The Arbitrage Pricing Theory as an Approach to Capital Asset Valuation E-Book

Christian Koch

• Herausgeber: GRIN Verlag
• Sprache: Englisch
• Veröffentlichungsjahr: 2009

Beschreibung

Diploma Thesis from the year 1996 in the subject Business economics - Banking, Stock Exchanges, Insurance, Accounting, grade: 1,3, European Business School - International University Schloß Reichartshausen Oestrich-Winkel, language: English, abstract: 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.

Leseprobe

Table of Content

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

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4. Arbitrage valuation of risky income streams 34

4.1 Commonly recognized valuation methods 34

4.2 Use of the arbitrage theory in the valuation process 35

5. Conclusion - some insights from the investment community 40

REFERENCES 41 APPENDIX 54

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ABBREVIATIONS

AMEX American Exchange APT Arbitrage Pricing Theory CAPM Capital Asset Pricing Model CCAPM Consumption-based Capital Asset Pricing Model CF Cash Flow CML Capital Market Line CRSP Center for Research in Security Prices DCF Discounted Cash Flow DDM Dividend Discount Model EBIT Earnings Before Interest and Taxes FL Factor Loading FLM Factor Loading Model GCAPM Generalized Capital Asset Pricing Model ICAPM Intertemporal Capital Asset Pricing Model IPO Initial Public Offering M&A Mergers and Acquisitions MVM Macroeconomic Variable Model NPV Net Present Value NYSE New York Stock Exchange OV Option Value p. Page SML Security Market Line WACC Weighted Average Cost of Capital Yrs Years

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SYMBOLS

bijSensitivity of assetito factorj(factor loading)

cov(Ri, Rm)Covariance between returns of assetiand returns of market portfoliom E(Ri)Expected return of asseti E(Rm)Expected return of market portfoliom E(Rp)Expected return of portfoliopFjExpected value of factorj~

FjRandom value of factorj~

fjRandom unexpected change in the value of factorj

kNumber of factorsnNumber of assets

~

RiRandom ex post return of asseti rfRisk-free rate var Variancez, xPercentage of wealth invested in risky assets

aiIntercept term of assetibiBeta coefficient of asseti

~ei

Random error term of assetirm, rfCorrelation coefficient of returns of market portfoliomand returns of risk-free raterfsStandard deviation of returns (of an asset or a portfolio)s2Variance of returns (of an asset or a portfolio)lCMLMarket price of risk in a CML context

0nn-zerovector0kk-zerovector0nk(n´k)-matrix of zerosn-vectorofkthfactorbkBn(n´k) matrix of factor loadings

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Dn(n´n) diagonal matrix of variancesenn-unitvector (n ones)fkk-vectorof factorsrnn-vectorof returnsTTransposexnn-vectorof assetsenn-vectorof error terms (residuals)Wn(n´n) variance-covariance matrix of residualsmnn-vectorof expected returns

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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.

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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.1The 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 versi ons have been devel oped releasing some of the restricti ons. 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 generatingk-factormodel 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.2Also, 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 toSharpe(1964),Lintner(1965), andMossin(1966) for the introduction of the CAPM. Based upon theMarkowitz(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

1The 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.

2See 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 thek-factorreturn

generating process. See Dybvig/Ross (1985), p.1181.

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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.52. 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 raterfat 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’sportfolio analysis showed that optimal portfolios plot on a positively sloped line referred to as theefficient frontier.Risk averse investors select these portfolios which offer the highest mean return for a given level of standard deviation (efficient set theorem).6In a[E(Rp),sp]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.7Through portfolio construction investors are able to diversify risk. If asset returns are less than perfectly correlated (rij< 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

3See 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”.

4See 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).

5This statement is based on a quadratic preference structure (utility function) or a normal probability distribution

of asset returns.