Deriving the APT when the Number of Factors is Unknown

This paper examines the use of proxies (or reference variables) for the true factors in the Arbitrage Pricing Theory (APT). It generalises the work of Reisman(1992) and Shanken(1992) and shows that, when there are more reference variables than the true factors, the APT still holds. The possibility of fewer reference variables than the true factors is also considered, but the APT is not shown to hold, in the same sense, for this case. This work builds on an earlier paper by Ingersoll(1984), and our propositions can be thought of as specialisations of his Theorems 1 and 6. Our work does not use the mathematics of Hilbert and Banach spaces (as used by Reisman(1992)) and, thus, is open to a much wider audience. Deriving the APT when the Number of Factors is Unknown 1) Introduction The Arbitrage Pricing Theory (APT)*, as proposed by Ross(1976), is an important tool for valuing risky assets. Originally, it assumed that the return from a large number of assets followed a strict factor structure i.e. a linear factor model with uncorrelated disturbances. Ross(1976) does not specify the number or nature of the required factors. Those chosen by researchers range from one to thirteen (and many more for practitioners’ models) and there is disagreement as to their identities. It is necessary, therefore, to consider using reference variables i.e. proxies for the true factors. Shanken(1982) shows that virtually any set of variables can be factors in a well-defined strict factor model. Chamberlain and Rothschild(1983) demonstrate that the Arbitrage Pricing Theory still holds even with an approximate factor model i.e. the disturbances need not be uncorrelated, although the eigenvalues of the residual covariance must be bounded as the number of assets tends to infinity. Ingersoll(1984), Theorem 6, derives a pricing restriction for the case where there are errors in measuring both the expected returns and the matrix of sensitivities to the true set of factors. Our work examines errors in identifying the factors, and, whilst this is mathematically similar to Ingersoll(1984), it is conceptually different. Specifically, we interpret the errors as mis-specifying the nature and number of factors. Thus, we can identify the errors in pricing in terms of characteristics of missing/included factors/proxies. Reisman(1992), Shanken(1992) and Nawalkha(1997) all investigate the situation where the true factors are unknown but the number of true unknown factors is known. To us, this appears to be implausible for the reasons shown below. We briefly present a sample from the many authors who have tried to identify the number of true factors from the data. Roll and Ross(1980) estimate that three factors are definitely priced and a fourth may also be present. Dhrymes, Friend and Gultekin(1984) appear to show that the “discovered” factors equate to approximately 10% of the number of the security group analysed, but they do not put it forward as a proposition, as the exercise was not repeated with many groups. Shanken(1987) finds that there is at least a 99% chance of there being more than ten factors in all cases. Conway and Reinganum(1988) find a cross-validation technique identified one dominant and one minor factor, whilst the likelihood ratio test identified from four to six factors. Connor and Korajczyk(1993) use an asymptotic test for an approximate factor structure and find evidence for one to six pervasive factors. Clare, Priestly and Thomas(1997) show that, by allowing for the existence of an approximate factor structure, five factors, plus a proxy for the market portfolio, are priced in the UK stock market. Chan, Karceski and Lakonishok(1998) find there is nothing to be gained by going beyond the first two or three principal components. From the above, it can be seen that, to date, research to determine the “correct” number of factors gives a range of one to in excess of ten. This work is further aggravated by the fact that, in constructing the model factors, by using factor analysis and/or principal components, there are possible variations in the process of estimating the factors. In this paper, we generalise the work of Reisman(1992) and Shanken(1992) by relaxing the assumption that the true number of factors is known. We show that more, or less, reference variables (proxies) can represent the factors. The number of proxies can be the same or more than the “true” factors and the APT remains valid (Propositions 1&2). We also show, in Proposition 3, that less reference variables can also lead to an APT relationship but that the order of magnitude (as a function of the number of stocks) of the squared pricing errors is higher. These orders of magnitude for the two cases of more or less proxies are interpreted in Proposition 4 in terms of the number of pricing errors that exceed a given bound. Our work shows that any number of proxy factors equal to or greater than the true number of factors allows the average pricing error to converge to zero. Apart from the obvious short-comings of factor analysis, this work may also go some way in helping to explain why so many researchers have come up with so many varying numbers of factors. Our proofs avoid the use of the mathematics of Hilbert and Banach spaces and involve only elementary linear algebra. Section 2 derives general forms of factor structures where proxy factors have replaced the true factors and derives the pricing relations from these factor structures. Section 3 contains some discussion and concluding comments.