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The Arbitrage Pricing Theory as an Approach to Capital Asset Valuation

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In fact, it is possible to view absence of arbitrage as the one concept that unifies all of finance. The law of one price is an immediate implication of the absence of arbitrage without being equivalent. By assumption, it is possible to run the arbitrage possibility at arbitrary scale [ At least one of the conditions will not hold in an arbitrage-free environment. Arbitrage opportunities in real world financial markets resulting from disparities in prices or rates of return will persist only momentarily.

The agent earns a riskless profit with no cash outlay. Prices are expected to adjust until no arbitrage remains. Hence, absence of arbitrage constitutes a necessary condition for market equilibrium in a pure exchange economy. The arbitrage argument relies on the assumption that there is at least one market agent who prefers more to less at any given scale of wealth formally, the relative risk aversion is uniformly bounded [36]. Thus, the absence of arbitrage is based on the individual rationality of a single agent.

In his seminal works he proved that a no-arbitrage environment implies the existence of a linear pricing rule which can be used to value all assets, marketed as well as non-marketed assets. The idea of arbitrage absence is particularly plausible and has not been seriously objected. Only by introducing market imperfections, such as restrictions on short sales or informational inefficiencies, arbitrage opportunities might become apparent to a certain extent. Factor models are statistical models that describe the random return generating process of assets.

The market model, mentioned earlier, is a one-factor model which assumes that asset returns are sensitive to only a single factor, namely the return on a market index. However, there is reason to believe that returns are influenced by more than one factor in the economy. Taken this into account, multiple-factor models are designed to capture various economic forces that systematically influence the movement of stock returns.

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The intuition is that changes in major macroeconomic variables like inflation or unemployment rate will impact all securities in the same way to some extent. Factor models implicitly assume that the returns of securities are commonly correlated to the factors specified in the model. Therefore, idiosyncratic return elements on one asset are supposed to be uncorrelated to other assets. The APT is based on a k -factor model with a small but unspecified number of unspecified factors.

It is assumed that ex post returns of n risky assets are generated by a model of the form linear multiple regression: E Ri is a constant term which represents the ex ante expected return on asset i when all factors and e i are zero. The realized returns Ri differ from their expected value E Ri by unexpected returns from pervasive factors weighted by their sensitivities and an unexpected residual return.

The k factors are assumed to capture systematic risk common to all assets. The noise term e i represents unsystematic risk which is idiosyncratic to the ith asset. It can be interpreted as random impact of information on asset i uncorrelated to other assets. The return generating equation can be rewritten in vector notation E.

It is assumed to have the following properties: The factors and the noise term reflect unexpected return components, hence their expected value is zero, E. W n is the variance-covariance matrix of the noise terms E. Its structure is crucial to the development of different versions of the APT. Ross , and Huberman initially assumed that the noise terms are sufficiently uncorrelated.

Chamberlain and Rothschild denoted factor models with this feature as a strict factor structure. Finally, n must be much greater than k E. For rather technical matters of a rigorous proof, assumptions on the boundedness of certain terms are made. The k -factor linearity of the return generating process is one of the primary assumptions from which the linear asset pricing rule was derived. It is consistent with the Arrow-Debreu state space preference approach of security pricing which will be discussed below.

Though intuitive in its application, the linear structure of asset returns was recently subject to critical remarks. Bansal and Viswanathan found this assumption as being unnecessarily restrictive and extended the APT to a non-linear version. The two fundamental principles and their implications apply. Agents have homogeneous expectations about the return generating process and are risk averse.

Expectations and risk aversion are bounded. Financial markets are competitive and frictionless: There is at least one asset with limited liability. The number of assets is countably infinite, modeled by an infinite sequence of economies with increasing sets of risky assets where each subsequence contains a finite number of assets. The derivation of the basic arbitrage condition follows primarily Ross , Roll and Ross , Huberman , and Dybvig and Ross A rigorous analysis and proof of the APT were elaborated in Ross , b.

The APT relies on the two fundamental principles stated above. Ross was the first who combined a linear return generating model with the absence of arbitrage in financial markets to derive a pricing formula for capital assets. The development of the APT starts with the return generating process. The random ex post return of the i th asset is generated by the k -factor equation:. The objective is to find an ex ante formula for the expected return E Ri on the i th asset representing an equilibrium condition.

The first step is the use of arbitrage portfolios connected to the return model. As defined in E. The APT asserts that arbitrage portfolios should not exist in well functioning markets. As no wealth is used and no risk encountered such a portfolio must earn a zero return in equilibrium, that is in absence of arbitrage. This means that long positions in an arbitrage portfolio are exactly financed by short positions. By definition, two vectors of the same order are orthogonal, when the inner product is zero. As a next condition, the arbitrage portfolio must be risk-free E.

The entire risk of the portfolio var x n Tr n , partitioned into systematic risk and unsystematic risk, has to be eliminated. Starting with unsystematic risk, according to E.

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Therefore, unsystematic risk can be represented by a diagonal variance matrix. Idiosyncratic risk can be eliminated only approximately. The portfolio must be sufficiently well diversified in order to apply the law of large numbers. Each element xi has to be kept quite small, while n must be sufficiently large. With these assumptions, the average variance of the error terms can be stated as follows:. The idiosyncratic disturbances have an expected value of zero E. It is to note, that the elimination of idiosyncratic risk is preliminary to the construction of an arbitrage portfolio.

In order to apply the law of large numbers, the model of an infinite economy is required. In an economy containing a finite number of assets it is impossible to diversify away specific risk elements. Thus, apriori, finite arbitrage opportunities are not available. In this sense, Huberman advanced the idea of asymptotic arbitrage opportunities. A sequence of economies with an increasing number of assets is considered.

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Let n increase to infinity. Formally, these models can be viewed as special cases of the Arrow-Debreu framework imposing restrictions on it. See Ross , p. See Merton , p. However, the assumptions of arbitrage theory and mean-variance theory are very different.

Unlike the CAPM, the APT, for instance, imposes restrictions on the dimensionality of systematic risk elements according to the k -factor return generating process. The concept of efficient portfolios is due to Markowitz Referring to the CML, Sharpe , p. For a discussion of these assumptions see Ross c.

The relative market value of securities is simply equal to the aggregate market value of the security divided by the sum of the aggregated market values of all securities. The intrinsic variance risk of an individual asset in a portfolio approaches asymptotically zero as the number of assets becomes larger. In a CAPM framework the factor is usually a market index. Factor models are discussed below in more detail. Accordingly, intertemporal analysis can be approached by two different models. The first is the discrete-time multiperiod model.

Breeden provides a list of major contributions to this approach. Ross b showed that either a risk-free asset or no restrictions on short sales are required for the CAPM. Brennan inquired the effects of corporate taxes. The CAPM formula was extended by a dividend yield term. Mayers derived a CAPM equation for expected returns on non-marketable risky assets. The basic mean-variance efficiency implications were obtained. Lintner showed that heterogeneous expectations are not critical to CAPM. Ross a described the conditions of stochastic return distribution resulting in the mutual fund separation theorem.

See Roll , p. For the following notation, vectors and matrices will be stated in bold, the superscript T indicates the transpose. In general terms, the first applies to an exchange economy, the latter to a commerce economy including production. For an extension, see Ross b and section 4. A non-linear pricing relation will be the consequence. They considered this assumption as too restrictive and developed an approximate factor structure. In the same spirit Ingersoll See also Reisman , ; Shanken In a corollary to this theorem, Ross showed that r is the risk-free rate of return if a riskless asset exists.

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This implies that the APT imposes a restriction on the expected returns of assets. BWL - Investition und Finanzierung. Business economics - Investment and Finance. VWL - Statistik und Methoden. GRIN Publishing, located in Munich, Germany, has specialized since its foundation in in the publication of academic ebooks and books. The publishing website GRIN. The risk-return combinations of all assets plot on the SML.

The market model [17] is usually used as the statistical return generating process in a CAPM context. Since the introduction of the classical CAPM, the number of contributions has been impressive. Arising from uncertain states of future investment opportunities for example changes in the risk-free rate , individuals face a set of risks against which they construct hedge portfolios.

This idea leads to the three-fund separation theorem. It is shown that optimal portfolios represent a linear combination of three mutual funds. In equilibrium, expected returns are then a function of systematic risk and risk of unfavorable changes of investment opportunities. It measures the asset sensitivity to changes in aggregate consumption. Levy considered transaction costs, Merton introduced various market segment specific SMLs, and Markowitz incorporated restrictions on short sales. Research was also done by investigating other market imperfections such as absence of a risk-free asset, taxes, non-marketable assets, heterogeneous expectations, and non-normal distributions.

Although it operates with the market portfolio, it is a step towards APT. In the overall picture, these tests using return data from the s to the s supported the implications of the CAPM.

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Based upon recent tests including more recent data, Fama and French , concluded the CAPM as empirically falsified. It seems that the last word has not yet been spoken. He argued that the CAPM implies the mean-variance efficiency of the market portfolio. All other implications such as the linear relationship between expected return and beta in E.

A valid test must rely on a true market portfolio including all individual assets in the economy. The identification and exact composition of the market portfolio for empirical tests is limited or impossible. And as long as it is limited the CAPM is not empirically testable.

Arbitrage Pricing Theory (APT): Tutorial on Implementation

The theoretical formulation of the CAPM is brilliant and it has enriched the knowledge about the functioning of financial markets. It has been the predominant paradigm in capital market theory over a long period of time and has been used in the investment community as a standard and widely accepted tool of security and portfolio analysis. However, there are indications that its primacy is declining.

The arbitrage theory of asset pricing will be inquired in the following section. The mean-variance efficiency of the CAPM relies on a set of strong restrictions such as normality in asset returns or quadratic preferences of investors. Theoretical as well as empirical considerations raised doubts about its ability to predict asset returns. It is indeed an appropriate and testable alternative to the CAPM. The two fundamental principles are absence of arbitrage and a linear k -factor model governing the random return generating process.

The arbitrage argument is the most powerful tool in positive financial economics. In fact, it is possible to view absence of arbitrage as the one concept that unifies all of finance. The law of one price is an immediate implication of the absence of arbitrage without being equivalent. By assumption, it is possible to run the arbitrage possibility at arbitrary scale [ At least one of the conditions will not hold in an arbitrage-free environment.

Arbitrage opportunities in real world financial markets resulting from disparities in prices or rates of return will persist only momentarily. The agent earns a riskless profit with no cash outlay. Prices are expected to adjust until no arbitrage remains. Hence, absence of arbitrage constitutes a necessary condition for market equilibrium in a pure exchange economy. The arbitrage argument relies on the assumption that there is at least one market agent who prefers more to less at any given scale of wealth formally, the relative risk aversion is uniformly bounded [36].

Thus, the absence of arbitrage is based on the individual rationality of a single agent. In his seminal works he proved that a no-arbitrage environment implies the existence of a linear pricing rule which can be used to value all assets, marketed as well as non-marketed assets. The idea of arbitrage absence is particularly plausible and has not been seriously objected. Only by introducing market imperfections, such as restrictions on short sales or informational inefficiencies, arbitrage opportunities might become apparent to a certain extent.

Factor models are statistical models that describe the random return generating process of assets.

Factor models implicitly assume that the returns of securities are commonly correlated to the factors specified in the model. Therefore, idiosyncratic return elements on one asset are supposed to be uncorrelated to other assets. The APT is based on a k -factor model with a small but unspecified number of unspecified factors.

The noise term e i represents unsystematic risk which is idiosyncratic to the ith asset. It can be interpreted as random impact of information on asset i uncorrelated to other assets. The return generating equation can be rewritten in vector notation E. It is assumed to have the following properties: The factors and the noise term reflect unexpected return components, hence their expected value is zero, E.

W n is the variance-covariance matrix of the noise terms E. Its structure is crucial to the development of different versions of the APT. Ross , and Huberman initially assumed that the noise terms are sufficiently uncorrelated. Chamberlain and Rothschild denoted factor models with this feature as a strict factor structure. Finally, n must be much greater than k E. For rather technical matters of a rigorous proof, assumptions on the boundedness of certain terms are made.

The k -factor linearity of the return generating process is one of the primary assumptions from which the linear asset pricing rule was derived. It is consistent with the Arrow-Debreu state space preference approach of security pricing which will be discussed below. Though intuitive in its application, the linear structure of asset returns was recently subject to critical remarks. Bansal and Viswanathan found this assumption as being unnecessarily restrictive and extended the APT to a non-linear version.

The two fundamental principles and their implications apply. Agents have homogeneous expectations about the return generating process and are risk averse. Expectations and risk aversion are bounded. Financial markets are competitive and frictionless: There is at least one asset with limited liability. The number of assets is countably infinite, modeled by an infinite sequence of economies with increasing sets of risky assets where each subsequence contains a finite number of assets. The derivation of the basic arbitrage condition follows primarily Ross , Roll and Ross , Huberman , and Dybvig and Ross A rigorous analysis and proof of the APT were elaborated in Ross , b.

Starting with unsystematic risk, according to E. Therefore, unsystematic risk can be represented by a diagonal variance matrix. Idiosyncratic risk can be eliminated only approximately. The portfolio must be sufficiently well diversified in order to apply the law of large numbers. Each element xi has to be kept quite small, while n must be sufficiently large. With these assumptions, the average variance of the error terms can be stated as follows:. The idiosyncratic disturbances have an expected value of zero E. It is to note, that the elimination of idiosyncratic risk is preliminary to the construction of an arbitrage portfolio.

A sequence of economies with an increasing number of assets is considered. Let n increase to infinity. Formally, these models can be viewed as special cases of the Arrow-Debreu framework imposing restrictions on it. See Ross , p. See Merton , p. However, the assumptions of arbitrage theory and mean-variance theory are very different. Unlike the CAPM, the APT, for instance, imposes restrictions on the dimensionality of systematic risk elements according to the k -factor return generating process.

The concept of efficient portfolios is due to Markowitz Referring to the CML, Sharpe , p. For a discussion of these assumptions see Ross c. The relative market value of securities is simply equal to the aggregate market value of the security divided by the sum of the aggregated market values of all securities.