Citation |

- Permanent Link:
- https://ufdc.ufl.edu/UF00098626/00001
## Material Information- Title:
- The arbitrage model of security returns an empirical evaluation
- Creator:
- Jordan, Bradford Dunson (
*Dissertant*) Pettway, R. H. (*Thesis advisor*) Cosslett, Stephen R. (*Reviewer*) Heggestad, A. A. (*Reviewer*) - Place of Publication:
- Gainesville, Fla.
- Publisher:
- University of Florida
- Publication Date:
- 1984
- Copyright Date:
- 1984
- Language:
- English
- Physical Description:
- v, 145 leaves ; 28 cm.
## Subjects- Subjects / Keywords:
- Arbitrage ( jstor )
Assets ( jstor ) Automatic picture transmission ( jstor ) Factor analysis ( jstor ) Finance ( jstor ) Modeling ( jstor ) Securities markets ( jstor ) Securities returns ( jstor ) Security portfolios ( jstor ) Statistics ( jstor ) Dissertations, Academic -- Finance, Insurance, and Real Estate -- UF Finance, Insurance, and Real Estate thesis Ph. D Securities -- Mathematical models ( lcsh ) - Genre:
- bibliography ( marcgt )
non-fiction ( marcgt )
## Notes- Abstract:
- Over the last two decades, the Capital Asset Pricing Model (CAPM) has emerged as the dominant theoretical basis for much of the research in financial economics. Because direct observation of the market portfolio is a pre-requisite for any valid application of the CAPM, it cannot serve as a theoretical basis for empirical research in securities markets. The Arbitrage Pricing Theory (APT) is a theoretical alternative to the CAPM in which the market portfolio plays no particular role. The purpose of this research is to develop and test a model of the security return generating process based on the APT. Particular emphasis is placed on two facets of the proposed arbitrage model. First, the central prediction of the APT is an absence of arbitrage opportunities, the empirical identification of which would lead to a rejection of the theory. Thus, the first use to which the model is put is the examination of abnormal performance for the securities individually and jointly. The second application involves an event study comparison of the arbitrage model and a popular variant of the market model. The objective of this comparison is to establish the stability and usefulness of the arbitrage model against a known benchmark. In light of the growing list of empirical anomalies associated with the market model and the difficulties in application of the CAPM, an empirically tractable and theoretically sound model of security returns would be a significant step forward in financial research. The data used in the study are daily returns for individual securities from the CRSP file and cover the period 1962 through 1979. The results indicate substantial support for the APT and the arbitrage model. Significant arbitrage opportunities are found to occur in less than 1% of the individual cases, and the hypothesis of jointly zero abnormal performance cannot be rejected in any case. In the event study comparison, the arbitrage model was found to work at least as well as the market model in all cases and was markedly superior in accounting for the January effect.
- Thesis:
- Thesis (Ph. D.)--University of Florida, 1984.
- Bibliography:
- Bibliography: leaves 139-144.
- General Note:
- Typescript.
- General Note:
- Vita.
- Statement of Responsibility:
- by Bradford Dunson Jordan.
## Record Information- Source Institution:
- University of Florida
- Holding Location:
- University of Florida
- Rights Management:
- Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
- Resource Identifier:
- 000437193 ( AlephBibNum )
11216712 ( OCLC ) ACJ7266 ( NOTIS )
## UFDC Membership |

Downloads |

## This item has the following downloads: |

Full Text |

THE ARBITRAGE MODEL OF SECURITY RETURNS: AN EMPIRICAL EVALUATION By BRADFORD DUNSON JORDAN A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1984 TABLE OF CONTENTS PAGE ABSTRACT . . . . . . . . . . . . iv CHAPTER I. ESSENTIALS OF THE ARBITRAGE MODEL . . . ... 1 Introduction ..................... 1 An Alternative to the CAPM . . .... .. . . .. 2 Testing the APT ......... 4 The Arbitrage Model as a Tool in Financial Research 7 Summary and Overview . . . . . . .... 9 II. PREVIOUS RESEARCH IN MULTI-FACTOR MODELS.. .. . . 11 Introduction . . . ..... .. . .. 11 Applications of Multivariate Statistical Techniques 11 Multiple Regression Models of Security Returns 18 Tests of the Arbitrage Theory . . . . . .. 19 Summary . . . . . . . . . . 23 III. THE ARBITRAGE MODEL: THEORY AND ESTIMATION . .. 25 Introduction ..... . . . . . . . . 25 The Arbitrage Pricing. Theory ......... .. 25 Estimating the Arbitrage Model . . . . . 31 Measuring the Risk Premia .. . . .. ... 35 Summary . . . . . . . . . . . 38 IV. TESTING THE ARBITRAGE THEORY . . . ... .. .40 Introduction . . . . . . . . . ... 40 Factor Analysis of Daily Security Returns ...... 42 Preliminary Analyses of the Arbitrage Model . .. .55 Univariate Results from the Arbitrage Model ..... 72 A Multivariate Test of the APT . . . . . 84 Summary .. ... . .. . . . . . 86 V. AN EVENT STUDY COMPARISON OF THE MARKET MODEL AND THE ARBITRAGE MODEL .......... 107 Introduction . . . . . . . . . . 107 Data for the Study . . . . . . . . 109 An Event Study Methodology . .. . . . 110 ii CHAPTER V. Impact of the Oil Embargo on the Petroleum Refining and Oil Field Services Groups . . . ... 114 Impact of the Con Ed Dividend Omission on the Electric Utility Group . . . . . 116 The Financial Services Group in the Period 8/73 - 9/74 . . . . . . . . . . 118 Some Results on the January Effect . . . . 123 Summary . . .. .. . . . . . . . 126 VI. RETURN, RISK AND ARBITRAGE: CONCLUSIONS . . .. 128 Introduction . . . . . . . . . 128 Testing the Arbitrage Theory . . . . . ... 129 Empirical Findings for the Arbitrage Model ... . 133 Implementing the Arbitrage Model . . . . .. 136 Conclusion . . . . . . . . . . . 138 REFERENCES . . . . . . . . . . . . . 139 BIOGRAPHICAL SKETCH . . .. . . .. .. . . . 145 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE ARBITRAGE MODEL OF SECURITY RETURNS: AN EMPIRICAL EVALUATION By BRADFORD DUNSON JORDAN April, 1984 Chairman: R. H. Pettway Major Department: Department of Finance, Insurance, and Real Estate Over the last two decades, the Capital Asset Pricing Model (CAPM) has emerged as the dominant theoretical basis for much of the research in financial economics. Because direct observation of the market port- folio is a pre-requisite for any valid application of the CAPM, it can- not serve as a theoretical basis for empirical research in securities markets. The Arbitrage Pricing Theory (APT) is a theoretical alternative to the CAPM in which the market portfolio plays no particular role. The purpose of this research is to develop and test a model of the security return generating process based on the APT. Particular emphasis is placed on two facets of the proposed arbi- trage model. First, the central prediction of the APT is an absence of arbitrage opportunities, the empirical identification of which would lead to a rejection of the theory. Thus, the first use to which the model is put is the examination of abnormal performance for the securities individually and jointly. The second application involves an event study comparison of the arbitrage model and a popular variant of the iv market model. The objective of this comparison is to establish the stability and usefulness of the arbitrage model against a known bench- mark. In light of the growing list of empirical anomalies associated with the market model and the difficulties in application of the CAPM, an empirically tractable and theoretically sound model of security returns would be a significant step forward in financial research. The data used in the study are daily returns for individual securities from the CRSP file and cover the period 1962 through 1979. The results indicate substantial support for the APT and the arbitrage model. Significant arbitrage opportunities are found to occur in less than 1% of the individual cases, and the hypothesis of jointly zero abnormal performance cannot be rejected in any case. In the event study comparison, the arbitrage model was found to work at least as well as the market model in all cases and was markedly superior in accounting for the January effect. CHAPTER I ESSENTIALS OF THE ARBITRAGE MODEL Introduction In the broadest sense, the primary concern of research in financial economics is the relationship between risk and return in well-organized markets. While security returns can generally be measured with relative ease, the determination of an appropriate measure of risk is a far more difficult question. Over the last two decades, the Capital Asset Pricing Model (CAPM) has emerged as the dominant theoretical basis for much of the research in this area. The fundamental result of the CAPM is straight-forward: the relevant riskiness for any asset is determined by the standardized covariance of its return with the return on the market portfolio, i.e., the portfolio consisting of all risky assets held in proportion to their value. As a theory, the CAPM is extremely powerful and broadly applicable; however, no valid test of its empirical content has appeared in the literature. For reasons discussed in Roll (1977), such a test requires that the return on the market portfolio be observed directly. Because it is not technologically possible to obtain the necessary data, it is unlikely that a valid test will be forthcoming. For the same reason, any attempt to estimate the parameters of the model introduces bias of unknown magnitude and direction. The impetus for this research stems from the need for a model free of these deficiencies. The purpose of this thesis is to develop and test an empirically tractable model of security returns which retains the 1 2 intuitive appeal of CAPM-based models without the need for the market portfolio in estimation. In the next section, the theoretical basis for such a model is outlined. An Alternative to the CAPM The CAPM is a general equilibrium model of perfect markets with homogeneous investor expectations. In such markets, the CAPM will hold if investors have quadratic preferences or asset returns possess multi- variate normal distributions. When these conditions are imposed, several important results follow. In particular: 1. An asset's expected return is independent of its own volatility; only that portion of its riskiness which cannot be diversified away is relevant. 2. All assets with the same non-diversifiable risk have the same expected return. 3. Asset returns contain two elements, one which is related to changes in the macro-economy and one which is unique to the particular asset. It is the unique portion which is eliminated by diversification. These propositions collectively form the basis for much of the modern theory of finance. Curiously, these propositions are often used in informal derivations to justify the CAPM (see, for example, Brigham (1983), pp. 158-169). However, if the validity of these results is assumed a priori, the CAPM is needlessly restrictive. If securities markets are characterized by risk-averse investors who make decisions based only on expected returns and risk, then the assets will be priced as substitutes and the first two results are no more than simple economic propositions. Any asset which offered compensation for diversifiable risk would have its price bid up until the premium was eliminated. If two assets possessing the same non-diversifiable risk had different expected returns, then investors would sell (or supply) the one with the lower return and demand the one with the higher return. The relative prices would adjust until the expected returns were equal. Moreover, these conditions would hold across any subset of securities. Finally, that the unique portion of security returns can be eliminated by diversi- fication is a property of any collection of imperfectly correlated variables. A certain portion will generally not be diversifiable simply because, to a greater or lesser extent, all asset returns depend on general economic conditions. Ross (1976, 1977) has formalized the kind of reasoning outlined above in his Arbitrage Pricing Theory (APT). The principal assumption of the APT is that investors homogeneously view the random return, r., on the particular set of assets under consideration as being generated by a k-factor model of the following form: r = E + + bil + + bikk + e i 1= ., n (1.1) where E. = the expected return on the ith asset S= the change in the pure interest rate. E[6 ] = 0. th 6j= The random value of the jth common factor. E[6.] = 0, j = 1, . ., k. bij = the sensitivity of the return on asset i to factor j. e. = the random (unsystematic) portion of r.. E[e.] = 0. also E[ei.1K1-0 E[e j 0 0 i j COV (ei, e ) = 2 1 2 .< "i =j el Intuitively, the APT models security returns as a linear function of Ross's formulation omits this term, implicitly assuming a constant risk-free rate. Including it allows for the absence of a risk-free asset, and is similar to Black's (1972) concept of a "zero-beta" portfolio. This issue is discussed in detail in Chapter III. some unspecified state variables plus a random component. By appeal to the law of large numbers, any well diversified portfolio will have virtually no unsystematic risk. It is interesting to note that any linear model (including the CAPM) is a special case of the APT. In this sense, the APT is, as Brennan (1981) has remarked, . a minimalist model since it predicts no more than the absence of arbitrage opportuni- ties . [and] is logically prior to our other utility-based models" (p. 393). Testing the APT Because the APT only predicts an absence of arbitrage opportunities, the identification of such opportunities would lead to a rejection of the theory. An arbitrage opportunity amounts to a constant non-zero portion of return not explained by the factors. In efficient markets, there are two fundamental no-arbitrage properties. First, portfolios with no net investment and no systematic risk must, on average, have no return. Second, portfolios with net positive investment and no systematic risk must have expected returns equal to the pure time value of foregone consumption. The return on such portfolios should equal the risk-free rate if such an asset exists; however, the existence of a risk-free asset is not a requirement of the APT. A test of the APT requires the estimation of the parameters of eq. (1.1). Referring to eq. (1.1), if it is assumed that the random portion of return is completely eliminated, then the no-arbitrage pro- positions imply the existence of k + 1 weights such that Ei = o + Ibil + + kbik' (1.2) where .j is the risk premium on the jth factor and o is the expected return on all portfolios with no systematic risk (this result is formally demonstrated in Chapter III). While the APT provides no insight as to the interpretation of the factor risk premia, it is possible to re-write (1.2) in a more useful form. Consider a portfolio formed such that bpl = bP2 = . = bPk = 0. If the portfolio has positive investment, its expected return is E = x o Next, a portfolio is formed with the property that its return is equal to the risk premium on the first factor; i.e., it is constructed such that bp1 = 1 and bp2 = . = bPk = 0. If it has no net investment, its expected return is E' = . Repeating this process for every factor, equation (1.2) can be written E = E Ebil + . + Ekbik. (1.3) Substituting (3) into (1) and defining Ei as Ei + '., then ri = E + E'bil + . + E bik + ei. (1.4) Equation (1.4) is an empirically useful representation of the APT: here the ex post return on the ith security is expressed as a linear combina- tion of the "zero beta" return (E") and the returns on the k arbitrage portfolios. Again, if k is taken to be one and E is interpreted as the market risk premium, then (1.4) is the ex post two-parameter CAPM (Black 1972)). Assuming that the returns on the k + 1 arbitrage portfolios can be determined (discussed in detail in Chapters III and IV), it is possible to test the APT. To accomplish this, the returns on n assets and the arbitrage portfolios are collected for some time period. Then for each security, a time-series regression is estimated of the form % o ^ 1 ^ r = bo + . kiE + ei (1.5) Because E measures the "zero beta" return, boi should equal unity. The intercept term, a, can be interpreted as a measure of abnormal performance and should not be significantly different from zero. A simple test of the predictions of the APT would consist of estimating the parameters of (1.5) subject to the constraints a. = 0 and boi = 1. The restricted estimate can then be compared to the unrestricted results using a standard F-test. If the constraint is binding in a substantial number of cases, then the APT may be rejected in that its predictions would be inconsistent with the data. Such a procedure, while intuitively appealing, suffers from at least two drawbacks. This approach has no objective decision rule. If the hypothesis were rejected in, say, 40% of the trials, would it then follow that the APT is invalid? Secondly, this approach requires that the contemporaneous residual covariances between returns be equal to zero. While this is formally an assumption of the theory, eq. (1.2) can be expected to hold as an approximation so long as the residuals are sufficiently independent for the law of large numbers to be operative. Hence, small, though significant correlations are not precluded. A number of large correlations would be indicative of an omitted factorss. For the reasons outlined above, a valid test of the APT requires that the cross-sectional dependence among the parameters be considered. Whether or not eq. (1.2) holds exactly is largely irrelevant (and probably untestable). With this theory, as with any theory, it is the extent to which its predictions are consistent with observed phenomena that is of interest. Brennan (1981) has remarked "[For an adequate test] . whatt is required is a test of the hypothesis that the intercept terms for all securities are equal to zero, though such a test may be difficult to construct" (p. 393). A consistent pattern of non-zero intercepts would be indicative of arbitrage opportunities, a result at odds with the arbitrage theory (and, for that matter, most of modern portfolio theory). Thus, a test of the APT amounts to testing whether the intercept terms are jointly different from zero, i.e., a pooled time-series and cross- sectional approach. Such a test is particularly appealing because, as shown in Chapter IV, it is formally equivalent to testing the following: H : there exists no well-diversified portfolio with zero systematic risk and zero net investment which earns a significantly non-zero return. vs. HA: such a portfolio exists. This is a powerful test; if, for any collection of assets, a single arbitrage portfolio (out of an arbitrarily large number) can be identified with a non-zero return, the APT will be rejected. This is a strongly positivist test as well. The null hypothesis is literally the central prediction of the theory; thus, it is strictly the content of theory which is examined, not the assumptions. On the other hand, the theory is tested against an unspecified alternative; moreover, the test is conditional on the measurement of systematic risk. As a result, rejecting the theory does not necessarily invalidate the model. If the view is adopted that "it takes a model to beat a model," then the return generating function of eq. (1-4) is interesting in its own right. In the next section, the use of the model as an alternative to current practice is discussed. The Arbitrage Model as a Tool in Financial Research In financial research it is often desirable to specify a model of security returns which controls for the differential riskiness of the assets. Once this is accomplished, it is possible to analyze the effect of other variables (e.g., dividend yields) or events (e.g., unanticipated information) on security returns. To this end, the so-called "market model" has been widely employed (see the June, 1983 Journal of Financial Economics for some recent examples). The return generating process specified by this model may be written ri = Ei + (rm E)bi + ei, (1-6) where (r Em) is the deviation of some broad-based market index from its expectation. The popularity of the market model can probably be traced to its simplicity, intuitive appeal, and similarity to the theoretical CAPM. However, as pointed out by Ross (1976) and more fully developed by Roll (1977, 1978), this similarity is more apparent than real. The model is in many ways closer to the APT than the CAPM; nonetheless, numerous shortcomings have been identified in the market model's ability to explain returns (e.g., Ball (1978), Banz (1981), Basu (1977), Reinganum (1981a)). The arbitrage model of eq. (1-5) is an empirical alternative to the market model. Unlike the market model, the arbitrage model has a solid theoretical basis while retaining a certain simplicity and intuitive appeal. Thus, a comparison of the usefulness of the arbitrage model with that of the market model is a logical step. One of the more popular uses of the market model has been the residual analysis methodology pioneered by Fama et al. (1969). Mandelker's (1974) study of the gains from mergers and Jaffee's (1974) research into the value of inside information are prime examples. The ability of this methodology to detect abnormal performance (systematic price changes unexplained by overall market movements) has been studied by Brown and Warner (1980). Their simulation results indicate that the procedure works quite well when the event date is known. Studies of stock price behavior around various types of events are based on market efficiency. In an efficient market, prices should adjust rapidly and fully to new information. In this study, the residual behavior of the two models is compared around several known events. This comparison addresses two issues. First, because the market model is known to perform well in this type of study, the substantive results from the arbitrage model should be similar. Second, the consistency of the two models with the concept of efficient markets is of interest. The more consistent model would show greater pre-event adjustment, more rapid adjustment about the event date, and less drift subsequent to the event. This comparison also addresses the issue of stability of the estimated parameters. To the extent that the arbitrage model provides better resolution of the information in the residuals, it may judged to be a superior model of the return generating process. Summary and Overview The objective of this dissertation is twofold. First, Shanken (1982) has argued that no truly valid test of any theory of asset returns has appeared in the literature. The methodology employed in Chapter IV to test the APT is free of the problems identified in previous work and is actually quite general. Similar approaches could have broad applicability. Second, the market model suffers from both theoretical and empirical deficiencies. An alternative model with a stronger theoretical foundation and better empirical properties would be a significant step forward in financial research. The present study is organized in six chapters. This chapter, the first, constitutes a brief outline of the need for research in this area and procedures by which it can be accomplished. Chapter II is a review of the relevant prior research in multi-factor models. Chapter III develops both the APT and the arbitrage model, as well as outlines the methodologies to be employed. In Chapter IV, the results of the tests a0 10 of the APT are presented. In Chapter V, the empirical properties of the model as an alternative to the market model are evaluated and reported. Chapter VI summarizes the major findings, suggests topics for future research, and concludes this study. CHAPTER II PREVIOUS RESEARCH IN MULTI-FACTOR MODELS Introduction The Arbitrage Pricing Theory outlined in Chapter I provides a theoreti- cal foundation for asset pricing without the stringent general equilibrium restrictions of the CAPM. Despite the theoretical justification and intuitive reasonableness of multi-factor models, empirical research has been dominated by the single-index "market" models. An extensive literature exists on the statistical properties of the model itself, and a number of authors have employed the model as a means of controlling for differential asset riskiness or general market conditions. Despite the popularity of this approach, research has been undertaken in three areas directly related to the arbitrage model. These areas are (1) purely empirical applications of multivariate statistical techniques (principally cluster and factor analysis), (2) multivariate regression models based on a priori assumptions as to the number and identity of the relevant factors, and (3) tests of the APT. Much of this research preceded the development of the APT and it is interesting to re-examine the empirical results obtained in an arbitrage model context. The next three sections examine this research and its implications for the arbitrage model. Applications of Multivariate Statistical Techniques When a group of variables exhibits a high degree a linear correlation or "redundancy," several dimension-reducing techniques are available to summarize the data in a more parsimonious fashion.1 Because security What follows is intended as a very brief, intuitive description. A good introduction to cluster analysis may be found in Elton and Gruber (1970). Factor analysis is taken up in detail in the next chapter. returns are often highly correlated, cluster and factor analysis have been applied in efforts to establish the existence of an underlying structure in the data. With either technique, it is hypothesized that the variables are elements of a k-dimensional subspace, where k is "small" relative to the numbers of variables. In either case, k is unknown a priori. With cluster analysis, the objective is to assign each variable to one of k homogeneous groups. In its simplest form, a cluster analysis of security returns begins with a full rank correlation matrix of returns. The two securities with the highest correlation are combined into a single variable, thereby reducing the rank of the correlation matrix by one. The correlation matrix is then recomputed with the new variable and the reamining n-2 securities. The two variables with the highest correlation in the new matrix are combined and so on. The process is continued in an iterative fashion until no significant correlations remain between some number of "clusters." However, no completely objective rule exists for determining the appropriate number of clusters. In the general factor analysis model, security returns are assumed to be characterized by a set of hypothetical or latent variables. The returns are expressed as a linear combination of these variables plus a random (or unique) portion.2 Like cluster analysis, factor analysis usually begins with the estimated correlation matrix. Using one of several techniques, an estimate of the percentage of total variance which is unique is obtained for each asset. The main diagonal of the correlation principal component analysis differs from factor analysis. In component analysis, no distinction is made between random and non-random portions. This point is discussed in the context of research which has used this approach. matrix (consisting of ones) is adjusted by subtracting this estimated "uniqueness." The result for a particular asset is an estimate of its "communality," i.e., that portion of its total return which is systematic. If the errors are assumed to be uncorrelated across securities, then the resulting adjusted correlation matrix can be interpreted as an estimate of the common intercorrelation. The next step is to construct an artificial variable which accounts for a maximum of the common variance. Next, a second variable (generally constrained to be orthogonal to the first) is constructed which accounts for a maximum of the remaining variance. This procedure is continued, yielding k variables which account for all the estimated common variance.3 Several objective criteria are available for determining k. A discussion of these is deferred to Chapter III. Both cluster and factor analysis are generally employed as explana- tory techniques and results obtained thereby are purely empirical. However, if the elements of a particular cluster have similar character- istics, it may be possible to formulate hypotheses for further testing. Similarly, if a given factor is particularly related to some group of securities, it may be possible to infer the identity of the factor. Regardless of the validity of such heuristics, the research examined below relates to the existence of multiple factors in security returns as well as the number of relevant dimensions. One of the earliest studies to employ dimension reducing techniques was that of Farrar (1962). Farrar applied the principal component approach to 47 industry groups in an effort to create a relatively small If the factor model fit perfectly, the reduced correlation matrix would be rank k. In practice, k is regarded as the approximate rank, allowing for measurement error and non-linearities. number of asset groups with low first-order correlations. He found that the first five components accounted for about 97% of the total joint variation among the industry groups, with the first component capturing 77% of the total. Examination of his results (p. 41) indicates the presence of a single, dominant factor with at least two additional significant factors. The principal component approach was also applied by Feeney and Hester (1967). The purpose of their research was to objectively develop weights for a stock market index. Using the 30 securities in the Dow Jones Index, they found that the first two components (of the covariance matrix) accounted for 90% of the total variance, with the first component accounting for 76%. Interestingly, the correlation between the Dow Jones Index and the first component was found to be in excess of .99. The results from the component analysis are nearly identical to those found by Farrar, despite the different samples and time periods employed. In 1966, King investigated the nature of the latent structure of security returns. His work is of particular importance because he recognizes both the presence of a market factor and the existence of unsystematic (or unique) effects. The explicit purpose of King's study was to determine whether inter-relationships among security returns could be attributed to a market factor and an industry factor correspond- ing to a two-digit SIC classification. Using a sample of 64 stocks in six industry groups, King performed both a mixed factor/cluster analysis and a multi-factor analysis. In the mixed analysis, he extracted the first factor (the market factor) and clustered the remaining variation. When the maximum correlation between groups dropped below .20, the group corresponded exactly to the SIC two-digit classifications. When a seven factor solution was obtained, the same pattern emerged; all securities were sensitive to the general market effect and an industry factor. Also, King found that the first factor accounted for 74% of the estimated total systematic variation; however, his results differ from those of Farrar and Feeney and Hester in that the subsequent factors (particularly the second and third) were not as pronounced. Also, the relative importance of the market factor in explaining the systematic or common variation was found to decline over time, from a high of 63% in the sub-period June 1927 to September 1935 to a low of 37% for the period August 1952 to September 1960. At the time of King's study (1966), the Sharpe (1963) single-index model was gaining popularity as a simplification to the general Markowitz (1959) portfolio problem. The validity of this model hinges on the absence of contemporaneous residual correlations among the assets. King's findings are at odds with this requirement. In a 1973 study, Meyers extended King's methodology to include less homogeneous industry groups, as well as the time period 1961-1967. After extracting the first princi- pal component, Meyers clustered the residual correlation matrix and found results generally supportive of King's; however, he does identify a weakening of the industry effects. Meyers then extracted six components from the residual correlation matrix and reported evidence of industry effects similar to King's, though with significantly less clarity. Meyers concludes that King's results overstate the importance of industry effects, but he concurs in the finding of residual covariance unexplained by a general market effect. The relative strength of industry effects was examined in 1977 by Livingston. In this study, a number of important issues are identified; in particular, Livingston documents that the principal components approach is inappropriate in that it tends to extract more common variance than actually exists. To determine the magnitude of industry effects, Livingston proceeded to regress returns from 734 securities (in over 100 industries) on the S & P Composite Index return. Next, the residual correlation matrix was examined for significantly non-zero correlations. Within industries, 20% of the correlations were found to be significantly different from zero, with very few negative elements. Across industries, 6% were significantly positive and 2% significantly negative. However, some of the industries examined showed little residual correlation. Livingston concludes that a single-index model ignores a significant portion of the co-movement in security returns and that the use of industry indices should improve the results. Such models have been constructed and are reviewed below. The most general conclusion which can be drawn from the King, Meyers, and Livingston research is that extra-market covariation does exist, but it is not clear whether the effect is related to industry classification per se. An alternative explanation could be offered to the effect that certain types of businesses are particularly sensitive to different macro-economic factors. If this proposition is correct for the members of a homogeneous industry group, then an "industry effect" will appear to exist. Because factors such as interest rates, foreign exchange rates, inflation, input prices (raw materials and wages), and so on do not move in lockstep, firms with particular dependencies on any one factor will exhibit "extra-market" influences. This is simply due to the averaging implicit in the construction of a market index.4 Studies by Farrell (1974, 1975) and Arnott (1980) have used cluster analysis to define groups of securities in terms of their return characteristics as opposed to industry classification. Farrell used a stratified (across industries) sample of 100 securities. He computed the residual correlation matrix from a single-index model. These residuals were clustered until no correlation above .15 remained. The results of this procedure were four clusters which Farrell labels as growth, stable, cyclical, and oil. Arnott used 600 securities and a somewhat less stringent rule to halt the clustering process. His results indicate five clusters which he labels quality growth, utilities, oil and related, basic industries, and consumer cyclicals. The results of the two studies are actually quite similar; the primary difference is that the Farrell study combines the utility, basic industry, and consumer cyclical into two clusters, the stable and cyclical. Both of these studies are generally supportive of a multi-factor model, where the factors are some set of macro-economic variables rather than simple industry effects. The multivariate studies reviewed in this section have, in varying degrees, a similar result: a single index model ignores potentially useful information about the co-movement of security returns. The techniques used in these analyses are all forms of correlation analysis; no model or theory is employed. In the next section, several models which attempt to incorporate extra-market information are examined. In the case of a value-weighted index, the averaging is in terms of the characteristics of the largest firms versus the most numerous in the case of an equal-weighted index. Multiple Regression Models of Security Returns Several authors have sought to improve the single-index model by including additional variables. In an early effort, Kalman and Pogue (1967) compared the ability of single and multiple index models to recreate the Markowitz efficient frontier and to predict correlation matrices. Their results indicate little, if any, benefit from a multi- index approach. Farrell (1974) criticizes the method used by Kalman and Pogue in constructing the multiple indices, attributing the lack of success to the high degree of collinearity among the industry indices. Using the relatively uncorrelated clusters (described in the previous section) in addition to a general market effect, he reports superior results when compared to a single index formulation.5 In another study examining the ability of various models to predict correlation matrices, Elton and Gruber (1973) test ten different models of security returns. They find that three models outperform all other techniques, including the single index and several multiple index models. The three models differ in their assumptions concerning the pattern of correlation coefficients. The overall mean model sets all coefficients equal to the average. The traditional industry mean sets all correlations within an industry equal to the industry average, and all inter-industry correlations are set equal to their average. The third model is the same as the traditional industry with the exception that industries are defined by a principal component solution ("pseudo-industries"). Elton Farrell extracts the market effect by regressing the cluster returns on a market index and using the residuals as "explanatory" variables. This procedure creates orthogonal indices by construction; however, such an approach is suspect on econometric grounds. It is difficult to justify the use of random noise (i.e., the residuals) from one estimation as "explanatory" variables in another. and Gruber's results indicate that superior forecasting is possible using information not produced by index models. Unfortunately, their multi-index models are based on principal component solutions and the assumption of zero residual correlations is inappropriate. Other studies have used information beyond a general market effect in estimation. Rosenberg (1974) assumed the general validity of the single index approach, but he used a number of firm-specific descriptorr" variables to obtain forecasts of the parameters. Lloyd and Schick (1977) have tested a two index model proposed by Stone (1974), where the additional index is composed of debt instruments. Langetieg (1978) adopted an approach similar to Farrell's, using orthogonalized industry indices to measure gains from mergers. All of these studies find benefits in the use of extra-market information, but they lack a theoretical underpinning. The arbitrage theory provides this missing element, and studies incorporating it directly are reviewed in the next section. Tests of the Arbitrage Theory The first published study of the APT is credited to Gehr (1975). Gehr constructed two samples of 360 monthly returns, one consisting of 24 industry indices and the other of 41 individual companies. He next obtained a three component solution for the 41 companies. The industry returns were then regressed on the components to estimate the sensitivity coefficients. A second-pass regression of the mean industry index returns against the coefficients was performed as the final step. Of the estimated risk premia, only one is found to be generally significant. An empirical anomaly associated with the market model has been investigated by Reinganum (1981b) and Banz (1981). When portfolios are formed based on firm size, small firms earn significantly greater rates of return, even after accounting for difference in estimated betas. Reinganum (1981a) has examined the same question using an arbitrage model. Essentially, Reinganum forms a set of control portfolios based on ranked factor loadings. Then, the returns on the control portfolios are subtracted from corresponding individual security returns. The resulting excess returns are ordered into deciles based on market equity values, and the average excess return is computed for each decile. Reinganum's results are similar to those found using the market model: portfolios of small firms offer a risk-adjusted return significantly greater than the portfolios of large firms. Thus, Reinganum rejects the arbitrage model as an empirical alternative to the simpler market model. Oldfield and Rogalski (1981) have examined the influence of factors estimated from Treasury bill returns on common stock returns. As a first step, they gather Treasury bill returns for 1 to 26 week maturities. The one week return is then subtracted from the subsequent maturities to calculate excess weekly returns. The one week rate is reserved for the risk-free rate. Next, the excess T-bill returns are factored and factor scores are computed.6 Next, individual common stock returns are regressed on the factor scores, yielding a set of sensitivity coefficients. The stocks are then randomly assigned to intermediate portfolios, and the covariance matrix of the returns among the portfolios is calculated. Using this covariance matrix, a minimum variance portfolio is calculated for each factor with the property that a particular portfolio is sensitive to that factor, with a zero loading on the others. Additionally, a minimum variance portfolio is formed with no sensitivity to any factor 6Factor scores are estimates of the population factors; hence they constitute a time-series of measurements of the factors. (a "zero-beta" portfolio). The weekly returns on these factor port- folios is computed, and these are used in time-series regressions to re-estimate sensitivity coefficients. Their first result from the procedure is that significant correlation exists between common stock returns and the factor portfolio returns. Next, the authors run a cross-sectional regression of the weekly inter- mediate portfolio returns and their factor loadings in each of 639 weeks. They then compare the mean regression coefficient for a particular factor with the mean return on the factor portfolio. They argue that the two should be equal, and find no statistical difference. By including an equal weighted market portfolio, the authors find that the significance of the factor portfolios is greatly diminished, a result which they attri- bute partially to the collinearity between the variables. The authors report that the estimated risk-free rate is significantly less than the corresponding T-bill rate, while the cross-sectional intercepts are not different from zero. Fogler, John, and Tipton (1981) have also attempted to relate the returns on debt and equity instruments in the context of the arbitrage theory. The basic data for this study were excess monthly returns on 100 securities divided into seven groups. The first four groups were selected on the basis of Farrell's cluster analysis, consisting of stocks classified as growth, stable, cyclical, and oil. The other three groups correspond to the pseudo-industries developed by Elton and Gruber (both studies are reviewed in a previous section). The authors next calculate excess monthly returns on a value-weighted market index, a three month Treasury bond index, and a long-term Aa utility bond index. The excess returns were calculated by subtracting the return on a one month Treasury bond. Next, the excess returns on the securities were regressed on the three indices; of the three, only the market index had generally significant coefficients. The authors report that some of the groups display consistent signs on other indices; however, no non-parametric results were included. In a second part of their study, Fogler, John and Tipton extract a principal component solution from the 100 securities, retaining the first three. They then examined the canonical correlation between the components and the three indices. From this analysis, one important result emerges: the correlation between the three components and the market index is near- ly perfect. Also, in some sub-periods there is a statistically signifi- cant relationship between the components and the three month Treasury bond yield. Whether or not the authors have achieved their goal of "imparting economic meaning to the stock returns factors" (p. 327) is difficult to say; yet they implicitly establish an important empirical result; namely, the return on the overall market can be decomposed without loss of information about the market while potentially including other relevant information. Thus, while their study is not actually a test of the APT, it nonetheless suggests a certain empirical rationale for the theory. A final study deserving of particular attention is that of Roll and Ross (1980). This study is a straightforward extension of Gehr's methodology. The authors form 42 portfolios of 30 securities each, using ten years of daily security returns. A factor solution is then obtained for each group. For each group, a cross-sectional GLS regression of mean returns on the factor loadings is estimated. The authors report that at least three factors of the five used are "priced" in the results. Next, an additional variable, the standard deviation of return, is included in the cross-sectional regressions. After correcting for the positive dependence between sample mean and sample standard deviation arising from the positive skewness in daily returns (by using non-overlapping samples), little support is found for the hypothesis that returns are related to total volatility. As a final test, Roll and Ross test for cross-sectional differences in the intercepts from the cross-sectional regressions. To do so, they employed Hotelling's T2 statistic to account for cross-sectional dependencies in the estimates. Their results indicate no significant difference, lending support to the APT. Summary The preceding three sections have reviewed research in three areas-- purely empirical analysis of stock market groups, multiple regression models based on a priori knowledge of the relevant variables, and studies testing the APT, either directly or indirectly. Of the multivariate studies, the results obtained from a variety of different approaches are consistent in that they generally indicate that a single index model ignores significant facets of security returns. This conclusion is reinforced by the multiple regression studies in that the additional variables specified add significant explanatory or predictive power despite their ad hoc nature. The APT offers, in principle, an empirical alternative. The studies published to date using it all suffer from serious methodological flaws; in addition, no tractable multi-index model based on the APT has been forthcoming. Because many of the methodological problems in the literature stem from a misapplication of factor analysis, a discussion of them is deferred to the next two chapters where the application of factor analysis to security returns is addressed. Problems also arise in the development 24 of testable hypotheses in an arbitrage pricing framework and with the nature of the appropriate return generating function. These three issues--factor analysis of security returns, testable hypotheses of the APT, and the structuring of a return generating function--are inter-related to the extent that the validity of any one of the three depends on the other two. In other words, the theoretical justification for a multi- factor return generating function obtained from a factor analysis is found in the APT. However, a test of the APT requires a return function obtained from a factor analysis procedure. Finally, a number of factor analysis procedures are available; the choice of a particular one depends on both the APT and the desired form of the empirical model derived therefrom. The next chapter considers each of the subjects independently before combining them into the arbitrage model. CHAPTER III THE ARBITRAGE MODEL: THEORY AND ESTIMATION Introduction In the previous two chapters, the need for an alternative model of security returns was established and evidence for the validity of a multi-factor representation was examined. In the first section of this chapter, the theoretical basis for such a model is illustrated. In the second section, the relationship between the APT and the general factor analysis model is developed. The results of these sections are used to derive an empirical model of returns and to establish the testable hypotheses of the APT. The Arbitrage Pricing Theory The APT was originally proposed by Ross (1976, 1977). A simplified approach was derived by Huberman (1982). The theory has been generalized and extended by Ingersoll (1982). The exposition in this section draws heavily from these three sources. The principal assumption of the APT is that investors homogeneously view the random returns, r, on the particular set of assets under consid- eration as being generated by a k-factor linear model of the following form: r = E + B6 + e, (3-1) where Strictly speaking, complete homogeneity of investor expectations is not required. Ross (1976) has established that the existence of non- negligible agents with upward bounded relative risk aversion and homogeneous opinions about expected returns are sufficient. As Ross notes, however, translating ex post occurences into ex ante anticipations will require homogeneity. En x 1 = B = nx k bJ changes in common factor j (factor loadings) k x 1 en x = It is also assumed that E[e] = 0 E[6] = 0 E[es'] = 0 _2 -2 E[ee'] = = ij < i= 0 i j. In other words, the deviation of the return on asset i from its expecta- tion is a linear combination of the random values of the k factors and a unique, residual component. The residuals are assumed to be independent of the factors and mutually uncorrelated. In the absense of a riskless asset with a constant certain return, eq. (3-1) may be written r = E + A + e, (3-2) where (k + 1) x = < .> = the random values of the k common factors S x with as the change in the "zero beta" return An (k+l) = i = <1> = the sum vector (a column vector of ones). Heuristically, the arguments underlying the APT begin with the considera- tion of a portfolio vector, x, chosen such that x'a = o. The components of x are the dollar amounts invested in each asset. Since the total invest- ment is zero by construction, all purchases (long positions) are financed 27 by sales (short positions).2 If x is a well-diversified portfolio with each xi of order 1/ in absolute magnitude, then by the law of large numbers, the dollar return on x is x'r = x'E + x'AS + x'e -x'E + (x'A)6. (3-3) If x is chosen to have no systematic risk as well, then the return is x'r x'E. (3-4) Taking a to be any non-zero scalar, then ax is an arbitrage portfolio. If it is assumed that the random portion of (3-3) can be completely eliminated by diversification, then (3-4) holds with equality and it must be the case that x'r = x'E = 0, (3-5) or unbounded certain profits are possible by increasing the scale (a) of the arbitrage operation. If this condition holds for all portfolios constructed in the manner described above, then there exist constants E = AX, (3-6) where A is the augmented factor loading matrix. Algebraically, (3-6) is simply the statement that all vectors orthogonal (perpendicular) to A are orthogonal to E if and only if E is in the span of the columns of A. This result and several others can be illustrated by introducing the following notation: 2In the absense of restrictions on short selling, such portfolios can always be constructed. Even with short selling restrictions, investors with positive net holdings can, in effect, engage in such activities by buying and selling. Letting w be the dollar amounts invested in the n assets (with w'e = W, the investor's net wealth), then, assuming no transactions costs, the difference between w and any other portfolio, w, is an arbitrage portfolio: w + x = w. Thus, an investor who changes his relative investments is implicitly purchasing an arbitrage portfolio. S = span {A}, where A is assumed to have full column rank S = set of all vectors orthogonal to S with orthogonal basis x = < .., x n-k-l>. By construction, then S IS = TR SnS = {0} x'A = 0 i = 1, ..., n-k-I S0 i = j j = 0 i j. Equation (3-6) follows from the no-arbitrage assumption; either EFS or arbitrage is possible. To see this, note that E can always be written E = AX + z, (3-7) where zeS But z is itself an arbitrage portfolio with return z'E = (z'A)X + z'z = z'z f 0. (3-8) So (3-6) must hold to prevent arbitrage. Following Huberman (1982), the results obtained above can be extended to the case where the residual portion of return is not completely elimi- nated. The objective is to establish an upper bound on the sum of the squared deviations from the pricing relationship (3-6). The APT considers a sequence of economies with increasing numbers of risky assets. The n economy has n risky assets whose returns are generated by a k-factor model, where k is a fixed number. Arbitrage is defined as the existence of a subsequence, z n, of arbitrage portfolios with the properties lim zE = (3-9) n-' lim var(z'E) = 0. (3-10) n-fm Intuitively, arbitrage possibilities exist whenever increasing profits at diminishing risk are obtainable as the number of assets grows. Put another 29 way, the reward to volatility ratio increases without limit. To preclude such occurrences, there must be an upper bound to (En AXnn)' (En An ), (3-11) the sum of squared deviations in the nth economy. Referring to (3-4) and assuming that z is scaled such that z'z increases to infinity with n, (the subscripts are understood) z'r = z'E + z'e, (3-12) substituting (3-7) for z'E z'r = z'z + z'e. (3-13) Letting a be a scalar between 1/2 and 1 and defining y = (z'z)- then Yz is an arbitrage portfolio with expected return and variance Yz'E = Yz'z = (z'z) 1-c = E[z] (3-14) and 2 -2 2 -2 1-2a z'E[ee']z = < a y (z'z) < (z'z) (3-15) By construction lim E[z] = = (3-16) n*= 2 lim a = 0. (3-17) n+~m For example, if a is taken to be 3/4, the expected return increases with the fourth root of z'z and the variance decreases with the square root. The reward to volatility ratio is E[z] > (z'z)1- (3-18) 1/2- = zl a a (z'z)1/2-a which does not have a lower bound unless z'z is bounded. From (3-7), z = E Ax; hence the foregoing suffices to show that unlimited deviations from the pricing relationship (3-6) give rise to arbitrarily large profits. If such profits are precluded by assumption then it follows that (E Ax)' (E Ax) < M. (3-19) Inequality (3-19) indicates that the permissible sum of squared deviations is less than some finite number for any number of assets. As a consequence, as the number of assets increases, the approximation improves. The reverse is, of course, for any finite set of assets, the approximation can be quite poor. The existence of a finite bound on the arbitrage pricing relationship suggests a natural test of the APT. For A and E given, it is possible to determine X such that the sum of the squared deviations from the pricing relationship is minimized, i.e., minimize (E AA)'(E AX). (3-20) Equation (3-20) is simply the OLS estimate of E on B. The resulting sum of squared errors could be compared to the bound in (3-19) and the APT rejected if the bound is exceeded. Such a test, however, requires that an a prior bound be established. In the absence of such a specification, several authors (notably Roll and Ross (1980)) have attempted to verify the APT by establishing the linearity of (3-6). As pointed out by Shanken (1982), the linearity of (3-6) is not literally an implication of the APT for any finite collection of assets. For any such collection, (3-6) is an approximation and will have a finite bound on the sum of the squared errors. To take a polar case, in the absence of any linear relationship for a particular set of assets, the bound would be equal to the sum of the squared expected returns. Such a result would not necessarily invalidate the APT because the permis- sible upper bound on the pricing relationship is not known a priori; hence, an examination of the degree of linearity in the pricing relationship is without power to reject the APT. The central prediction of the APT is the absence of arbitrage oppor- tunities where arbitrage is defined as a non-zero return on a well 31 diversified portfolio with no net investment and no factor risk. The empirical identification of such opportunities would lead to a rejection of the theory. In the next section, the estimation of a k-factor model to be used in testing the APT is discussed. Estimating the Arbitrage Model The APT is a theory of the structure of asset returns. In the theoretical development of the previous section, it was assumed that the matrix of factor loadings (and the number of factors) was known. Because the APT provides no insight into the nature or number of the factors, it will be necessary to infer both from observed security returns. Techniques for accomplishing this fall under the general heading of factor analysis. In this section, a particular type of factor analysis due to Lawley (1940) is outlined, following Joreskog (1967) and Lawley and Maxwell (1971). It is easiest to conceive of factor analysis as a form of linear regression in which the number and identity of the regressors is unknown. Factor analysis is then a technique by which (for k regressors) the coefficients and variables of the linear model are simultaneously deter- mined. Based on the assumptions of the previous section, the covariance matrix of returns is E[rr'] = E[B5 + e)(B6 + e)'] = BE(66')B' + E[ee'] = B B; + 4, (3-21) where i is the covariance matrix of the factors. Let Q be a matrix satisfying Q'Q = I and Q Q' = I.3 Equation (3-21) can be written 3Such a matrix exists for any symmetric matrix of full rank. The columns of Q are the eigenvectors of 4 scaled by the square root of their respective eigenvalues. See Frieberg et al. (1979). E[rr'] = B(Q'Q) 6 (Q'Q)'B' + = (BQ')(Q'B)' + = B*B*' + p. (3-22) Thus, without loss of generality, it can be assumed that the factors are mutually uncorrelated with unit variances. When the factors have this relationship, the insertion of any full rank orthonormal matrix leads to a mathematically equivalent solution. Thus there is an infinity of mathematically equivalent factor loading matrices. A linear transformation of this type is termed a rotation and geometrically amounts to a rigid motion of the factor axes to a new set of coordinates. This indetermi- nateness is not a problem with the APT since no interpretation of the factors is necessary; researchers are free to choose any convenient orientation. Naturally, this lack of uniqueness would make the task of "identifying" the factors difficult and any interpretations of them suspect. If it is assumed that the vectors 6 and e follow multivariate normal distributions, then the elements of the sample covariance matrix S have a Wishart distribution with t 1 degrees of freedom (t is the number of observations). In this case it is possible to obtain maximum likelihood estimates for B and t (again, for a given k). Following the usual technique of maximizing the log-likelihood function, it can be shown (Lawley and Maxwell (1971), p. 26) that an equivalent procedure is to minimize the function F(B,P) = Inl Z + tr(S-I1) In s n, (3-23) where Z is the hypothetical covariance matrix and n is the number of variables. No direct solution exists, so numerical techniques are used to find the minimum value and the resulting estimates B and . The principal attraction of the maximum likelihood approach is that it allows a test for the number of common factors. Denote by L(n) the 33 maximum of the likelihood function for k unrestricted and let L(; ) be the maximum under the null hypothesis of exactly k factors. If X is the ratio of the restricted maximum to the unrestricted, it is well known (e.g., Mendenhall and Schaeffer (1973)) that -21nx converges in distribu- tion to x with degrees of freedom equal to the number of parameters or functions of parameters assigned specific values under the null hypothesis. In the case of a factor analysis, the number of parameters estimated in the unrestricted model is the sum of n variances and the 1/2(n2 n) unique covariances, for a total of 1/2n(n + 1). There are nk unknowns in B and n unknowns in P. Without further restrictions, the matrix B is not uniquely defined. For computational reasons, it is convenient to require that B'PB be a diagonal matrix. This has the effect of imposing 1/2k(k-1) restrictions; hence the total number of free parameters is nk + n 1/2k(k-1) and the degrees of freedom are 1/2n(n + 1) nk n + 1/2k(k 1) = 1/2[(n-k)2 (n + k)]. (3-24) 2 In fact, the value of the computed X statistic is simply t times the minimand of eq. (3.23), explaining its use in the estimation procedure. The maximum likelihood procedure provides for a test of k = k , where ko is a prespecified number of factors. In essence, the hypothetical covariance matrix constructed using only k0 factors is compared to the saturated (sample) covariance matrix and if the discrepancy is found to be sufficiently small, the hypothesis is not rejected. In the usual exploratory case, it is not possible to specify a pre-determined value of k; instead, what is desired is an estimate of the dimension of the model. The procedure adopted is to begin with a small hypothesized value of k. If the hypothesis is rejected, k is increased by one and the test repeated. The dimension of the model is taken to be the smallest value of k which 2 yields a non-significant x at a predetermined significance level. Because the ultimately determined value of k depends on a sequence of prior tests, the assumptions of the classical Neyman-Pearson theory are violated. Thus, the test can only be interpreted as a test of sufficiency, and practice has shown that the value of k arrived at by this procedure using conventional significance levels is greater than the number of relevant factors; hence, the test should be regarded as conservative in the sense that it is unlikely to lead to an underestimate of the true number of factors (Lawley and Maxwell (1971), Harman (1976), Horn and Engstrom (1979)). With this approach, it is important to note 2 that significant x values lead to the fitting of more factors; hence, greater significance levels will lead to fewer factors being retained. Horn and Engstrom's (1979) results from a related criterion indicate that, for large samples, significance levels in excess of .999 are warranted. Because of the tendency for the maximum likelihood approach to over-estimate the number of substantially important dimensions, two other criteria will be examined. Akaike (1973, 1974) has proposed an information theoretic loss function as an extension to the likelihood approach and Schwarz (1978) has developed a large-sample Bayesian criterion. All three criteria are related. Schwarz (p. 461) indicates that his and Akaike's approach amount to adjusting the maximum likelihood estimator. If M(kj) is the value of the likelihood function for k factors, then Akaike's procedure results in the selection of k such that InM(k-) Inkj is largest and Schwarz's criterion results in the selection of k such that InM(k.) 1/2kjlnt is maximized. In large samples, the three criteria can lead to very different estimates of k. Schwarz's criterion will lead to smallest estimate and the maximum likelihood procedure the largest. The results from applying each of these standards to daily security returns are reported in the next chapter. Several other points about the maximum likelihood approach are of interest. A consequence of the weighting scheme implicit in the procedure is that the resulting estimates are scale-free; hence, correlations may be used instead of covariances. In regard to the distributional assump- tions underlying the approach, Howe (1955) has demonstrated that the same loadings and residual variances result from maximizing the determinant of the estimated common correlation matrix; thus the approach is valid as a descriptive measure regardless of the underlying distribution. The principal drawback relates to the computational resources required to obtain the solution. The CPU time required varies exponentially and appears to be proportional to the fourth or fifth power of the number of variables. Sample sizes are thus limited. Also, convergence of the numerical algorithms employed need only be local; thus the solution obtained may not be the global maximum. In any event, once the number of factors and their associated loadings are determined, it is still necessary to measure the factor risk premia. This topic is the subject of the next section. Measuring the Risk Premia The final step in constructing a multi-factor model of security returns is to use the information from the factor analysis to estimate the time-series behavior of the factor risk premia. In this section, the estimates of the factor loading matrix, B, and the diagonal residual covariance matrix, i, are taken to be fixed; hence the carats are dropped. Also, it is assumed that the estimates of B and P are obtained from the covariance matrix, so the typical element of B is the covariance of the return on security i with common factor j. The typical element of P is 36 the uniqueness for asset i multiplied by its variance, i.e., the portion of its variance not associated with the common factors. The approach adopted here is similar to methods used by Oldfield and Rogaliski (1981), though with some important differences. The general technique employed is to partition the observations into two groups: an estimation (base) period and a test period. The data from the base period are used to obtain a factor solution. Next, for each factor, a portfolio is constructed with unit sensitivity to that factor subject to the constraints that it use no wealth and that it have no correlation with the other factors. Also, a zero beta portfolio is formed possessing positive investment and no correlation with the factors. Without further constraints, the weight vectors for each of the portfolios are not unique; for any particular one there would n-k-l linearly independent choices plus any number of linear combinations. For a particular factor, then, the weight vector is chosen such that its unsystematic portion is minimized. For example, the weights for the zero beta portfolio, x are the solution to minimizing x'px subject to x' = 1 and x'b = x'b = . x'b = 0, 0 0 0 0 0 k ok where the b. are the loadings on the ith factor. This program is then repeated for each factor. More generally letting ci be the standard unit vector, the weights for the k + 1 portfolios, x, are the solutions to Min x'.Wx. i = 0, . ., k x. s.t. xA = ci. The solution is found by introducing k + 1 Lagrange multipliers and minimizing L(xi, xi) = xi.xi 2\i (x.A ci), (3-25) where Xi is the row vector of the multipliers. Thus L(xi,i) xi = 0 = > xi A = 0. (3-26) L(xi Xi)/ri = 0 = > xIA ci = 0. (3-27) Defining x as solutions for all k + 1 weight vectors can be written as ' A A= 0 (3-26a) x'A I = 0. (3-27a) Eliminating the multipliers and solving for x yields X = -1A(A'-1A)-1. (3-28) Once the portfolio weights are obtained as in (3-28), the data from the base period are discarded. This is done to avoid the circularity inherent in using the same data to fit and test the model. In particular, it is desirable for testing purposes to have estimates with known distributions. For example, one of the tests performed by Roll and Ross (1980) consists of regressing mean returns cross-sectionally on the factor loadings. A standard t-test is used to evaluate the significance of the estimated risk premia. Clearly, any results obtained by such a procedure depend on the previously determined factor solution and especially on the number of factors determined to be significant. Such a procedure seems little dif- ferent from determining a one factor solution and running such a regression, then obtaining a two-factor solution and repeating the regression, and so on until the kth factor is insignificant. A discussion of the problems with such pre-test estimators may be found in Judge et al. (1980, pp. 54-94). Once the weights are obtained, the returns on the arbitrage port- folios are calculated using the test period data. Letting R be the n x t matrix of individual security returns, the arbitrage portfolio returns are R'X = R'-1A'(A'~-1A)-1. (3-29) Estimates obtained in this fashion have some interesting features. First, 38 they are equivalent to the estimates obtained from running cross-sectional GLS regressions of the security returns on the factor loadings for each day, and the mean returns on the arbitrage portfolios are equivalent to regression coefficients of mean security returns on the factor loadings. There is no question of "statistical" difference as considered by Oldfield and Rogalski (1981). Second, the estimated weights are actually just factor scoring coefficients estimated with Bartlett's approach, and are unbiased estimates of the true factors (Lawley and Maxwell (1971), p. 109). Bartlett's method also produces factor estimates that are univocal, i.e., uncorrelated with the other factors (Harman (1976), p. 385). Finally, the use of the inverse of the residual variances in (3-29) has the effect of correcting for the heteroscedasticity of the residuals and, all other things being equal, places greater weight on those securities with greater common variances. The final step in estimating the arbitrage model is to use the factor risk premia as independent variables in time-series regressions of the form rt = + bo + bji1 + . + bjkbk + ejt, (3-30) for each security. Equation (3-30) is the basis for the tests of the next chapter and the event study comparison with the market model in Chapter V. Summary In this chapter, the theory and estimating procedure underlying the arbitrage model were discussed. The result is a model in which security returns are linearly related to a set of unspecified, though measurable, latent variables. The next step is to compare the empirical results obtained from its use with the predictions of the underlying theory. To the extent that they are not in accord, the model loses one of its prime justifications. The last step in developing the arbitrage model is of a 39 more practical nature. Theoretical consideration aside, the arbitrage model is somewhat more involved than other models (and more expensive to use), so its performance relative to simpler models is a subject worth investigating. CHAPTER IV TESTING THE ARBITRAGE THEORY Introduction In this chapter, attention turns to the empirical issues of the arbitrage model. Previous chapters have addressed the need for a multi- factor model and examined evidence suggesting its appropriateness. In the last chapter, an arbitrage model was specified as an empirical analog to the APT. Because the content of the model stems directly from the predictions of the arbitrage theory, it becomes a natural vehicle for establishing the general validity of the APT. For all of its simplicity and intuitive appeal, the arbitrage theory is rather limited; all that is indicated is that the return generating process has an approximate linear dimension less than the number of risky assets in the economy. It is important to realize, however, that the dimension reduction is the content of the theory, not the factors them- selves. If the dimension of the structure of security returns is known, the factors are implicit in that structure. In this sense, they are best viewed as continuous versions of Arrow-Debreu pure or "primitive" securities. In the continuously distributed case, the number of states is equal to the number of securities; hence, n linearly independent securities are necessary to exactly span the state space. The arbitrage theory amounts to the assertion that the state space is approximately spanned by k + 1 linearly independent vectors, and that the degree of approximation improves as the number of securities increases. Interest in the theory, then, should not stem from the possibility of interpreting 40 the factors as some collection of macro-variates. In fact, there is nothing in the theory which suggests that such an interpretation exists, certainly the theory does not require it. The APT is not a causal theory; security returns are merely associa- ted with some collection of measurements. These measurements are best interpreted as indices, the level of which have no intrinsic meaning. The best that can be said is that the indices are aggregate measures of the information sets and expectations of market participants. Changes in the levels of the factors are consequences of the arrival and assimila- tion of information and its role in the formulation of market expecta- tions, a process about which little is known. While the theory is mute in regard to the nature and number of the factors, estimates derived therefrom must have certain properties if the theory is to be empirically testable or practically usuable. Formally, all that is required from the theory is that lim k = 0. (4-1) n-*o- n Thus, for any finite collection of assets, the number of factors can be quite large, and that number can change as assets are added. In the non-stable case, where the number of factors and the factor loadings change as assets are added, the theory is probably of limited practical interest. What is hoped instead is that the economy has a fixed, finite number of sources of risk and that k< theory in this light is inversely related to the ratio in (4-1), at least for a particular market. The arguments above suggest that the practical interest in the APT is related to the degree of empirical parsimony possible through its use. 42 This subject is of interest in its own right and is taken up in the next section. Once the dimension of the model is fixed, subsequent sections report various univariate results obtained from the model, and, finally, a multivariate test of the APT. While a fixed dimension and relative stability of the factor loadings across securities are sufficient for research in this area to be interesting, one additional requirement is that the estimates contain a degree of intertemporal stability sufficient to justify their use vis-a-vis other, simpler models. Evidence relating to this stability is a by-product of the results of this chapter; however, a discussion of the implications is deferred to Chapter V. Factor Analysis of Daily Security Returns The purpose of this section is to arrive at an estimate of the number of relevant factors. The basic data consist of daily holding period returns including dividends on nearly 5000 New York Stock Exchange and American Stock Exchange listed securities extracted from the Center for Research in Security Prices (CRSP) (1983) daily returns file. The computations are performed on the University of Georgia IBM 370 (MVS/OS) using the maximum likelihood factor routine in the Statistical Analysis System (1982), 1982b version. The first analysis performed is similar to that of Roll and Ross (1980). Returns for the first 1250 trading days in the CRSP file (7/3/62 6/19/67) were assembled for the first 30 securities (alphabetically) with a complete return series over the period. A maximum likelihood solution was obtained for one factor, two factors, and so on up to eight factors. The results are summarized in Table 4-1. TABLE 4-1 SUMMARY INFORMATION FACTOR ANALYSIS RESULTS Number of Securities: 30 Number of Observations: 1250 Sample Period: 7/3/62 6/19/67 Number Comouted Degrees Probability Schwarz's Akaike's of x2 of K Factors Bayesian Information Factors Value Freedom Sufficient Criterion Criterion 1 526.78 405 <.0001 479.98 652.10 2 447.97 376 .0063 543.69 630.74 3* 371.70 348 .1830 605.09 609.86 4* 321.38 321 .4836 675.99 613.15 5* 282.11 295 .6953 748.92 625.58 6* 243.94 270 .8709 818.80 637.07 7*+ 214.85 246 .9248 889.70 655.73 8*+ 187.45 223 .9600 957.87 674.06 TABLE 4-1 (continued) COMMUNALITY ESTIMATES Number Total Percentage of Total Estimated Communality Attributaole to of Estimated Each Factor Factors Communality (Percent) 1 2 3 4 6 7 3 1 17.60 100 2 19.57 91.47 8.53 3* 19.34 88.24 11.76 4- 21.11 81.34 10.70 7.96 5* 22.44 76.81 9.94 7.40 5.84 6' 21.92 70.97 12.21 9.31 7.51 7* 20.73 64.27 15.46 10.53 9.75 8* 83.39 56.66 19.47 13.37 5.89 1.87 SQUARED CANONICAL CORRELATIONS Number Squared Canonical Correlations for Each Factor with the Variables of I (Percent) Factors 1 2 3 4 5 6 7 8 1 84.08 2 84.30 33.37 3* 100.00 78.96 33.33 4" 100.00 79.40 33.65 27.38 5* 100.00 79.69 33.67 27.44 22.98 6* 100.00 100.00 73.78 32.62 25.95 22.94 7* 100.00 100.00 100.00 65.70 31.54 23.88 22.52 8- 100.00 92.79 81.56 75.24 57.25 29.81 26.34 22.93 Indicates a Heywood Case + Indicates a lack of convergence after 15 iterations, convergence is approximate. 45 Several interesting results emerge from this analysis. First, at the 5 percent level, the hypothesis that three factors are sufficient cannot be rejected (Table 4-1). Akaike's criterion reaches its minimum at three as well, but Schwarz's criterion only picks up the dominant first factor. The three factor result agrees with that ultimately obtained by Roll and Ross (1980). A problem that can arise with the algorithm employed is the potential for Heywood cases. A Heywood case occurs when the factor model is a perfect fit for one or more of the variables (i.e., a communality of one). The likelihood function is dis- continuous at such points. The solution adopted here is to delete the offending variable(s) and fit k 1 (or k minus the number of eliminated variables) factors. Heywood cases arise from the numerical algorithm; at each iteration, the securities are weighted by the reciprocal of their uniqueness. Variables with greater communalities are thus given greater weight. If there are too many factors relative to the number of variables, the uniqueness can approach zero, assigning an extremely large weight to a particular security. For example, in the three factor solution, the communality for one of the securities moves from .49 to 1.00 in five iterations. It is removed from the sample and convergence is established in two additional iterations, where covergence requires that no changes in the communality estimates exceed .001 in absolute magnitude. Once the variable is deleted, two factors are fit to the remaining 29 variables. The communality estimates in Table 4-1 refer to the variance explained by the factors collectively and individually. They are obtained in the following way: the hypothetical population correlation matrix is repro- duced using k factors after weighting each variable by the reciprocal of its uniqueness. The percentage of variance explained by the factors is just the trace of reproduced correlation matrix divided by the trace of 46 the sample correlation matrix (which is simply the number of variables). So, in the two factor solution, an estimated 19.57 percent of the total variance is systematic. Of that systematic portion, the first factor accounts for 91.47 percent with the second accounting for the remaining 8.53 percent. Information regarding the adequacy of the sample size is contained in Table 4-1. The squared canonical correlations for each factor with the variables are measures of the extent to which the factors can be predicted from the variables and can be interpreted as squared multiple correlation coefficients. Inspecting Table 4-1, the first 2 factor can be predicted with reasonable accuracy (R = 88.24%), but there are insufficient variables to accurately measure the others. The perfect correlations stem from the Heywood cases. The Heywood cases and low multiple correlations for factors beyond the first indicate a need for a larger sample. In an attempt to improve the results, the sample size was doubled to 60 securities, selected sequentially from the CRSP file beginning with the thirty-first security with complete returns data. The results are reported in Table 4-2. With 60 securities, the results are not drastically different from those obtained with the smaller sample. Again, Schwarz's criterion points to the single dominant factor. Based on Akaike's criterion, a five factor solution is optimal, though only slightly better than a four factor representation, and, depending on the significance level chosen, the maximum likelihood criterion would also indicate five factors at the most. The communality estimates obtained with the five factor solution are quite similar to those obtained with three factors in the smaller sample, indicating again that about 20 percent of the total variance is systematic. TABLE 4-2 SUMMARY INFORMATION FACTOR ANALYSIS RESULTS Number Computed Degrees Probability Schwarz's Akaike's of x2 of K Factors Bayesian Information Factors Value Freedom Sufficient Criterion Criterion 1 2235.56 1710 <.0001 1566.12 2516.54 2 1920.02 1651 <.0001 1616.36 2314.28 3 1779.84 1593 .0007 1752.23 2288.44 4 1638.73 1536 .0339 1883.99 2259.50 5 1522.11 1480 .2180 2024.60 2253.39 6 1432.15+ 1425 .4419 2175.20 2272.38 7 1344.32+ 1371 .6915 2323.26 2291.43 8 1264.53 1318 .8515 2471.82 i 2316.62 i Number of Securities: 60 Number of Observations: 1250 Sample Period: 7/3/62 6/19/67 TABLE 4-2 (continued) COMMUNALITY ESTIMATES .lumoer Total Percentage of Total Estimated Communality Attributable to Eacn of Estimated Factor Factors Communality (Percent) 1 2 3 4 5 6 7 8 S14.25 100.00 2 16.27 90.17 9.83 3 17.47 85.12 9.35 5.53 4 18.66 80.56 8.91 5.35 5.18 5 19.86 76.81 8.58 5.22 4.96 4.43 6+ 21.66 72.25 8.41 6.79 4.58 4.43 3.56 7+ 22.20 70.47 8.02 5.62 4.56 4.41 3.53 3.40 8 22.99 68.37 7.83 5.C6 1.60 4.27 3.41 3.38 3.08 SQUARED CANONICAL CORRELATIONS Number Squared Canonical Correlations for Each Factor Aith the Variables or (Percent Factors 1 2 3 4 5 6 7 8 1 89.53 2 89.80 48.98 3 89.92 49.50 36.69 4 90.02 49.96 37.45 36.70 5 90.15 50.56 38.35 37.13 34.55 6+ 90.37 52.21 46.86 37.31 36.54 31.62 7+ 90.37 51.64 42.80 37.77 37.01 31.96 31.15 8 90.41 51.93 41.10 38.81 37.37 32.01 31.80 29.79 +Indicates a lack of convergence after 15 iterations, convergence is approximate. 49 As indicated in Table 4-2, there is a significant increase in the squared canonical correlation for the second factor, rising from about 33 percent to 50 percent. The increased sample size also eliminates the troublesome Heywood cases, with no communalities exceeding .50. The algorithm converges rapidly for up to five factors, generally requiring no more than five iterations. Although convergence was achieved for the eight factor solution (fifteen iterations), the six and seven factor attempts did not converge. The conclusion from this analysis is that no more than five factors are needed to account for the systematic inter- correlation for sixty securities. These analyses were repeated for several different samples with the same general results: not more than five factors are needed. As a final check, the two samples were combined with ten other securities for a total of one hundred. The results in Table 4-3 indicate that the five factor solution originally indicated is insufficient. As indicated in Table 4-3, factor solutions for up to six factors converge rapidly (as few as three iterations); however, the maximum likeli- hood criterion indicates that even a seven factor solution is inadequate. With eight or nine factors, the algorithm was unable to find even an approximate solution. Since Horn and Engstrom's (1979) results indicate the use of a larger significance level with the larger sample, an eleven or even ten factor solution is indicated. Akaike's criterion indicates six factors with Schwarz's criterion again identifying only the dominant factor. The use of the larger sample results in general improvement in the predictability of the first several factors; however, the communality totals are relatively unchanged. The results from the different sized samples contain some similari- ties. First, the Bayesian criterion always indicates a single factor. 4 C 01 i &r 4) o C CO oj r- cO o cM 00 CD m 3u (nCD -0 o io I 0 o L0O,44 44*.- n^ 00%T 0400 0^ 040400o s_. ^ 04 )0 . . . .- .4 .0 . 04u tr a o to o o to to to to to to S n)-^ u3 C^ y3 C^ in O~ < I^ CM J O N .- 0 % 0 0 t 0 t ut- l 0 o no - i- no- n- to 04 L 0044 to 04 to 04 II @0 040 t~o O .'01> 0 04 0) L0 0 040 04 i 04 C 04 UU 0~ 0 0 0 0 0 0) LI to to Ct aO- 00 0 000 00 0 0 O o 4 V4VVVV4oL ~~ ~C)C 0.. Uy f, . 4' 0 O 0 f 0a 0.-~ n to O I) ^- i) LI) to o 4-0 LI I Ol o t O @0 04 L. 0U on n- nn nn no ^ ^ T O i O LI 3 0 o 0 0 L. to @04 0.4'- 01 '- 0 G i- 0) to to i- 04J n 'f 0 O M 04 O to oa- t 04 q O ^ 0 0 o r*. 0 0 n^ ^ @0 4 1 0 4 0)i- CT L- I. + (U 0 .1. + *H .0444- + *K 0 i- 04 0) E Ot. i-0 es dn ^o rn K i i i 0 0 (U l - CM p 0I 4- 0 0 00 I.. C. .> .0 0. i-I r^N o- rj c t r - <3- C i- tj CM cij CM 0 N- CO- to 0 C t w o m CM c 'Jr~ i N -N 0 to a o- m- N^ i.- (1 CM Cj -M 0 m tO t r- 09 CO L 0 rc .- h. r o N .- . U 2 (0~~ ~ ____________ LU ------------------------------ 0 4-CT CT (^ r. 4^ 3 ou ^ 0 i- o-n no n .1 o^<5ClC ~ E r^, co i> d-m e ~ ~ N l iD m o (' +j mh h ~ . . . LU O r" Ci cJ cJ en CM o *- r-. 0 COi U, e 0 CM m r- r^. Ll *- O o ) F N- c r r2~ z 2; -9 z LA O 1 C C9 1 4 0 C c o 0 CT c'J Ci CO Lf+ 4f-' E c- U, r- CO O (NJ (f tO r. rC 0, 0 -4 + * + O M 04 SO U CM cf B U t N- ,- ^- i- ,, s Z" Ll- ~ ~ d~ 0 0 0 0 0 CMS r-- 0 ^ 0 C o oC 1^ in o o t CO r- ^T 0 0- o N0CJ 0- c o tc o 0 0 l to 0 0 n n m o g 8 MS 1 0 0 0 0 0 to CO0 NJ C ^ff S O 0 to N- 00 002 n NJ 0 NJ 00 tOtN N- 0 00 ON 0J CC0 CJ C0 to NJ CO 00 COh O in 0 LO LO L r^ Fia N~ . .. (U N cr U ii^ """ m n ^ a-r- c~~o o ^i < o 00 NJNJ O CIO 00 N- 0NJ tnnoto t 0 0 0 0 VIa + 0 + + 1 ;-+0 + 0 0 *- NM C 0 C, S-C SCt XS- 00 C, 00L 0 04 000u 00m vrrr C~ 0 000IT O44 U -o 0~ I- 0 Based on the description in Chapter III, this is not surprising; all other things equal, large numbers of observations (1250 in this case) will tend to greatly reduce the number of significant factors in this approach. The classical maximum likelihood estimate behaves in a relatively predict- able fashion as well with the number of factors determined to be significant varying with the sample size. In fact, the number of factors appears to be roughly proportional to the number of securities at 8 to 10 percent. Akaike's criterion is relatively stable, indicating five factors with sixty securities and five to six with one hundred. An important similarity between the results is the behavior of the estimated total communality; one conclusion which can be drawn is that the portion of daily security returns which is systematic is approximately 20 percent. Of this portion, the market factor accounts for about 75 percent or 15 percent of the total variance. If the one factor solution is viewed as the best possible single index, then the inclusion of multiple indices can result in a 33 percent (.05/.15) improvement in systematic risk estimates. Moreover, inspection of the factor loadings indicates that this potential gain is not uniformly spread among the securities; in some cases, the variance explained by factors beyond the first is greater than the single factor explanatory power--evidence of significant "extra- market" sources of risk. As a final attempt at objectively determining the approximate dimension of the model, the analysis of the one hundred securities was extended to include an additional five hundred observations per security. The results were virtually identical with those obtained in the previous trial and are not reported here. The very similar results stem from the fact that there is little difference in the sample correlation matrices despite the different lengths of time. When the sample size is increased to 150 securities, the computational requirements become prohibitive, with a six factor solution requiring more than forty minutes of CPU time. In comparison, the six factor solution with 100 securities required only twelve minutes. This section began in an attempt to arrive at an objective determina- tion of the number of factors. Based on analyses of different sample sizes, only one criterion is completely consistent regarding the number of factors. Schwarz's Bayesian criterion indicates that the basic intuition of the market model is correct; namely, there is a single dominant factor present in security returns. On the other hand, the communality estimates indicate significant gains to be had from additional information, particularly for certain securities. The fact that the variance explained by factors beyond the first is concentrated in a subset of the securities improves the case for a multi-index approach. If the securities in the subset are relatively homogeneous, then single-index models may systematically mis-price their riskiness. This potential bias could account for anomalies associated with the market model. Some evidence for the validity of this speculation is offered in Chapter V. For the reasons discussed above, the dimension of the arbitrage model must be specified subjectively. The decision was made to continue this research using six factors, the number indicated by Akaike's criterion. This choice also agrees with the five extra-market clusters identified by Arnott (1980). If anything, six factors would appear to be more than enough. Referring to Table 4-3, factors beyond the fourth individually account for less than one percent of the total variation and less than five percent of the estimated communality. Other techniques are available for arriving at a decision in regard to the number of factors. The two most popular are Cattell's (1966) scree _~ chart and Kaiser's (1960) eigenvalue criterion. Both approaches are based on the relative magnitudes of the eigenvalues of the sample correla- tion matrix. Cattell's method amounts to inspecting a chart of the eigenvalues plotted in order of decreasing magnitude and looking for breaks in the pattern. Because the average eigenvalue of a correlation matrix is unity, Kaiser's rule is to retain all eigenvalues with values greater than one. Neither of these approaches is especially enlightening for the securities data; the only clear break in the ordered eigenvalues occurs at the second factor and the number of eigenvalues exceeding one is quite large (23 out of 100 in one case). In summary, there is sufficient noise in the ex post realized returns that it is impossible to objectively determine the number of factors beyond the first. Grouping techniques would no doubt reduce the noise content, but the results of this section indicate that randomly formed portfolios would probably swamp the extra-market components. It appears likely that the number of factors present ex ante is less than six, and evidence for this will be forthcoming in the tests of the APT. In the next section, the time periods and sampling techniques used in this study are described along with some properties of the six factor solutions in different samples. Preliminary Analyses of the Arbitrage Model The basic data used to test the APT once again consist of daily security returns from the CRSP file. Three non-overlapping base periods were chosen for the factor analyses, each covering 1250 trading days (about five years). Within each of three base periods, three samples of 100 securities each were created by taking every tenth security with a complete returns series from the CRSP file beginning with the first, second and third listed securities. Interval sampling was used to avoid 56 undue concentration in certain industries; for example, alphabetic selection would result in one group being dominated by financial institutions--over thirty have names beginning with the letters of "Fi." To avoid confusing the samples, the notational convention of designating the time periods as 1, 2, and 3 and the samples as a, b, and c is adopted; thus, sample 2b is the second sample in the second time period. The exact dates for the base periods are Base Period 1: 7/3/62 6/19/67 Base Period 2: 7/20/67 7/11/72 Base Period 3: 7/12/72 6/23/77. Once the data were obtained, a six factor maximum likelihood solution was obtained for each of the nine groups. Table 4-4 summarizes the results. The total communality estimates in the first two base periods are all between 20 and 25 percent; however, the third period totals are greater; between 26 and 32 percent of the total variance is systematic. The market factor accounts for 71 to 80 percent of the total communality, with increased importance in the third period. As discussed in Chapter III, any orthogonal linear transformation of the initial solution generates a mathematically equivalent result. Because the APT is designed to explain cross-sectional differences in security returns, a weighted Varimax rotation (Cureton and Mulaik, 1975) was used to increase the cross-sectional variation in the factor loadings. With this rotation, the factor loadings for the individual securities are first weighted by reciprocal of their uniqueness estimates; then, an orthogonal transformation matrix is determined such that the variance of the loadings on a particular factor is maximized. By maximizing the variance of the column loadings, the larger estimates are increased and the smaller estimates decreased. The weighted Varimax rotation also TABLE 4-4 SUMMARY INFORMATION Base Period 1: 7/3/62 6/19/67 Base Period 2: 6/20/67 7/11/72 Base Period 3: 7/12/72 6/23/77 Trading Days per Period: 1250 Samples per period: 3 Number of Securities per Sample: 100 INITIAL SOLUTION BASE PERIOD 1 BASE PERIOD 2 BASE PERIOD 3 SAMPLE a b c a b c a b c TOTAL COMMUNALITY 22.05 24.10 21.41 21.13 22.45 21.90 26.10 31.75 23.07 PERCENTAGE OF TOTAL a b1 c a b c a b FOR EACH FACTOR 1 70.97 76.65 75.41 76.44 77.15 75.58 78.37 78.48 79.48 2 10.00 7.77 7.94 8.05 6.75 9.16 7.23 7.43 6.87 3 7.44 5.18 5.67 4.59 5.47 4.69 4.36 4.87 4.16 4 4.27 3.79 4.53 4.05 3.95 4.01 4.07 3.36 3.58 5 3.86 3.32 i 3.44 3.51 3.44 3.41 3.11 3.19 3.19 6 3.46 3.09 1 3.02 3.36 3.25 3.15 2.86 2.67 2.72 increases the "gain" on factors two through six at the expense of the first factor. As indicated in Table 4-5, the total communality is spread fairly evenly over the first four factors as a result. This effect should make it possible to measure the factors beyond the first with greater accuracy. The three test periods in this study are, for each sample, the five hundred trading days subsequent to the base periods. The exact dates are Test Period 1: 6/20/67 7/25/69 Test Period 2: 7/12/72 7/08/74 Test Period 3: 6/24/77 6/18/79. To calculate the returns on the arbitrage portfolios, the weights are obtained using eq. (3-28) X = 4-A(A-1A)1 (3-28) where A is the augmented factor loading matrix and is the diagonal matrix of residual variances. The arbitrage portfolio returns over the test periods are calculated using eq. (3-29) R'X = R'V-1A(A -A)-1, (3-29) where R is the 100 by 500 matrix of daily returns. As noted in Chapter III, the arbitrage portfolio returns calculated in this fashion are identical to a time-series of coefficients obtained from 500 GLS cross- sectional regressions, with the zero beta return as the time series of estimated intercept terms. Summary univariate statistics are reported for the nine samples in Tables 4-6 through 4-14. In the first test period, covering 7/67 7/69, the zero beta return is insignificantly different from zero in all three samples. Point esti- mates of the average daily return range from a negative 2.6 basis points to a positive 3 basis points, with standard deviations of 60 to 70 basis points. Averaging the three point estimates and annualizing the results, TABLE 4-5 VARIMAX ROTATED SOLUTION BASE PERIOD 1 BASE PERIOD 2 BASE PERIOD 3 SAMPLE a b c a b c a b c TOTAL COMMUNALITY 22.05 24.10 21.41 21.13 22.45 21.90 26.10 31.75 28.07 PERCENTAGE OF TOTAL a b c a b c a b c FOR EACH FACTOR 1 17.62 32.99 25.94 26.11 22.94 33.65 23.31 29.64 27.53 2 17.20 27.19 19.23 25.03 20.23 29.14 21.03 24.88 23.16 3 21.37 15.00 20.06 19.73 16.42 19.73 19.91 18.79 15.02 4 19.42 13.64 14.21 16.95 19.55 9.35 17.78 14,24 15.34 5 14.57 6.12 15.04 7.40 12.77 4.80 12.07 8.50 10.50 6 9.82 5.07 4.50 4.78 5.06 3.34 5.90 3.95 8.46 i ----------_____ ____ ____ ____ ___ \ ____ ____ ____ ____I a return in the neighborhood of 1 percent is indicated, at a time when money market rates varied from four to six percent (Federal Reserve Bulletin, 1970). In the second test period (7/72 6/74), the zero beta returns are larger and vary from 2 to 7 basis points per day. An approximate annual yield of 18 percent is obtained by averaging the three. Once again, the standard deviations of the estimates are large relative to the point values. Also, the standard deviations are similar to each other, ranging from 89 to 106 basis points. A low estimate of .4 basis points per day was obtained in the third period (7/77 6/79) with a high estimate of 5 points. The average of the three is about 2 points per day or a 7.2 percent annual yield which compares favorably with the 7.19 percent yield on 90-day Treasury Bills for the year 1978 (Federal Reserve Bulletin, 1979). The standard deviations range from 49 to 63 basis points. Because of the variability in the zero beta return, the estimates are not especially reliable; however, the average estimates obtained in the first and third periods are fairly close to the then prevailing interest rates. One problem with the GLS estimates as opposed to OLS estimates is that unbiasedness is achieved at the expense of greater variance in the estimated factor scores. When the zero beta returns are estimated using OLS, the averages in basis points per day for the three periods are 2.2, 3.53, and 1.47 respectively. The estimate in the first period is quite close to the 1968 daily return on 90-day Treasury Bills which averaged 2.18 points per day (Federal Reserve Bulletin, 1970). The estimate in the second period is still high, though less so than with the GLS estimate, and the third period estimate appears low. It is difficult to generalize from these results; however, there appears to be no tendency for the estimates to be consistently higher than some measure of the risk-free rate, unlike TABLE 4-6 SUMMARY INFORMATION Test Period: la Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .00006 .0067 .199 .842 Arbitrage *1 .1128 2.0702 1.219 .224 Arbitrage 2 .0317 1.2539 .566 .572 Arbitrage 43 -.1503 1.0789 -3.115 .002 Arbitrage A4 .0223 1.1348 .440 .660 Arbitrage C5 -.0120 1.3648 .197 .844 Arbitrage '6 .0577 1.3576 .950 .343 Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta #1 #2 #3 A4 05 '6 :ero Beta 1.000 Arbitrage 1i .750 1.000 Arbitrage ': .190 .109 1.000 Arbitrage -3 .074 .036 .280 1.000 Arbitrage -1 .084 .006 .109 .044 1.000 Arbitrage 5 .192 .002 .166 .201 .041 1.000 Arbitrage 6 .160 .220 .152 .141 .111 .175 1.000 TABLE 4-7 SUMMARY INFORMATION Test Period: Ib Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic Zero Beta -.00026 .0061 .944 .345 Arbitrage 41 -.1180 .9780 -2.698 .007 Arbitrage 42 .1947 2.0155 2.160 .031 Arbitrage '3 .0143 1.3830 .232 .818 Arbitrage 44 .0232 1.0805 .480 .631 Arbitrage 45 .0186 1.3815 .300 .764 Arbitrage 6 .0933 1.6222 1.286 .199 CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta #1 #2 #3 4 45 =b Zero Beta 1.000 Aroitrage #1 .265 1.000 Arbitrage 2 .765 .301 1.000 Arbitrage #3 .347 .058 .233 1.000 Arbitrage '4 .171 .255 .090 .059 1.000 Arbitrage -5 .116 .075 .191 .007 .041 1.000 Arbitrage #6 .308 .113 .236 .223 .209 .112 1.000 TABLE 4-8 SUMMARY INFORMATION Test Period; Ic Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic Zero Beta .0003 .0064 1.054 .293 Arbitrage l -.0787 1.0224 -1.721 .086 Arbitrage :2 .0566 1.7255 .734 .463 4rbitrage :3 .1169 1.5585 1.677 .094 Arbitrage '4 .00C6 1.4533 .010 .992 Arbitrage cS -.0873 1.3578 -1.440 .151 Arbitrage 46 -.0363 1.4333 .566 .571 CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta #1 #2 03 #4 45 si :ero Beta 1.000 Arbitrage Cl .351 1.CO0 Arbitrage 2 .623 .310 1.000 Arbitrage e3 .039 .003 .013 1.C00 Arbitrage e4 .312 .221 .103 .001 1.0CO Arbitrage :S .324 .383 .161 .406 .1C1 1.000 Arbitrage 76 .138 .070 .160 .168 .108 .098 1.000 TABLE 4-9 SUMMARY INFORMATION Test Period: 2a Vumber of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .0002 .0106 .465 .642 Arbitrage 1 .1659 1.9190 1.933 .054 Arbitrage -2 -.0016 1.5070 .024 .981 Arbitrage ;3 .0110 1.4200 .173 .863 Arbitrage *4 .0175 1.2895 .303 .762 Arbitrage 5 .0271 1.5663 .386 .699 Arbitrage e6 .0345 1.5421 .501 .617 Portfolio Zero Arbitrage Arbitrage Arbitrage Aritrage Aroitrrage Arbitrage Beta #1 #2 O3 4 *5 =C :ero Beta 1.000 Arbitrage CI .695 1.000 Arbitrage C2 .327 .270 1.000 Arbitrage 43 .370 .301 .136 1.000 Arbitrage *4 .082 .101 .151 .233 1.000 Arbitrage #5 .168 .014 .050 .002 .086 1.000 Arbitrage 6 .066 .089 .047 .088 .362 .040 1.000 TABLE 4-10 SUMMARY INFORMATION Test Period: 2b Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta #1 #2 #3 '4 '5 'b Zero Beta 1.000 Arbitrage -1 .438 1.000 Arbitrage 2 .417 .113 1.000 Arbitrage 43 .144 .262 .154 1.000 Arbitrage 44 .088 .002 .053 .319 1.000 Arbitrage i5 .329 .042 .074 1.75 .039 1.000 Arbitrage 46 .192 .250 .077 1.82 .075 .119 1.000 Average Standard Computed Significance PortDolio Dally Deviation T Level Return Statistic :ero Beta .0006 .0095 1.426 .154 Aroitrage I .0079 1.3987 .126 .900 Arbitrage 2 .1534 1.5485 2.215 .027 Arbitrage #3 .0191 1.3076 .327 .744 Arbitrage '4 -.0111 1.3230 .188 .851 Aroitrage 45 -.0421 1.5497 .607 .544 Arbitrage i6 .0136 1.2597 .241 .810 -- TABLE 4-11 SUMMARY INFORMATION Test Period- 2c Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD Average Stanaard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .0007 .0089 1.786 .075 Arbitrage -1 .0953 1.4644 1.455 .146 Arbitrage p: .0626 1.3501 1.037 .300 Arbitrage #3 .0138 1.0822 .284 .776 Arbitrage 4 -.0893 1.5654 -1.275 .203 Arbitrage 05 .0647 1.5594 .928 .354 Arbitrage #6 -.0674 1.6286 .925 .355 CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Portfolio 2ero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta FI #2 S3 a4 '5 .1 Zero Beta 1.000 Arbitrage I .219 1.000 Arbitrage ': .123 .255 1.000 Arbitrage 03 .108 .107 .399 1.000 Arbitrage '4 .387 .047 .090 .112 1.000 Arbitrage t5 .219 .033 .095 .192 .085 1.000 Arbitrage -6 .048 .025 .336 .149 .170 .1029 1.000 TABLE 4-12 SUMMARY INFORMATION Test Period: 3a Number of Trading Days: 500 Number of Securlties: 100 UNIVARIATE STATISTICS FOR TEST PERIOD Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .00007 .0063 .253 .800 Arbitrage l .0474 .9139 1.160 .247 Arbitrage #2 .1099 1.0036 2.450 .016 Arbitrage 03 -.0104 1.0073 .217 .829 Arbitrage '4 -.0287 .9802 .655 .513 Arbitrage 45 -.0053 .9601 .123 .902 Arbitrage #6 -.0649 .9340 -1.550 .122 CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS ?ortfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta 1 #2 53 i '0 "b :ero Beta 1.000 Arbitrage 1 .128 1.000 Arbitrage '= .251 .094 1.000 Arbitrage '3 .737 .020 .018 1.000 Arbitrage 4 .302 .053 .125 .263 1.000 Arbitrage S .291 .110 .114 .160 .264 1.300 Arbitrage '6 .163 .005 .102 .155 .158 .132 1.000 TABLE 4-13 SUMMARY INFORMATION Test Period: 3b Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .0005 .0068 1.535 .125 Arbitrage al .0843 1.0709 1.760 .079 Arbitrage #2 -.0087 .9648 .201 .841 Arbitrage #3 -.0623 .9539 -1.460 .145 Arbitrage #4 -.0228 .9241 .551 .582 Arbitrage #5 -.0104 .9786 .239 .811 Arbitrage #6 -.0050 .9335 .120 .905 Portfolio Zero Arbitrage Arbitrage Arbira bitrae Arbitrage Arbitrage Beta #1 #2 #3 4 s5 -6 Zero Beta 1.000 Arbitrage '1 .189 1.000 Arbitrage '2 .185 .102 1.000 Arbitrage #3 .742 .053 .278 1.000 Arbitrage .210 .123 .035 .043 1.000 Arbitrage c5 .554 .035 .084 .401 .044 1.000 Arbitrage 6 .100 .014 .003 .014 .018 .130 1.000 TABLE 4-14 SUMMARY INFORMATION Test Period: 3c Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE STATISTICS FOR TEST PERIOD Average Standard Computed Significance Portfolio Daily Deviation T Level Return Statistic :ero Beta .0004 .0049 .181 .856 Arbitrage 41 .0812 .9404 1.930 .054 Arbitrage =: .0132 .9073 .325 .745 Arbitrage 03 -.0329 1.0015 .734 .453 Arbitrage 4 -.0059 .8027 .164 .870 Arbitrage 5 .0296 .9363 .706 .480 Arbitrage 06 .0194 .9542 .454 .650 CORRELATIONS AMONG ARBITRAGE PORTFOLIO RETURNS Portfolio Zero Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Arbitrage Beta 01 42 '3 -4 45 *6 :ero Beta 1.000 Arbitrage i .365 1.000 Arbitrage 1: .109 .177 1.000 Arbitrage =3 .224 .208 .020 1.000 Arbitrage 41 .029 .004 .122 .201 1.000 Arbtrage 05 .440 .120 .085 .015 .022 1.000 Arbitrage =6 .165 .098 .032 .157 1.,6 .074 1.000 70 the same estimates reported in studies of the market model (e.g., Fama and MacBeth, T973). Also reported in Tables 4-6 through 4-14 are the estimated returns on the arbitrage portfolios and their standard deviations. In the first period, one portfolio in sample a has a significantly non-zero return while b and c each have two such portfolios (at a 10% level). Samples 2a and 2b each contain one significant return, and 2c contains none. In the third period, each sample has one portfolio with a statistically non-zero return. In the base periods, the hypothetical factors were constructed such that they were mutually uncorrelated with unit variances. Estimates of the factor scores will in general possess neither property exactly. In the first two periods, the variances generally exceed one, while in the third period they are relatively close. More troublesome are the correlations of the arbitrage portfolio returns with the zero beta return; in every case, there is at least one arbitrage portfolio with a substantial negative correlation. This unanti- cipated result is difficult to explain. However, if the risk premia are defined as excess returns above the zero beta return, then an inverse relationship implies that the nominal premia are not constant and decrease when the zero beta return increases. Alternatively, noise in the ex post data may give rise to differential measured sensitivities to the zero beta return. In this case, an extra factor may appear to exist and the appro- priate dimension of the model would be over-estimated. The extra factor would not be priced, however. A purely empirical explanation is readily apparent when the structure of the arbitrage portfolios is examined. The portfolios with large negative correlations correspond to factors which tend to be dominated by public utilities, a group traditionally considered to be interest rate sensitive. This should not necessarily be interpreted as evidence for an interest rate factor. The utilities are regulated, they have substantial dividend yields, and they tend to be large. Thus, the phenomenon could relate to regulatory risk or lag, taxation of dividends, a size effect, or some other common characteristic. Whatever the explana- tion, the phenomenon is persistent and the collinearity between the zero beta return and the arbitrage returns may lead to econometric difficulty. The results of this section indicate considerable volatility in the estimated zero beta returns and the risk premia, most of which are not statistically different from zero. It is important to emphasize here that no conclusions concerning the APT can be drawn at this point. The statis- tical significance (or lack thereof) of the risk premia cannot be inter- preted as evidence for or against the APT for two reasons. First, on a theoretical level, such evidence relates to the linearity of the pricing relationship which for reasons discussed in Chapter III is not the relevant issue. Second, the risk premia are not uniquely determined empirically. The factor analysis uniquely determines the space into which returns are projected; the orientation of the factors within that space is only unique up to non-singular transformation. In practical terms, the significance of the risk premia are a function of the rotation chosen, using an oblique (non-orthogonal) rotation will generally result in greater numbers of "significant" factors at the expense of greater correlations among the risk premia. In fact, with the Promax rotation (Lawley and Maxwell, 1971), the degree of factor inter-correlation is to some extent controllable, and it can simply be increased until most of the premia are significant. This inherent indeterminateness renders attempts at identifying the number of relevant factors by cross-sectional regression meaningless. In sum, the requisite degree of linearity in the pricing relationship is unknown a priori and empirically indeterminate. The result of a factor analysis is a unique estimate of a common factor space and the residual variances for the securities. Testing the APT requires examination of this unique information. Because the zero beta portfolio is constructed to be a member of the space perpendicular to the common factor space, it is unaffected by rotations of the factors within that space. When the additional requirement of minimum residual variance is imposed, the zero beta portfolio is uniquely defined. In fact, in the hypothetical population, the zero beta portfolio is the global minimum variance portfolio, and is thus mean-variance efficient.1 The implications of this fact are pursued in the next section. Univariate Results from the Arbitrage Model In this section, the portfolio returns created in the previous section are used as independent variables in time-series regressions of the form specified in (3-30) rjt = _j + bjooot + bjlt + . + bj666t + ejt, (3-30) j = 1, . ., 100 t = 1 ...., 250 251, . ., 500. The intercept term is a measure of abnormal performance and will not differ from zero unless arbitrage opportunities exist. The coefficient of the zero beta return should be one, and the other coefficients should be generally significant. Mathematically, the coefficient of the zero beta return would be exactly equal to one if the testing were done in 1This fact is pointed out by Ingersoll (1982). the base period using the actual residual covariance matrix to obtain the weights. In this case, the portfolio is mean-variance efficient in the sample and has an exactly zero correlation with any arbitrage port- folio. Because of this the estimated coefficient, bo is equal to the covariance of the return on security j with the zero beta return, divided by the variance of the zero beta return. It is easily verified (e.g., Roll, 1976) that the global minimum variance portfolio has the property that its covariance with any non-arbitrage portfolio is equal to its own variance; hence, an "estimated" coefficient would be unity with a zero error. As it is employed here, the zero beta portfolio return is an estimate of the unobservable minimum variance portfolio for all risky assets of the type under consideration.2 In the actual estimation, each of the three test periods is divided into two sub-periods, each of which covers 250 trading days. This was done in order to examine the stability of the estimates and the extent to which they deteriorate over time. The total number of estimated equations is 1800, consisting of nine samples of 100 securities each and two subperiods per sample. The results are reported in Tables 4-15 to 4-23. In each of the tables, several items are tabulated. The first is the number of securities for which significant intercepts were found. The second and third are the number of estimates of b which differ from zero and one respectively. In the first 250 trading days of sample la, three of the 100 securities had significant intercepts, and 85 had significant coefficients on the zero beta portfolio, eight of which were significantly This argument is extendable to all risky assets in the economy if it is possible to hedge any type of systematic risk using the subset of assets from which the factors were obtained. TABLE 4-15 SUMMARY INFORMATION Test Period: la Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE TESTS OF APT Hypothesis Rejections ccO S=0 0 b = 0 bc=1 3 85 8 Hypothesis Rejections = 0 1 b = 0 88 1 16 : 1 16 0 NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Second 250 Trading Days Trading Days k # of eq. k # of eq. 0 1 0 2 1 9 1 13 2 12 2 20 3 18 3 14 4 21 4 22 5 23 5 17 6 16 6 12 Note: Significance level is .05 for two-tailed T-test. First 250 Trading Days Second 250 Trading Days TABLE 4-16 SUMMARY INFORMATION UNIVARIATE TESTS OF APT Second 250 Trading Days Hypothesis Rejections S0 0 b= 0 90 b = 1 18 o1 NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Second 250 Trading Days Trading Days k # of eq. 0 0 1 8 2 17 3 24 4 22 5 20 6 9 k # of eq. 0 1 1 11 2 20 3 25 4 17 5 19 6 7 Note: Significance level is .05 for two-tailed T-test. Test Period: lb Number of Trading Days: 500 Number of Securities: 100 First 250 Trading Days Hypothesis Rejections a= 0 2 b =0 87 b =1 11 o TABLE 4-17 SUMMARY INFORMATION Test Period: Ic Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE TESTS OF APT Second 250 Trading Days Hypothesis Rejections a= 2 b = 0 95 b =1 18 0 NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) Second 250 Trading Days k # of eq. 0 0 1 4 2 23 3 26 4 23 5 18 6 6 Note: Significance level is .05 for two-tailed T-test. First 250 Trading Days First 250 Trading Days k # of eq. 0 2 1 11 2 22 3 19 4 21 5 21 6 4 TABLE 4-18 SUMMARY INFORMATION UNIVARIATE TESTS OF APT Second 250 Trading Days NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) Second 250 Trading Days k # of eq. k # 0 1 0 1 4 1 2 15 2 3 20 3 4 25 4 5 23 5 6 12 6 Note: Significance level is .05 for two-tailed T-test. Test Period: 2a Number of Trading Days: 500 Number of Securities: 100 First 250 Trading Days Hypothesis Rejections = 0 0 S= 0 94 b 1 24 0 ___________ Hypothesis Rejections a= 0 0 = 0 89 S b 1 13 L 0 First 250 Trading Days TABLE 4-19 SUMMARY INFORMATION Test Period: 2b Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE TESTS OF APT Second 250 Trading Days Hypothesis Rejections a: 0 0 b = 0 88 b = 1 19 0__ _ NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Trading Days Second 250 Trading Days k # of eq. k 0 1 0 1 8 1 2 10 2 3 24 3 4 21 4 5 19 5 6 17 6 Note: Significance level is .05 for two-tailed T-test. First 250 Trading Days Hypothesis Rejections = 0 0 b 0 96 bo 1 13 TABLE 4- 20 SUMMARY INFORMATION Test Period: 2c Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE TESTS OF APT Second 250 Trading Days NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) Second 250 Trading Days k # of eq. 0 0 1 2 2 13 3 13 4 28 5 30 6 14 Note: Significance level is .05 for two-tailed T-test. First 250 Trading Days Hypothesis Rejections = 0 0 b 0 95 b 1 21 0 Hypothesis Rejections S 0 0 b= 0 86 b =1 15 0 First 250 Trading Days k # of eq. 0 1 1 7 2 7 3 25 4 21 5 23 6 15 TABLE 4-21 SUMMARY INFORMATION UNIVARIATE TESTS OF APT First 250 Trading Days Second 250 Trading Days Hypothesis Rejections S= 0 b 0 b =1 0 0 87 15 NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Second 250 Trading Days Trading Days k # of eq. k # of eq. 0 1 0 2 1 6 1 5 2 22 2 7 3 19 3 28 4 20 4 19 5 14 5 21 6 18 6 18 Note: Significance level is .05 for two-tailed T-test. Test Period: 3a Number of Trading Days: 500 Number of Securities: 100 Hypothesis Rejections a=0 2 b 0 85 b 1l 20 o TABLE 4-22 SUMMARY INFORMATION UNIVARIATE TESTS OF APT First 250 Trading Days Second 250 Trading Days Hypothesis Rejections S0 0 b = 0 89 b = 1 17 0 NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Trading Days Second 250 Trading Days k # of eq. k # 0 2 0 1 11 1 2 17 2 12 3 14 3 2: 4 22 4 2E 5 25 5 2: 6 9 6 1( Note: Significance level is .05 for two-tailed T-test. Test Period: 3b Number of Trading Days: 500 Number of Securities: 100 Hypothesis Rejections a=0 0 b= 0 79 b = 1 11 ob ________ TABLE 4-23 SUMMARY INFORMATION Test Period: 3c Number of Trading Days: 500 Number of Securities: 100 UNIVARIATE TESTS OF APT Second 250 Trading Days Hypothesis Rejections a = 0 0 b 0 87 S=1 22 So NUMBER OF EQUATIONS WITH k SIGNIFICANT FACTORS (not including zero beta portfolio) First 250 Second 250 Trading Days Trading Days k # of eq. 0 0 1 5 2 10 3 20 4 23 5 23 6 19 Note: Significance level is .05 for two-tailed T-test. First 250 Trading Days Hypothesis Rejections S=0 0 b 0 83 b = 1 12 0 k # of eq. 0 3 1 7 2 13 3 17 4 24 5 23 6 13 different from one. Also reported are the number of equations which were found to have various numbers of significant coefficients on the arbi- trage portfolios. The majority of securities have from two to five such coefficients. The results are in substantial, though not complete, agreement with the predictions of the APT. Significant intercepts occur less than 1% of the time; the zero beta return is significant in about 90% of the trials, and differs from one in about 16% of the trials. Over 90% of the securities have significant coefficients on three or more of the arbitrage portfolios. There does not appear to be any significant deterioration in explanatory power over the second sub-periods. Moreover, the average R2's agree with the initial communalities; for example, the communality estimated in sample la was 22.05% and the average R2 from the first sub-period was 20.04%. These summary measures are to some extent misleading in that they understate the degree of conformity of the results with the APT. There is a definite tendency for the model to work quite well for the majority of the securities and work poorly for a minority. Typically, the intercept is insignificant at any conventional level, b0 is within one standard error of its predicted value, and several other coefficients are significant at any conventional level. The coefficient estimates are frequently in excess of three standard errors away from zero. The results in this section are generally in accord with the APT. That the model occasionally works poorly is not surprising; the theory is itself an approximation and is expected to have low explanatory power for some subset of the securities under consideration. While the predictions of the APT appear to be supported by the data, the results in this section do not account for cross-sectional dependencies in the estimates, and no general conclusion can be drawn about the central prediction of the theory, namely, 84 an absence of arbitrage opportunities. This is the subject of the next section. A Multivariate Test of the APT In this section, consideration turns to the question of whether the intercepts and zero beta coefficients are jointly different from their predicted values. A good reference for the multivariate techniques employed in this section may be found in Timm (1975). The individual time-series estimates of eq. (3-30) can be brought together in matrix notation as R = AB + E, (4-2) where R = 6= B = E = It is assumed that E[E] = 0 E[R] = AE V[R] = It 0 E. In this case B = (AA )- 'R, and an unbiased estimate of c is = (R AB)1 (R B) t-k-l (4-3) = R'(I A(A' A) A' )R. (4-4) t-k-l The columns of B in (4-3) are the usual univariate OLS estimates. Even though the equations are distrubance related, the regressors are identical ---- 85 in each equation and thus OLS on each equation is efficient (Thiel (1971), p. 309). The hypothesis of interest is H : b = 0, 1 (4-5) O -0 vs. HA: a, b o 0, 1 where a is the first row of B and b is the second. If it is assumed that -o the rows of R possess multivariate normal distributions, then this hypothe- sis can be tested by comparing restricted and unrestricted sum of squares and cross-products. The hypothesis can be written as H : CBM = r, where C is the matrix of restrictions operating within the individual equations with M operating across equations. From Timm (1975), the error sums of squares and cross-products are Q = M'R'[I A(A' )-1 A] RM, and the sums due to the hypothesis are Qh = (CBM r)' [C(A'A)1C']-1 (CBM r). The total sums of squares and cross-products under the null hypothesis is Qt = Qe + Qh With these definitions, four statistics based on the eigen- values of Q e-Qh or Qt1 Qh are available. Letting yi be the ordered eigen- values of Q e1 h' then the following statistics are estimated: Wilk's Lambda = |Qe1 Qt = 1 +n Pillai's Trace = tr(QQt-) = E i 1 + Yi Hotelling-Lawley Trace = tr(Q e-Qh) =Q -i Roy's Greatest Root = yl. No general preference for any one of the four can be established; however, in the case where the null hypothesis is of the form of (4-5), the F approximation for Wilk's Lambda is exact and Roy's criterion is an upper bound. When the hypothesis is of the form H : a = 0 or H : b = 1, all o o -o four criteria are equivalent. The results from the securities data for the three hypotheses S= 0, b = 1, and a, b = 0, 1 are reported in Tables 4-24 to 4-32. For first two hypotheses only Wilk's Lambda is reported because of the equiva- lence of the four criteria. Examining the results, the restriction a = 0 is not binding in any of the trials; thus it does not appear to be possible to form portfolios with no factor risk or net investment with significantly non-zero returns. On the other hand, the restriction b = 1 is binding in -o- every case. This suggests the existence of portfolios with no factor risk and returns that exceed the zero-beta return. Such an operation requires a positive investment, however. This finding may be attributable to the difficulties in estimating the zero beta return described above. When the two hypotheses are combined, the results indicate some dependency between the estimates; however, no unambiguous conclusion concerning the validity of the hypothesis can be reached. Depending on the significance level and test statistic, the hypothesis appears to be binding in about one third of the trials, indicating significant though not complete agreement with the predictions of the APT. Summary The purpose of this chapter has been to elevate the APT from a relatively unstructured theory to a concrete model of security returns. Also developed were testable hypotheses of the APT with the emphasis placed on the conformity of the uniquely determined parts of the estimated structure with the predictions of the theory. The results of this positiv- ist approach are strongly supportive of the basic content of the theory: it is not possible to form portfolios with no net investment and no factor TABLE 4-24 SUMMARY INFORMATION Test Period: ia Number of Trading Days: 500 Number of Securities: 100 FIRST 250 TRADING DAYS TEST OF Ho: 0 o TEST OF Ho: 1 0 -0 - TEST OF H : _, = 0,1 Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic CF OF Level dilk's Lambda .361 1.072 186 300 .296 Pillai's Trace .781 1.041 186 302 .377 Hoteliing-Lawley 1.376 1.102 186 298 .226 Roy's Greatest Root 1.575 1.575 93 151 .007 Note: F Statistic for Roy's Greatest Root is an upper bound. F Statistic for Wilk's Lamba is exact. TABLE 4-24 (continued) SECOND 250 TRADING DAYS TEST OF HO: = o TEST OF H : b= 1 0 -0 - TEST OF H: I, = 0,1 --st I Comuted Computed Numerator Denominator Significance Statistic Value F Statistic OF DF Level .ilk's Lambda .281 1.429 186 300 .003 Pillai's Trace .879 1.274 186 302 .031 Hotelling-Lawley 1.986 1.591 186 298 .001 Roy's -reatest Root 1.638 2.660 93 151 ..001 TABLE 4-25 SUMMARY INFORMATION Test Period: lb Number of Trading Days: 500 Number of Securities: 100 FIRST 250 TRADING DAYS TEST OF H : = o TEST OF HF: I 1 0 -0 - TEST OF Ho: a, = .,1 Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic DF DF Level Wilk's Lambda .308 1.291 186 300 .025 Pillai's Trace .863 1.233 186 302 .054 Hotelling-Lawley 1.686 1.350 186 298 .011 Roy's Greatest Root 1.236 2.006 93 151 .001 Note: F Statistic for Roy's Greatest Root is an upper bound. F Statistic for Wilk's Lamba is exact. TABLE 4-25 (continued) SECOND 250 TRADING DAYS TEST OF H: o TEST OF H: b- 1 o-0 - TEST OF Ho: a, = 0,1 es I Computed Computed Numerator Cenominator Significance Statistic Value F Statistic OF DF Level Wilk s 'Labda .287 1.398 186 300 .005 Pillai's Trace .874 1.260 186 302 .038 Hotelling-Lawley 1.925 1.542 IB6 298 .001 Roy's greatestt 0oot 1.568 2.546 93 151 .001 TABLE 4-26 SUMMARY INFORMATION Test Period: Ic Number of Trading Days: 500 Number of Securities: 100 FIRST 250 TRADING OAYS TEST OF H : a = 0 TEST OF Ho: b = 1 0 .-, - TEST OF H : a, b = o,1 0 _0 - Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic OF CF Level Wilk's Lambda .287 1.399 186 300 .005 Pillai's Trace .908 1.350 186 302 .010 Hotelling-Lawley 1.809 1,449 186 298 .002 Roy's Greatest Root 1.278 2.075 93 151 .001 Note: F Statistic for Roy's Greatest Root is an upper bound. F Statistic for Wilk's Lamba is exact. TABLE 4-26 (continued) SECOND 250 TRADING DAYS TEST OF H : = o TEST OF Ho: b - TEST OF H: 3, bO = 0,1 0 -s- it Computed Computed Numerator Denominator Significance Statistic V alue F Statistic DF OF Level Ailk's Lambda .286 1.420 186 300 .005 Pillai s Trace .905 1.341 186 302 .012 Hotellig-Lawley 1.828 1.464 186 298 .002 Roy's Greatest Root 1.324 2.149 93 151 .001 TABLE 4-27 SUMMARY INFORMATION Test Period: 2a Number of Trading Days: 500 Number of Securities: 130 FIRST 250 TRADING JAYS TEST OF H : a= TEST OF Ho: b = 0 ~ TEST OF H : I, -* ,1 Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic OF OF Level Wilk's Lambda .274 1.469 186 300 .002 Pillai's Trace .855 1.213 186 302 .069 Hotelling-Lawley 2.179 1.745 186 298 .001 Roy's Greatest Root 1.936 3.143 93 151 .001 Note: F Statistic for Roy's Greatest Root is an upper bound. F Statistic for Wilk's Lamba is exact. TABLE 4-27 (continued) SECOND 250 TRADING DAYS TEST OF H : = o Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic OF OF Level Wilk's Lambda .789 .431 93 150 I.999 TEST OF Ho: = 1 Test Computed Computed Numerator Denominator Signifizance Statistic Value F Statistic OF OF Level Wilk's Lambda .466 1.846 93 150 .001 TEST OF Ho: i, = c,1 SComputed Computed Numerator Denominator Significance Statistic Value F Statistic DF OF Level Wilk's Lanbda .372 1.033 186 300 .399 Pillai's Trace .737 .947 186 302 .657 Hotel!1ng-Liwley 1.399 1.121 186 298 .190 Roy's greatest Root 1.145 1.859 93 151 .001 TABLE 4-28 SUMMARY INFORMATION Test Period: 2b Number of Trading Days: 500 Number of Securities: 100 FIRST 250 TRADING DAYS TEST OF H : a = o TEST OF H : = 1 a -' - TEST OF He: p, b 0,1 Test Computed Computed Numerator Denominator Significance Statistic Value F Statistic OF OF Level Wilk's Lambda .384 .988 186 300 531 Pillai's Trace .706 .886 186 302 .816 Hotelling-Lawley 1.366 1.094 186 298 .245 Roy's Greatest Root 1.163 1.889 93 151 .001 Note: F Statistic for Roy's Greatest Root is an upper bound. F Statistic for Wilk's Lamba is exact. ------- |