guarantees an infinite wealth with probability 1. An infinite certainty equivalent wealth is not guaranteed, however.
As a counterexample, consider a sequence of random portfolio returns ˜Znwith realiza-tions{1 − n, 1 + n, 1} with probabilities {n−3, n−3, 1 − 2n−3}. Then for u(Z) = −e−z,
Z¯n= 1 + n−3[1 − n + 1 + n − 2] = 1, (7.25a) σ2n= n−3[(−n)2+ n2] = 2
n → 0, (7.25b)
Eu( ˜Zn) = − 1
n3(en−1+ e−n−1− 2e−1) − e−1 ∼ − 1
n3en−1, (7.25c)
7.7 Arbitrage Pricing of Assets with Idiosyncratic Risk 125
which implies that in the limit the certainty equivalent wealth is−∞. Although it would be difficult to construct an economy in which a sequence of arbitrage portfolios yielded this sequence of returns, the point is made that the opportunity to increase wealth without any investment commitment, even with vanishingly small risk, does not guarantee an increase in expected utility. Certain investors may refrain from taking advantage of some asymptotic arbitrage opportunities.
General conditions sufficient to guarantee that any asymptotic arbitrage opportunity will lead to bliss,u(∞), are not known. Concave utility functions which are bounded below will possess this property. Restrictions such as bounds on the assets’ returns will be insufficient because portfolios copied still have unbounded returns.
7.7 Arbitrage Pricing of Assets with Idiosyncratic Risk
Given that there are no asymptotic arbitrage opportunities of the type described in (23), we can prove the following result.
Theorem 1 If the returns on the risky assets are given by a K factor linear model with bounded residual risk and there are no asymptotic arbitrage opportunities, then there exists a linear pricing model which gives expected returns with a mean square error of zero. That is, there are factorsλ0, λ1, . . . , λKdependent onn such that
vi ≡ ai− λ0− XK k=1
bikλk,
n→∞lim 1 n
Xn i=1
v2i ≡ lim
n→∞
1
n kvnk2 = 0.
(7.26)
Proof. Arbitrarily select n of the assets and number them 1 to n. “Regress” their ex-pected returnsai on thebik and a constant. Call the regression coefficientsλk. (This “re-gression” is a thought experiment performed not on observations of the returns but on the true expectations. Mathematically, it is a projection of the vector a into the space spanned by B and the vector 1. If the regression cannot be performed because of multicollinear-ity, arbitrarily prespecify a sufficient number of the λk to remove the collinearity.) The
“residuals,”vi, of this regression are given by
ai = λ0+ XK k=1
bikλk+ vi. (7.27)
Thevi will be the same vector elements defined in (26). From the orthogonality property of regressions,
Xn i=1
vi = 0, (7.28a)
Xn i=1
vibik = 0, allk. (7.28b)
Consider the arbitrage portfolioωi = vi/kvnk√ The last line follows from (28b). The expected profit is
(√ The first two sums in the second line of (30) are zero from (28a), (28b). The variance of the profit is
¡nkvnk2¢−1Xn
1
vi2s2i 6 S2
n . (7.31)
Now suppose that the stated theorem is false; then the expected profit remains nonzero while its variance goes to zero as n is increased. But this would represent an arbitrage op-portunity of the type defined in (23). Hence,kvnk2/n must vanish, and (26) obtains. Q.E.D.
The derived no arbitrage condition is 1
Each term in this infinite sum is nonnegative. The average term is zero; therefore most of the terms (i.e., all but a finite number of them) must be negligible. To be precise, if the assets are ordered by their absolute pricing errors so that|v1| ≥ |v2| ≥ · · · ≥ |vi| ≥ · · · then for anyδ no matter how small, there exists an N such that fewer than N of the assets are mispriced by more thanδ:
|v1| ≥ · · · ≥ |vN −1| ≥ δ > |vN| ≥ |vN +1| ≥ · · · . (7.33) This means that the linear pricing model
Z¯i = λ0+ XK
1
bikλk (7.34)
prices “most” of the assets “correctly,” and all of the assets together with a negligible mean square error. However, it can be arbitrarily bad at pricing a finite number of the assets.
It may seem somewhat strange that the magnitudes of the residual variances do not enter the pricing bound in (26). Intuitively, we would expect that assets with little residual variation should be priced quite closely whereas those with high residual variances could deviate by more. In particular, we know that among assets with zero residual risk the pricing must be exact. This intuition is correct.
Theorem 2 Under the same conditions as in Theorem 1 the pricing errors must satisfy
n→∞lim
7.8 Risk and Risk Premiums 127
Proof. The proof is left to the reader. [Hint: Consider a weighted least squares “regres-sion.”]
As promised, this theorem provides the intuition, lacking in the previous theorem, that the pricing deviation permitted each asset depends critically on its residual variation. It does more than simply clarify this issue, however. It provides a stronger condition which must be met in economies with no arbitrage, as the following example demonstrates.
Consider an economy with no common factors; ai = i−1, s2i = i−2 for i even; and
vi2/n goes to zero, and Theorem 1 is satisfied. Equation (35) is violated, however, since an arbitrage opportunity with an expected profit and variance of
Xn
If there is an upper boundS2on the residual variances as we have assumed, then Theo-rem i is a special case of TheoTheo-rem 2 sinceP
vi2 6 S2P vi2/s2i.
The earlier theorem is still useful, of course, because the residual variances need not be known for its application. If residual variances are not bounded, then (35) does not imply (26); however, the proof of Theorem 1 is no longer valid either. If residual variances are bounded away from zero as well, then Theorems 1 and 2 are equivalent.
In these two theorems the factor premiums may change at each step along the way. It is also possible to prove that the mean square error goes to zero even for a fixed vector of factor premiums.
7.8 Risk and Risk Premiums
Unless something further about the nature of the factors in the linear generating model can be specified, no economic significance can be attached to the signs or magnitudes of theλk. In fact for any arbitrage-pricing-model economy, it is possible to create a different set of K uncorrelated factors with only one receiving a positive factor risk premium. The arbitrage model can be written as
˜z = a + B˜f + ˜ε, (7.39a)
a≈ λ01+ Bλ. (7.39b)
Now consider a new set of factors created by an orthogonal transformation of the original set:
ˆf ≡ T0f. (7.40)
T is an orthogonal matrix with TT0 = T0T= I, so f = Tˆf.
Then
˜z = a + BTˆf + ˜ε ≡ a + ˆBˆf+ ˜ε; (7.41) that is, B≡ ˆBT0. Substituting for B in (39b) gives
a≈ λ01+ ˆBT0λ. (7.42)
To substantiate our claim, we now need only show that T0λ is a vector with only one nonzero element and that (41) is a proper linear generating model for the factors ˆf. The latter is easily accomplished, since for any orthogonal matrix T,
E(εˆf0) = E(εf0T) = E(εf0)T = 0, and
E(ˆfˆf0) = E(T0ff0T) = T0E(ff0)T = T0IT0 = T0T= I. (7.43) To establish the former, choose the heretofore arbitrary transformation matrix
T=¡
λ(λ0λ)−1/2, X¢
, (7.44)
whereX is any K by K − 1 matrix of mutually orthogonal unitary columns all orthogonal toλ. By construction T is an orthogonal matrix; therefore
I= T0T=¡
T0λ(λ0λ)−1/2, T0X¢
(7.45) and
(1, 0, . . . , 0)0 = T0λ
(λ0λ)1/2, ((λ0λ)1/2, 0, . . . , 0)0 = T0λ. (7.46) The fact that a K factor model can be “reduced” to a different model with only one priced factor must not be construed as a criticism of the original model. This “simplifi-cation” cannot be performed until all the original factor risk premia are known. (Aside:
We have already seen something similar in a different context. In Chapter 4 we saw that assets could be priced in terms of two mean-variance efficient portfolios; however, these portfolios could not be identified unless all expected rates of return were known.)
The principal lesson to be learned is that in virtually any equilibrium only a portion of the uncertainty brings compensation. This can be true even when an unpriced risk is common to many or all assets. No meaningful economic statement has been made, however, until the priced and unpriced sources of risk are identified. The CAPM is one case in which this identification is made. We will encounter others in Chapters 13 and 15.
Even if certain factors are unpriced, it is useful to know the asset loadings on that factor despite the fact that they do not affect expected returns. For example, in an event study it would be useful to remove the common unpriced component as well as the common priced component of an asset’s return to reduce the variation in the residual.
7.9 Fully Diversified Portfolios
Even though the factors may not be identified with any specific macroeconomic variables, the factor premiums may be given portfolio interpretations under certain conditions. Before doing so, however, we must introduce a new concept.
7.9 Fully Diversified Portfolios 129
A fully diversified portfolio is the limit of a sequence of positive net investment portfolios whose weights satisfy
n→∞lim n Xn
1
wi2(n) ≤ C < ∞. (7.47)
It is obvious thatwimust vanish for each asset in a fully diversified portfolio. In particular, wi must beO(n−1) for “most” assets. Fully diversified portfolios have no residual risk in the limit since
There are also less than fully diversified portfolios with no residual risk. One example is the limit of the sequence in whichwi(n) = n−3/4for the firstn/2 assets and 2/n − n−3/4 for the second half. In this case
Xw2i(n) = n For largen then the residual risk is less than S/√
n, so it vanishes in the limit. Nevertheless, the limit in (47) is√
n, which is not bounded.
Theorem 3 The expected returns on all fully diversified portfolios are given correctly with zero error by any linear pricing model satisfying Theorem 1.
Proof. From Equation (27) the expected return on the portfolio in the nth step of the sequence is
λ0+X X
wibikλk+X
wivi, (7.50)
so the theorem is proved if the third term vanishes. By the Cauchy-Schwarz inequality
³Xwivi
But the first term on the right hand side is bounded, and the second vanishes in the limit.
Q.E.D.
If we can create a fully diversified portfolio with bpk = 1 and bpi = 0 for all other factors, thenλ0+ λk is the expected return on this portfolio. Similarly,λ0 is the expected return on a fully diversified portfolio with no factor risk (if such is feasible), but since it has no residual risk either,λ0 = R.
If there is only one factor and if the market portfolio is fully diversified, then bMλ = Z¯M − R, and the no arbitrage condition may be written as
¯
andwiM ands2M are zero in the limit for fully diversified portfolios.
Despite the formal identity between the relation in (52) and the equilibrium in Equation (39) of Chapter 6 for a market whose securities’ returns are members of a “separating
distribution,” there is a fundamental difference in the underlying economics of these two models. The separating distribution (CAPM) relation is a market equilibrium, whereas the linear model relation is a “no arbitrage” condition. Now although there obviously can be no arbitrage possibilities in a market at equilibrium, the converse is not true. The absence of arbitrage opportunities does not necessarily imply that the market is in equilibrium.
There are two consequences of this distinction which are important for empirical work.
The first is that the APT relation (52) will hold for a subset of asset returns which meets its
The first is that the APT relation (52) will hold for a subset of asset returns which meets its