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# Dominant theory in the field of empirical asset pricing

## Introduction

The Capital Asset Pricing Model (CAPM) that was proposed by Sharpe (1964) and Lintner (1965) has been the dominant theory in the field of empirical asset pricing for more than thirty years. Although the CAPM managed to withstand intense econometric investigation for decades from dozens of researchers, recent tests indicate that this model is not quite enough for explaining expected returns on stocks. Under these circumstances, conditional linear factor models and nonlinear Arbitrage Pricing Theory (APT) models, have received particular attention. The Arbitrage Pricing Theory starts with specific assumptions on the distribution of asset returns and relies on approximate arbitrage arguments. In this assignment we present the prevailing studies dealing with the validity and the role of the APT (Ross, 1976).

## Review of the Arbitrage Pricing Theory

The APT (Ross, 1976) is a financial pricing model which attempts to explain expected returns on assets. The arbitrage model was proposed as an alternative to the mean variance CAPM that has become the major analytic tool for explaining phenomena observed in capital markets for risky assets. The APT states that the expected return of a financial asset can be modeled as a linear function of various macro-economic factors or theoretical market indices, where sensitivity to changes in each factor is represented by a factor-specific beta coefficient. The model-derived rate of return will then be used to price the asset correctly. The asset price should equal the expected end of period price discounted at the rate implied by model. If the price diverges, arbitrage should bring it back into line (Ingersoll, 1984).

The APT along with the CAPM is one of two influential theories on asset pricing. The APT differs from the CAPM in that it is less restrictive in its assumptions. It allows for an explanatory (as opposed to statistical) model of asset returns. It assumes that each investor will hold a unique portfolio with its own particular array of betas, as opposed to the identical "market portfolio". In some ways, the CAPM can be considered a "special case" of the APT in that the securities market line represents a single-factor model of the asset price, where beta is exposed to changes in value of the market.

Additionally, the APT can be seen as a "supply-side" model, since its beta coefficients reflect the sensitivity of the underlying asset to economic factors. Thus, factor shocks would cause structural changes in assets' expected returns, or in the case of stocks, in firms' profitability.

On the other side, the CAPM is considered a "demand side" model. Its results, although similar to those of the APT, arise from a maximization problem of each investor's utility function, and from the resulting market equilibrium (investors are considered to be the "consumers" of the assets).

The most significant difference between the two models is that while APT uses “approximate arbitrage" to approximately price almost “all" assets the CAPM as an arbitrage-free pricing uses strict arbitrage to price assets that can be replicated exactly. Another difference, arising from the use of arbitrage, between CAPM and arbitrage pricing theory is that CAPM has a single non-company factor and a single beta, whereas arbitrage pricing theory separates out non-company factors into as many as proves necessary. Each of these requires a separate beta. The beta of each factor is the sensitivity of the price of the security to that factor. Unfortunately the potentially large number of factors means more betas to be calculated. There is also no guarantee that all the relevant factors have been identified. This added complexity is the reason arbitrage pricing theory is far less widely used than CAPM.

Specifically, risky asset returns are said to follow a factor structure if they can be expressed as follows:

Where,

E(rj) is the jth asset's expected return,

Fk is a systematic factor (assumed to have mean zero),

bjk is the sensitivity of the jth asset to factor k, also called factor loading,

and εj is the risky asset's idiosyncratic random shock with mean zero.

The APT states that if asset returns follow a factor structure then the following relation exists between expected returns and the factor sensitivities:

Where,

RPk is the risk premium of the factor and

rf is the risk-free rate,

That is, the expected return of an asset j is a linear function of the assets sensitivities to the n factors.

There are some assumptions and requirements that have to be fulfilled for the latter to be correct: There must be perfect competition in the market, and the total number of factors may never surpass the total number of assets (in order to avoid the problem of matrix singularity) Ross (1976).

One of the key advantages of the APT is that it derives a simple linear pricing relation approximating that in the CAPM without some of the latter's objectionable assumptions. In general the model that APT is based on gives a reasonable description of return and risk as its factors seem plausible. Finally a great advantage of APT is that there is no need to measure market portfolio correctly.

Perhaps the key disadvantage of the APT is that it provides no clues as to what might be important factors but also as to how to interpret the factor premiums which appear in the pricing equation. That is because the APT model itself does not say what the right factors are. In addition factors of the APT model can change over time. Finally a significantly hindering drawback is that of using APT is that estimating multi-factor models requires more data.

To sum up Arbitrage Pricing Theory does not rely on measuring the performance of the market. Instead, APT directly relates the price of the security to the fundamental factors driving it. The problem with this is that the theory in itself provides no indication of what these factors are, so they need to be empirically determined. Obvious factors include economic growth and interest rates. For companies in some sectors other factors are obviously relevant as well - such as consumer spending for retailer

## Validity and role of APT

Discussing the validity of APT Ingersoll (1984) tested two theorems to provide stronger no-arbitrage conditions to the APT. Also he developed sufficient conditions for the APT pricing relation to be unique and to have a multi-beta asset pricing model interpretation. Specifically Ingersoll (1984) used a linear factor model to derive a stronger version of Huberman's (1982) "preference free" pricing theorem. Then Ingersoll (1984) relates the pricing result of the expected return on an asset to its factor responses and the covariance structure of the residuals. According to Ingersoll (1984) this relation must characterize any infinite asset economy in which no arbitrage opportunities are present whether or not a linear factor model with uncorrelated residuals is the appropriate returns generating mechanism. Finally, Ingersoll (1984) provides an interpretation of the factor premiums and gives a valid asset pricing model interpretation to the APT that overcomes the disadvantages of the APT.

Dybvig, P. H. and S. A. Ross (1985) considered the testability of the APT and pointed out the irrelevance for testing of the approximation error. They also explained the testability of the APT on subsets, and explored the relationship between the APT and the CAPM. Both the CAPM and the APT are closely linked to separation theory. The researchers argued that testing the APT on a subset is typically valid. Dybvig, P. H. and S. A. Ross (1985) refute Shanken's objections and state that Shanken's analysis of the APT has little relevance for actual empirical tests.

Shanken J. (1982) in his study challenges the view that the APT is inherently more susceptible to empirical verification than the CAPM. He based his theory on the fact that the CAPM is not truly testable on a strict sense. He also argues that the usual formulation of the testable implications of the APT are proved to be inadequate as it precludes the very expected return differentials which the theory attempts to explain. To reach his conclusions he based his tests on observation of the return on the true market portfolio.

Chinhyung, Cheol and Senbet (1986) in their paper tested the APT in an international setting. They used Inter-battery factor analysis to estimate the international common factors and the Chow test to test the validity of the APT. The inter-battery factor analysis results showed that the number of common factors between a pair of countries ranges from one to five, while the cross-sectional test results lead the researchers to the rejection of the joint hypothesis that the international capital market is integrated and that the APT is internationally valid. With their results, however, they did not rule out the possibility that the APT holds locally or regionally in segmented capital markets. Specifically the inter-battery factor analysis results of their research have shown that there are about three or four worldwide common factors and that the number of common factors between two countries ranges from one to five depending on the degree of their economic integration. On the other hand the cross-sectional test they conducted led the researchers to the rejection of the joint hypothesis that the international capital market is integrated and that the APT is valid internationally. The researchers however were not able to determine whether rejection of the joint hypothesis reflects segmentation of capital markets or the failure of the international APT.

Chang and Shanker (1987) apllied the APT to option pricing to derive a new and simple option pricing formula and to reproduce some important existing option pricing ones. The purpose of Chang and Shanker (1987) was to illustrate that the APT can be applied to the valuation of options and that under either certain return distribution assumptions or the assumption that there is only one common factor, the underlying asset of an option is the sole risky factor that explains the cross sectional variation in expected returns on all of the options that can be derived from it (Chang and Shanker, 1987).

Reinganum (1981) empirically investigated whether a parsimonious arbitrage pricing model can account for the differences in average returns between small firms and large firms listed on the New York and American Stock Exchanges. The logic for this investigation was that if a parsimonious APT can explain these differences, then using the APT as an empirical replacement for the CAPM would be more reliable. However, if a parsimonious APT failed to account for differences in average returns, then the model should be rejected. Reinganum (1981), finally concluded that the APT offers a parametric alternative to the simple one period CAPM. Having proposed that a minimum requirement for an alternative model should be that it accounts for empirical anomalies that arise within the CAPM he showed that a parsimonious APT failed that test. However while the evidence in his study does not support the APT, the conducted tests do not pinpoint exactly the source of error Reinganum (1981).

Kan and Wang (2000) reexamined the empirical performance of the conditional CAPM and the nonlinear APT model. Specifically using a nonparametric approach to estimate the conditional CAPM they looked at the models’ abilities to price the cross-section of time-varying expected returns of stock portfolios. The researchers found that the conditional CAPM does a much better job in explaining the time-varying expected returns on the stock portfolios. On pricing the conditional expected returns, the conditional CAPM has smaller pricing errors than the nonlinear APT for each and every one of our test portfolios. Kan and Wang (2000) also compare cross-sectional forecasts using a continuously updated window estimation approach. Using a pricing kernel approach to estimate the conditional CAPM, the researchers found that the model significantly outperforms the nonlinear APT model in explaining the cross-section of time-varying expected returns. Moreover, they provided simulation evidence that the conventional measure of pricing errors can be misleading when analyzing a series expansion nonlinear APT model. The researchers summarize their work in that conditional linear factor models, which have been convincingly advocated by many, are still promising candidates for further development (Kan and Wang, 2000).

## Conclusion

In conclusion, we could make the general statement that APT compared to CAPM, uses fewer assumptions but is harder to use. The conditional CAPM and the nonlinear APT are two important extensions of the Sharpe (1964) and Lintner (1965) constant beta CAPM. One prevailing question however is which one is empirically more successful and testable. APT seems to be a fruitful alternative to the utility based models that have so far been developed with the CAPM leading. Specifically in certain respects it resembles CAPM but still has some subtle and substantial differences. Most importantly it produces approximately the same linear pricing relation with the CAPM without the latter’s objectionable assumptions. Literature however is yet to be conclusive on the reliability of the two theories and for that reason research on this field is an ongoing battle. In this assignment we presented a brief literature review emphasizing this dilemma as there are numerous studies suggesting that the nonlinear APT is empirically more successful than the conditional CAPM but there are also some studies that conclude that APT do not outperform CAPM’s reliability. It is certain that the reliability of both asset pricing models is a serious focal point in the finance literature