Cost of equity estimation

Transcription

1 MSc in Finance & International Business Authors: Anna Kwiatkowska Magdalena Mazuga Academic Advisor: Frank Pedersen Cost of equity estimation Application of the Capital Asset Pricing Model on the Warsaw Stock Exchange August

2 Table of contents Table of contents...2 Abstract...3 Introduction...5 Chapter 1 Cost of equity framework...9 Theories literature review...9 Capital Asset Pricing Model...9 Arbitrage Pricing Theory...18 Multifactor models...23 Market inefficiencies risk premiums...25 Estimating cost of equity...38 Overview of methods...38 Methods used by Polish practitioners...39 Chapter 2 Study description...43 Hypothesis statement...43 Sampling...43 Analysis presentation...45 Data...46 Chapter 3 Methodology...48 Simple regression analysis...48 Multiple regression analysis...51 Coefficient of multiple determination...53 Residual analysis...54 Testing for significance of the multiple regression model...55 Chapter 4 Industry effect...58 Industry betas...58 Industry premium...72 Construction...72 IT...86 Telecommunication...91 Food...96 Banking Chapter 5 Size effect Bibliography

3 Abstract The thesis elaborates on the accuracy to apply the Capital Asset Pricing Model in order to measure risk of the companies traded on the Warsaw Stock Exchange. The CAPM provides a technique to estimate the cost of equity capital, which is the rate of return required by investors for the risk they bear, captured quantitatively by the beta in the model. It was interesting to research application of the CAPM on the emerging Polish capital market. First the authors of the thesis found the evidence that the above-mentioned modern finance theory is popular and commonly used by the Polish practitioners. It was accomplished by means of the survey carried out among the financial directors of publicly traded companies on the Warsaw Stock Exchange. According to the CAPM, which is the rational asset pricing theory, only systematic risk, captured by beta, is priced by investors. The unsystematic risk is diversified away and it does not require any additional premium. It means that the CAPM equation should accurately describe the relationship between the required rate of return and risk, measured by the beta. The Warsaw Stock Exchange is still young and developing market, so the authors hypotheses are that the investors are not sufficiently diversified, so the unsystematic risk, in form of the industry risk and the small-sized company risk, may be priced on the Polish Capital Market. The research was carried out on the sample stocks traded on the Warsaw Stock Exchange, which were representatives of the following industries: construction, telecommunication, IT, food and banking. The researched period ranged 5 years, starting from 1st July 2000 up to 30th June First the CAPM equation was constructed for each sample stock by means of a simple regression and statistical significance of the model was checked. In almost every created model the regression fit, measured by the coefficient of 2 determination ( R ), was below 50%. It means that in practice less than 50% of variation in the sample stock s returns was driven by the systematic risk factors captured by beta. A substantial portion of variation in stock s returns remained 3

4 to be explained by the other risk factors. The highest 2 R was observed for large, liquid stocks and the lowest for rather illiquid, smaller stocks with the fragmented trading history. In order to improve regression fit and descriptive power of the asset pricing model, other variable that is the industry premium and its slope, the industry beta, were added to the CAPM equation. The industry beta is a measure of the unsystematic risk. In most cases, the rebuilt model as a whole, and the second explanatory variable, were statistically significant. The stocks with large capitalization and the most considerable share in the industry capitalization has shown the regression fit to be substantially improved, so that 2 R exceeded 80%. In other cases the regression fit was not substantially improved, but the descriptive power of the industry risk, was noticed in the Capital Asset Pricing Model, which proved one of the stated hypotheses. Further on the small-sized company study was performed, which scrutinized the relationship between the size of the company (as measured by market capitalization) and the coefficient of determination of the CAPM model for the tested companies. The relationship was tested regressing the coefficients of determination against market capitalizations of the respective companies and in addition regressing the R 2 against natural logs of market capitalizations. It was shown that the higher the market capitalization the better the fit of the CAPM model to the real data. Proving the significance of the relationship of the market capitalization (either core market capitalization or its natural log) and the R 2 of the CAPM model implies that returns predicted by CAPM for the smaller companies are usually far from reality. 4

5 Introduction The cost of capital is one of the most important concepts in finance. It plays a foremost role in capital budgeting or investment performance evaluation. The cost of capital is a quantitative measure of risk, for each adequate rate of return is required. In 1964 the capital asset pricing model (CAPM) marked the birth of asset pricing theory and for the first time provided a technique to measure the cost of equity capital. It gave theoretical background of the relationship between the expected return (cost of equity capital) and risk. Up to date, the CAPM is still widely used in practice. It is interesting to research the application of the CAPM on the emerging Polish capital market. The first chapter provides the overview of literature on the cost of equity capital issues. Its central point refers to the Capital Asset Pricing Model, its assumptions and main conclusions. According to the above mentioned rational asset pricing theory only systematic risk, captured by beta, is priced. The unsystematic risk is diversified away and it does not require an additional premium by investors. Afterwards, the supporting theory, the Arbitrage Pricing Model, is reviewed. Additionally multifactor generalization of afore-named models is shown. The multifactor models are goaled to improve statistical significance and descriptive power of the originally one-factor models. The considerable part of the chapter is devoted to the market anomalies, like size, liquidity and industry effects. Some of the empirical findings are presented that discovered that firm-specific (unsystematic) risk in form of size, illiquidity or industry risk was awarded a premium in the real world. It implies that some of the assumptions of the theoretical models are violated in practice and unsystematic risk is not diversified away. The first chapter is concluded by the evidence that the CAPM - modern finance theory, is popular and commonly used by the practitioners on relatively young capital market in Poland. The results of the survey, carried out among the financial directors of publicly traded companies on the Warsaw Stock Exchange 5

6 (WSE), are presented. The way in which the cost of equity is estimated in practice is prefaced by the theoretical overview of possible methods. The second chapter introduces the empirical study, performed in the thesis. The following hypothesis is stated: the industry risk and the small-sized company risk, is priced on the Warsaw Stock Exchange. If hypothesis is proven to be true, we believe it implies that the firm-specific risks, like industry or size risk, are rewarded a premium. Such finding does not contradict the CAPM, but is an effect of violating the assumption that all portfolios of investors present on the market are sufficiently diversified. The chapter also contains a description and a discussion of the sample of companies as well as data, taken to the empirical studies. The second chapter embodies a description of the models, used to search occurrence of the industry effect and the size effect on the WSE. In the industry study, first the CAPM equation for each sample security is set up by means of a simple regression. The derived betas are used to calculate the industry betas. Then the industry effect model is tested by means of the multiple regression analysis. The statistical significance of a second explanatory variable, added to the CAPM equation, and which is the premium for bearing nonsystematic industry risk, is analyzed. The chapter also includes the description of the study of the relationship between the size of the company (as measured by market capitalization and, subsequently, by the natural log of market capitalization) and the coefficient of determination of the CAPM equation for the tested companies. The relationship is tested by means of regressing the coefficients of determination against market capitalizations of the respective companies and in addition regressing the R 2 against natural logs of market capitalizations. The regressions are then analyzed in order to determine how significant the found relationships are. The chapter 3 presents the methodology applied in the industry and size effect studies. It contains formal derivation of the models, used in the empirical analysis. It also explains the single linear regression and multiple regression analysis, and in particular the statistical tests applied to test the hypotheses, a measure of a regression fit, as well as the regression s residuals analysis. 6

7 The chapter 4 shows the outcome of the analysis of industry effect on the WSE. First in the empirical study the systematic risk in form of beta was measured for each security of the sample. For that purpose the CAPM equations were derived by means of a regression model. In every created model, with only two exceptions, the regression fit, measured by the coefficient of determination 2 ( R ), was below 50%. It means that in practice less than 50% of variation in the sample stock s returns was driven by the systematic risk factors captured by beta. A substantial portion of variation in stock s returns remained to be explained by the other risk factors. The highest 2 R was observed for large, liquid stocks and the lowest for rather illiquid, smaller stocks with the fragmented trading history. In most cases the regression models were statistically significant. The models for the stocks with very fragmented trading track record appeared not to be statistically significant and were excluded from the further analysis (in industry study as well as the study on the size effect). The lowest 2 R for the statistically significant model was observed at the level of 1.96%. In order to improve regression fit, other variable that is the industry premium and its slope, the industry beta, were added to the CAPM equation. In most cases, the rebuilt model as a whole, and the second explanatory variable, were statistically significant. The large capitalization stocks with the most considerable share in the industry capitalization have shown the regression fit to be substantially improved, so that 2 R exceeded 80%. In other cases the regression fit was not substantially improved, but the descriptive power of the industry risk, which is the unsystematic risk, was noticed in the Capital Asset Pricing Model. This way one the stated hypotheses was proved to be true. The chapter 5 presents the outcome of the analysis of the size effect on the WSE. It is possible to show that the higher the market capitalization the better the fit of the CAPM model to the real data. For larger companies the actual returns are determined to a higher degree by the market risk premium and their betas, which represents the systematic risk (as given by the CAPM model), or put it differently the results are clustered around the capital market line. However in our sample (as specified in the Chapter 5) there are just two companies where the coefficient of determination exceeded 50%. More than a 7

8 half of the companies (58%) had their R 2 above 10%. For smaller companies listed on the WSE, the explanatory power of the CAPM for the cross section of the experience returns was not satisfactory. Even though the model was statistically significant, its R 2 was low. The analysis of the relationship between the market capitalization and the coefficient of determination is performed by means of a single factor regression, where the market capitalization is the independent variable and the R 2 is the dependent (explained) variable. The relationship between the market capitalization and the R 2 is found to be statistically significant. Due to wide range of capitalizations of the companies listed on the Warsaw Stock Exchange and included in our sample, it has been found that use of natural log of market capitalization as opposed to the market capitalization itself proved to be explaining more variation of the R 2. Proving the significance of the relationship of the market capitalization (either core market capitalization or its natural log) and the R 2 of the CAPM model implies that returns predicted by CAPM for the smaller companies are usually far from reality. 8

9 Chapter 1 Cost of equity framework Theories literature review Capital Asset Pricing Model Harry Markowitz laid down the foundation of modern portfolio management in The CAPM was developed 12 years later in the articles of William Sharpe 1 and John Lintner 2. The Capital Asset Pricing Model builds on the portfolio theory developed by Markowitz and develops the model for pricing all risky assets, allowing for determining the required rate of return for any risky asset. Because it builds on the Markowitz portfolio model, the Capital Asset Pricing Model adopts the same assumptions, extending it by some additional ones. The optimal investment policy. The theory assumes that all investors are efficient and will intend target points on the efficient frontier. The efficient frontier, called by Sharpe the investment opportunity curve is a set of investments, which guarantee the best possible combinations of the risk and return. The investment is said to be efficient if it exhibits the best expected return for a given risk or lowest risk for possible expected return. 1 Sharpe W., (1964), Capital Asset Prices: A Theory of Market Equilibrium, Journal of Finance 2 Lintner J., (1965), The valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets, Review of Economics and Statistics 9

10 U 3 U 2 Figure 1 The Efficient frontier A Expected return C B Standard deviation Source: Sharpe, Capital Asset Prices For the figure above the investment B is inefficient and dominated by the investment A, because A for given expected return it ensures lower standard deviation and by the investment C, because for the given risk, measured by the standard deviation, the investment C ensures higher expected return 3. Their specific location on the frontier will depend on the investors individual risk return preferences. Because the investors exhibit risk aversion the curves relating their expected future wealth and standard deviation are upward sloping. Figure 2 Utility (indifference) curves U 1 Expected return Standard deviation Source: Sharpe, Capital Asset Prices 3 Reilly F. K. Braun K. C., 2003: An introduction to asset pricing model, Investment analysis and portfolio management, 7th edition, Thomson South-Western, USA, p

11 For an investor, who is more risk averse than average the utility curves will be steeper than depicted in the figure above, which would show, that for the more risk-averse investor to be willing to bear an additional unit of risk the compensation in additional return would have to be greater. The pure rate of interest. The CAPM assumes that investors can borrow and lend money at the risk free rate of return or the pure rate of interest. The risk free asset has the expected return of rf and the standard deviation (risk) of 0. Since the risk free asset has a zero standard deviation and the correlation of 0 with each risky asset, all combinations including any risky asset or combination of such and the risk free asset must have expected rate of return and standard deviation, which lie along a straight line between the points representing the two components. Figure 3 Combinations of risky assets and risk free portfolio; Capital Market Line Expected return C B M RFR Standard deviation Source: Sharpe, Capital Asset Prices One can construct a portfolio by placing w Rf in the risk free asset and the remainder of 1 wrf in some risky asset. In this case the expected return of such a portfolio would then be as given by the Equation 1. Equation 1 The expected return of a portfolio composed of a risk free asset and a risky asset E R ) = w E( R ) + (1 w ) E( R ) ( P Rf f Rf i where, 11

12 E( RP ) - the expected return on the portfolio E( R f ) - the pure rate of interest E ( Ri ) - the expected return on the asset i wrf - the percentage invested in the risk free investment Equation 2 The standard deviation of a portfolio composed of a risk free asset and a risky asset σ Rp = w Rfσ Rf + ( 1 wrf ) σ R + 2ρ f, iwrf (1 wrf ) where, wrf σ i R i - the percentage invested in the risk free investment σ Rf - the standard deviation of returns of the risk free asset σ Ri - the standard deviation of returns of the risky asset ρ f,i - the correlation between the risky asset and the risk-free asset Since the σ Rf and ρ f, i equal zero, the Equation 2 simplifies in the following way: σ Rp = ( 1 wrf ) σ R. i Rf σ On the Figure 3 above, the possible combinations of a risky asset (or combination of such) and the risk free asset can be seen along the line RFR-B. However, all the combinations, which plot along the line RFR-M dominate the portfolios on the line RFR-B. The point M represents a combination in which the weight of the risk free asset wrf equals 0. The RFR-M line represents the Capital Market Line (CML). One of the assumptions adopted by the CAPM is that the borrowing at the pure rate of interest is also possible. In reality even though an investor can lend at the risk free rate meaning investing in some risk free asset, e.g. buying government bonds it is difficult, if not impossible for him/her to borrow at that prime rate. Such portfolios, constructed when investor borrows at the risk free rate and invests the proceeds in the combination of risky assets plot on the capital market line to the right of the point M (the tangency point, or the market portfolio). 12

13 Under the assumptions described above, each investor will view his investment options in the same way. As far as the implications for the Capital Market Line are concerned, each investor will chose some combination of the market portfolio (M) and the risk free asset, however the positioning on the CML will depend on each investor s individual risk-return preferences, as given by the set indifference curves. Equilibrium in the capital market. The above mentioned combination of risky assets must include all risky assets available on the market. If one asset is not included in the combination, its price would fall, causing the expected return to increase and move closer to the CML. Likewise, if assets are included in the combination desired by each market participant, their price would rise, cutting the expected return and moving the point below the line. The process continues until all the assets are included in the market portfolio. The key point to note is that the CML includes many alternative combinations of risky assets, but since they plot along one line they are perfectly correlated (but the individual assets are not perfectly correlated). This is the condition when the capital markets are in equilibrium, which means that all investments are priced in accordance with their risk, or put it differently, there is a simple linear relationship between the expected return and standard deviation of return for efficient combinations of risky assets. Other assumptions adopted by the CAPM. The model assumes that all investors have homogenous expectations, meaning that they think of possible returns in terms of some probability distribution or they are assumed to agree on the prospects of various investments their expected values, standard deviations and correlations 4. All investors in theory have the same one-period horizon (e.g. one month, year etc). All investments are indefinitely divisible, which means that it is possible to sell/buy fractions of any shares contained in the portfolio. In the CAPM framework there are no taxes or transaction costs related to selling or 4 Sharpe W. (1964), Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, The Journal of Finance, Vol. 19, No. 3, pp

14 buying any assets. There is also no inflation or changes in interest rates or inflation are fully anticipated. Conclusions of the Capital Asset Pricing Model for asset prices. The main implication of the model framework, as concluded independently by Sharpe and Lintner is that only systematic risk is priced. Higher expected returns can be expected of assets of which returns are more responsive (have a higher beta) to changes in the returns of market portfolio. Beta is the predicted response of return on a given asset to changes in return in combination and for all assets included in the combination the expected returns exhibit linear relationship with their betas. Prices will adjust so that assets, which are more responsive to changes in the returns of the combination portfolio, will have higher expected returns than of those, which are less responsive 5. Beta (the magnitude of this responsiveness) should be directly priced. All other types of risk can be mitigated by diversification and are therefore awarded no premium. Figure 4 Security Market Line, relationship between beta and expected return Expected return Pure rate of interest Beta Source: Sharpe, Capital Asset Prices The model has been tested by several researches and support for its assumptions came from Fama and MacBeth 6. The basis for the study was the statement that the expected return on security i is the expected risk free rate of return plus the 5 Sharpe, p Fama E., MacBeth J., (1973) Risk, Return and Equilibrium: Empirical Tests, The Journal of Political Economy, p

15 risk premium multiplied by the beta; E R ) = E( R ) + [ E( R ) E( R )]β. The ( i f m f authors tested three implications of the model: (1) the expected return on a security and its risk in an efficient portfolio is linear; (2) beta is the complete measure of risk of a security in an efficient portfolio, and (3) higher risk should be associated with higher return. They developed a stochastic model for returns of the stock listed on the New York Stock Exchange, which took up the following shape: Equation 3 The model for returns, Fama and MacBeth R it γ γ β γ β γ s + η 2 = 0t + 1t i + 2t i + 3t i 7 it, where, Rit - one period return on security i βi - beta of security i, si - a measure of risk of security i, not related to β i, ηit - disturbance, assumed to have zero mean and to be independent of all other variables in the equation The linearity of the security s risk and return is tested by testing the hypothesis that E( γ 2t ) = 0 (meaning that the return on a security is unrelated to the squared beta), the hypothesis used to test the condition (2) is that E γ ) = 0 (there is ( 3t it no other risk capable of explaining returns) and the condition (3) is that E( γ 1t ) is positive, or E ( Rm ) E( R f ) > 0. The authors of the research were unable to reject the hypothesis that the investors keep efficient portfolios, meaning that on average there is a positive trade-off between risk and return. They also were unable to reject the hypothesis testing the condition (1), and they concluded that the risk return relationship is linear. The condition (3) was also verified in favor of the model; the authors did 7 Fama E., MacBeth J., (1973) Risk, Return and Equilibrium: Empirical Tests, The Journal of Political Economy, p

16 not reject the hypothesis that there is no other measure of risk (other than portfolio risk) that systematically affected returns. The adequacy of the capital asset pricing models of Sharpe, Lintner and Black as empirical representations of capital market equilibrium has been however seriously challenged by a number of scientific researchers. Many of them found that there are measures of risk, which, in addition to beta help explaining the average returns; that would imply that the investors do not attempt to hold efficient portfolios. Assuming that market portfolio is efficient, there are risks, which are awarded a premium and which do not contribute to the risk of an efficient portfolio 8. In 1981 Marc Reinganum attempted to investigate empirically whether securities with different betas had different rates of return. The study was based on a sample consisting of companies listed on the New York Stock Exchange and American Stock Exchange. The test results confirmed that different betas were not systematically related to average returns across securities, specifically the high beta stocks did not exhibit higher returns than low beta stocks 9. The systematic risk input for an individual asset is derived from a regression model, referred to the asset s characteristic line, R it = i i Mt α + β R + ε, which is the best fit through a scatter plot of rates of return for the individual risky asset and for the market portfolio of risky assets over some designated past period. 8 Douglas, W., (1969) Risk in the Equity Markets: An Empirical Appraisal of Market Efficiency, Yale Economic Essays 9, p Reinganum M., A new empirical perspective on CAPM, The Journal of Financial and Quantitative Analysis, p

17 Figure 5 Scatter plot of rates of return R-Rf Rm-Rf An important decision to make while computing the characteristic line is which time series to use as a proxy for the market portfolio. In theory the market portfolio should contain all risky assets. Due to the limited availability of such time series most researchers used the S&P index or some other NYSE stock series, which is constrained only to the US stocks and includes approximately 20% of all risky assets in the global economy. Most of the academic researches recognize the problem however assume it not to be very serious. Ross however (1977) questioned the testability of the Capital Asset Pricing Model due to the problem with selection of the market portfolio. Roll had referred to it as a benchmark problem. According to him, if the market portfolio were mistakenly specified, both beta and the security market line would be wrong. The figure below illustrates the situation when the true portfolio risk (beta B ) is underestimated (beta A ) because of the proxy used to compute the beta. Using that beta ((beta A ) would imply that the portfolio s performance is superior (appears above the SML) whereas it could be inferior to the true SML, which would plot above. 17

18 Figure 6 Consequences of misspecified market portfolio for portfolio performance evaluation Expected return A B Beta A Beta Beta B According to Roll, the test of the CAPM requires an analysis of whether the proxy chosen for the market portfolio is mean-variance efficient, i.e. whether it is positioned on the efficient frontier. Roll had proven that a positive and exact cross-sectional relation between ex ante expected returns and betas must hold if the market index against which the betas are computed lies on the positively sloped segment of the mean variance efficient frontier 10. Arbitrage Pricing Theory The Capital Asset Pricing Model has provided the first quantitative definition of equilibrium in a capital market. The arbitrage pricing theory (APT), developed by Stephen Ross 11 (1976), provides further support and interpretation of pricerisk relationship in market equilibrium. The APT provides rationale behind equilibrium in a capital market, which is a result of price movements to role out arbitrage opportunities. Such opportunity arises when an investor can contract a zero investment portfolio that will yield a profit. To construct such a portfolio one has to be able to sell short at least one 10 Roll R., (1977), A critique of the asset pricing theory's tests, Journal of Financial Economics 4, Ross S.A., The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory, Dec 1976, p ; S. A. Ross, Return, Risk and Arbitrage, Risk and Return in Finance, Cambridge, Mass, Ballinger, 1977, p

19 asset and use the proceeds to purchase one or more assets. An arbitrage opportunity can appear on a single asset when it is traded at different prices in two markets or on a portfolio of assets. A portfolio owing to diversification can give higher proceeds then all assets that combine a portfolio if they are purchased separately. The APT explains that when arbitrage opportunity exists on a mispriced asset, investors will promptly purchase it, which will move the asset s price to equilibrium. In macro scale the investors tilt their portfolios toward the underpriced and away from the overpriced assets. The CAPM and APT equilibrium can be obtained in the efficient market 12. The efficiency condition is met when the prices reflect all currently available information and all market players have identical access to information. All assets in such environment must be appropriately priced. In an efficient market, information about a mispriced asset is commonly accessible and an arbitrage opportunity, it creates, is immediately used and eliminated by investors. In contrary situation, when information about some of the mispriced assets is not known, the market will not achieve equilibrium in a sense of the CAPM and APT. The Ross model, which describes rationale behind the APT, leads to the similar results as the CAPM. In the model uncertainty in asset returns has two sources: a macroeconomic (common) and a firm-specific. The model is presented by the following equation. Equation 4 The APT model r i = E( r ) +β F + ξ i i i where, E ( r i ) - the expected return on stock i F the deviation of the common factor from its expected value β i - the sensitivity of firm i to the common factor ξ i - the firm-specific disturbance 12 Ross S., Westerfield R., Jordan B., Essentials of Corporate Finance, McGraw-Hill/Irwin, New York, 2001, p

20 The model states that the actual return on stock i will equal its initially expected return plus a random amount attributable to unanticipated economywide events, which has zero expected value, plus another (zero expected value) random amount attributable to firm-specific events, which also has zero expected value. In the APT model, similarly to the CAPM, if a portfolio is well diversified, its firm-specific risk can be diversified away, which is proven below. If an n-stock portfolio, with weights of return on this portfolio is as follows: w ( w = 1), is constructed, then the rate i n i= 1 i r p = E( r ) +β F+ ξ, where p p p n β p = i= 1 wβ i i and ξ p = wξ i i. n i= 1 The variance of the portfolio can be divided into systematic and nonsystematic sources σ = β σ + σ ( ξ ) where p p F σ p - the portfolio s variance, 2 σ F - the variance of the factor F, p 2 σ ( ) - the variance of the nonsystematic (firm-specific) risk of the portfolio. ξ p Farther on, n σ ( ξ ) = variance( wξ ) = w σ ( ξ ). p i= 1 i i n i= 1 i i The variances of the firm-specific risk are uncorrelated and hence the variance of the sum of nonsystematic elements is the weighted sum of the individual nonsystematic variances with the square of the weights. If the portfolio is equally weighted, w i = 1/ n and n number of asstes, then the nonsystematic variance is: 20

21 n n σ ( ξ i ) σ ( ξ p, wi = ) = ( ) σ ( ξ i ) = = n n n n i= 1 i= σ ( ξi ). n When the portfolio contains substantial number of securities, and n gets large, the nonsystematic variance approaches to zero. The well-diversified portfolio is defined as one that is diversified over a large enough number of securities with proportions, w, each small enough that the nonsystematic variance, σ 2 ( ), is i negligible. Thus, for a well-diversified portfolio only systematic risk commands a risk premium in market equilibrium. The graphical presentation below shows superiority of holding the welldiversified portfolio as oppose to a single asset, according to the APT. ξ p Figure 7 Returns on a well-diversified portfolio and a single stock according to the APT A. Well-diversified portfolio B. Single stock Rp Rp = E(Rp) + Bp * F Ri E(Rp) E(Ri) 20 F Source: Arbitrage Pricing Theory The graph A presents the returns of a well-diversified portfolio for various realizations of the common factor. The well-diversified portfolio s return is determined completely by the systematic factor. In comparison, the graph B shows a single undiversified stock. It is subject to nonsystematic risk, which is seen in a scatter of points around the line. Such situation provides arbitrage opportunities, as a well-diversified portfolio outperforms a single stock. Thus, in the APT equilibrium, investors hold well-diversified portfolio, all lying on the same line to preclude arbitrage opportunities. 21

22 To show, what is the relationship between the expected return on a welldiversified portfolio and its β, which expresses portfolio sensitivity to the macroeconomic factor, suppose there are two well-diversified portfolios (U and V) that are combined into a zero-beta portfolio (Z). Their weights in a zero-beta portfolio are presented in the below table. Table 1 Zero-beta portfolio characteristics Portfolio Expected return Beta Portfolio weight U r ) V r ) E ( U β U E ( V B V V β V β β β V U U β β U Source: Arbitrage Pricing Theory β Z βv β U = wuβ U + wvβ V = βu + βv β β β β V U V U = 0 Portfolio Z is risk-free, as it has no diversifiable risk because it is well diversified and it has no exposure to the systematic risk because its beta equals 0. To rule out arbitrage, its rate of return must be the risk-free rate and this observation is implemented to the below equations. E ( rz ) = wu E( ru ) + wv E( rv ) β β V U = E ( ru ) + E( rv ) = βv βu βv βu r f, where r f - risk free rate. After arithmetic conversion: E( ru ) rf E( rv ) r = β β U V f, which implies that the risk premiums of all well-diversified portfolios in market equilibrium are proportional to their betas. If a well-diversified market portfolio (M) is taken together with another well diversified portfolio (P), the last equation can be rewritten as follows: 22

23 E( rp ) rf E( rm ) r = β β P M f and β = 1 M P f [ E( r M r f ] β P E( r ) = r + ) No arbitrage condition of the APT has lead to the Security Market Line of the CAPM. The APT has given the more flexible approach to the expected return - beta equation, because any well-diversified portfolio can serve as a market benchmark instead of true market portfolio that combines all assets on the market, like it is required in the CAPM. In practical implementation, the market index portfolio is used as a proxy for the true market portfolio. It was one of the concerns of CAPM analysts whether input data was precise enough to make the theory valid in practice. According to the APT, if the index portfolio is well diversified, the expected return-beta relationship holds. The APT can be generalized to a single asset. If expected return-beta relationship is satisfied by all well diversified portfolios, it must be satisfied by all individual securities that are components of those portfolios. Multifactor models The CAPM and APT in their primary versions were the single-factor models, which means that the excess expected return on an asset or a well-diversified portfolio depends on one macroeconomic factor, which mirrors market risk. However the empirical evidence has not shown the satisfactory statistical validity of the models. The fraction of the variation in stock s returns explained by the variations of the explanatory variable, measured by 2 R in a regression analysis, was perceived as not sufficient. Researchers have tried to constitute a multifactor model, which could serve as a useful refinement of one factor model. The systematic risk arises from a number of macroeconomic sources. The improved model would include sensitivities of a security s return or a return of a well-diversified portfolio to each empirically identified risk factor, instead of one market risk variable. The multifactor model would also have the improved descriptive power as investors could simply interpret what sources of risk they bear exposure to and what they demand premiums for. 23

24 The CAPM and the APT have obtained a multifactor generalization. A multifactor APT model is expressed in the equation below. Equation 5 The multifactor APT model r i = E... ( ri ) + β i1 F1 + β i2f2 + + βin Fn + ξi, where: E ( r i ) - the expected return on stock i F n - n factor driven by a source of systematic risk that significantly effects stock returns β in - - the sensitivity of firm i to the common factor ξ i - the firm-specific disturbance Similarly to a single-factor APT, each macro factor has a zero expected value and the firm-specific component of unexpected return, F n ξ i, also has zero expected value. Each systematic factor is a well-diversified portfolio constructed to have a beta of 1 on one of the factors and a beta of 0 on any other factor. Factor portfolios serve as the benchmark portfolios for a multifactor security market line. As an example, one of the multifactor models was proposed by Chen Roll and Ross 13, who constructed the following set of macroeconomic indicators. Equation 6 The multifactor model of Roll and Ross R it = α + β IP + β EI + β UI + β CG + β GB + ξ, i iip t iei t iui t icg t igb t it where: IP percentage change in industrial production EI percentage change in expected inflation UI percentage change in unanticipated inflation CG excess return of long-term corporate bonds over long-term government 13 Chen N., Roll R., Ross S., Economic Forces and the Stock Market, Journal of Business 59, 1986, p

25 bonds GB excess return of long-term government bonds over T-bills t holding period. Early research suggests that the multiple APT explains expected rates of return more precisely than does the univariate CAPM 14. However the multifactor models have not attracted wide practical usage. Their foremost shortcoming is that the determinants of the risk premiums are not specified and they must be empirically researched for each company for each specific time. Further more the statistical method to verify several factors coefficients - a multiple regression, is much more complicated than simple regression analysis, used in a single-factor model. Market inefficiencies risk premiums The validity of the CAPM, after its failure in empirical tests, was questioned 15. Insufficient statistical significance of the CAPM has lead researchers to look for its cause and for an improved, alternative model. The extensive research was carried out and some of the most appealing findings are mentioned hereafter. First empirical contradiction of the security market line was the size effect discovered by Banz (1981) 16. He found that size, measured by market equity (ME) explains the cross-section of average returns provided by market betas. Average returns of small (low ME) stocks were too high given their beta estimates and average returns on big stocks were too low. Another contradiction was the positive relation between leverage and average return documented by Bhandari (1988) 17. He found that leverage is a significant 14 Copeland T., Koller T., Murrin J., Valuation: Measuring and Managing the Value of Companies, John Wiley&Sons, Inc., New York, 2000, p Fama E. F., French K. R., The CAPM is Wanted, Dead or Alive, Journal of Finance, Volume 51, 1996, p Banz R. W., The relationship between return and market value of common stocks, Journal of Financial Economics 9, 1981, p Bhandari Laxmi Chand, Debt/equity ratio and expected common stock returns: Empirical evidence, Journal of Finance 43, 1988, p

26 explanatory variable in the multifactor model, which includes size as well as beta. Stattman 18 (1980) and Rosenberg, Reid, Lanstein 19 (1985) found that average returns on the U.S. stocks are positively related to the ratio of a firm s book value of common equity (BE) to its market value (ME). Chan, Hamao and Lakonishok 20 (1991) found that book-to-market equity (BE/ME) explains the cross-section of average returns on Japanese stocks. Size, leverage, book-to-market equity are firm-specific characteristics and according to rational pricing models (CAPM, APT) these factors should not be awarded a special premium over the market risk premium. The unexplained area has inspired further studies. The most commonly known research was performed by Fama and French. They believed in rational asset pricing in an efficient market, but they tried to prove that the CAPM beta is not sufficient to explain variability of stocks expected returns in equilibrium. They supposed that discovered market anomalies, like size or leverage effect, may proxy for fundamental determinants of risk, which added as variables to the asset pricing model, may improve its statistical significance. First study by Fama and French (1992) 21 has searched the US stock s behavior in the period of They run regressions with the following explanatory variables: size, beta, leverage, E/P, book-to-market equity. They found that the relation between average returns and firm size is negative and statistically significant. There was also strong relation between average returns and book-tomarket equity. Fama and French stratified firms into 10 groups according to book-to-market ratios and examined the average monthly rate of return of each of the groups. The decile with the highest book-to-market ratio had an average 18 Stattman D., Book Values and Stock Returns, The Chicago MBA: A Journal of Selected Papers 4, 1980, p Rosenberg B., Reid K. and Lanstein R., Persuasive Evidence of Market Inefficiency, Journal of Portfolio Management 11, 1985, p Chan L., Hamao Y., Lakonishok J, Fundamentals and stock returns in Japan, Journal of Finance 46, p Fama E. F., French K. R., The cross-section of expected stock returns, Journal of Finance 47, 1992, p

27 monthly return of 1.65%, while the lowest-ratio decile averaged only 0.72% per month. The dependence of returns on book-to-market ratio was independent of beta. This finding has suggested that high book-to-market ratio is serving as a proxy for a risk factor that affects equilibrium expected returns. Next Fama and French tried to create an asset pricing model as an alternative to the CAPM and build in it the empirically found variables: the market equity (ME) and the ratio of book equity to market equity (BE/ME) that capture much of the cross section of average stock returns. They used the multifactor APT concept that risk factors, additional to market risk, can be captured by well-diversified portfolios, which returns are not correlated to returns of portfolios served as other variables in a model. Fama and French proposed a three-factor model 22 (1993), which is presented by the below equation. Equation 7 The three-factor model of Fama and French i f i [ E( Rm ) R f] + si E( SMB) + hi E( HML + i E( R ) R = b ) ξ, where SMB small minus big, the difference between the returns on a portfolio of small stocks and a portfolio of big ones HML high minus low, the difference between the returns on a portfolio of high book-to-market-equity (BE/ME) stocks and a portfolio of low BE/ME stocks A security s expected return depends on the sensitivity of its return to the market return and the returns on two portfolios meant to mimic additional risk factors. The mimicking portfolios are small minus big (SMB), and high minus low (HML). Empirical evidence proved that a three-factor asset pricing model that includes a market factor and risk factors related to size and BE/ME captures the crosssection of average returns on the US stocks. In order to provide economic 22 Fama E. F., French K. R., Common risk factors in the returns on stocks and bonds, Journal of Financial Economics 33, 1993, p

28 explanation of the model, Fama and French studied whether the behavior of stock prices (returns), in relation to size and book-to-market-equity, is consistent with the behavior of earnings 23. Rational stock prices are discounted expected future earnings (net cash flows). If the size and BE/ME risk factors in returns (unexpected changes in stock prices) are the result of rational pricing, they must be driven by common factors in expected earnings that are related to size and BE/ME. The authors confirmed that, as predicted by simple rational pricing model, BE/ME is related to earnings. High BE/ME (a low stock price relative to book value) signals sustained low earnings on book equity. Low BE/ME (a high stock price relative to book value) is typical of firms with high average returns on capital (growth stocks), whereas high BE/ME is typical of firms that are relatively distressed. The evidence showed that size is also related to profitability. Small stocks tend to have lower earnings on book equity than do big stocks. The authors further tried to search whether size and book-to-market factors in earnings mirror those in returns. The tracks of the size factor in earnings were clear in returns. However there was no evidence that the book-to-market factor in earnings drives the book-to-market factors in returns, which the authors suggest is due to noisy measures of shocks to expected earnings. Some still argue that the Fama and French three-factor pricing model is empirically inspired and lacks strong theoretical foundations. Fama and French have not yet proved empirical superiority of their model over the CAPM. As an example, they estimated the cost of equity for industries in the USA 24. They found that estimates were imprecise and standard errors of more than 3.0% per year were typical for both the CAPM and the three-factor model. They interpreted these large standard errors as a result of uncertainty 23 Fama E. F., French K., Size and Book-Market Factors in Earnings and Returns, Journal of Finance 50, 1995, p Fama E. F., French K., Industry costs of equity, Journal of Financial Economics, Vol.43, Iss. 2; Feb. 1996, p

29 about true factor risk premiums, and imprecise estimates of the loadings of industries on the risk factors. Up to date the CAPM is widely used by practitioners and it was not yet substituted in applications by any other model. The attraction of the CAPM originates from its powerful, simple and intuitive predictions about how to measure risk. It is similar to democracy, which is not a perfect system in practice but the best ever applied 25. However noticeable market anomalies were observed, which were much larger variability of stocks returns than it is implied by the rational pricing theories. There is another direction to interpret them. They may be caused by violation of market efficiency condition. Further empirical research is presented, which evidenced that small, illiquid stocks or industry characteristics of a security were awarded a special premium. In those cases the firm-specific risks, namely size, illiquidity, or industry risk were not diversified away and were priced. In the CAPM expected return - beta equation, firm-specific premium is additive to the common risk premium, what is presented below. Equation 8 The CAPM with the firm-specific premium i f i [ E( rm rf ] + Pi i E( r ) r = β ) + ξ, where P i - a special premium that measures the effect of market inefficiency. Existence of the small-firm effect (size effect) was originally discovered by Banz (1981). The attempts to quantify the size effect were undertaken in the USA by using the rate of returns database developed at the University of Chicago Center for Research in Security Prices. The results of two independent empirical studies are presented: Ibbotson Associates Studies and Standard & Poor s Corporate Value Consulting Studies (formerly the 25 Many forms of Government have been tried, and will be tried in this world of sin and woe. No one pretends that democracy is perfect or all-wise. Indeed, it has been said that democracy is the worst form of government except all those other forms that have been tried from time to time. Winston Churchill 29

30 PricewaterhouseCoopers studies) 26. Both of them were based on the returns of stocks traded on the New York Stock Exchange (NYSE), American Stock Exchange (AMEX), and Nasdaq Stock Market (NASDAQ). The researchers have divided stocks into portfolios of similar size and calculated the return estimates, by means of the CAPM, and the realized returns. The difference between predicted and realized returns has appeared to increase with a decreasing stocks size. The results implied a size premium. Ibbotson Associates has broken down stocks returns into deciles by size, as measured by the market value of the common equity in the period of The table below details the calculation of the size premium for each decile portfolio. Additionally the table presents the size premiums for the mid-, low-, and micro-cap size grouping. Table 2 Long-term returns in excess of CAPM estimation for decile portfolios of the NYSE/AMEX/NASDAQ ( ) Decile Beta Arithmetic mean return Realized return in excess of riskless rate Estimated return in excess of riskless rate Size premium (return in excess of CAPM) 1-Largest % 6.84% 7.03% -0.20% % 8.36% 8.05% 0.31% % 8.93% 8.47% 0.47% % 9.38% 8.75% 0.62% % 9.95% 9.03% 0.93% % 10.26% 9.18% 1.08% % 10.46% 9.58% 0.88% % 11.38% 9.91% 1.47% % 12.17% 10.43% 1.74% 10-Smallest % 15.67% 11.05% 4.63% Mid-Cap, % 9.23% 8.65% 0.58% Low-Cap, % 10.52% 9.45% 1.07% Micro-Cap, % 13.18% 10.56% 2.62% Betas are estimated from monthly portfolio total returns in excess of the 30-day US Treasury bill total return versus the S&P 500 total returns in excess of the 30-day US Treasury bill, January December Historical riskless rate is measured by the 75-year arithmetic mean income return component of 20-year government bonds (5.22 percent). 26 Pratt S.P., Cost of Capital, Estimation and Applications, John Wiley & Sons, Inc., New Jersey, 2002, p.90 30