The Arbitrage Pricing Theory as an Approach to Capital Asset Valuation

CONTENTS

ABBREVIATIONS

SYMBOLS

1. Introduction

2. Overview of asset pricing in modern capital market theory
2.1 Classification
2.2 The Capital Asset Pricing Model
2.2.1 Assumptions
2.2.2 Capital Market Line and Security Market Line
2.2.3 Major implications
2.2.4 Extensions of the Capital Asset Pricing Model
2.3 Empirical relevance of mean-variance efficiency models

3. The Arbitrage Pricing Theory
3.1 The original Arbitrage Pricing Theory of Ross
3.1.1 Fundamental principles
3.1.1.1 Absence of arbitrage
3.1.1.2 Linear k -factor model
3.1.2 Formal statement of the theory
3.1.2.1 Assumptions
3.1.2.2 The basic arbitrage condition
3.1.3 Intuition
3.2 Contributions
3.2.1 Classification
3.2.2 Traditional Arbitrage Pricing Theory
3.2.3 Equilibrium Arbitrage Pricing Theory
3.3 Empirical evidence and economic factors
3.3.1 Testability and testing methods
3.3.2 Results from capital markets
3.3.3 Identification and interpretation of macroeconomic factors

4. Arbitrage valuation of risky income streams
4.1 Commonly recognized valuation methods
4.2 Use of the arbitrage theory in the valuation process
4.2.1 Traditional valuation approaches
4.2.2 Innovative valuation approaches
4.2.2.1 The general arbitrage approach to asset valuation
4.2.2.2 The binomial option pricing approach to asset valuation: a prospective

5. Conclusion - some insights from the investment community

REFERENCES

APPENDIX

ABBREVIATIONS

AMEX American Exchange

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SYMBOLS

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1. Introduction

A “few surprises” could be the trivial answer of the Arbitrage Pricing Theory if asked for the major determinants of stock returns. The APT was developed as a traceable framework of the main principles of capital asset pricing in financial markets. It investigates the causes underlying one of the most important fields in financial economics, namely the relationship between risk and return. The APT provides a thorough understanding of the nature and origins of risk inherent in financial assets and how capital markets reward an investor for bearing risk. Its fundamental intuition is the absence of arbitrage which is, indeed, central to finance and which has been used in virtually all areas of financial study. Since its introduction two decades ago, the APT has been subject to extensive theoretical as well as empirical research. By now, the arbitrage theory is well established in both respects and has enlightened our perception of capital markets. This paper aims to present the APT as an appropriate instrument of capital asset pricing and to link its principles to the valuation of risky income streams. The objective is also to provide an overview of the state of art of APT in the context of alternative capital market theories. For this purpose, Section 2 describes the basic concepts of the traditional asset pricing model, the CAPM, and indicates differences to arbitrage theory. Section 3 constitutes the main part of this paper introducing a derivation of the APT. Emphasis is laid on principles rather than on rigorous proof. The intuition of the pricing formula and its consistency with the state space preference theory are discussed. Important contributions to the APT are classified and briefly reviewed, the question of APT’s empirical evidence and of its risk factors is attempted to be answered. In Section 4, arbitrage theory is linked to traditional as well as to innovative valuation methods. It includes a discussion of the DCF method, arbitrage valuation and previews an option pricing approach to security valuation. Finally, Section 5 concludes the paper with some practical considerations from the investment community.

2. Overview of asset pricing in modern capital market theory

2.1 Classification

The two prevailing paradigms of capital asset valuation under uncertainty are the CAPM and the APT.^[1] The CAPM is based on mean-variance efficiency of the market portfolio in order to derive equilibrium rates of return for risky assets. While imposing strong assumptions on the preference structure of investors and the return distribution, it provides a simple and intuitive linear relation between expected return and the beta coefficient of an asset. A large number of CAPM versions have been developed releasing some of the restrictions. The APT was introduced as an alternative to mean-variance capital market theory. The price of asset risk is determined by the asset’s sensitivity to a small number of common factors representing systematic risk. The two very general assumptions of the theory are a linear return generating k -factor model and absence of arbitrage. APT and CAPM are different by conception. Nevertheless, it is possible to view the CAPM as a special case of APT with a single factor structure (where the market portfolio is a proxy for the single factor). On empirical grounds, the market model and a one-factor APT are equivalent.^[2] Also, the CAPM was extended to a multi-beta setting and thus approaching APT from the other side. A classification of asset pricing theories is provided in Appendix 1. For a better understanding of the differences to the APT, the CAPM will be reviewed in this section.

2.2 The Capital Asset Pricing Model

Credit is usually given to Sharpe (1964), Lintner (1965), and Mossin (1966) for the introduction of the CAPM. Based upon the Markowitz (1952) approach to portfolio analysis it describes the basic equilibrium conditions of asset prices in capital markets. Accordingly, the expected return of an asset depends linearly on its systematic risk which is captured by the asset’s covariance with the market portfolio. The beta coefficient is an alternative risk measure. Central to the CAPM is the mean-variance efficiency of the market portfolio, all other implications follow immediately.^[3]

2.2.1 Assumptions

The CAPM is based on the following assumptions:^[4]

1. Investments are evaluated on the basis of mean (defined as expected returns) and variance (defined as risk) of the probability distribution of returns over a one-period horizon.^[5]
2. Investors are risk averse, nonsatiated and act as price takers (in competitive markets). They have homogeneous expectations about asset returns over the same one-period time horizon.
3. Financial markets are frictionless. There are no taxes, transaction costs, or other restricting regulations. Information is freely and simultaneously available to all investors. Assets are marketable, divisible, and fixed in quantities.
4. There is a risk-free rate rf at which investors can lend or borrow money. It is the same for all investors and constant over the time period.

2.2.2 Capital Market Line and Security Market Line

Markowitz ’s portfolio analysis showed that optimal portfolios plot on a positively sloped line referred to as the efficient frontier. Risk averse investors select these portfolios which offer the highest mean return for a given level of standard deviation (efficient set theorem).^[6] In a [ E(Rp), s p ] space and in absence of riskless assets, an investor will realize a point at which his utility indifference curve is tangent to the efficient frontier.^[7] Through portfolio construction investors are able to diversify risk. If asset returns are less than perfectly correlated (r ij < 1), the portfolio risk can be substantially lower than the average risk of single assets included in the portfolio. Rational, risk averse investor will therefore hold portfolios.

Tobin (1958) extended this model by including a risk-free asset and built the bridge to the CAPM. A risk-free asset offers, by definition, a rate of return which is certain (variance of zero). If investors can lend and borrow money at the risk-free rate, portfolio choices change profoundly.^[8] The Markowitz efficient frontier is then dominated by a linear graph, known as the capital market line (CML). In a [ E(Rp), s p ] diagram, the CML usually starts at the geometrical point of the risk-free rate and is tangent to the efficient frontier.^[9] The tangency portfolio P is mean-variance efficient. Investors will combine risk-free lending or borrowing with the portfolio of risky assets. The proportion of risky assets to total investment determines the level of risk an investor is willing to assume. The point at which the individual utility indifference curve is tangent to the CML represents the optimal risk-return relation for a particular investor. This means that the optimal investment position is formed in two steps: (1) investors select an optimal diversified portfolio of risky assets, independently of risk-return preferences, and then (2) they determine the desired risk level of the overall position by borrowing or lending at the risk-free rate. This implication of the CAPM is referred to as the separation theorem.^[10]

As investors have the same set of return expectations (homogeneous expectations) and use the same risk-free rate, they all calculate the same CML and invest in the same portfolio. If all investors proportionally realize portfolio P, then P must include all marketable risky assets because assets with positive market values must be owned by at least one investor. In equilibrium, all assets must have a positive proportion of the tangency portfolio. Therefore, portfolio P is the market portfolio.^[11] The CML can formally be described as follows:

illustration not visible in this excerpt

In market equilibrium, the CML represents a linear relationship between the expected return E(Rp) and the standard deviation s p as the relevant risk measure for an efficient portfolio position. The term l CML can be interpreted as the market price of risk for a fully diversified positions.^[12] Investors who are willing to take higher risk can expect a higher rate of return. However, the CML gives no information about the relation of expected return and risk for single securities. For this analysis, the model has to be extended.

It can be shown that the risk of the market portfolio, measured by the variance s 2 m, is equal to the weighted average of the covariances cov(Ri,Rm) of all securities with the market portfolio.^[13] The relevant risk that an additional security contributes to a portfolio is thus the covariance risk (and not the standard deviation of the asset). Securities with larger cov(Ri,Rm) should generate proportionately larger expected returns, if security prices are in equilibrium. Based upon this intuition the security market line (SML) can be derived. The relation of the risk premium to covariance risk should be the same for all assets and portfolios in market equilibrium:^[14]

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The SML as the essence of the CAPM represents a linear relationship between the expected return of an asset and its relevant risk. As can be seen in (E.4), the expected return of a single security depends on (1) the risk-free rate, (2) the market priced premium per risk unit of the market portfolio, and (3) the risk contribution of a security to a well diversified portfolio. The covariance risk cannot be eliminated through diversification, it is said to be systematic risk. The beta coefficient in (E.5) is an alternative measure for systematic risk of a security.

2.2.3 Major implications

The following implications of the CAPM have become “conventional wisdom” of capital market theory and will be summarized. The total risk of any individual asset can be partitioned into unsystematic (idiosyncratic or specific) risk and systematic (market or macroeconomic) risk. The market rewards the investor with a risk premium only for systematic risk because unsystematic risk can be avoided by diversification.^[16] Systematic risk is defined by the asset covariance relative to the market portfolio or, alternatively, by the asset beta. In market equilibrium, the CAPM states a linear relationship between expected return and systematic risk: the excess return on a risky asset equals to the excess return on the market portfolio multiplied by the individual beta. 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.

2.2.4 Extensions of the Capital Asset Pricing Model

Since the introduction of the classical CAPM, the number of contributions has been impressive. As some CAPM versions are close to the ideas inherent to APT, a brief survey of extensions will be given.^[18]

One disadvantage of the Sharpe-Lintner-Mossin CAPM is the static one-period framework. To overcome this problem, an intertemporal CAPM (ICAPM) has been developed.^[19] The Merton (1973) model is based on stochastic processes in a continuous-time setting, relating to the assumption that individuals make consumption and investment decisions constantly and within a lifetime perspective. 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. The multi-beta structure of Merton ’s ICAPM version is empirically close to the APT factor structure.^[20] Combining this model with endogenous production and random technological change, Cox, Ingersoll and Ross (1985) derive a partial differential equation for asset prices in general equilibrium. Breeden ’s (1979) consumption-based CAPM (CCAPM)^[21] reduces the different state variables of the ICAPM to a single risk factor, the consumption beta. It measures the asset sensitivity to changes in aggregate consumption. The generalized CAPM (GCAPM) lessens some of the strict assumptions of the classical CAPM. Levy (1978) considered transaction costs, Merton (1987) introduced various market segment specific SMLs, and Markowitz (1990) 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.^[22] In the Sharpe (1977) multi-beta CAPM systematic risk is partitioned over several betas, taking into consideration that risk may have more than one source. Although it operates with the market portfolio, it is a step towards APT. Other efforts were undertaken to combine APT and CAPM.^[23]

2.3 Empirical relevance of mean-variance efficiency models

Early empirical tests of the CAPM were conducted by Black, Jensen and Scholes (1972), Miller and Scholes (1972), Fama and MacBeth (1973), and Blume and Husic (1973). In the overall picture, these tests using return data from the 1930s to the 1960s supported the implications of the CAPM.^[24] However, other studies^[25] found empirical anomalies (according to which the return depended on firm size, seasonality, price/earnings ratio, or dividend yields) which could not be explained by the CAPM. Based upon recent tests including more recent data, Fama and French (1992, 1993) concluded the CAPM as empirically falsified.^[26] Challenged, Chan and Lakonishok (1993) and Black (1993) responded to these results and found empirical data in support of the CAPM.^[27] They concluded that the announcement of “beta’s death” may be premature. It seems that the last word has not yet been spoken.

Important critique on the testability of the CAPM was formulated by Roll (1977). 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.5) are deduced from it and are not susceptible to independent testing. 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.^[28]

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.^[29] Ross proposed in the mid-seventies the APT as a theoretically well founded and empirically testable alternative for the CAPM. The arbitrage theory of asset pricing will be inquired in the following section.

3. The Arbitrage Pricing Theory

3.1 The original Arbitrage Pricing Theory of Ross

3.1.1 Fundamental principles

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.^[30] In contrast, the APT is based on few intuitive assumptions. 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.

3.1.1.1 Absence of arbitrage

The arbitrage argument is the most powerful tool in positive financial economics. “Most of modern finance is based on either the intuitive or the actual theory of the absence of arbitrage. In fact, it is possible to view absence of arbitrage as the one concept that unifies all of finance.”^[31] Its central idea is straightforward: if no arbitrage opportunities exist then perfect substitutes in (perfect) financial markets must have the same price. The law of one price is an immediate implication of the absence of arbitrage (without being equivalent).^[32] Dybvig and Ross (1989) provide a working definition of arbitrage:

“An arbitrage opportunity is an investment strategy that guarantees a positive payoff in some contingency with no possibility of a negative payoff and with no net investment. By assumption, it is possible to run the arbitrage possibility at arbitrary scale [...].”^[33]

Formally, an arbitrage opportunity is defined as a portfolio represented by a n- vector x n T º{ x1,x2,...,xn }, where each of the components xi is the money amount invested in asset i as a fraction of total wealth, such that:^[34]

(E.6) x n Te n = 0, meaning the portfolio x n is costless, and

(E.7) var(x n Tr n) = 0, portfolio x n has no risk (measured by variance), and

(E.8) E (x n Tr n) > 0, the expected (certain) return is positive.

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. As soon as it is discovered, an arbitrageur will exploit this difference instantly by a selling/buying or lending/borrowing strategy. 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.^[35] Theoretically, arbitrage profits are not possible. In other words, there is “no free lunch on Wall Street”.

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.^[37] Ross (1976, 1977) deduced by preclusion of arbitrage the fundamental theorem of asset pricing, which inaugurated a new paradigm in finance. 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.^[38] This is consistent with an optimal demand for at least one agent in the market who prefers more to less.^[39] A derivation of the pricing formula will be given below. 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.^[40]

3.1.1.2 Linear k -factor model

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. 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.^[41] Return changes unexplained by the factors are assumed to be specific or idiosyncratic. 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):^[42]

illustration not visible in this excerpt

[illustration not visible in this excerpt] are random variables.^[43] Ri is the ex post return of asset i, fj denotes the mean zero factor j common to the returns of all n assets and e i is the mean zero error term of asset i. 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 coefficient bij denotes the sensitivities (or factor loadings) of the ith asset’s return to the movements in the common factors fj. 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.^[44] Thus, the distinction between market risk and idiosyncratic risk of an asset is presumed apriori by the structure of the return generating model.

The return generating equation can be rewritten in vector notation (E.12a). It is assumed to have the following properties:^[45]

(E.12a) r n = m n + B n f k + e n.

(E.13) E (e n) = 0 n, e n T º { e 1 ,..., e n }, 0 n T º {0,...,0}.

(E.14) E (f k) = 0 k, f k º { f 1 ,..., fk }, 0 k º {0,...,0}.

(E.15) E (e n e n T) = W n, W n: n ´ n variance-covariance matrix of residuals.

(E.15a) W n = D n D n:

(E.16) E (e n f k T) = 0 nk, 0 nk: n ´ k zero-matrix.

(E.17) n >> k.

The factors and the noise term reflect unexpected return components, hence their expected value is zero, (E.13) and (E.14). W n is the variance-covariance matrix of the noise terms (E.15). Its structure is crucial to the development of different versions of the APT. Ross (1976, 1977) and Huberman (1982) initially assumed that the noise terms are sufficiently uncorrelated. Therefore, (E.15) takes the form of a diagonal matrix as stated in (E.15a). Chamberlain and Rothschild (1983) denoted factor models with this feature as a strict factor structure.^[46] Furthermore, the factors are uncorrelated with the idiosyncratic error terms (E.16). Finally, n must be much greater than k (E.17) in order to apply the law of large numbers. For rather technical matters of a rigorous proof, assumptions on the boundedness of certain terms are made.^[47]

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 (1993) found this assumption as being unnecessarily restrictive and extended the APT to a non-linear version.^[48] Nonetheless, Roll and Ross (1980) clarified that research on the return generating patterns can be separated from testing asset pricing hypotheses.^[49]

3.1.2 Formal statement of the theory

3.1.2.1 Assumptions

The APT relies on the following assumptions:^[50]

1. The two fundamental principles and their implications apply.
2. Agents have homogeneous expectations about the return generating process and are risk averse. Expectations and risk aversion are bounded.
3. Financial markets are competitive and frictionless: no transaction costs, no taxes, no short sale restrictions, assets are divisible. There is at least one asset with limited liability.
4. 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.

3.1.2.2 The basic arbitrage condition

The derivation of the basic arbitrage condition follows primarily Ross (1977), Roll and Ross (1980), Huberman (1982, 1989) and Dybvig and Ross (1989). A rigorous analysis and proof of the APT were elaborated in Ross (1976, 1978b).^[51]

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:

[illustration not visible in this excerpt], for: i = 1,..., n; j = 1,..., k.

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.6) through (E.8), an arbitrage portfolio is a zero net investment, zero risk and nonzero return combination of assets. 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. In order to change a current portfolio into an arbitrage portfolio an investor will combine n positions by purchasing or selling risky assets, where x­i represents the dollar amount of the ith asset as a fraction of the total investments. The arbitrage portfolio x n T uses no net investment, therefore xi must be chosen as such that the sum of the x­i proportions is:

(E.21) x n Te n º [illustration not visible in this excerpt].

This means that long positions in an arbitrage portfolio are exactly financed by short positions. The vector x n T º { x 1,..., xn } is orthogonal to the constant vector e n º {1,...,1}, both are of order n. By definition, two vectors of the same order are orthogonal, when the inner product is zero.^[52] This applies in (E.21) and will be important later.

As a next condition, the arbitrage portfolio must be risk-free (E.7). 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.15a) the random terms e i are mutually independent. 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:

(E.22) var(x n T e n) º [illustration not visible in this excerpt], for n ® ¥.

The idiosyncratic disturbances have an expected value of zero (E.13) and a variance of approximately zero (E.22), consequently they become negligible. 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 (1982) advanced the idea of asymptotic arbitrage opportunities. A sequence of economies with an increasing number of assets is considered. Let n increase to infinity. “Think of arbitrage in this environment as the opportunity to create a sequence of arbitrage portfolios whose expected returns increase to infinity while the variances of returns decrease to zero.”^[53] In practical terms however, the elimination of asset specific risk in portfolios can only be an approximation.

[...]

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^[1] The preeminent state space preference approach to general equilibrium pricing under uncertainty developed by Arrow (1964) and Debreu (1959) is the ground upon which CAPM and APT are built. Formally, these models can be viewed as special cases of the Arrow-Debreu framework imposing restrictions on it. See Ross (1977), p.190. The state preference model has never gained the popularity of CAPM (or APT) as it was too general to be testable. See Merton (1977), p.143-145.

^[2] See Ross (1977), p.205; Copeland/Weston (1989), p.83; Ross/Westerfield/Jaffe (1996), pp. 304-309. However, the assumptions of arbitrage theory and mean-variance theory are very different. In a purely theoretical perspective, the CAPM cannot be asserted being a special case of the APT. Unlike the CAPM, the APT, for instance, imposes restrictions on the dimensionality of systematic risk elements according to the k -factor return generating process. See Dybvig/Ross (1985), p.1181.

^[3] See Roll (1977), p.130; Ross (1977), p.193; Sharpe (1991), p.498. The concept of efficient portfolios is due to Markowitz (1952). Referring to the CML, Sharpe (1970), p.101, defined a portfolio to be efficient “[...] if (and only if) its expected return equals the pure interest rate plus the product obtained by multiplying the risk involved times the price of risk”.

^[4] See Sharpe (1964), p.433; Lintner (1965), p.15; Bicksler (1977), p.81; Copeland/Weston (1983), p.186; Sharpe/Alexander/Bailey (1995), p.262; Ross/Westerfield/Jaffe (1996), pp.275n. For a discussion of these assumptions see Ross (1978c).

^[5] This statement is based on a quadratic preference structure (utility function) or a normal probability distribution of asset returns.

^[6] See Sharpe/Alexander/Bailey (1995), p.194; Appendix 2 illustrates Markowitz portfolio selection.

^[7] See Brennan (1989), p.93; for illustration see Sharpe (1970), pp.30-32.

^[8] See Ross/Westerfield (1988), p.207.

^[9] An illustration and derivation of the CML is provided in Appendix 3.

^[10] See Sharpe/Alexander/Bailey (1995), p.263; Ross/Westerfield/Jaffe (1996), p.274.

^[11] See Sharpe/Alexander/Bailey (1995), p.264: “The market portfolio is a portfolio consisting of all securities where the proportion invested in each security corresponds to its relative market value. 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.” See also Ross (1978c), p.887.

^[12] See Drukarczyk (1993), p.236.

^[13] See Sharpe/Alexander/Bailey (1995), p.269; Ross/Westerfield/Jaffe (1996), p.267. The intrinsic variance risk of an individual asset in a portfolio approaches asymptotically zero as the number of assets becomes larger.

^[14] See also Drukarczyk (1993), p.238.

^[15] An illustration and a more formal derivation of the SML is given in Appendix 4.

^[16] Appendix 5 illustrates the effect of diversification on idiosyncratic risk. A study by Evans/Archer (1968) showed that 20-30 securities are sufficient to reduce substantially specific risks in a portfolio.

^[17] The market model is a single-factor model. In a CAPM framework the factor is usually a market index. Factor models are discussed below in more detail.

^[18] Refer to Appendix 1 for a classification of CAPM-extensions.

^[19] Breeden (1989) presents a summary of “Intertemporal Portfolio Theory and Asset Pricing”. 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. The second model, the continuous-time model, is due to Merton (1973), Cox/Ingersoll/Ross (1985) and Breeden (1979). The above presented ICAPM is based on the second approach.

^[20] See Brennan (1989), p.99.

^[21] See Breeden (1979); see also Lucas (1978).

^[22] Black (1972) replaced the risk-free rate by a zero-beta portfolio and developed a two-factor CAPM, stating that a riskless asset is not necessary. Ross (1977b) showed that either a risk-free asset or no restrictions on short sales are required for the CAPM. Brennan (1970) inquired the effects of corporate taxes. The CAPM formula was extended by a dividend yield term. Mayers (1972) derived a CAPM equation for expected returns on non-marketable risky assets. The basic mean-variance efficiency implications were obtained. Lintner (1969) showed that heterogeneous expectations are not critical to CAPM. Ross (1978a) described the conditions of stochastic return distribution resulting in the mutual fund separation theorem. See also Dybvig/Ingersoll (1982).

^[23] See, for instance, Connor (1984); Sharpe (1984); Shanken (1985); Wei (1988); Merville (1993).

^[24] Some found, though, that the empirical SML was too “flat”, i.e. low beta (bi < 1) assets have higher returns as predicted, high beta (bi > 1) assets had lower returns as predicted by the model.

^[25] See Blume/Friend (1973), Basu (1977); Litzenberger/Ramaswamy (1979); Banz (1981); Reinganum (1981); Keim (1983).

^[26] Möller (1988) came to similar results for the German capital market.

^[27] However, the evidence supporting the CAPM is not very strong.

^[28] Using proxies for the market portfolio, as a broad market index, is subject to other theoretical difficulties. See Roll (1977), p.130; see also Ross (1978c).

^[29] See Roll/Ross (1984a), p.14; Schneller (1990), p.1; Sharpe/Alexander/Bailey (1995), p.333; Steiner/Nowak (1994), p.347.

^[30] See Ross (1976), p.341; Roll/Ross (1980), p.1073; Huberman (1982), p.183.

^[31] Dybvig/Ross (1989), p.67; see also Ross (1978b, 1987a, 1989). The absence of arbitrage requirement has been broadly used to develop models in corporate finance (Miller/Modigliani), option pricing (Black/Scholes, Merton, Cox/Ross/Rubinstein), pricing of forward contracts (Black), interest rate parity (Keynes), etc.

^[32] Dybvig/Ross (1989), p.58.

^[33] Dybvig/Ross (1989), p.57.

^[34] See Huberman (1982), p.186; Ingersoll (1984), p.1023. For the following notation, vectors and matrices will be stated in bold, the superscript T indicates the transpose. Refer also to “Symbols”.

^[35] The absence of arbitrage is similar to the zero economic profit condition for a firm. In general terms, the first applies to an exchange economy, the latter to a commerce economy including production. For more insights see Dybvig/Ross (1989), pp. 57/58.

^[36] See Ross (1976), p.346.

^[37] See Dybvig/Ross (1989), p.57; Jarrow (1988), pp. 20-25.

^[38] This requires, in fact, that the assets are spanned by the same set of states. See section 3.1.2 and Dybvig/Ross (1989), p.60. For an extension, see Ross (1978b) and section 4.2.2.

^[39] See Dybvig/Ross (1989), p.60; Ross (1978b), p.458.

^[40] Miller (1991), p.75, comments that due to technical problems of short-selling in some cases arbitrage portfolios with zero investment cannot be created. A non-linear pricing relation will be the consequence.

^[41] See Sharpe/Alexander/Bailey (1995), p.294.

^[42] See Sharpe/Alexander/Bailey (1995), pp. 305/341; Ross (1976), p.347; Ross (1977), p.194; Roll/Ross (1980), p.1078; Huberman (1982), p.184.

^[43] For ease of notation the random symbol tilde: ~ will be omitted in the following paragraphs.

^[44] See Chen/Roll/Ross (1986).

^[45] See Ross (1976), pp.346-354; Ross (1977), p.212; Huberman (1982), p.186; Ingersoll (1984), p.1022.

^[46] See Chamberlain/Rothschild (1983), p.1282. They considered this assumption as too restrictive and developed an approximate factor structure. In the same spirit Ingersoll (1984). See also Reisman (1988, 1992); Shanken (1992).

^[47] For assumptions on the boundedness of certain terms see Ross (1976), pp.346-351; Huberman (1982), p.186; Ingersoll (1984), p.1022.

^[48] See Bansal/Viswanathan (1993). Handa/Linn (1993) consider the case that investors have incomplete information on the return generating process.

^[49] See Roll/Ross (1980), p.1077.

^[50] See Ross (1976, 1977); Huberman (1982); Jarrow (1988).

^[51] Ross (1976), p.350, proved the theorem that if the no-arbitrage condition holds in an approximate sense then the sum of the squared deviations is uniformly bounded as n increases to infinity: . In a corollary to this theorem, Ross showed that r is the risk-free rate of return if a riskless asset exists. This implies that the APT imposes a restriction on the expected returns of assets. See also Admati/Pfleiderer (1985) for a simple approach. Alternative proofs also based on an infinite economy were presented by Chamberlain/Rothschild (1983) and Ingersoll (1984). Dybvig (1983) and Grinblatt/Titman (1983) developed finite versions of the APT. See section 3.2 Contributions.

^[52] See Yamane (1968), pp.419/420.

^[53] Huberman (1982), p.185.