Testing the APT

Tests of the APT are plagued by the same conceptual and econometric problems as tests of the CAPM. The model is formulated in expectations but has to be tested with observed realizations. An additional difficulty is that APT does not say which, or even how many, factors should be included. Within these limitations, tests of the APT can be performed with a two-pass regression procedure similar to the one used for the CAPM. Suppose we want to test a factor model that uses a number of industry indices, rather than one total market index. Then the first-pass regression is a time series analyses of each asset’s risk premium on the industry indices’ risk premia I_kt − r_{f t}:

rit− rf t =αi+ β_1i(I_1t − rf t)+ β_2i(I_2t− rf t)+ .. + βKi(IKt− rf t)+εit

The result could be called the ‘characteristic hyperplane’ analogous to the characteristic line in the CAPM. The estimated β coefficients are the sensitivities. The β’s are used as explanatory variables in the second-pass, cross section regression in which the factor risk premia are estimated:

rp_i= γ₀+ γ₁β_1i+ γ₂β_2i+ .. + γ_kβ_Ki+u_i where rp_i = 

T(r_it − r_{f t})/T, asset i’s average risk premium. APT predicts that the intercept γ₀should be zero and the gammas should be equal to the averaged risk premia of the industry indices over the observation period: I₁− rf, I₂− rf, etc.

Industry indices are readily observable and they describe the market completely, i.e.

the weighted industries’ returns add up to the total market return. This makes the results easy to interpret: the estimated risk premia can be compared with the observed returns

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15 The construction of this portfolio is included in the exercises.

of the indices. This is not the case if the factors are defined in terms of business char-acteristics, such as book-to-market value and price-earnings ratio, or general economic variables, such as inflation, productivity and the oil price. With these variables it is diffi-cult (or impossible) to be complete, i.e. it is not clear how many factors should be used.

Moreover, the results are less easy to interpret because there are no observed returns for these factors. In the absence of observed factor returns, directional predictions (i.e. returns will increase or decrease with a factor) are often resorted to. Another possibility is to con-struct factor portfolios and use their returns. That approach is followed in the well-known Fama–French three-factor model.

The empirical basis of Fama and French’s (1993) three-factor model are long times series (1963–1991) of monthly returns of stocks on the three US stock exchanges.¹⁶ These stocks are grouped into portfolios in various ways. The first way is according to size. In each year the stocks are ranked on size and split into two portfolios, one of small and one of big companies. For each month, the difference between the returns of the portfolios of small and big stocks (SMB, small minus big) is calculated. SMB is meant to mimic the risk premium of a size-related risk factor in returns. The second way of grouping is according to book-to-market value of equity. The stocks are assembled into three portfolios, one with high (top 30 per cent), one with medium (middle 40 per cent) and one with low (bottom 30 per cent) book-to-market values of equity. Again, for each month the difference between the returns of the portfolios of high and low book-to-market stocks (HML, high minus low) is calculated. HML is meant to mimic the risk premium of a book-to-market related risk factor in returns. Their construction makes these two portfolios almost pure factor portfolios, i.e. SMB is largely free of the influence of book-to-market equity and HML is largely free of the size factor in returns. The third factor represents the market as a whole and is measured as the return of a value weighted port-folio of all stocks minus the risk-free interest rate (rmt− rf t). The returns to be explained are those of 25 portfolios formed on size and book-to-market equity. The first-pass, time series regression estimates the portfolios’ sensitivities (or factor loadings) for the three factor returns:

r_it− r_{f t} =a_i+ b_i(r_mt− r_{f t})+s_iSMB_t + h_iH ML_t+ε_it (3.19) where b_i,s_i and h_i are the sensitivities of portfolio i,ε_it is the error term anda_i is the intercept. Notice that SMB and HML already are risk premia (of small over large and high over low book-to-market) and, hence, are not taken in excess of the risk-free rate.

The Fama–French three-factor model can then be formulated in expectations as:

E(r_i)− rf =a_i+ b_i[E(rm)− rf] +s_iE(SMB)+ h_iE(H ML) (3.20) As we have just seen, APT predicts that the intercept in (3.20),ai, should be zero and Fama and French (1993) find values close to zero in almost all cases. They also claim that the three-factor model in (3.20) explains much of the cross-sectional variation in average stock returns. The model is widely used to calculate expected returns of portfolios in situations where size and value effects can play a role. It should be noted, however, that later research (to be discussed in the next chapter) indicates that the model’s relevance has diminished over time.

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16 These are the New York Stock Exchange, American Stock Exchange and Nasdaq.

89 3.4 Arbitrage pricing theory

Instead of using characteristics of the business, industry or economy, factors can also be obtained with a statistical technique called factor analysis.¹⁷ Consider a data matrix of T (t = 1, 2, ., T) returns for I (i = 1, 2, ., I) assets. It is always possible to completely describe the variance of the T returns with T new variables (the factors) which are linear combinations of the original variables (= returns):

F₁= α₁₁r₁+ α₁₂r₂+ .. + α_1Tr_T F₂= α₂₁r₁+ α₂₂r₂+ .. + α_2Tr_T

.... ...

F_T = αT1r₁+ αT2r₂+ .. + αT Tr_T

where F are the new factors, r are the returns and α are coefficients to be calculated. The rationale of the method lies in the way the factors are constructed (i.e. how the coefficients α are calculated). The most frequently used variant of factor analyses, called principal components analysis, constructs the first factor in such a way that it makes a maximal contribution to the sum of the variances of the T returns. The second factor is constructed such that it is orthogonal to the first and makes a maximal contribution to the remaining sum of the variances of the T returns, etc. In this way, the first few factors describe the bulk of the variance and the other factors can be omitted without much loss of information.

Arguably, factors selected in this way are very close to the factor concept in APT.

A disadvantage of principal component analysis is that it is difficult to decide which factors should be included and which not. Nor does principal component analysis give any economic meaning to the factors; they are simply linear combinations of the original variables (i.e. returns). Factors can be interpreted by inspecting their correlations with characteristics of the business, industry or economy.

Like the CAPM, APT has been tested extensively; Roll and Ross (1980) and Chen et al. (1986) are among the most frequently cited publications. The former study uses factor analysis while the latter analyzes a number of macroeconomic factors. As we have seen, Fama and French’s (1993) three-factor model defines factors by firm characteristics.

3.4.5 Underlying assumptions

APT is built on the assumption that asset returns are generated by a multi-factor pro-cess. The number of factors is not specified, but it has to be small relative to the number of assets, so that portfolios can be formed in which the idiosyncratic risk is completely diversified away.¹⁸APT relies on arbitrage to bring about equilibrium in the form of arbitrage-free prices. This requires markets to be perfect, for otherwise arbitrage would be hampered by market imperfections such as taxes or transaction costs. It also requires that market participants make use of arbitrage opportunities when they occur, which means that market participants have to be greedy (prefer more to less). It further requires that market participants ‘see’ the same opportunities, which implies that they

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17 Explanations of factor analysis can be found in most textbooks on financial econometrics, e.g. Tsay (2005). No longer modern but still very readable is Harman (1967).

18 Strictly speaking, this implies that APT is only asymptotically valid as ε → 0 and only for diversified portfolios.

This reduces APT’s testability, as has been pointed out in the literature.

have homogeneous expectations regarding asset returns and that they use the same factor model.

Clearly, the behavioural assumptions of APT are much weaker than those of the CAPM, which assumes that investors maximize their utility by choosing investments based on their mean-variance characteristics. Moreover, arbitrage concerns the prices of assets rel-ative to each other and not relrel-ative to a market portfolio containing the entire investment universe. So APT ‘works’ on subsets of the investment universe, which makes the char-acteristics of the market portfolio irrelevant. All this underlines that APT is much more general than the CAPM. However, the price of this generality is paid in terms of preci-sion. While the CAPM gives a clear measure and price of risk, APT gives neither. APT does not specify what or how many risk factors to use, nor what the risk premia on the factors are.