The residual market factor,the APT,and mean-variance efficiency

The rapidly increasing volume of both published and unpublished work on the arbitrage pricing theory (APT) of Ross (1976) has given rise to a number of misunderstandings at the interface of theoretical and econometric work. In this article we extend the theoretical structure of our previous work (McElroy and Burmeister, 1985, 1988; Burmeister and McElroy, 1987, 1988) to provide a broad yet rigorous framework both for econometric estimation and for better economic interpretation of new empirical results. We begin with the case where allK factors are observed, and then present the second case ofK−1≡J observed APT factors and one unobserved factor, theresidual market factor introduced in McElroy and Burmeister (1985). The economic interpretations for equivalent specifications of this model are discussed, and we enumerate several immediate payoffs to these specifications. The main new results are concerned with the sometimes intricate relationships among APT models withK factors and APT models withK factors that are constrained to satisfy mean-variance efficiency restrictions. These results are not only of theoretical interest, but more importantly they provide the basis for econometric estimation and testing of nested hypotheses. These econometric issues are discussed in detail.     

Joint Estimation of Factor Sensitivities and Risk Premia for the Arbitrage Pricing Theory

The APT is represented as a multivariate regression model with across-equations restrictions. Both observed and unobserved (latent) macroeconomic factors are included, thus generalizing and unifying two previous strands of literature. Large portfolios representing unobserved factors are treated as endogenous, and nonlinear 3SLS estimates are shown to differ sharply from estimates that ignore this endogeneity. Using monthly stock returns and six factors, we cannot reject January effects. The following results are invariant with respect to the inclusion of January effects: we reject the CAPM in favor of the APT; however, we cannot reject the APT restrictions on the linear factor model.     

Estimation of the underlying structure of systematic risk with the use of principal component analysis and factor analysis

We present an improved methodology to estimate the underlying structure of systematic risk in the Mexican Stock Exchange with the use of Principal Component Analysis and Factor Analysis. We consider the estimation of risk factors in an Arbitrage Pricing Theory (APT) framework under a statistical approach, where the systematic risk factors are extracted directly from the observed returns on equities, and there are two differentiated stages, namely, the risk extraction and the risk attribution processes. Our empirical study focuses only on the former; it includes the testing of our models in two versions: returns and returns in excess of the riskless interest rate for weekly and daily databases, and a two-stage methodology for the econometric contrast. First, we extract the underlying systematic risk factors by way of both, the standard linear version of the Principal Component Analysis and the Maximum Likelihood Factor Analysis estimation. Then, we estimate simultaneously, for all the system of equations, the sensitivities to the systematic risk factors (betas) by weighted least squares. Finally, we test the pricing model with the use of an average cross-section methodology via ordinary least squares, corrected by heteroskedasticity and autocorrelation consistent covariances estimation. Our results show that although APT is very sensitive to the extraction technique utilized and to the number of components or factors retained, the evidence found partially supports the APT according to the methodology presented and the sample studied.     

Yes,The APT Is Testable

The Arbitrage Pricing Theory (APT) has been proposed as an alternative to the mean-variance Capital Asset Pricing Model (CAPM). This paper considers the testability of the APT and points out the irrelevance for testing of the approximation error. We refute Shanken's objections, including his assertion that Roll's critique of the CAPM is applicable to the APT. We also explain the testability of the APT on subsets, and we explore the relationship between the APT and the CAPM.     

International Arbitrage Pricing Theory: An Empirical Investigation

In this paper, we test the arbitrage pricing theory (APT) in an international setting. Inter-battery factor analysis is used to estimate the international common factors and the Chow test is used in testing the validity of the APT. Our inter-battery factor analysis results show that the number of common factors between a pair of countries ranges from one to five, and our cross-sectional test results lead us to reject the joint hypothesis that the international capital market is integrated and that the APT is internationally valid. Our results, however, do not rule out the possibility that the APT holds locally or regionally in segmented capital markets. Finally, the basic results of both the inter-battery factor analysis and the cross-sectional tests are largely invariant to the numeraire currency chosen.     

Valuation before and after tax in the discrete time, finite state no arbitrage model

We establish necessary and sufficient conditions for a linear taxation system to be neutral—within the multi-period discrete time “no arbitrage” model—in the sense that valuation is invariant to the exact sequence of tax rates, realization dates as well as immune to timing options attempting to twist the time profile of taxable income through wash sale transactions.

“In the study of investments, taxes are largely a source of embarrassment to financial economists.” (Introduction to Dybvig and Ross 1986) “Accordingly, my approach in this chapter is to examine the restrictions on the income measurement rules applicable to financial instruments implied by the requirement that the rules be linear. . . . Linearity is a desideratum of a tidy tax system.” (Bradford 2000, p. 373–374)

     

Some comments on the APT

Abstract

The paper reviews some aspects of arbitrage pricing theory (APT). It derives an improved version of the model and examines it in view of the APT debate, adding some new observations in favour of the model. The topics examined include: (a) model testability; (b) implications of approximate APT pricing to Fama-Macbeth testing methodology; and (c) a comparison between APT pricing and approximate exact pricing.     

No Arbitrage and Arbitrage Pricing: A New Approach

We argue that arbitrage-pricing theories (APT) imply the existence of a low-dimensional nonnegative nonlinear pricing kernel. In contrast to standard constructs of the APT, we do not assume a linear factor structure on the payoffs. This allows us to price both primitive and derivative securities. Semi-nonparametric techniques are used to estimate the pricing kernel and test the theory. Empirical results using size-based portfolio returns and yields on bonds reject the nested capital asset-pricing model and linear APT and support the nonlinear APT. Diagnostics show that the nonlinear model is more capable of explaining variations in small firm returns.     

An intertemporal equilibrium beta pricing model

This article develops an intertemporal, discrete-time, competitiveequilibrium version of the arbitrage pricing theory, (APT) andexplores the econometric implications of this model under variousrestrictions on investor preferences and on the dynamic behaviourof dividends. We describe conditions under which the econometrictechnique typically used for estimating and testing the APTcan be shown to be consistent with our economic model. We relateour intertemporal version of the APT to the static APT and toMerton's intertemporal capital asset pricing model.     

A Semiautoregression Approach to the Arbitrage Pricing Theory

This paper developes a semiautoregression (SAR) approach to estimate factors of the arbitrage pricing theory (APT) that has the advantage of providing a simple asymptotic variance-covariance matrix for the factor estimates, which makes it easy to adjust for measurement errors. Using the extracted factors, I confirm the finding that the APT describes asset returns slightly better than the CAPM, although there is still some mispricing in the APT model. I find that not only are the factors “priced” by the market, but the factor premiums move over time in relation to business cycle variables.     

An Empirical Test of the Arbitrage Pricing Theory on Australian Stock Returns 1974-85

This paper examines empirically, issues concerning the Arbitrage Pricing Theory (APT). Firstly, in the spirit of Chamberlain and Rothschild [1983], the existence of an approximate factor structure is explored. Secondly, following Beggs [1986] and employing a principal components approach, a test of arbitrage pricing and the importance of the error of approximation, is conducted. Finally, using a non nested framework, the APT and CAPM are tested against each other. The results show mixed support for the APT having up to 3 priced factors.     

Simulating and Forecasting Utility Stock Returns: Arbitrage Pricing Theory vs. Capital Asset Pricing Model

Just when the capital asset pricing model (CAPM) has become accepted by public utility regulators as a method for estimating a utility's screening rate, academic criticism of the model's theoretical and empirical shortcomings has led to empirical testing of the alternative arbitrage pricing theory (APT). This paper expands on recent APT-CAPM performance comparisons by simulating returns of public utility stocks using versions of both models, as was done by Bower, Bower, and Logue in a 1984 paper. In addition, the models are used for ex-post forecasting of returns in a subsequent time period. The Litzenberger-Ramaswamy method is used to correct for errors-in-variables in the CAPM cross-sectional equation. This allows for estimating the security market line using firm betas. The same methodology is used in the APT stages. Three different criteria—the Theil inequality, the sources of mean square error, and Chen's estimated weights of expected return-are used to compare CAPM and APT simulation and forecasting of the equity screening rates. Tested on a sample of 128 public utility companies, results show that neither model is clearly dominant. There is a tendency for reversal of performance. The model that is superior for simulating returns tends to be inferior for forecasting them, and vice-versa.     

Equity prices and fundamentals: a DDM–APT mixed approach

This paper focuses on the linkage between equity prices and fundamentals for 27 individual shares belonging to the French stock price index (CAC40). To assess fundamental value, the traditional dividend discount model (DDM) is coupled with the arbitrage pricing theory (APT), which assumes that investors hold efficient portfolios. This yields a simple equity valuation relationship for which the APT determines the long-term risk premium included in the DDM. Accordingly, equity risk premia are determined by common factors reflecting the non diversifiable risk. These factors are not a priori identified by the theory, and therefore must be exhibited through an empirical analysis. Four domestic and three international common factors are found, all being among those identified by empirical analyses of the APT in the literature. While studies related to stock price indices showed that DDM fundamental values are very smooth compared to stock indices, our DDM–APT model reproduces both trends and major fluctuations of share prices. Further, as for studies based on stock indices, a mean-reverting process of equity prices towards fundamentals is highlighted, but the linear error correction model that was considered contains shortcomings suggesting a more complex adjustment process.     

Optimal importance sampling with explicit formulas in continuous time

In the Black–Scholes model, consider the problem of selecting a change of drift which minimizes the variance of Monte Carlo estimators for prices of path-dependent options. Employing large deviations techniques, the asymptotically optimal change of drift is identified as the solution to a one-dimensional variational problem, which may be reduced to the associated Euler–Lagrange differential equation. Closed-form solutions for geometric and arithmetic average Asian options are provided. The authors acknowledge the support of the National Science Foundation under grants DMS-0532390 (Guasoni) and DGE-0221680 (Robertson) at Boston University.     

An explicit bound on individual assets' deviations from APT pricing in a finite economy

Ross's Arbitrage Pricing Theory (APT) is a tractible and reasonable alternative to the mean-variance model. Nonetheless, understanding of the theory has been obscured by the complexity of the sequence economy models used for motivation. By contrast, we give a simple and direct derivation of the APT in a finite economy. Using an explicit bound on the deviations from APT prices across assets, a coarse calculation shows that theoretical deviations from APT pricing are negligible in our economy.     

Tests of financial models in the presence of overlapping observations

A general approach to testing serial dependence restrictionsimplied from financial models is developed. In particular, wediscuss joint serial dependence restrictions imposed by randomwalk, market microstructure, and rational expectations modelsrecently examined in the literature. This approach incorporatesmore information from the data by explicitly modeling dependenciesinduced by the use of overlapping observations. Because theestimation problem is sufficiently simple in this framework,the test statistics have simple representations in terms ofonly a few unknown parameters. As a result, relatively goodsize properties are attained in small samples. In addition,the benefit to overlapping observations and the advantage ofexamining multiperiod time series are explicitly quantified.     

Co-monotonicity of optimal investments and the design of structured financial products

We prove that, under very weak conditions, optimal financial products on complete markets are co-monotone with the reversed state price density. Optimality is meant in the sense of the maximization of an arbitrary preference model, e.g., expected utility theory or prospect theory. The proof is based on a result from transport theory. We apply the general result to specific situations, in particular the case of a market described by the Capital Asset Pricing Model or the Black–Scholes model, where we derive a generalization of the two-fund-separation theorem and give an extension to APT factor models and structured products with several underlyings. We use our results to derive a new approach to optimization in wealth management, based on a direct optimization of the return distribution of the portfolio. In particular, we show that optimal products can (essentially) be written as monotonic functions of the market return. We provide existence and nonexistence results for optimal products in this framework. Finally we apply our results to the study of bonus certificates, show that they are not optimal, and construct a cheaper product yielding the same return distribution.     

Factor pricing in a finite economy

Prior theoretical derivations of the Arbitrage Pricing Theory (APT) bound an aggregate measure of the deviation of mean asset returns from that predicted by a linear pricing equation. It is conceivable, given this bound, that some assets might be badly mispriced by the model. In this paper, a more intuitive derivation of the factor pricing equation is presented which describes the deviation on an asset by asset basis. The deviation is shown to be small for assets in a realistic finite economy and is arbitrarily close to zero for those assets with arbitrarily small size relative to aggregate wealth. It follows that the linear pricing equation provides a good approximation for the mean returns of all traded assets.     

Security factors as linear combinations of economic variables

A new framework is proposed to find the best linear combinations of economic variables that optimally forecast security factors. In particular, we obtain such combinations from Chen et al. (Journal of Business 59, 383–403, 1986) five economic variables, and obtain a new GMM test for the APT which is more robust than existing tests. In addition, by using Fama and French's (1993) five factors, we test whether fewer factors are sufficient to explain the average returns on 25 stock portfolios formed on size and book-to-market. While inconclusive in-sample, a three-factor model appears to perform better out-of-sample than both four- and five-factor models.