The APT framework versus a single-factor alternative

Reinganum (1981) states that although the APT framework is a plausible alternative to the single-factor CAPM framework, the reliance upon a more complicated model is justified only if it conveys more information relative to a simpler model. By this reasoning, the APT framework must provide a better description of returns relative to the CAPM to be considered a replacement and viable alternative. It is within this context that Chen (1983) investigates how the APT model fares against the CAPM. To investigate the cross-sectional explanatory power of the APT and CAPM, factor loadings are used to explain expected returns within the APT model and betas estimated using the Standard & Poor’s (S&P) 500 Index, CRSP value-weighted and equally-value-weighted stock indices are used to explain expected returns within the CAPM. The resultant R s are considered as indicators of explanatory power for the ² respective models. The R for the APT model employing factor loadings is found to be ²

16 The other sub-periods span the February 1970 to December 1975 and the January 1976 to December 1980 periods.

almost double that of the CAPM employing market betas,¹⁷ suggesting that the multifactor APT model is superior in explaining the cross-section of expected stock returns relative to the single-factor CAPM. To establish which model best describes actual returns, (r , actual _i) returns are regressed on expected returns (rˆ ) generated by the APT model and the CAPM:_i ¹⁸

, , variation in r is fully explained by the APT model. Results for each of the sub-periods and _i indices used to estimated market betas are reported below:

Table 2.1: Estimated weights of the expected return from APT and CAPM

Period S&P 500 α

Although, Chen (1983) notes that in a number of instances the estimated α differs from 1, it is clearly evident that the αs are all very close to one in all sub-periods and regardless of the market proxy used. These results are further supported by posterior odds analysis which overwhelmingly favours the APT model over the CAPM. This confirms that the multifactor APT model provides a more adequate description of expected returns relative to the single-factor CAPM. Furthermore, this implies that a multisingle-factor model of the return generating process is also superior in describing returns; if the APT model is superior in describing

17 The average R² for the APT is 0.12 and 0.076 for the CAPM over all sub-periods and market proxies. The average R²for the APT model is defined as the sum of R²over the periods divided by the number of sub-periods. For the CAPM, the average R² is defined as the sum of R² over the sub-periods and market proxies divided by the number of sub-periods multiplied by the number of market proxies used.

18 See Chen (1983: 1398). APT (r_i =λ +λbˆ_i +...λbˆ_ik+ε_it

expected returns and the APT model reflects the underlying return generating process, it may be inferred that the underlying return generating process is superior in describing the time series behaviour of stock returns.

Bower et al. (1984) argue that it is undesirable to adopt a single-factor approach if it can be shown that a multifactor approach provides a better indication of asset risk. While the single-factor framework proposed by the CAPM has furthered the understanding of expected returns on assets, the APT model not only contributes to the understanding of expected returns but also offers a systematic link between expected returns and the return generating process.

Unlike Chen (1983) who only considers the performance of the APT model against an alternative, Bower et al. (1984) consider both aspects of the APT framework. In the first test, the authors use returns on the CRSP value-weighted index to estimate a single-factor model of the return generating process underlying the CAPM. A four-factor model of the return generating process underlying the APT model is also estimated using factor scores.¹⁹ The returns to be explained are returns on stock and bond portfolios over the January 1971 to December 1979 period.²⁰

Returns on each portfolio are regressed on the CRSP value-weighted index and the factor scores. The results indicate that the average R for the multifactor return generating process ² underlying the APT model is 0.869 whereas the average R for the single-factor model of the ² return generating process underlying the CAPM is approximately 0.605. The implications of these results are best summarized by Bower et al. (1984: 1046) who state that “these findings are consistent with a conclusion that the APT provides a better description of the return generating process than does CAPM.” A second test is conducted whereby factor scores are used to explain returns on a holdout sample consisting of the securities of electric utilities, gas companies, telecommunication providers and industrials.²¹ As before, results indicate that the R is greater for 80 percent of individual stocks when the four APT factors are used to ²

19 Factor scores are the values of a given factor at time t derived through factor analysis (Blume, M. Gultekin &

N. Gultekin, 1986). Bower et al. (1984) refer to the return generating processes underlying the APT model and CAPM as the “characteristic line.” This is another name for the return generating process (Ruppert, 2011).

20 Twenty-six portfolios consisting of stocks and four bond portfolios. Another four stock portfolios, consisting of electric utilities, gas companies, telecommunication utilities and industrials are excluded from the estimation procedure and are treated as a holdout sample to test the predictive ability of each model in further tests.

21 The factor scores are derived from a sample which excludes securities in the holdout sample. This addresses the criticism that the high level of explanatory power observed for the APT framework is the result of using factor scores derived from the very same return series that these factor scores are used to explain (see Bower et al., 1984).

model the return generating process relative to the single-factor model characterizing the CAPM. The average R for the multifactor model is 0.323 whereas for the single-factor ² model, the average R is 0.263. Based upon these findings, Bower et al. (1984) again ² acknowledge that the multifactor APT framework is better at explaining the return generating process relative to the single-factor CAPM framework. Although, the authors warn that this superior explanatory power may be attributed to the use of factor scores derived from the returns that they are used to explain, this criticism is addressed by the use of a holdout sample.

It is desirable to extend the finding of superior explanatory power to the cross-sectional APT model, as this will indicate that both aspects of the APT framework - the return generating process and the APT model - are superior relative to a simpler framework. If the APT model reflects the superior explanatory power of the underlying return generating process, then it may be concluded that the superior cross-sectional explanatory power observed in studies focusing upon modelling returns in equilibrium is indicative of the explanatory power of the multifactor return generating process underlying the APT model. This indeed appears to be the case; Bower et al. (1984) find that the four-factor APT model explains over 40 percent of cross-sectional variation in expected returns whereas the CAPM explains just under 30 percent. The authors conclude that although the case for the APT is not absolute, the APT framework appears to be better at explaining both the time series and cross-sectional variation in returns. Moreover, it is demonstrated that both aspects of the APT framework - the return generating process and the APT model - are superior relative to a single-factor model in terms of explanatory power. A linkage between the superior explanatory power of the APT model and the underlying multifactor return generating process is demonstrated.

Beenstock and Chan (1986) compare the adequacy of the CAPM and APT models within-sample and out-of-within-sample by comparing the R of the two models. The APT model ² significantly outperforms the CAPM model in-sample; the average R for the APT model ² over the two sub-periods and three sub-samples is 0.263 whereas the average R for the ² CAPM is negligible.²² Out-of-sample tests are conducted by estimating market betas and factor loadings over a ten-year period and running cross-sectional regressions over each month in the subsequent year (subsequent to estimation sample) starting in 1973. As before,

22 0.009 to be precise.

the multifactor APT model outperforms the CAPM; the average R for the CAPM is 0.023 ² whereas the average R for the APT model is substantially greater at 0.18. Expected returns ² are then regressed on returns predicted (fitted) by the APT model and the CAPM (equation (2.6)). As in Chen (1983), the data favours the multifactor APT model over the CAPM both in-sample and out-of-sample (see Beenstock & Chan, 1986: Table 8). Both Chen’s (1993) and Beenstock and Chan’s (1986) studies suggest that the APT model is superior and better suited to explaining the cross-sectional variation in expected returns relative to the CAPM.

Similarly to Bower et al. (1984), Elton and Gruber (1988) consider both aspects of the APT framework, although the emphasis is on explaining the time series variation in returns and comparing the performance of a four-factor model against the performance of a single-factor model of the return generating process. A four-factor return generating process specification is estimated for returns on size sorted portfolios together with a single-factor model utilizing returns on the NRI 400 stock index as the only explanatory factor. The average R reveals ² that while the single-factor model explains 55 percent of the time series variation in returns, the four-factor model explains 78 percent of the time series variation in returns. These results suggest that a multifactor model is superior relative to a single-factor model in explaining the return generating process of securities. Elton and Gruber (1988) also investigate the consistency of the explanatory power of the single-factor and the four-factor models and find that it differs across the differently sized portfolios. Whereas the single-factor model incorporating returns on the NRI 400 stock index explains between 14 percent and 90 percent of the variation in returns on the smallest and largest portfolios, the explanatory power of the four-factor model lies within a narrower range of between 66 percent and 82 percent respectively. This implies that a multifactor model is far more consistent and uniform in its ability to explain the time series variation in returns. Although Elton and Gruber (1988) also conduct cross-sectional tests of the APT model, unlike Chen (1983), Bower et al. (1984) and Beenstock and Chan (1986), the cross-sectional explanatory power of the APT model is not compared against that of a single-factor alternative. In conclusion, the authors suggest that a four-factor model is sufficient in explaining the return generating process. Notably, the study introduces another criteria upon which to judge the appropriateness of a model; consistency in explanatory power. In this regard, the APT framework is more consistent relative to a single-factor alternative.

The above studies suggest that the APT framework is a superior framework for modelling the return generating process and expected returns. Chen (1983) and Beenstock and Chan (1986) show that the APT model is more adequate and superior in explaining expected returns relative to the CAPM. Bower et al. (1984) show that the APT framework is better at explaining the time series variation in returns and the cross-section of expected returns. Elton and Gruber (1988) find that although only a single-factor is priced in cross-sectional analysis, the multifactor return generating process specification underlying the APT model is superior in a number of respects relative to a single-factor model relying upon an aggregate index.

These findings suggest that the more complex APT framework is superior in explaining return behaviour and therefore, this multifactor framework should serve as a conceptual basis for investigating the return generating process or explaining equilibrium returns.