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

31 Estimated return in excess of riskless rate was calculated in the context of the CAPM by multiplying the long-horizon equity risk premium by beta. The equity risk premium is estimated by the annual arithmetic mean total return of the S&P 500 (12.98 percent) minus the annual arithmetic mean income return component of 20-year government bonds (5.22 percent) from All data have been rounded for presentation purposes and any calculation discrepancies are due to rounding. Source: Stocks, Bonds, Bills and Inflation Valuation Edition 2001 Yearbook, Ibbotson Associates, Inc. The research results show that with the decreasing size of companies, their beta and the predicted return increase. However the CAPM beta does not fully explain the total returns of smaller stocks as there is added a size premium. The exhibit below graphically shows the size effect. Figure 8 Security Market Line versus size-decile portfolios of the NYSE/AMEX/NASDAQ ( ) 25% 10 Arithmetic Mean Return 20% 15% 10% 5% Riskless rate S&P % Beta Source: Stocks, Bonds, Bills and Inflation Valuation Edition 2001 Yearbook, Ibbotson Associates, Inc. Standard & Poor s Corporate Value Consulting Studies has broken down stocks returns in the period of into 25 size categories, according to eight measures of size that are: market value of equity, book value of equity, five-year average net income, market value of invested capital, total assets, five-year average earnings before interest, taxes, depreciation and amortization (EBITDA), sales, number of employees. From the universe of researched stocks, those with the following characteristics were excluded: American Depository Receipts, nonoperating holding companies, all financial companies, companies lacking five years of publicly traded price history, companies with sales below 1 million USD in any of the 31

32 previous five years, companies with a negative five-year EBITDA, companies considered high financial risk. The computed size premiums over the CAPM estimates in Standard & Poor s Corporate Value Consulting Studies are presented in the table below. 32

33 Table 3 Long-term returns in excess of CAPM estimation (size premiums) for 25 portfolios divided by size according to different categories on a sample of stocks traded on the NYSE/AMEX/NASDAQ ( ) Market value of equity Book value of equity 5-Year average net income Market value of invested capital Portfolio rank Average Arithmetic aver. Premium Average Arithmetic aver. Premium Average Arithmetic aver. Premium Average Arithmetic aver. Premium by size [mil. USD] premium over CAPM [mil. USD] premium over CAPM [mil. USD] premium over CAPM [mil. USD] premium over CAPM % 1.1% % 0.8% % 0.8% % 0.6% % -0.9% % -0.4% % 0.8% % -0.9% % -1.4% % 0.8% % 0.6% % -0.5% % 0.0% % -0.3% % -0.1% % -0.9% % -1.3% % 1.0% % -0.2% % 0.0% % 0.3% % 1.7% % 2.3% % 1.5% % 1.0% % 1.7% % 1.0% % 1.0% % 1.7% % 0.4% % 0.8% % -0.1% % -0.8% % 0.6% % 1.7% % 0.6% % -0.1% % 0.3% % 0.3% % 0.8% % 1.5% % 2.3% % 2.7% % -1.1% % 1.6% % 2.2% % 1.8% % 0.9% % 0.4% % 0.8% % 1.8% % 2.1% % 2.7% % 2.5% % 1.1% % 1.4% % 1.6% % 1.8% % 2.6% % 2.4% % 1.0% % 2.5% % 1.1% % 2.4% % 1.1% % 1.5% % 2.2% % 1.8% % 3.0% % 1.2% % 2.0% % 1.0% % 0.3% % 1.2% % 2.2% % 0.8% % 2.1% % 3.6% % 2.7% % 2.7% % 3.3% % 2.3% % 3.6% % 2.7% % 2.4% % 4.2% % 4.4% % 3.5% % 3.4% % 3.3% % 3.9% % 2.4% % 4.2% % 2.9% % 3.5% % 3.6% % 7.0% % 4.8% % 4.8% % 6.8% Arithmetic average premium states for an average portfolio premium, derived by subtracting the average income return earned on long-term Treasury bonds (Ibbotson data) from an average rate of return for each of 25 portfolios. Premium over CAPM is the difference between the arithmetic average premium and CAPM estimated premium. Source: Standard & Poor s Corporate Value Consulting, Risk Premium Report 2000 (formerly the PricewaterhouseCoopers Risk Premium Study 2000) 33

34 Table 4 Long-term returns in excess of CAPM estimation (size premiums) for 25 portfolios divided by size according to different categories on the sample of stocks traded on the NYSE/AMEX/NASDAQ ( ) Total assets 5-year average EBITDA Sales Number of employees Portfolio rank Average Arithmetic aver. Premium Average Arithmetic aver. Premium Average Arithmetic aver. Premium Average Arithmetic aver. Premium by size [mil. USD] premium over CAPM [mil. USD] premium over CAPM [mil. USD] premium over CAPM [mil. USD] premium over CAPM % 0.1% % 1.0% % 0.8% % -0.9% % -0.6% % -0.1% % -0.6% % 0.1% % 1.0% % 1.1% % 1.3% % 1.0% % 0.0% % 0.7% % 2.3% % 2.2% % 2.2% % 1.5% % 2.6% % 2.2% % 1.7% % 1.8% % 1.5% % 1.5% % 0.7% % 0.9% % 1.2% % 0.6% % 0.0% % 0.5% % 1.5% % 1.2% % 0.9% % 0.9% % -1.1% % 2.3% % 0.8% % 1.3% % 1.4% % 1.1% % 1.6% % 2.0% % 1.1% % 1.9% % 0.2% % 2.3% % 3.6% % 1.7% % 1.1% % 3.4% % 3.6% % 1.5% % 3.6% % 1.0% % 1.8% % 0.9% % 2.7% % 1.0% % 1.8% % 3.5% % 2.4% % 3.2% % 0.8% % 1.8% % 2.7% % 3.0% % 3.3% % 1.2% % 2.4% % 2.1% % 1.4% % 1.5% % 2.9% % 3.0% % 2.1% % 0.4% % 2.2% % 1.7% % 2.2% % 3.8% % 0.8% % 2.5% % 2.2% % 1.5% % 2.2% % 4.0% % 2.2% % 3.9% % 3.6% % 4.0% % 3.8% % 3.2% % 3.8% % 3.9% % 3.2% % 4.2% % 5.2% % 4.4% % 5.1% % 4.8% Arithmetic average premium states for an average portfolio premium, derived by subtracting the average income return earned on long-term Treasury bonds (Ibbotson data) from an average rate of return for each of 25 portfolios. Premium over CAPM is the difference between the arithmetic average premium and CAPM estimated premium. Source: Standard & Poor s Corporate Value Consulting, Risk Premium Report 2000 (formerly the PricewaterhouseCoopers Risk Premium Study 2000) 34

35 The Standard & Poor s Corporate Value Consulting study has supported findings of Ibbotson Associates that risk captured by the company s size was awarded a special premium over the market premium. The observation was especially visible in the smallest 4% of the companies. The size effect can be explained by the violation of one of the conditions, describing an efficient market, namely that market prices comprise all available information. In such environment all assets are appropriately priced. When information set is not available to investors, it may cause lack of awareness of market disequilibrium and asset s mispricing. Information deficiency about small firms compared to large ones may result from their worse transparency and comparatively little coverage by the investment analysts. Market efficiency is assured by the analysts that spend considerable resources on research of any news to improve their investment performance. Grossman and Stiglitz 27 argue that investors have an incentive to spend time and resources to analyze and uncover new information only if such activity is likely to generate higher investment returns. Small firms tend to be neglected by the institutional traders, who professionally monitor the market and gather high-quality information. Such observed occurrence may cause market inefficiency. Another market anomaly concerns liquidity effect. It results from violation of one of the CAPM assumptions - costless trading. In reality all trades involve some transaction costs, but considerable discrepancies in this area were observed among securities. Investors prefer more liquid assets with lower transaction costs. One of the studies, performed by Amihud and Mendelson 28, has shown that less-liquid assets generate substantially higher returns. The researchers attempted to quantify illiquidity payoff. They studied New York Stock Exchange issues over the years Liquidity was defined in terms of bid-ask spreads as a percentage of overall share s price. The spread is 27 Grossman S. J., Stiglitz J. E., On the Impossibility of Informationally Efficient Markets, American Economic Review 70, June Amihus Y., Mendelson H., Asset Pricing and the Bis-Ask Spread, Journal of Financial Economics 17, December 1986, p and Liquidity, Asset Prices, and Financial Policy, Financial Analysts Journal 47, November/December 1991, p

36 generated by brokers, which take a position of market makers. A relatively illiquid stock means there s not enough flow of orders from customers to buy it so a broker has to bear transaction costs to sell it to a new investor. A market maker demands higher fee in form of bid-ask spread to service a deal and thus a spread widens. Amihud and Mendelson discovered that liquidity spreads measured as a percentage discount from the stock s total price range from less than 0.1% to as much as 5%. The widest-spread group was dominated by smaller, low-priced stocks. Amihud and Mendelson showed that these stocks show a strong tendency to exhibit abnormally high risk-adjusted rates of return. Their study found that the least-liquid stocks outperformed the most-liquid stocks, on average, by 8.5 percent annually over the 20-year period. The illiquidity effect is presented at the below figure. Figure 9 The relationship between illiquidity and average returns for the New York Stock Exchange stocks over the years Average monthly return (% per month) 1.2% 1.0% 0.8% 0.6% 0.4% 0.2% Bid-ask spread (% of share 0.0% price) 0.0% 0.5% 1.0% 1.5% 2.0% 2.5% 3.0% 3.5% Source: Derived from Y. Amihud and H. Mendelson, Asset Pricing and the Bid-Ask Spread, Journal of Financial Economics 17 (1986), p It is worth noting that there are two more elements of transaction costs in addition to the brokerage fees and the bid-ask spread. One of them is the price impact, which is difficult to measure, to say the least. The price impact is the difference between the execution price for a transaction on a security and the price, which was binding when the (successfully executed) buy order was put in. It shows the cost of revealing the interest in buying a stock to the market. The opportunity cost occurs when a buy order is not completely executed (due to e.g., lack of supply) and can be measured as the difference between the price of 36

37 a security on the next day minus the price valid when the buy order was put in. The price impact and the opportunity costs can account for a large percentage of total transaction costs. Therefore illiquid stocks have to offer comparatively higher returns in order to attract attention and to compensate for all those costs. Another firm-specific risk that should be diversified, according to rational pricing theory, is industry risk. However there is evidence that industry premium existed. The Ibbotson Associates in its 2001 Yearbook quantified it for 300 industries classified in accordance to American Standard Classification (SIC) code down to three-digit level 29. The method Ibbotson Associates used in formation of industry premiums relies on a risk index for each industry that can be used to compare risk level of an industry with that of the market as a whole. The industry risk premium methodology uses the following equation. Equation 9 The expected industry risk premium IRP i = ( RI ERP) ERP, where: i IRP i - the expected industry risk premium for industry i, or the amount by which investors expect the future return of the industry to exceed that of the market as a whole RI i - the risk index for industry i ERP the expected equity risk premium. For an industry with a risk index of 1 (the same as that of the market), the expected industry risk premium would be 0. For those industries with a risk index greater than 1, the industry premium will be positive, and for those with a risk index less than 1, it will be negative. Some argue that the market anomalies never existed but were a result of data mining. Nevertheless similar abnormalities, concerning the size effect or illiquidity premiums, were observed in many markets in different time. However, they are of little practical use, as the noticed pattern from rerunning 29 Pratt S.P., Cost of Capital, Estimation and Applications, John Wiley & Sons, Inc., New Jersey, 2002, p

38 the computer database with past returns cannot be applied to predict future returns. Estimating cost of equity It was interesting to find out how estimation of cost of equity looks in practice in emerging Polish capital market and whether the modern finance theories are applied. First methods of cost of equity estimation are overviewed, which is followed by presentation of the survey, carried out among Polish practitioners. Overview of methods There are three commonly used methods of estimating the cost of equity capital: dividend discount model, earnings models and the Capital Asset Pricing Model. The dividend discount model (DDM) was popularized by Gordon 30 (1962). It provides the present equity value in form of a sum of the future discounted dividends that a company is expected to pay its shareholders. The discount rate used in DDM serves as the cost of equity. The dividend model is presented in the equation below. Equation 10 The dividend discount model D0 (1+ g) EV0 =, where k g EV 0 - the equity value at present D 0 - the present value of yearly dividends g- the dividend growth rate k- the discount rate (the cost of equity capital). However usage of DDM is limited to the companies that lead a transparent and stable dividend policy, which in consequence enables the analysts to predict future dividend payments. The DDM is often used for valuation and estimation 30 Gordon, M.J., The Investment, Financing and Valuation of the Corporation, Homewood, Illinois, Irwin,

39 of cost of capital for the banking sector in Poland, which exhibit most stable dividend policies. The second method - an earnings model can be employed to estimate the cost of equity capital by taking the inverse of the price-earnings ratio. A price-earnings multiple (P/E) is the ratio of price per share to earnings per share. P/E ratio is perceived as a useful indicator of expectations concerning company s growth opportunities. A firm with good investment opportunities, which is to implement projects promising the return on equity (ROE) higher than the cost of equity, is rewarded with a higher P/E multiple by a market 31. However, while using P/E ratio, care needs to be taken to exclude exceptional events affecting earnings per share. The third method, developed from rational asset pricing theory is the Capital Asset Model, established by Sharpe (1964) and Lintner (1965). Methods used by Polish practitioners The article Practitioners perspectives on the UK cost of capital 32 was the inspiration to carry out the similar survey among the financial directors of publicly traded companies on the relatively young Warsaw Stock Exchange (WSE). The Warsaw board was established in 1991, after more than 40 years of socialistic economy in Poland. The first trading took place on 12 April 1991 with stocks of five companies available. At the end of 2005 there were 255 companies traded on the Warsaw Stock Exchange and their overall capitalization amounted to PLN 425bn 33. In June 2005, after 14 years of WSE opening, the study, that researched applied methodology of cost of equity estimation in Poland, was carried out. It was 31 Bodie Z., Kane A., Marcus A.J., Investments, McGraw-Hill, Irwin, New York, 2002, p McLaney E., Pointon J., Thomas M., Tucker J., Practitioners perspectives on the UK cost of capital, The European Journal of Finance, Volume 10, April 2004, p The Warsaw Stock Exchange statistics, 39

40 limited in scope compared to the UK survey, which questioned the cost of capital methods. In Polish survey the questionnaire was sent to 157 financial directors of the companies traded on the Warsaw Stock Exchange. The operating profile of the researched companies was production (industries: food, metal, building products, textile, electronic machinery & automation, wood, chemical) and service (industries: information technology, retail). Fifteen companies (10% of the peer group) answered the survey. Six of them informed that they do not calculate the cost of equity capital. One of them was a subcontractor and it did not need to gather financial sources for investments itself and took care of cost of capital issues. One of them explained it had too short trading history on the stock exchange to be able to calculate the cost of capital. One of them stated that it was too small and did not have a person who was responsible for such detailed financial issues. Twenty-two companies could not answer the survey as the cost of capital issues were treated as confidential under their information policy. The reason for some of them not to answer the survey was their bankruptcy status. The first observation is that despite of public status and high expected transparency of such companies by their shareholders, there are ones among them that neglect monitoring and managing the cost of equity capital. The table below presents the survey s results. Table 5 Methods the companies traded on the Warsaw Stock Exchange used to calculate the cost of equity capital Number of answers Dividend model with a growth factor P/E CAPM Number % Number % Number % 1 6.7% % % Source: Survey carried out among financial directors of companies traded on Warsaw Stock Exchange The most popular methods to calculate the cost of equity capital are the P/E multiplier and the CAPM. Three of the companies have regularly paid dividends for last three-four years. One of them used the dividend model with a growth 40

41 factor to calculate the cost of equity capital, two others used the CAPM. One of the companies estimated the cost of equity capital by means of both, the P/E and the CAPM. Two of the companies used other methods. One enlarged the risk free rate by arbitrary set risk premium, other as quoted: The basis is a longterm risk free rate (e.g. average return on 10-year Treasury bonds), which is adjusted for the premium for the shareholders (average return on investments over risk free rate that is average return for main indexes on the stock exchange) and beta which explains risk of industry in relation to the market risk plus country risk measured by the difference between T-bonds of particular country and US T-bonds.(Cost of equity capital = Long-term risk free rate + Market premium * Beta + Country risk). The results of the cost of equity estimation were gathered for relatively small sample of public companies traded on the Warsaw Stock Exchange, but they gave insight into the methods used by Polish practitioners. It appeared that different modern finance theories are applied, namely the CAPM and the P/E ratio. The dividend discount model is not very popular. One of the companies used modified version of the CAPM but the methodology was not very clear. The special premium was added for the country risk. A country risk could be perceived as an unsystematic risk in a context of international pricing model. As any other undiversified unsystematic risk it is priced in form of a special premium. The international generalization of the CAPM or the APT states that the capital markets are globally integrated and market equilibrium should be achieved at the international level. Ibbotson, Carr and Robinson 34 defined a broad world market portfolio and calculated betas of equity indexes of several countries against a world equity index. The assets, which are less accessible to foreign investors, carry higher risk than the international CAPM would predict which implies a country risk premium. 34 Ibbotson R.G., Carr R.C., Robinson A.W., International Equity and Bon Returns, Financial Analysts Journal, July-August

42 For valuation purposes, financial institutions as well as consulting agencies in Poland mostly use the CAPM for estimating the cost of equity capital. From our experience, however, little attention is directed towards precise estimation of the market portfolio, the market risk premium is usually chosen arbitrarily. As a risk free rate, usually a 5-year or 10-year (depending on the valuation horizon) Polish government bond yield is used. For the purpose of valuation of privately held companies, occasionally 1 percentage point premium is added for lack of liquidity or due to the company s small size. 42

43 Chapter 2 Study description Hypothesis statement We state the hypothesis that the industry risk and the small size risk, is priced on the Warsaw Stock Exchange. If hypothesis is proven to be true, we believe it implies that firm-specific risk, like industry or size, is rewarded a premium, which is inconsistent with the CAPM. However such finding does not contradict the rational pricing theory, but just shows that the assumption of full diversification of investors present on the market is violated. The industry premium may imply that there are not enough assets within an industry group to diversify an industry risk away. In terms of size effect, we believe, it would be a result of less coverage of small firms by the investment analysts, which are mainly informers of the institutional investors. This may cause not sufficient presence of small firms in the portfolios of institutional investors, thus small size risk would not be diversified away by them. That could also be due to the fact that the market does not fulfill the requirement of the second-degree efficiency (which would imply that all publicly available information is instantly reflected in stock prices). Small companies, which are not broadly covered by the analysts, might then exhibit mispricing for a substantial period of time. We do not support the explanation of Fama and French (1992) that size might proxy for a determinant of systematic risk, which affects equilibrium expected returns. Sampling Our research sample contains the securities traded on the Warsaw Stock Exchange, which represent sectors with largest representation of stocks and for which the sector indexes are being counted. These are: construction, telecommunication, IT, food and banking. The asset allocation to a certain 43

44 sector is consistent with the classification applied by the Warsaw Stock Exchange 35. The researched period ranges 5 years, starting from 1 st July 2000 up to 30 th June From the sample described above the securities with the following characteristics were excluded: Its trading history does not cover the researched period, A company is filled under bankruptcy, A company recorded a negative book value at the end of any year within the researched period. The following table presents the group of companies within each industry, which data was taken to the research. Table 6 List of the sample companies Construction Telecommunication IT Food Banks 1 Budimex Elektrim Comarch Beefsan BOS 2 Budopol-Wrocław Netia Computerland Ekodrob BPH 3 Echo Investment Telekomunikacja Polska CSS Indykpol BRE 4 Elbudowa Elzab Jutrzenka BZ WBK 5 Elektromontaż Północ Internet Group Kruszwica DZ Polska 6 Elektromontaż Warszawa Macrosoft Mieszko Fortis 7 Elektromontaż Południe Prokom Pepees Handlowy 8 Energopol-Południe Softbank Rolimpex ING 9 Hydrobudowa Śląsk Sokolow Kredyt Bank 10 Instal Kraków Strzelec Nordea 11 Instal Lublin Wawel Pekao 12 Mostostal Eksport Wilbo 13 Mostostal Płock Zywiec 14 Mostostal Warszawa 15 Naftobudowa 16 Pemug 17 Polimex-Mostostal Siedlce 18 Polnord 19 Projprzem

45 Analysis presentation The aim of the analysis is to scrutinize whether the unsystematic risk in form of industry (industry effect) or small firms risk (size effect) was rewarded a premium on the sample securities traded on the Warsaw Stock Exchange. The industry study starts from derivation of the relationship between expected return and beta according to the CAPM equation for each sample stock. The statistical significance of the equation set up by means of a simple regression is inseparably studied. Having estimated the betas of all sample securities, the sector betas are calculated as the weighted average of betas for stocks belonging to the particular industry. The stocks capitalization is used for weighting. The discrepancies among industry peers betas are also presented. Next a two-factor model is created for each sample security. The premium rewarded for an industry risk is added as a second explanatory variable to the CAPM equation. The industry variable is uncorrelated to the market variable, which serves as a determinant of systematic risk. If the added factor is statistically significant, the firm-specific risk is priced on the sample assets traded on the Warsaw Stock Exchange. In order to study the hypothesis, multiple regression analysis is carried out. The size effect is simplified and includes only the companies representing the five mentioned sectors for which the regressions as given by the CAPM equations are significant. We examined the relationship between the market capitalizations (market equity) and subsequently the natural log of the market capitalizations versus the coefficients of determination for the models used for derivation of the relationship between expected return and beta according to the CAPM equation for all sample securities 45

46 Data The regression analyses were preformed in the 5 years period ( ). A weekly return was a basic data unit, which resulted in 260 holding periods, and the same number of observations for the regression. It is a sufficient data set to provide statistical significance of the analysis. The stocks prices were gathered from the Polish financial news service, Parkiet (web site: It contains the archive of historical trading record of the companies listed on the Warsaw Stock Exchange. The Warsaw index WIG, was taken as a proxy of the market portfolio. It covers shares, which construct 99% of the stock exchange market capitalization, which is defined as current market price times shares outstanding plus the dividend payments. The industry portfolios were proxied by the industry indexes, namely WIG-bud for construction, WIG-info for IT, WIG-telekom for telecommunication, WIG-spoz for food, WIG-bank for banking industry. The values of market and industry indexes were taken from financial service, Parkiet (web site: The 52-weeks Polish Treasury bills rate of return was applied as a risk free rate in the study. The prices of T-bills issues were gathered from the Polish Ministry of Finance web site ( The 52-weeks T-bills were chosen as a risk-free rate because their prices are relatively stable compared with other short-term treasury securities. The tender for 52-weeks T-bills was taken place on Tuesdays almost every week in the researched period. When there was no data available for 52-weeks T-bills in a particular week, because of lack of trading, such week was excluded from the analyses. The prices for a security, the market index (WIG), and an industry index were gathered the same day as the tender of 52-weeks T-bills took place, usually on Tuesday. On the basis of the gathered prices, weekly rate of returns for a security, WIG, and an industry index were calculated. When there was no trading of a security in a day of a T-bill tender, the price from the previous day was taken. When there was still no data available, the rate of return 46

47 from the previous week was applied. Additionally the mechanism to exclude the extreme observations was used. When a security s price was deviated from the price of the prior period by more then 20%, such observation was excluded from the regression analysis. The information of the security s book value and number of stocks traded of a particular firm was withdrawn from the companies financial statements, publicly available at each company website. 47

48 Chapter 3 Methodology Simple regression analysis The simple regression analysis was applied in order to derive the securities betas, which describe the relationship between the excess return of any sample asset and the excess return of market index, as specified in CAPM and to test the hypothesis of existence of the size effect on the WSE. CAPM model describes the relation between the expected returns on individual securities and the beta of the security. The beta is the covariance between the security i's and the market return, divided by the variance of the market return. CAPM delivers an expected value for security i's excess return (over the risk free rate) that is linear in the beta, which is security specific. In this case the ordinary least squares (OLS) regression is used to estimate the beta coefficients for each security. The regression is used according to the following equation. i fi [ E( RWIG i rfi] i E( R ) r = α + β ) + ξ Where R i - i th observation of the stock return r fi - i th observation of the risk-free rate R WIG - i th observation of the return on WIG (Warsaw Stock Exchange s market i index) β -the beta coefficient related to the market premium ξ i - the error of the i th estimate or diversifiable, nonsystematic or idiosyncratic risk [ E ( R WIGi ) rfi )] β - nondiversifiable, systematic risk In this model, the beta is the ratio of covariance to the variance of the market return. The alpha is the intercept in the regression. The equation is not exactly a CAPM equation but one that allows to estimate the stock s beta coefficient; according to CAPM the alpha is zero. The only type of risk for bearing which 48

49 the investors are rewarded a premium, according to CAPM is the systematic risk, which cannot be diversified away. For each given security, the OLS regression is used in order to estimate the alpha and beta, being the intercept and the slope of the line shown in the picture below. Figure 10 Scatter plot of excess returns of the asset and the WIG index and the asset s characteristic line R-Rf Rm-Rf Source: Own calculations, based on parkiet.com, bossa.pl The error in the regression, ξ i, is the distance from the line to each point on the graph. Upon estimating the asset characteristic lines the analysis of the regression is performed in order to determine whether the line is a good fit for the scatter of the actual data and if the independent variable (here market excess return) is significant in explaining the variation of the dependent variable (here: security excess return over the risk free rate). The key statistics are the coefficient of 2 determination ( R ), F-statistic, t-statistics. The graph of residuals is examined for potential existence of autocorrelation and heteroskedasticity and to ensure whether the assumptions of the regression are not violated, which could spur the test results. In the graph presented below the examination of the distribution of the residuals brings to an initial conclusion that the residuals are not correlated across observations and they do not exhibit heteroskedasticity. 49

50 Figure 11 Sample residuals plot Residuals Independent variable Source: Own calculations, based on parkiet.com, bossa.pl Ordinary least square regression is also performed in order to verify the existence of the size effect on the Warsaw Stock Exchange. In this case the regression is used to estimate the coefficients of the Equation 11. Equation 11 The relation between the market capitalization and the coefficient of determination 2 R ( ) t = α + β MCAP t, 2004 Where: 2 Rt - coefficient of determination for the regression of security t s excess returns over the risk free rate against the market (WIG index) excess returns over the risk free rate α - the intercept term of the regression β - the slope of the regression MCAP t, market capitalization of security t as at the end of

51 Since the market capitalizations of the securities used in the sample are very dispersed 36, instead of using the market capitalization as the independent variable, the natural log of the capitalization value was used, and the regression equation transformed to the following: Equation 12 The relation between the natural log of market capitalization and the coefficient of determination 2 R = α + β ln( MCAP ) t Where: t 2 R t, α, β - as previously t t ln( MCAP t, 2004) - natural log of the market capitalization of security t as at the end of 2004 For both regressions, the analysis is performed in order to determine, how much of the variation of the dependent variable is explained by the variation of independent variable (log of or market capitalization) and the significance of the regression estimates is tested. Multiple regression analysis The multiple regression analysis is a scientific method applied in order to study the statistical significance of the two-factor model, which proves existence of the industry effect on the Warsaw Stock Exchange. The two-factor model being studied is presented by the multiple regression equation. Equation 13 Multiple regression model for the purpose of industry study E R ) r = a+ [ E( R ) r ]+ ( i fi β WIG i fi β sec [ R WIG ) i rfi] + i E adj ξ ( sec 36 The market capitalizations range from over PLN 6m PLN to PLN 23bn as at the end of

52 where: a - intercept R i - i th observation of the stock return r fi - i th observation of the risk-free rate R WIG - i th observation of the return on WIG (Warsaw Stock Exchange s i market index) β -the beta coefficient related to the market premium R sec - i th observation of the return on an adjusted industry index WIG adj i β sec - the sector beta coefficient related to the industry premium ξ i - the error of the i th estimate. According to the above model, the expected return on a security depends on two explanatory variables: the market premium, which is a reward for bearing systematic, un-diversifiable risk and the industry premium, which is a price for not-diversified unsystematic risk. The market premium is an excess return on WIG over the risk-free rate. The industry premium is a part of an excess return on an industry index over the risk-free rate, which is not captured by the market premium, hence not correlated with the market premium. Some of the factors that drive the market index s returns are similar to ones that influence the industry index s returns, and that is why the returns of those indexes are in general correlated. In order to exclude a part of variation in the industry premium that is explained by the market premium, the following steps are undertaken. First the single linear regression, which is presented below, is run. Equation 14 Regression model with market premium explaining the industry premium [ E( R r ] + ζ E R ) r = a+ b ) E ( sec, where WIG R WIG ) r f f WIG ( sec - the industry (sector) premium E( R WIG ) r f - the market premium f 52

53 a the intercept term b- the slope coefficient ζ - regression residuals. The residuals of above regression state for variation in the sector premium that is not explained by the market premium. It is an input to the two-factor model, Equation 13, as a second explanatory variable, adjusted industry premium. E R WIG ) r adj ( sec, the The multiple regression analysis 37 is performed based on the coefficient of 2 multiple determination ( R ), residual analysis, and tests for the significance of the model. The detailed equations of the computations that can be derived from the MS Excel outcome in form of the analysis of variance, () table, are not presented hereafter. Coefficient of multiple determination f 2 The coefficient of determination ( R ) in the two-factor model represents the proportion of the variation in excess stock s return over the risk-free rate that is explained by both market premium and industry premium. In the two-factor model the adjusted after adding second explanatory variable in comparison to model. The adjusted 2 R is used, which shows the regression fit 2 R of the one-factor 2 R reflects the number of explanatory variables in the model and the sample size. It is computed as follows. Equation 15 Adjusted 2 R 2 R adj = 1 (1 R 2 n 1 ) n k 1, where 37 Levine D. M., Ramsey P. P., Smidt R.K., Applied Statistics for engineers and scientists, Prentice-Hall Inc., New Jersey, 2001, p

54 2 R -the coefficient of determination taken from the table n- the sample size k- the number of explanatory variables in the regression equation Residual analysis Regression analysis relies on number of assumptions concerning the residuals. When these assumptions are violated, the inferences drawn from the model become questionable. The primary violations of assumptions concerning residuals are: heteroskedasticity, serial correlation or while the residuals are not normally distributed. In order to observe if they appear, the residuals plot is analyzed. The residuals are assumed to be homoskedastic, which means they have constant variance across all observations. Heteroskedasticity occurs when the variance of the residuals is not the same across all observations in the sample. This can be observed at the residual plot, when in one time interval the errors terms have low values and are close to the x-axis, while in other period they have noticeably higher values, thus they are more spread out over the x-axis. The second assumption concerning the residuals states that they are independently distributed or interchangeably stated that the error for one observation is not correlated with that of another observation. Serial correlation, also known as autocorrelation, refers to the situation in which the residual terms are correlated with one another. Autocorrelation can be seen at the residuals plot when there appears to be a pattern in the error terms. Positive serial correlation is seen when positive (negative) errors tend to be followed by positive (negative) errors and negative serial correlation occurs in the odd situation. The last assumption, which can be checked, based on the residual plot is that the error terms are normally distributed. 54

55 Testing for significance of the multiple regression model Tests for significance in the multiple regression involve testing whether: Individually, the independent variables contribute to the explanation of the dependent variable, Together, some or all of the individual independent variables contribute to the explanation of the dependent variable, The additional explanatory variable contributes to the explanation of variation in the dependent variable. The methodology of testing for significance of the multiple regression is presented on the general example of the regression model with two explanatory variables. Y 1 = 0 + β1 X 1i + β 2 X 2i β + ξ i A test of significance for individual slope coefficients is conducted to determine if the independent variables make a significant contribution toward the explanation of the dependent variable. A t-test for that purpose is used. The null and alternative hypotheses are set as follows. H : β 0 The slope coefficient for the j th independent variable equals zero, 0 j = thus there is no statistical linear relationship between the j th variable and the dependent variable, independent H : β 0 There is statistical relationship between the j th independent variable 1 j and the dependent variable. The decision rule is to reject H 0 at the α level of significance if t > tα / 2, n k 1, otherwise do not reject H 0. t is the outcome of t- test and t α / 2, n k 1 is the twotailed critical value on the t distribution with n-k-1 degrees of freedom. The F test is applied to test the statistical significance of the entire multiple regression model. It determines whether there is a significant relationship 55

56 between the dependent variable and the set of explanatory variables, as a group. That means, the F-statistics is used to test whether at least one independent variable in a set of independent variables explains a significant portion of the variation in the dependent variable. The null and alternative hypotheses are set as follows. H β =β 0 There is no linear relationship between the dependent variable 0 : 1 2 = and the explanatory variables, H : At least one β 0 There is a linear relationship between the dependent 1 j variable and the explanatory variables. The decision rule is to reject H 0 at the α level of significance if F > F k, n k 1, otherwise do not reject H 0. F is the outcome of F test and F k, n k 1 is the critical value on the F distribution with k and n-k-1 degrees of freedom. The partial F-test criterion is performed to determine the contribution of an explanatory variable to the multifactor model. In developing a multiple regression model, the objective is to utilize only those explanatory variables that are useful in predicting the value of a dependent variable. If an explanatory variable is not helpful in making this prediction, it could be deleted from the model. The partial F-test criterion involves determining the contribution to the regression sum of squares made by each variable after all the other explanatory variables have been included in the model. The null and alternative hypotheses are set as follows. H 0 : Variable X 2 does not significantly improve the model once variable X 1 has been included, H 1 : Variable X 2 significantly improves the model once variable X 1 has been included. 56

57 The contribution of an explanatory variable to be included in the model can be determined by taking into account the regression sum of squares of a model, SSR that includes all explanatory variables except the one of interest. Thus the contribution of variable X 2 given that variable X 1 is already included, can be determined from the following equation. Equation 16 SSR, contribution of the explanatory variable X 2 given that variable X 1 is already included in the two-factor model SSR( X 2 / X 1) = SSR( X 1andX 2 ) SSR( X 1), where SSR ( X 1 andx 2 ) - the sum of squares for regression that includes both explanatory variables ( table) SSR ( X 1) - the sum of squares for regression that includes only the explanatory variable X 1 ( table) The partial F-test criterion for determining the contribution of the independent variable, X 2, is calculated according to the equation below. Equation 17 The partial F-test criterion for determining the contribution of variable X 2 SSR( X 2 / X 1) F partial =, where MSE SSR X 2 / X ) - the sum of squares taken as an outcome from Equation 16 ( 1 MSE- the average sum of squared errors adjusted for degrees of freedom for regression that includes both explanatory variables ( table) The decision rule is to reject H 0 at the α level of significance if F, otherwise do not reject H 0. partial > F k, n k 1 F partial is an outcome of the partial F-test criterion, calculated in the Equation 17 and F k n k 1 is the critical value on the F distribution with k and n-k-1 degrees of freedom., 57

58 Chapter 4 Industry effect The chapter 4 presents the analysis of the industry effect on the WSE. First sector risk is captured in form of the sector beta for the following industries: construction, telecommunication, IT, food, and banking. Then further analysis is undertaken in order to check whether industry risk, perceived as an unsystematic risk, was awarded an industry premium by investors. Industry betas An industry risk depends on risk of companies that operate within a certain industry. Thus an industry beta is based on betas of the sample securities that construct a particular sector. A beta of each sample security is derived from a CAPM equation, which is found by means of a single regression. Budimex is taken as a representative stock for the construction industry and the regression analysis for its CAPM equation is presented below. Equation 18 The CAPM equation of Budimex (single regression model) [ E( R r ] ζ E R ) r = a+ β ) + ( Bu dim ex f Bu dim ex ex f WIG f, where E( R Bu dim ) r - the excess rate of return, that is the expected rate of return for Budimex over the risk-free rate E ( - the market premium R WIG ) r f a the intercept term β Bu dim ex - the slope coefficient, the Budimex beta ζ - regression residuals. The model data were input into Excel spreadsheet and the outcome of the regression analysis is presented below. 58

59 Figure 12 Graphic presentation of Budimex single regression model. 0.2 y=r-rf y = x R 2 = x=r(wig)-rf Source: The analysis of the thesis s authors Table 7 Microsoft Excel output for the regression analysis of Budimex Multiple R R square Adjusted R square Standard error Observations 242 Regression E-14 Residual Total Intercept Slope (B Budimex) E The Analysis of variance () table shows the coefficients of the intercept and slope as well as their standard errors and t-statistics for the level of significance equal to α =

60 The t-statistics is used to test the statistical significance of the coefficients in the regression model. In order to test the slope, following the methods, described in the Chapter 3 Methodology, the null and alternative hypotheses are set as follows. H β 0 The slope coefficient equals zero, thus there is no statistical 0 : Bu dim ex = linear relationship between the market premium and the expected rate of return for Budimex, H β 0 There is statistical relationship between the market premium 1 : Bu dim ex and the expected rate of return for Budimex. The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as t-stat is greater than t critical ( >1.96). The beta for Budimex in the analyzed regression is statistically significant. Subsequently the t-test for the intercept is carried out. The null and alternative hypotheses are set as follows. H : a 0 The intercept equals zero, 0 = H : a 0 The intercept is different from 0. 1 The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is not rejected as t-stat is less than t critical ( <1.96). The intercept in the analyzed regression is not statistically significant. The regression model implies the equation for Budimex. 60

61 E( RBu ex ) rf = Bu dim ex [ E( RWIG ) rf] as: E R ) = r + [ E( R ) r ] dim β, which can be presented ( Bu dim ex f β Bu dim ex WIG f. The outcome has given the CAPM equation, where the beta is a measure of a systematic risk. Budimex beta was approximately equal The coefficient of determination, 2 R, for a model equals It means that approximately 20.9% of the variation in Budimex excess rate of return can be explained by the market premium. Such a result implies low regression fit and possibility that other risk factors explain Budimex returns. Finally the analysis of the regression residuals is carried out. Residuals are studied in order to check whether they meet the following assumptions: homoskedasticity, independence, normal distribution. Figure 13 Budimex residuals plots obtained from MS Excel Normal probability plot Y Sample percentile 61

62 Residuals versus explanatory variable Residuals Explanatory variable Source: The analysis of the thesis s authors From the above figure one can see that the residuals are normally distributed, and they are homoskedastic and independent, so all assumptions regarding residuals are met. The Budimex regression analysis shows that the model is statistically significant and beta is significantly related to the stock s return. This is the prerequisite that the Budimex is taken to the further analysis, as one of the securities that serve as a base to calculate the sector beta, and to carry out the industry study. Similar analysis is performed for every sample stock. Overall compiled results are presented below and the detailed MS Excel outcomes ( table, regression graph) are enclosed in the appendix 1. 62

63 Table 8 Outcome of regression analysis for the sample stocks of the construction industry Parameters value T values T critical Significance Lack of trading data compared to WIG full trading history Name a b a b Tc a b CAPM model R2 [%] Budimex N Y Y 21% 0% Budopol N N N 3% 26% Echo N Y Y 14% 2% Elbudowa N Y Y 13% 1% Elektromontaz Polnoc Y N N 0% 16% Elektromontaz Warszawa N Y Y 4% 6% Elektromontaz Poludnie N Y Y 5% 6% Energopol N N N 0% 20% Hydrobudowa N N N 0% 22% Instal Krakow N N N 0% 62% Instal Lublin N Y Y 4% 14% Mostostal Eksport N Y Y 15% 0% Mostostal Plock N Y Y 7% 3% Mostostal Warszawa E N Y Y 12% 3% Naftobudowa Y Y Y 6% 1% Pemug N N N 1% 47% Polimex Mostostal N Y Y 11% 2% Polnord N N N 1% 6% Projprzem N Y Y 7% 13% Y-yes, N-no 63

64 Table 9 Outcome of regression analysis for the sample stocks of the IT industry Parameters value T values T critical Significance Lack of trading data compared to WIG full trading history Name a b a b Tc a b CAPM model R2 [%] Comarch N Y Y 45% 0% Computerland N Y Y 46% 0% CSS N Y Y 8% 4% Elzab N Y Y 4% 3% IBSystem Y N N 1% 21% InternetGroup Y Y Y 9% 2% Macrosoft Y Y Y 24% 2% Prokom N Y Y 47% 0% Softbank N Y Y 34% 0% Table 10 Outcome of regression analysis for the sample stocks of the telecommunication industry Parameters value T values T critical Significance Lack of trading data compared to WIG full trading history Name a b a b Tc a b CAPM model R2 [%] Elektrim Y Y Y 10% 0% Netia Y Y Y 28% 0% TPSA N Y Y 61% 0% 64

65 Table 11 Outcome of regression analysis for the sample stocks of the food industry Parameters value T values T critical Significance Lack of trading data compared to WIG full trading history Name a b a b Tc a b CAPM model R2 [%] Beefsan N Y Y 2.0% 25% Ekodrob Y N N 1.1% 26% Indykpol N Y Y 2.0% 2% Jutrzenka N Y Y 23.1% 1% Kruszwica N Y Y 10.9% 1% Mieszko N N N 0.9% 2% Pepees N Y Y 9.8% 5% Rolimpex N Y Y 24.6% 0% Sokolow N Y Y 26.1% 0% Strzelec N Y Y 2.0% 1% Wawel Y Y Y 2.3% 7% Wilbo N Y Y 16.3% 0% Zywiec N Y Y 15.0% 2% 65

66 Table 12 Outcome of regression analysis for the sample stocks of the banking industry Parameters value T values T critical Significance Lack of trading data compared to WIG full trading history Name a b a b Tc a b CAPM model R2 [%] BOS N N N 0% 24% BPH N Y Y 46% 0% BRE E N Y Y 35% 0% BZ WBK N Y Y 42% 0% DZ Polska N N N 1% 15% Fortis N Y Y 4% 18% Handlowy N Y Y 20% 0% ING N Y Y 18% 0% Kredyt Bank N Y Y 15% 0% Nordea N Y Y 3% 48% Pekao N Y Y 51% 0% 66

67 The CAPM equation was not statistically significant for rather illiquid stocks with the fragmented trading history. They were marked grey in the above tables and they were excluded from the further analysis. 2 The regression fit, measured by the coefficient of determination ( R ), for statistically significant models, 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 lowest 2 R for the statistically significant model was observed at the level of 1.96% for Strzelec. In few cases, e.g. Naftobudowa or Ekodrob, the t-statistic test showed significance of the intercept (a) in the regression. When beta was also significant, then the whole model was taken as statistically significant as the value of intercept was very low, and then could be neglected, so the CAPM equation holds. Subsequently the industry betas were calculated. Each of the sector betas was taken as the weighted average of the betas for stocks taken to the further analysis that construct a certain industry. The weights were set on the basis of average stock s capitalization in Average yearly capitalization is a multiple of number of shares at the end of a year and average stock price in a year. The dispersion among stocks betas within the industry group was also computed by means of the dispersion coefficient (a standard deviation divided by an arithmetic mean). 67

68 Table 13 Industry beta - construction Name Beta Capitalisation [2004] in mln PLN Share in industry beta Budimex ,131,123, % Echo ,173, % Elbudowa ,574, % Elektromontaz Warszawa ,698, % Elektromontaz Poludnie ,058, % Instal Lublin ,151, % Mostostal Eksport ,217, % Mostostal Plock ,205, % Mostostal Warszawa ,729, % Naftobudowa ,590, % Polimex Mostostal ,888, % Projprzem ,307, % Industry beta ,809,719,743 Total Standard deviation (S) Aritmetic mean (A) Dispersion coefficient (S/A) 23.3% The construction industry had relatively big representation on the WSE, however three stocks made for about 85% of the sector capitalization. The largest stocks had similar betas, between 0.6 and 0.7. The beta for the construction industry amounted to It was representative for the sector, as the beta s dispersion among the analyzed securities was not large. The industry beta is less than 1, which means that the construction stocks returns are less volatile than the market returns; hence the stocks of the construction industry are less risky than the average stock on the market. 68

69 Table 14 Industry beta - IT Name Beta Capitalisation [2004] in mln PLN Share in industry beta Comarch ,754, % Computerland ,139, % CSS ,057, % Elzab ,856, % InternetGroup ,072, % Macrosoft ,503, % Prokom ,342,901, % Softbank ,189, % Industry beta ,141,475,694 Total Standard deviation (S) Aritmetic mean (A) Dispersion coefficient (S/A) 41.4% The IT sector had representation of 8 stocks from the sample. The dispersion of stocks betas was substantial as three of them recorded low betas, below 1, while the rest of the securities showed betas above 1. The three non representative stocks were little and had small influence on the industry beta, which amounted to Not surprisingly the IT sector was more risky than the market. Similarly to the construction industry, there were few stocks, Prokom and Computerland that made for 75% of the sector capitalization, and thus tremendously influenced the industry beta. It is worth noticing that their betas were similar. Table 15 Industry beta - telecommunication Name Beta Capitalisation [2004] in mln PLN Share in industry beta Elektrim ,777, % Netia ,516,637, % TPSA ,194,392, % Industry beta ,279,807,182 Total Standard deviation (S) Aritmetic mean (A) Dispersion coefficient (S/A) 23.5% The telecommunication sector is not very diversified on the WSE. There are three stocks in the category and the industry is mostly represented by one of the 69

70 biggest companies on the market, TPSA. The telecommunication beta was 1.338, which implies that the sector is more risky than the market. The Elekrim s beta was lower than 1, but the company led diversified operations and faced serious financial problems, so it is not very representative for the industry. Table 16 Industry beta - food Name Beta Capitalisation [2004] in mln PLN Share in industry beta Beefsan ,331, % Indykpol ,780, % Jutrzenka ,482, % Kruszwica ,795, % Pepees ,054, % Rolimpex ,380, % Sokolow ,134, % Strzelec ,000, % Wawel ,500, % Wilbo ,120, % Zywiec ,123,419, % Industry beta ,105,997,400 Total Standard deviation (S) Aritmetic mean (A) Dispersion coefficient (S/A) 46.9% The food industry on the Warsaw Stock Exchange was represented mainly by Zywiec, which accounted for 72.1% of the sector capitalization in the end of The average sector beta amounted to 0.505, which generally accords with the common sense the food stocks are less risky than the market and are usually countercyclical and should not have high correlation with the broad market. However, some stocks, such as Rolimpex had beta of slightly below 1 and generally beta values were closer to one. The median beta for the industry amounted to The sectors dispersion was rather high, amounting to slightly below 47%. 70

71 Table 17 Industry beta - banking Name Beta Capitalisation [2004] in mln PLN Share in industry beta Bank BPH ,645,277, % BRE Bank ,273,296, % BZ WBK ,077,120, % Fortis Bank ,070,516, % Handlowy ,375,280, % ING Bank Slaski ,095,899, % Kredyt Bank ,567,176, % Nordea ,781, % Pekao ,974,474, % Industry beta ,722,823,700 Total Standard deviation (S) Aritmetic mean (A) Dispersion coefficient (S/A) 44.2% The banking industry is by far the largest capitalization sector represented on the Warsaw Stock Exchange. Only 9 stocks are present in the table above; the largest Polish bank, PKO BP was excluded from the analysis due to the fact that it did not have enough data as it was listed since BOS and DZ Polska were included in the sample for testing but not included in the calculations of the mean sector betas due to the fact that in estimating the betas, the coefficients of determination turned out to be insignificant as did the whole regression. This is due to their low liquidity and probably the vast amount of non-trading days for the stocks. The main index of the Warsaw Stock Exchange (capitalization weighted, composed of 20 stocks, revised quarterly) is represented by five largest banks (including PKO BP) only because each sector is limited up to 5 companies representing a given sector. The weighted average beta for the banking industry (with the exceptions stated above) amounts to 0.907, which implies that the banks are almost as risky as the broad market. This is in line with the experience; the banks are usually highly cyclical during economic upturns when companies become profitable and the wealth and financial surpluses of the individuals increase, the banks experience positive developments e.g. in lower risk provisions along with expansions in lending volumes, which impact their stock prices performance. The median (0.513) is lower than the weighted average; this is due to the fact that the weighted 71

72 average is dominated by three large banks (BPH, BZ WBK, Pekao) with betas close to 1, which have a joint share in industry beta of 68%, whereas most banks betas are lower. For instance Handlowy s beta equals 0.511, which is said to be due to the specifics of its business (the Bank is perceived as the most countercyclical in its league due to its business and policy). The dispersion of the observed betas around the arithmetic mean is quite large; it amounts to 44.2%. Industry premium In the industry study the second explanatory variable, which is the industry premium, and its slope, that is the industry beta, were added to the CAPM equation. If the industry premium is statistically significant variable then the hypothesis that unsystematic risk was priced on the WSE, is proven to be true. The industry study is carried out for each of the researched sectors. Construction Budimex is taken as a representative stock for the construction industry and its two-factor model, which includes the industry premium ( E( R WIG rf ), is presented below. ) bud adj Equation 19 The two factor model of Budimex (multiple regression model) E ( R ) r = a+ β [ E( R ) r ]+ Bud i Bud _ bud fi Bud WIG i [ E RWIG i rfi] ξi + β ) + where: a - intercept ( bud _ adj R Bud i - i th observation of the Budimex return r fi - i th observation of the risk-free rate fi 72

73 R WIG - i th observation of the return on WIG (Warsaw Stock Exchange s i market index) β Bud - the Budimex beta coefficient related to the market premium R _ - i th observation of the return on an adjusted industry index WIG bud adj i (WIG-bud) [construction- budownictwo in Polish] β Bud _ bud - the Budimex beta for construction related to the industry premium ξ i - the error of the i th estimate. Adjusted industry premium On the graph below the Budimex price trend can be observed in the researched period, juxtaposed with the price trend of the market index (WIG) and the construction index (WIG-bud). The prices were rebased to 1 at the beginning of the track period. One can see that volatility of Budimex price is higher than the one of the market index or the industry index. However the prices follow the same pattern, which implies they are highly correlated. 73

74 Figure 14 Budimex price and volume in the researched period Price [PLN] 2.5 Volume [thousand shares] Budimex WIG WIGbud Volume /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/2005 Source: The analysis of the thesis s authors Some of the factors that drive the market index s returns are similar to ones that influence the industry index s returns, and that is why the returns of those indexes are in general correlated. In order to gather data for the two factor model in form of industry premium observations, which are uncorrelated with the market premium observations - E ( R WIG i rf, the following steps are undertaken, according to the ) bud adj i Chapter 3 Methodology. First the single linear regression, which is presented below, is run. 74

75 Equation 20 Regression model with market premium explaining the industry premium for construction [ E( R r ] + ζ E R ) r = a+ b ) bud (, where WIG E R WIG ) r bud f f WIG ( - the industry (sector) premium E( R WIG ) r f - the market premium a the intercept term b- the slope coefficient ζ - regression residuals. f Figure 15 Graphic presentation of regression (market premium explaining industry premium for construction) 0.12 y=rwigbud-rf y = x R 2 = x=rwig-rf Source: The analysis of the thesis s authors 75

76 Table 18 Microsoft Excel output for analysis of regression (market premium explaining industry premium for construction) Multiple R R square Adjusted R square Standard error Observations 243 Regression E-31 Residual Total Coefficients Standard error t Stat P-Value Lower 95% Upper 95% Intercept (a) Slope (b) E The coefficient of determination, 2 R, for the model equals circa It means that approximately 43.5% of the variation in industry premium can be explained by the market premium. The residual analysis implies that the needed assumptions are met, because the residuals are homoskedastic, independent and normally distributed. 76

77 Figure 16 Residuals plots for the regression (market premium explaining industry premium for construction) Normal probability plot Y Sample percentile Residuals versus explanatory variable Residuals Explanatory variable Source: The analysis of the thesis s authors In order to test the statistical significance of the slope, the null and alternative hypotheses are set as follows. H : b 0 The slope coefficient equals zero, thus there is no statistical linear 0 = relationship between the industry premium for construction and the market premium, H : b 0 There is statistical relationship between the industry premium for 1 construction and the market premium. 77

78 The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as t-stat is greater than t critical ( >1.96). The slope coefficient in the analyzed regression is statistically significant. Subsequently the t-test for the intercept is carried out. The null and alternative hypotheses are set as follows. H : a 0 The intercept equals zero, 0 = H : a 0 The intercept is different from 0. 1 The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is not rejected as t-stat is less than t critical ( <1.96). The intercept in the analyzed regression is not statistically significant. The regression analysis implies the statistical dependence between the industry premium for construction and the market premium, which is described by the E( R ) r = E( R ) r +. bud following equation: [ ] ζ WIG f WIG f Thus, E( RWIG ) rf = = E( RWIG ) rf [ E( RWIG rf ] ζ. bud adj bud The residuals of the regression represent the variation in the sector premium that is not explained by the market premium. It is an input to the two-factor model as a second explanatory variable, the adjusted industry premium. 78

79 Multiple regression analysis Table 19 Microsoft Excel output for the multiple regression analysis of Budimex two factor model Multiple R R square Adjusted R square Standard error Observations 242 Regression E-43 Residual Total Intercept Slope 1 (B Bud) E Slope 2 (B Bud_bud) E The Analysis of variance () table shows the coefficients of the intercept and the slopes as well as their standard errors and t-statistics for the level of significance equal to α = The adjusted coefficient of determination, 2 R, for the model equals approximately It means that approximately 55.4% of the variation in Budimex excess rate of return over the risk-free rate can be explained by the market premium and the industry premium. Such a result implies much higher 2 regression fit than in case of one-factor model ( R =0.209). Industry premium added substantial explanatory power to the Budimex model. The analysis of the residual plots implies that they are homoskedastic, independent and normally distributed. Hence all assumptions regarding the regression residuals are met. 79

80 Figure 17 Budimex residuals plots of the two-factor model Y Normal probability plot Sample percentile Residuals versus first explanatory variable Residuals First explanatory variable Residuals versus second explanatory variable Residuals Second explanatory variable Source: The analysis of the thesis s authors 80

81 Tests for significance in the multiple regression involve testing whether: Individually, the independent variables contribute to the explanation of the dependent variable, Together, some or all of the individual independent variables contribute to the explanation of the dependent variable, The additional explanatory variable contributes to the explanation of variation in the dependent variable. A test of significance for individual slope coefficients is conducted to determine if the independent variables make a significant contribution toward the explanation of the dependent variable. In order to test the statistical significance of the slope 1, the null and alternative hypotheses are set as follows. H : β 0 The slope coefficient equals zero, thus there is no statistical linear 0 Bud = relationship between the market premium and the expected rate of return for Budimex, H : β 0 There is statistical relationship between the market premium and 1 Bud the expected rate of return for Budimex. The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as t-stat is greater than t critical ( >1.96). The β Bud in the analyzed regression is statistically significant. 81

82 In order to test the statistical significance of the slope 2, the null and alternative hypotheses are set as follows. H β 0 The slope coefficient equals zero, thus there is no statistical 0 : Bud _ bud = linear relationship between the industry premium and the expected rate of return for Budimex H β 0 There is statistical relationship between the industry premium 1 : Bud _ bud and the expected rate of return for Budimex. The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as t-stat is greater than t critical ( >1.96). The Bud _ bud statistically significant. β in the analyzed regression is Subsequently the t-test for the intercept is carried out. The null and alternative hypotheses are set as follows. H : a 0 The intercept equals zero, 0 = H : a 0 The intercept is different from 0. 1 The critical value on the t distribution is 1.96 for 5% level of significance and 239 degrees of freedom. The null hypothesis is not rejected as t-stat is less than t critical ( <1.96). The intercept in the analyzed regression is not statistically significant. The F test is applied to test the statistical significance of the entire multiple regression model. It determines whether there is a significant relationship 82

83 between the expected rate of return for Budimex and the market premium together with the industry premium. The null and alternative hypotheses are set as follows. H 0 : Bud = Bud _ bud = β β 0 There is no linear relationship between the expected rate of return for Budimex and the market premium together with the industry premium, H : At least one β 0 There is a linear relationship between the expected 1 j rate of return for Budimex and the market premium together with the industry premium. The critical value on the F distribution is 3.00 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as F is greater than F critical (150.89>3.00). The multiple regression model is statistically significant. The partial F-test criterion is performed to determine the contribution of the second explanatory variable (the industry premium) to the multifactor model. The null and alternative hypotheses are set as follows. H 0 : Variable 2 industry premium (X2), does not significantly improve the model once variable 1 (X1) market premium, has been included, H 1 : Variable 2 industry premium significantly improves the model once variable 1 market premium, has been included. 83

84 Equation 21 SSR, contribution of the explanatory variable X 2 given that variable X 1 is already included in the two-factor model of Budimex SSR( X 2 / X 1) = SSR( X 1andX 2 ) SSR( X 1) = = 0.162, where SSR ( X 1 andx 2 ) = taken from the Table 19 SSR ( X 1) = the sum of squares for regression that includes only the explanatory variable X 1 taken from the Table 7 Equation 22 The partial F-test criterion for determining the contribution of variable X 2 SSR( X 2 / X 1) F partial = = 188.8, where MSE SSR X 2 / X ) = the outcome from the Equation 21 ( 1 MSE = the average sum of squared errors for regression that includes both explanatory variables taken for the Table 19 The critical value on the F distribution is 3.00 for 5% level of significance and 239 degrees of freedom. The null hypothesis is rejected as F is greater than F critical (188.8>3.00). The industry premium significantly improves the model once the market premium has been included in the two-factor model for Budimex. The compiled results of the multiple regression for the representative stocks of the construction industry are presented below. The detailed MS Office outcomes in form of the tables are enclosed in the appendix 1. 84

85 Table 20 Outcome of two-factor regression analysis for the sample stocks from construction industry One factor model Two factor model Parameters value T values F Significance Name B R2 R2 a B (b1) B_bud (b2) t (a) t (b1) t (b2) F a b1 b2 F Two factor model Budimex % 55% N Y Y Y Y Echo % 30% N Y Y Y Y Elbudowa % 14% N Y Y Y Y Elektromontaz Warszawa % 6% N Y Y Y Y Elektromontaz Poludnie % 6% N Y Y Y Y Instal Lublin % 4% N Y N Y N Mostostal Eksport % 18% N Y Y Y Y Mostostal Plock % 7% N Y N Y N Mostostal Warszawa % 12% N Y N Y N Naftobudowa % 5% Y Y N Y N Polimex Mostostal % 17% N Y Y Y Y Projprzem % 9% N Y Y Y Y Y-yes, N-no In 8 out of 12 cases, the model with the industry premium as a second explanatory variable was statistically significant. In general 2 the regression fit ( R ) in those models was not improved substantially except few cases: Budimex, Echo, Polimex Mostostal. It is worth noticing that those stocks has the highest share in the industry capitalization, namely Budimex 40.3%, Echo 30.6% and Polimex Mostostal 14.7%. Other sample stocks had insignificant share in the sector s capitalization, not exceeding 5%. 85

86 IT Prokom is taken as a representative stock for the IT industry and the simplified analysis of its two-factor model is presented. First it is shown that the prices of the market index (WIG) and the IT index (WIGinfo) followed the same pattern, which implies correlation between them. Figure 18 Prokom price and volume in the researched period Price [PLN] 1.6 Volume [thousand shares] Prokom WIG WIGinfo volume /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/2005 Source: The analysis of the thesis s authors The correlation is proved by finding the dependence between the market premium and the premium for IT industry. The regression equation is used in order to derive the adjusted industry premium. 86

87 Figure 19 Graphic presentation of regression (market premium explaining industry premium for IT) 0.25 y=rwiginfo - Rf 0.2 y = x R 2 = x=rwig - Rf Source: The analysis of the thesis s authors Table 21 Microsoft Excel output for analysis of regression (market premium explaining industry premium for IT) Multiple R R square Adjusted R square Standard error Observations 242 Regression E-51 Residual Total Coefficients Standard error t Stat P-Value Lower 95% Upper 95% Intercept (a) E Slope (b) E

88 Next multiple regression of Prokom two factor model is run, which showed the statistical significance of the model and its coefficients. Table 22 Microsoft Excel output for the multiple regression analysis of Prokom two factor model Multiple R R square Adjusted R square Standard error Observations 240 Regression E-108 Residual Total Intercept Slope 1 (B Pro) E Slope 2 (B Pro_info) E The residuals plots prove that assumptions concerning the regression s error terms are met. The residuals are homoskedastic, independent and normally distributed. Figure 20 Prokom residuals plots of the two-factor model Normal probablility plot Y Sample percentile 88

89 Residuals versus first explanatory variable Residuals First explanatory variable Residuals verus second explanatory variable 0.1 Residuals Second explanatory variable Source: The analysis of the thesis s authors The compiled results of the multiple regression for the representative stocks of the IT industry are presented below. The detailed MS Office outcomes in form of the tables are enclosed in the appendix 1. 89

90 Table 23 Outcome of two-factor regression analysis for the sample stocks from IT sector One factor model Two factor model Parameters value T values F Significance Name B R2 R2 a B (b1) B_bud (b2) t (a) t (b1) t (b2) F a b1 b2 F Two factor model Comarch % 58% N Y Y Y Y Computerland % 59% N Y Y Y Y CSS % 8% N Y N Y N Elzab % 3% N Y N Y N InternetGroup % 9% Y Y N Y N Macrosoft % 27% Y Y Y Y Y Prokom % 88% N Y Y Y Y Softbank % 51% N Y Y Y Y Y-yes, N-no In 5 out of 8 cases the model with the industry premium as a second explanatory variable was statistically significant. The adjusted 2 industry premium improved substantially the regression fit ( R ) in those models, especially in case of Prokom the largest company with the biggest capitalization within the industry. Prokom capitalization constituted 56.6% of the IT industry size. 2 R for the Prokom s two-factor model amounted to 88%, while 90

91 Telecommunication TPSA is taken as a representative stock for the telecommunication industry and the simplified analysis of its two-factor model is presented. First it is shown that the prices of the market index (WIG) and the telecommunication index (WIGtelekom) followed the similar pattern, which implies correlation between them. Figure 21 TPSA price and volume in the researched period Price [PLN] 1.6 Volume [thousand shares] TPSA WIG WIGtelekom volume /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/ /07/ /09/ /11/ /01/ /03/ /05/2005 Source: The analysis of the thesis s authors The correlation is proved by finding the dependence between the market premium and the premium for the telecommunication industry. The regression equation is used in order to derive the adjusted industry premium. 91

92 Figure 22 Graphic presentation of regression (market premium explaining industry premium for telecommunication) y=rwigtelekom - Rf y = x R 2 = x=rwig - Rf Source: The analysis of the thesis s authors Table 24 Microsoft Excel output for analysis of regression (market premium explaining industry premium for telecommunication) Multiple R R square Adjusted R square Standard error Observations 242 Regression E-68 Residual Total Coefficients Standard error t Stat P-Value Lower 95% Upper 95% Intercept (a) Slope (b) E

93 Next multiple regression of TPSA two factor model is run, which showed the statistical significance of the model and its coefficients. Table 25 Microsoft Excel output for the multiple regression analysis of TPSA two factor model Multiple R R square Adjusted R square Standard error Observations 241 Regression E-95 Residual Total CoefficientsStandard error t Stat P-value Lower 95% Upper 95% Intercept Slope 1 (B TPSA) E Slope 2 (B TPSA_telekom) E The residuals plots prove that assumptions concerning the regression s error terms are met. The residuals are homoskedastic, independent and normally distributed. Figure 23 TPSA residuals plots of the two-factor model Normal probability plot Y Sample percentile 93

94 Residuals versus first explanatory variable Residuals First explanatory variable Residuals verus second explanatpry variable Residuals Second explanatory variable Source: The analysis of the thesis s authors The compiled results of the multiple regression for the representative stocks of the telecommunication industry are presented below. The detailed MS Office outcomes in form of the tables are enclosed in the appendix 1. 94

95 Table 26 Outcome of two-factor regression analysis for the sample stocks from telecommunication industry One factor model Two factor model Parameters value T values F Significance Name B R2 R2 a B (b1) B_bud (b2) t (a) t (b1) t (b2) F a b1 b2 F Two factor model Elektrim % 18% Y Y Y Y Y Netia % 31% Y Y Y Y Y TPSA % 84% Y Y Y Y Y Y-yes, N-no In every case of three representative stocks of the telecommunication industry, the two-factor model was statistically significant. The most considerable improvement of the regression fit after adding the second explanatory variable to the CAPM equation was noticed for the biggest company within the sector, TPSA. constituted 91.4% of the industry size. 2 R of the two-factor model for TPSA amounted to 84%, while its capitalization 95

96 Food Sokolow is taken as a representative stock for the Polish food industry and below the simplified analysis of its two factor model is presented. On the figure below the performance of the stock price and the performance of the WIG index as well as the sector index for the food industry are shown. First it is shown that the prices of the market index (WIG) and the food industry index (WIG spoz ) followed the similar pattern, which implies correlation between them. The correlation coefficient of the time series for Sokolow and for the WIG index amounts to 0.916, whereas the correlation of weekly returns of the stock and the weekly returns experienced by the index (excluding the observations when weekly return was more than 20%) amounted to The correlation is then apparent. Figure 24 Sokolow price and volume in the researched period Price (PLN) 3 Volume Volume Sokolow WIG WIGspoz Source: The analysis of the thesis s authors The correlation is proved by finding the dependence between the market premium (the excess return over the risk free rate) and the premium for the food industry (the surplus of the return of the industry index and the risk free rate of return). The regression is then run based on those two series of data. The 96

97 regression equation resulting from that is subsequently used to estimate the adjusted industry premium. Figure 25 Graphic presentation of regression (market premium over the risk free rate explaining industry premium above the risk free rate for the food industry) 0.2 y=rwigspoz-rf 0.1 y = x R 2 = x=rwig-rf Source: The analysis of the thesis s authors Table 27 Microsoft Excel output for analysis of regression (market premium explaining industry premium for the food industry) Multiple R R square Adjusted R square Standard error Observations 259 Regression E-24 Residual Total Intercept (a) Slope (b) E From the Figure 25 as well as the regression output presented in Table 27, it can be inferred that the risk premium of the market index and the sector index are fairly highly correlated, the correlation coefficient amounts to and the coefficient of determination, R 2 of the regression specified above is 0.326, which implies that 32.6% of the variation in the food industry risk premium is 97

98 explained by the variation of the broad market risk premium, represented by the WIG. The residual analysis shows that the necessary assumptions are met, because the residuals are homoskedastic, independent and normally distributed. The slope for the regression (0.525) is statistically significant, which can be seen its t- value for the slope of , which is above the tcritical of We thus reject the null hypothesis H : b 0 (no statistical linear relationship between the 0 = industry premium for food industry premium and the market premium). We are however unable to reject the null H : a 0 (the intercept equals 0), since the 0 = absolute value of the t-stat for the intercept is not higher than t critical of The regression analysis implies the statistical dependence between the industry premium for food industry and the market premium, which is described by the following equation: E R ) r = [ E( R ) r ] + ζ spoz (. WIG f WIG f Thus, E( RWIG ) rf = = E( RWIG ) rf [ E( RWIG rf ] ζ spoz adj spoz As for the sectors described in the previous sections, multiple regression of Sokolow two-factor model is run. The two variables used for explanatory variables are the market risk premium and the adjusted industry premium, or the residuals of the regression of the market risk premium against the sector risk premium. 98

99 Table 28 Microsoft Excel output for the multiple regression analysis of Sokolow two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression E-29 Residual Total Intercept (a) Slope 1 (B Sokolow) E Slope 2 (B Sokolow_food) E Based on the analysis of variance in the above table it can be inferred that both slope coefficients are not equal to zero at significance level ofα = We cannot reject the hypothesis that the intercept is equal to zero at the significance level of The adjusted coefficient of determination, 2 R, for the model equals approximately It means that approximately 40% of the variation in Sokolow excess rate of return over the risk-free rate can be explained by the market premium and the industry premium. Such a result implies much higher 2 regression fit than in case of one-factor model ( R =0.261), implying that the industry premium added substantial explanatory power to the Sokolow model. The residuals plots prove that assumptions concerning the regression s error terms are met. The residuals are do not exhibit heteroskedasticity, are independent and normally distributed across observations. 99

100 Figure 26 Sokolow residuals plots of the two-factor model Y Normal probability plot Sample percentile Residuals Residuals versus first explanatory variable First explanatory variable Residuals Residuals versus second explanatory variable Second explanatory variable Similarly, as for the previously described sectors, various tests for significance of the outcome of the multiple regression are conducted, in particular, relating to contribution of the individual variables to the explanation of the dependent variable, contribution of all of the variables to the explanation of the security s excess returns and the additional explanatory variable contributes to the explanation of variation in the dependent variable. 100

101 A test of significance for individual slope coefficients is composed of testing the following null hypotheses: H : coefficient 0 the alternative hypotheses 0 = being H : coefficients 0. The t-tests are conducted to test the hypothesis. 1 Both slope coefficients, β 1 (0.852) and β 2 (0.838) are statistically significant since both of their t-statistics (10.593, 7.861, respectively) are greater than the t critical. Therefore we are able to reject the hypothesis of no statistical linear relationship between the variables (market risk premium and the adjusted industry risk premium) and the explained variable. We are, however, unable to reject the null hypothesis that H : a 0, the intercept equals zero, as 0 = t a = 0.283, which is below the t critical. The F test is applied to test the statistical significance of all the independent variables (the risk premium and the adjusted industry risk premium) in explaining the variation of the excess return of Sokolow. The null and alternative hypotheses are the following: H 0 : β1 =β 2 = 0 and H 1 that at least one coefficient is not equal to 0. Failing to reject the null would mean that there is no linear relationship between the expected rate of return for Sokolow and the market premium together with the industry premium. The critical value on the F distribution is 3 for 0.05 level of significance and 256 degrees of freedom. Since the F of is greater than 3, the model is statistically significant. Performing the partial F-test criterion also brings to the conclusion that the model is statistically significant as calculated F is greater than the critical value of 3. The results of the multiple regressions for stocks representing the food industry on the WSE are presented below. The numbers generated in the course of analysis with use of MS Excel in form of the tables are enclosed in the appendix

102 Table 29 Outcome of two-factor regression analysis for the sample stocks from the food industry One factor model Two factor model Parameters value T values F Significance Name B R2 R2 a B (b1) B_food (b2) t (a) t (b1) t (b2) F a b1 b2 F Two factor model Beefsan % 2% N N Y Y Y Indykpol % 1% N Y N N N Jutrzenka % 24% N Y Y Y Y Kruszwica % 11% N Y N Y N Pepees % 10% N Y N Y N Rolimpex % 32% N Y Y Y Y Sokolow % 40% N Y Y Y Y Strzelec % 3% N Y Y Y Y Wawel % 3% Y Y N Y N Wilbo % 17% N Y Y Y Y Zywiec % 53% N Y Y Y Y Y-yes, N-no In 4 out of 11 cases the coefficient at the adjusted industry premium was insignificant. For those companies, the adjusted 2 R for the twofactor model was not much higher than for the one factor model (taking only the market risk premium). However, in case of all four mentioned companies (Indykpol, Kruszwica, Pepees and Wawel) the F-test shows that the model is statistically significant. There are only few cases where the adjusted coefficient of determination for the two-factor model was substantially higher than for a one-factor model (Rolimpex from 25% to 32% and Zywiec (from 15% to 53%). In case of Indykpol, the adjusted for just one factor model, employing the market premium. In other cases the improvement was neglible. 2 R turned out to be lower than 102

103 Banking Bank BPH is chosen a representative stock for the Polish banking industry. All other banks were tested and analyzed in the similar manner as Bank BPH. The summary of the analysis of BPH follows. The stock s performance is highly correlated with the index. On the figure below the performance of the stock price and the performance of the WIG index as well as the sector index for the banking industry are shown. The prices of the market index (WIG) and the banking industry index (WIG bank ) followed the similar pattern, which implies correlation between them. The correlation coefficient of the time series for Bank BPH and for the WIG index amounts to 0.945, while the correlation of weekly returns on the stock and the weekly returns experienced by the index amounted to The correlation between the two mentioned time series is strong. Figure 27 Bank BPH price and volume in the researched period Price (PLN) 3.5 Volume 120, , ,000 60,000 40,000 20,000 Volume BPH WIG WIGbank Source: The analysis of the thesis s authors The correlation is proven by regressing the banking industry risk premium (the surplus of the return of the banking industry index and the risk free rate of 103

104 return) against the market risk premium. The resulting regression equation is used, identically as in previous cases, to estimate the adjusted industry premium. Figure 28 Graphic presentation of regression (market premium over the risk free rate explaining industry premium above the risk free rate for the banking industry) y=rwigbank-rf y = x R 2 = x=rwig-rf Source: The analysis of the thesis s authors Table 30 Microsoft Excel output for analysis of regression (market premium explaining industry premium for the banking industry) Multiple R R square Adjusted R square Standard error Observations 260 Regression E-71 Residual Total Intercept (a) Slope (b) E

105 Based on the scatter chart presented in the Figure 28 and Table 30 we infer that the variations of the market risk premium explain 70.75% of the total variation of the industry risk premium over the risk free rate. The residuals are homoskedastic, independent and normally distributed. The slope for the regression (0.890) is statistically significant, as the value of its t- stat of , is above the tcritical of 1.96 allows us to reject the null hypothesis H : b 0 (no statistical linear relationship between the industry premium for 0 = banking industry premium and the market premium). We are however unable to reject the null H : a 0 (the intercept equals 0), since the absolute value of the 0 = t-stat (1.4) for the intercept is not higher than tcritical of 1.96, which means that the intercept is not statistically significant. The regression analysis implies the statistical dependence between the industry premium for food industry and the market premium, which is described by the E( R ) r = E( R ) r +. bank following equation: [ ] ζ WIG f WIG f Therefore: E( RWIG ) rf = = E( RWIG ) rf [ E( RWIG rf ] ζ. bank adj bank For Bank BPH, as previously, we run two-factor regression. The two variables used for explanatory variables are the market risk premium and the adjusted industry premium, or the residuals of the regression of the market risk premium against the sector risk premium, as given by the equation above. 105

106 Table 31 Microsoft Excel output for the multiple regression analysis of Bank BPH two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-55 Residual Total Intercept (a) Slope 1 (B BPH) E Slope 2 (B BPH_banking) E Based on the analysis of variance in the above table it can be inferred that both slope coefficients are not equal to zero at significance level ofα = We cannot, however, reject the hypothesis that the intercept is equal to zero at the significance level of The model appears to be statistically significant; as indicated by the adjusted coefficient of determination, 2 R, approximately 62.0% of the variation in BPH s excess rate of return over the risk-free rate is explained by the market premium and the industry premium. In one factor regression model, where the sole explanatory variable was the market risk premium, the explained variation. 2 R was 46%, so the two factor model improves the percentage of The scatter plots of residuals shows that the residuals exhibit no heteroskedasticity, are independent and normally distributed across observations. 106

107 Figure 29 Bank BPH residuals plots of the two-factor model Normal distribution plot Y Sample percentage Residuals versus first explanatory variable Residuals Firs explanatory variable Residuals versus second explanatory variable Residuals Second explanatory variable As for the previously described sectors, the two factor model is then tested for significance by means of various tests, in particular, relating to contribution of the individual variables to the explanation of the dependent variable, contribution of all of the variables to the explanation of the security s excess 107

108 returns and the additional explanatory variable contributes to the explanation of variation in the dependent variable. The tests of significance for individual slope coefficients show that both slope coefficients, β 1 (1.05) and β 2 (1.102) are statistically significant since both of their t-statistics ( and , respectively) are greater than the t critical. Therefore we are able to reject the hypothesis of no statistical linear relationship between the variables (market risk premium and the adjusted industry risk premium) and the explained variable. We are, however, unable to reject the null hypothesis that H : a 0, as the intercept equals zero, as t = , which is below the t critical. 0 = a In order to test the statistical significance of all the independent variables of the model (the risk premium and the adjusted industry risk premium) in explaining the variation of the excess return of Bank BPH the F-test is applied. The null: H 0 : β1 =β 2 = 0 is rejected in favor of the H 1 that at least one coefficient is not equal to 0, because the value of F-statistic is , which is above the critical value of 3 (for 0.05 significance level and 2 and 257 degrees of freedom). The partial F-test also shows that the model is statistically significant as calculated F of is greater than the critical value of 3. The results of the multiple regressions for all stocks representing the banking industry on the WSE are presented below. The detailed numbers generated in the course of analysis with use of MS Excel in form of the tables are enclosed in the appendix

109 Table 32 Outcome of two-factor regression analysis for the sample stocks representing the banking industry One factor model Two factor model Parameters value T values F Significance Name B R2 R2 a B (b1) B_food (b2) t (a) t (b1) t (b2) F a b1 b2 F Two factor model BPH % 62% N Y Y Y Y BRE % 37% N Y Y Y Y BZ WBK % 50% N Y Y Y Y Fortis % 4% N Y N Y N Handlowy % 20% N Y N Y N ING % 25% N Y Y Y Y Kredyt Bank % 17% N Y Y Y Y Nordea % 3% N Y N Y N Pekao % 69% Y Y Y Y Y Y-yes, N-no In 3 out of 9 companies the coefficient at the adjusted risk premium was insignificant. Among those cases there were models where the other coefficient (at the market risk premium was statistically significant). For Bank Handlowy, even though the b 2 coefficient was insignificant, the b 1 was significant and the F-stat was higher than its critical value. For the three companies the adjusted 2 R upon adding another independent variable was not much higher, or even lower than for the one-factor model, containing only market risk premium (case of Fortis). Only larger, more liquid stocks experience visible improvement in the explanatory power of the model; Pekao (the coefficient of determination was 18 percentage points higher for the two-factor model as compared with the one factor model); ING (the improvement amounted to 7pp), BPH 16pp and BZ WBK 8pp. 109

110 Chapter 5 Size effect Historically, small firms have exhibited greater risk and experienced higher returns than large capitalization firms 38. This is due to the fact that the investment in smaller firm commands premium for increased risk, which is inherent in such an investment. Many researches were trying to explain this phenomenon in relation to CAPM, which if it holds claims that such risk should not be priced as this is only related to unsystematic risk. As mentioned in Chapter 1 the work on the relationship between size of the company (size effect or small firm effect) and average returns was widely documented. For example in 1981 Banz examined the topic and concluded that there was strong negative relation between average return and firm size 39. The main result of the study carried out by Fama and French in brought the result that for the examined period, 1963 through 1990 size for the US stocks, book to market value were significant for explaining the cross section of returns. The relation between average returns and firm size was shown to be negative and statistically significant, the relation between average returns and book-to-market equity was also strong and positive. As described in Chapter 1 in the section covering market inefficiencies, Fama and French divided firms into groups according to book-to-market ratios and found different average monthly returns for different deciles based on their book to market values. Moreover, that relation of returns on book-to-market ratio was independent of beta. Based on that finding the authors have suggested that high book-to-market ratio is serving as a proxy for a risk factor that affects equilibrium expected returns. The explanation of such finding was that firms with low market capitalization were more likely to have poor prospects and therefore low stock prices and lower Price to Book ratios (high Book/Market). On the other hand, the large capitalization stocks are prone to have stronger 38 Pratt, S., Cost of Capital, Estimation and Applications, 2 nd edition, p Banz R. W., The relationship between return and market value of common stocks, Journal of Financial Economics 9, 1981, p Fama E. F., French K. R., The cross-section of expected stock returns, Journal of Finance 47, 1992, p

111 prospects and lower Book to Market values, higher prices and lower average stock returns 41. Fama and French found that the size effect appeared to be weaker in the subperiod The size element as a factor explaining the cross section of returns also showed up in later work of Fama and French, in 1993 when they proposed a three factor model for estimating the cost of equity. Equation 23 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 Based on this equation, 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 explanation of the model, in 1995 Fama and French examined whether the behavior of stock prices (returns), in relation to size and book-to-market-equity, provides any explanatory input for the earnings 42. The studies confirmed that, as predicted by simple rational pricing model, BE/ME is related to properties of earnings. High book to market (or, in today s valuation terminology, low P/BV ratio) signals sustained low earnings on book equity, or is related to low ROE, which is typical of low profitability or distressed stocks. Conversely, stocks 41 Fama E. F., French K. R., The cross-section of expected stock returns, Journal of Finance 47, 1992, p Fama E. F., French K., Size and Book-Market Factors in Earnings and Returns, Journal of Finance 50, 1995, p

112 with low book to market value (or high P/BV ratio) are typical of companies that enjoy high return on equity capital highly profitable 43. The evidence showed that size is also related to profitability. Small stocks tend to have lower earnings on book equity than do big stocks. This property is often used in practice for the purposes of valuation of financial institutions in Poland (mainly represented by the banks) and worldwide. Figure 30 Estimated return on shareholders equity (ROE, on sustainable earnings, adjusted for one-offs and differences in costs of risk) versus estimated price to book ratio (P/BV) for Polish banks y = 19.15x R 2 = 0.74 Getin BZ WBK PKO BP Pekao 3.5 BPH P/BV BOS Millennium BRE Kredyt Bank ING 1.5 Handlowy 1.0 0% 5% 10% 15% 20% 25% ROE Source: Own calculations based on the Banks financial statements and Bloomberg Comparisons of banks in Poland based on their Price-Book Value ratios need to be done in correspondence with their ROE ratios. Generally, when adjusted for one-off items and differences in risk costs (due to considerable bad debt recoveries and lately increasing popularity of securisations), the higher the estimated return on shareholders equity the higher the future (and justified) P/BV ratio (or lower book to market, in Fama and French terminology). 43 Fama and French relate to the studies conducted by Penman in Penman, S. (1991), An evaluation of accounting rate of return, Journal of Accounting, Auditing, and Finance 6, p

113 The authors researched the topic whether the size and book to market factors would explain cross section of 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. Fama and French found that short-term variations in profitability have little connection with returns and book to market ratio; the firms that they have researched had lower ratios if they experience inferior profitability for approximately eleven years. An attempt to quantify the size effect was undertaken independently by Ibbotson Associates Studies and Standard & Poor s Corporate Value Consulting Studies 44. As it was presented in more detail in Chapter 1, both researchers calculated the size premium as the difference between predicted (based on the CAPM) and realized returns. That was negatively related to the stock size (it was larger for smaller stocks). Ibbotson associates study has brought the conclusion that the smaller the company the larger the beta and the predicted return and that the beta from CAPM did not explain the total returns of smaller stocks due to the existence of the size premium. The other test conducted by the Standard & Poor s Corporate Value Consulting provides supporting material for the findings of Ibbotson Associates, namely the higher risk associated with the smaller company is awarded a special premium over the market premium. The study. We were inspired by the results of the research on the size effect and tried to verify whether such an effect existed for the companies listed on the Warsaw Stock Exchange. The conclusions of the previous research were that the smaller the company the less of the variation of the cross section of returns the CAPM beta was able to explain. We examined the relationship between the market capitalization as at the end of 2004 for the companies chosen for the 44 Pratt S.P., Cost of Capital, Estimation and Applications, John Wiley & Sons, Inc., New Jersey, 2002, p

114 tests. For the testing we utilized single factor regression model as given by the equation below. Equation 24 The relation between the market capitalization and the coefficient of determination 2 R ( ) t = α + β MCAP t, 2004 Where: 2 Rt - coefficient of determination for the regression of security t s excess returns over the risk free rate against the market (WIG index) excess returns over the risk free rate α - the intercept term of the regression β - the slope of the regression MCAP t, market capitalization of security t as at the end of 2004 The dispersion of the observation for the regression set in the way described above is high. The minimum capitalization amounted to PLN 6.15m and the highest one PLN 22.97bn, while the median stood at PLN 412.9m. The largest companies represented on the WSE are the Polish banks, and the monopolistic fixed line telecommunication company, TPSA. Therefore we decided to use the natural log of the market capitalization as the explanatory variable, modifying the regression equation accordingly. Equation 25 The relation between the natural log of market capitalization and the coefficient of determination 2 R ln( ) t = α + β MCAP t, 2004 Where: 2 R t, α, β - as previously ln( MCAP t, 2004) - natural log of the market capitalization of security t as at the end of

115 The sample consists of 41 observations. We concentrated on the companies that satisfied the same criteria as the ones taken for the industry study, meaning, we have included the companies representing five sectors on the Warsaw Stock Exchange (construction, IT, telecommunications, food and banking). Companies fulfilling any of the following criteria were excluded: Companies without trading history covering the entire sample period, The bankrupt companies, Companies that recorded a negative book value at the end of any year within the researched period. Moreover, we excluded the companies, for which when the CAPM regressions were run, we were unable to reject the hypothesis that the slope coefficient (beta) was equal to zero, or, put it alternatively, the absolute values of their F- stat and t-stats were lower than the F critical and t critical, respectively, for the 0.05 significance level and the respective number of degrees of freedom. Presentation and discussion of results. The scatter plot of observations ( R versus the market capitalization) is presented in the figure below. 2 Figure 31 Graphic presentation of regression (the coefficient of determination, versus the market capitalization) 2 R, 70% 60% R 2 y = 0.00x R 2 = % 40% 30% 20% 10% 0% Market Capitalization (PLNbn) Source: The analysis of the thesis s authors 115

116 Due to the high dispersion of the observations, the Figure 31 shows that most of them are positioned in the area below PLN 5bn (the median amounted to PLN 413m). Therefore for the analysis of variance we decided to use the natural log of market capitalization. The resulting scatter plot, along with the fitted line follows. Figure 32 Graphic presentation of regression (the coefficient of determination, versus the natural log of market capitalization) 2 R, 70% 60% R2 y = 0.05x R 2 = % 40% 30% 20% 10% 0% Natural log of Market Capitalization Source: The analysis of the thesis s authors The fit of the line appears to be superior for the regression given in this way. 2 Table 33 Microsoft Excel output for the regression analysis ( R versus ln( MCAP ) t, 2004 Multiple R R square Adjusted R square Standard error Observations 41 Regression E-07 Residual Total Intercept (a) E Slope (b) E

117 The analysis of variance shows that the variation of the independent variable ( MCAP ) ) explains as much as 49% of the variance of the dependent ln( t, 2004 variable). The F-test and the t-test demonstrate that the model is statistically significant. For the slope coefficient, which amounts to 0.05 the t-stat amounts to At the 0.05 significance level and for 39 degrees of freedom, the t critical = The absolute value of the t-statistic of the slope coefficient is greater than 2.023, we reject the hypothesis H : β 0. Also, since F-stat = 0 = is greater than the F critical =4.091 (at 0.05 significance level, df 1 =1, and df 2 =39), we reject the hypothesis of the equality of all slope coefficients to 0. Unsurprisingly, in this case, since there is only one explanatory variable, the F- test results and t-test are the same. Also, we are able to reject the null hypothesis that H : α 0, as the absolute value of t-stat for the intercept (4.907) is also 0 = larger than t = This can also be seen by examining the P values for critical both the intercept and the slope coefficient. They both are lower than adopted 0.05 significance level, which means that we reject both null hypotheses described above. Also, significance F is lower than our assumed Based on the above analysis we can infer the conclusion that the natural log of market capitalization is a variable significant for the explanation of variations in the 2 R and that relation is linear. The larger the company, the more of the variation in the excess returns is explained by the CAPM beta. The results are consent with the common sense. The larger companies move closer with the broad market while the returns of smaller companies are more independent of the behavior of the market as a whole. For example for smaller construction companies present on the WSE the coefficient of determination amounts to less than 10%, meaning that the excess returns for those securities were not that much dependent on the market risk premium and the sensitivity to that premium as captured by the CAPM beta. That could be explained by the fact that there is an additional risk premium required by the investors who would be willing to undertake such investment. Smaller companies are not researched by the investment analysts, therefore it is possible that all the information is not immediately reflected in the price. Smaller stocks can be subject to periods of 117

118 mispricing. Furthermore, smaller stocks are usually illiquid and therefore expected returns must reflect the transaction costs, which encompasses brokerage, bid-ask spread, price impact and opportunity costs. CAPM beta does not cover those risks. With imperfect diversification of the investors, returns of smaller companies as predicted by the CAPM are far from reality. 118

119 Bibliography Amihus Y., Mendelson H., 1986: Asset Pricing and the Bid-Ask Spread, Journal of Financial Economics, 17 Amihus Y., Mendelson H., 1991: Liquidity, Asset Prices, and Financial Policy, Financial Analysts Journal, 47 Banz R. W., 1981: The relationship between return and market value of common stocks, Journal of Financial Economics, 9 Bhandari Laxmi Chand, 1988: Debt/equity ratio and expected common stock returns: Empirical evidence, Journal of Finance, 43 Bodie Z., Kane A., Marcus A.J., 2002: Investments, McGraw-Hill, Irwin, New York Chan L., Hamao Y., Lakonishok J, Fundamentals and stock returns in Japan, Journal of Finance, 46 Chen N., Roll R., Ross S., 1986: Economic Forces and the Stock Market, Journal of Business, 59 Copeland T., Koller T., Murrin J., 2000: Valuation: Measuring and Managing the Value of Companies, John Wiley&Sons, Inc., New York DeFusco R. A. McLeavy D. W. Pinto J. E., Runkle D. E., 2001: Quantitative Methods for Investment Analysis, Association for Investment Management and Research, Baltimore Douglas, W., 1969: Risk in the Equity Markets: An Empirical Appraisal of Market Efficiency, Yale Economic Essays, 9 Fama E. F., French K. R., 1992: The cross-section of expected stock returns, Journal of Finance,

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122 Appendix 1 The appendix includes the MS Excel outcomes ( table, regression graph) for the single regression model of each researched stock. For the stocks which single regression model was statistically significant, the appendix also contains the table for the two-factor regression model. Construction industry Budimex Figure 1 Graphic presentation of Budimex single regression model. 0.2 y=r-rf y = x R 2 = x=r(wig)-rf Source: The analysis of the thesis s authors

123 Table 1 Microsoft Excel output for the single regression model of Budimex Multiple R R square Adjusted R square Standard error Observations 242 Regression E-14 Residual Total Intercept Slope (B Budimex) E Table 2 Microsoft Excel output for the multiple regression analysis of Budimex two factor model Multiple R R square Adjusted R square Standard error Observations 242 Regression E-43 Residual Total Intercept Slope 1 (B Bud) E Slope 2 (B Bud_bud) E

124 Budopol Figure 2 Graphic presentation of Budopol single regression model R-rf y = 0.417x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 3 Microsoft Excel output for the single regression model of Budopol Multiple R R square Adjusted R square Standard error Observations 233 Regression Residual Total Intercept Slope (B Budopol)

125 Echo Figure 3 Graphic presentation of Echo single regression model. 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 4 Microsoft Excel output for the single regression model of Echo Multiple R R square Adjusted R square Standard error Observations 241 Regression E-09 Residual Total Intercept Slope (B Echo) E

126 Table 5 Microsoft Excel output for the multiple regression analysis of Echo two factor model Multiple R R square Adjusted R square Standard error Observations 241 Regression E-19 Residual Total Intercept Slope 1 (B Echo) E Slope 2 (B Echo_bud) E Elbudowa Figure 4 Graphic presentation of Elbudowa single regression model R-rf y = 0.59x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

127 Table 6 Microsoft Excel output for the single regression model of Elbudowa Multiple R R square Adjusted R square Standard error Observations 241 Regression E-09 Residual Total Intercept Slope (B Elbudowa) E Table 7 Microsoft Excel output for the multiple regression analysis of Elbudowa two factor model Multiple R R square Adjusted R square Standard error Observations 241 Regression E-09 Residual Total Intercept Slope 1 (B Elb) E Slope 2 (B Elb_bud)

128 Elektromontaz Polnoc Figure 5 Graphic presentation of Elektromontaz Polnoc single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 8 Microsoft Excel output for the single regression model of Elektromontaz Polnoc Multiple R R square Adjusted R square Standard error Observations 228 Regression Residual Total Intercept Slope (B Elekt. Pln)

129 Elektromontaz Warszawa Figure 6 Graphic presentation of Elektromontaz Warszawa single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 9 Microsoft Excel output for the single regression model of Elektromontaz Warszawa Multiple R R square Adjusted R square Standard error Observations 221 Regression Residual Total Intercept Slope (B Elekt. War.)

130 Table 10 Microsoft Excel output for the multiple regression analysis of Elektromontaz Warszawa two factor model Multiple R R square Adjusted R square Standard error Observations 221 Regression Residual Total Intercept Slope 1 (B Elektr.War.) Slope 2 (B Elektr. War._bud) Elektromontaz Poludnie Figure 7 Graphic presentation of Elektromontaz Poludnie single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

131 Table 11 Microsoft Excel output for the single regression model of Elektromontaz Poludnie Multiple R R square Adjusted R square Standard error Observations 221 Regression Residual Total Intercept Slope (B Elekt. War.) Table 12 Microsoft Excel output for the multiple regression analysis of Elektromontaz Poludnie two factor model Multiple R R square Adjusted R square Standard error Observations 221 Regression Residual Total Intercept Slope 1 (B Elektr.War.) Slope 2 (B Elektr. War._bud)

132 Energopol Figure 8 Graphic presentation of Energopol single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 13 Microsoft Excel output for the single regression model of Energopol Multiple R R square Adjusted R square Standard error Observations 233 Regression Residual Total Intercept Slope (B Energopol)

133 Hydrobudowa Figure 9 Graphic presentation of Hydrobudowa single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 14 Microsoft Excel output for the single regression model of Hydrobudowa Multiple R R square Adjusted R square Standard error Observations 236 Regression Residual Total Intercept Slope (B Hydrobudowa)

134 Instal Krakow Figure 10 Graphic presentation of Instal Krakow single regression model R-rf y = x R 2 = 5E-06 Rm(WIG)-rf Source: The analysis of the thesis s authors Table 15 Microsoft Excel output for the single regression model of Instal Krakow Multiple R R square E-06 Adjusted R square Standard error Observations 231 Regression E E Residual Total Intercept Slope (B Instal Kr.)

135 Instal Lublin Figure 11 Graphic presentation of Instal Lublin single regression model. 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 16 Microsoft Excel output for the single regression model of Instal Lublin Multiple R R square Adjusted R square Standard error Observations 237 Regression Residual Total Intercept Slope (B Instal Lub.)

136 Table 17 Microsoft Excel output for the multiple regression analysis of Instal Lublin two factor model Multiple R R square Adjusted R square Standard error Observations 237 Regression Residual Total Intercept Slope 1 (B Intal Lub.) Slope 2 (B Instal Lub._bud) Mostostal Eksport Figure 12 Graphic presentation of Mostostal Eksport single regression model. 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

137 Table 18 Microsoft Excel output for the single regression model of Mostostal Eksport Multiple R R square Adjusted R square Standard error Observations 238 Regression E-10 Residual Total Intercept Slope (B Most. Eks.) E Table 19 Microsoft Excel output for the multiple regression analysis of Mostostal Eksport two factor model Multiple R R square Adjusted R square Standard error Observations 238 Regression E-11 Residual Total Intercept Slope 1 (B Most. Eks.) E Slope 2 (B Most. Eks._bud)

138 Mostostal Plock Figure 13 Graphic presentation of Mostostal Plock single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 20 Microsoft Excel output for the single regression model of Mostostal Plock Multiple R R square Adjusted R square Standard error Observations 239 Regression E-05 Residual Total Intercept Slope (B Most. {Plock) E

139 Table 21 Microsoft Excel output for the multiple regression analysis of Mostostal Plock two factor model Multiple R R square Adjusted R square Standard error Observations 239 Regression E-05 Residual Total Intercept Slope 1 (B Most. Plock) E Slope 2 (B Most. Plock_bud) Mostostal Warszawa Figure 14 Graphic presentation of Mostostal Warszawa single regression model R-rf y = x + 7E-05 R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

140 Table 22 Microsoft Excel output for the single regression model of Mostostal Warszawa Multiple R R square Adjusted R square Standard error Observations 235 Regression E-08 Residual Total Intercept E Slope (B Most. War.) E Table 23 Microsoft Excel output for the multiple regression analysis of Mostostal Warszawa two factor model Multiple R R square Adjusted R square Standard error Observations 235 Regression E-07 Residual Total Intercept Slope 1 (B Most. War.) E Slope 2 (B Most. War._bud)

141 Naftobudowa Figure 15 Graphic presentation of Naftobudowa single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 24 Microsoft Excel output for the single regression model of Naftobudowa Multiple R R square Adjusted R square Standard error Observations 226 Regression Residual Total Intercept Slope (B Naftobudowa)

142 Table 25 Microsoft Excel output for the multiple regression analysis of Naftobudowa two factor model Multiple R R square Adjusted R square Standard error Observations 226 Regression Residual Total Intercept Slope 1 (B Naftobudowa) Slope 2 (B Naftobudowa_bud) Pemug Figure 16 Graphic presentation of Pemug single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

143 Table 26 Microsoft Excel output for the single regression model of Pemug Multiple R R square Adjusted R square Standard error Observations 226 Regression Residual Total Intercept Slope (B Pemug) Polimex Mostostal Figure 17 Graphic presentation of Polimex Mostostal single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

144 Table 27 Microsoft Excel output for the single regression model of Polimex Mostostal Multiple R R square Adjusted R square Standard error Observations 239 Regression E-07 Residual Total Intercept Slope (B Polimex Most.) E Table 28 Microsoft Excel output for the multiple regression analysis of Polimex Mostostal two factor model Multiple R R square Adjusted R square Standard error Observations 239 Regression E-10 Residual Total Intercept Slope 1 (B Polimex Most.) E Slope 2 (B Polimex Most._bud) E

145 Polnord Figure 18 Graphic presentation of Polnord single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 29 Microsoft Excel output for the single regression model of Polnord Multiple R R square Adjusted R square Standard error Observations 240 Regression Residual Total Intercept Slope (B Polnord)

146 Projprzem Figure 19 Graphic presentation of Projprzem single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 30 Microsoft Excel output for the single regression model of Projprzem Multiple R R square Adjusted R square Standard error Observations 241 Regression E-05 Residual Total Intercept Slope (B Projprzem) E

147 Table 31 Microsoft Excel output for the multiple regression analysis of Projprzem two factor model Multiple R R square Adjusted R square Standard error Observations 241 Regression E-06 Residual Total Intercept E Slope 1 (B Projprzem) E Slope 2 (B Projprzem_bud) IT industry Comarch Figure 20 Graphic presentation of Comarch single regression model. 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

148 Table 32 Microsoft Excel output for the single regression model of Comarch Multiple R R square Adjusted R square Standard error Observations 240 Regression E-33 Residual Total Intercept Slope (B Comarch) E Table 33 Microsoft Excel output for the multiple regression analysis of Comarch two factor model Multiple R R square Adjusted R square Standard error Observations 240 Regression E-45 Residual Total Intercept Slope 1 (B Com) E Slope 2 (B Com_info) E

149 Computerland Figure 21 Graphic presentation of Computerland single regression model 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 34 Microsoft Excel output for the single regression model of Computerland Multiple R R square Adjusted R square Standard error Observations 242 Regression E-33 Residual Total Intercept Slope (B Computerland) E

150 Table 35 Microsoft Excel output for the multiple regression analysis of Computerland two factor model Multiple R R square Adjusted R square Standard error Observations 242 Regression E-47 Residual Total Intercept Slope 1 (B Comput.) E Slope 2 (B Comput._info) E CSS Figure 22 Graphic presentation of CSS single regression model. 0.2 R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

151 Table 36 Microsoft Excel output for the single regression model of CSS Multiple R R square Adjusted R square Standard error Observations 238 Regression E-05 Residual Total Intercept Slope (B CSS) E Table 37 Microsoft Excel output for the multiple regression analysis of CSS two factor model Multiple R R square Adjusted R square Standard error Observations 238 Regression E-05 Residual Total Intercept Slope 1 (B CSS) E Slope 2 (B CSS_info)

152 Elzab Figure 23 Graphic presentation of Elzab single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 38 Microsoft Excel output for the single regression model of Elzab Multiple R R square Adjusted R square Standard error Observations 235 Regression Residual Total Intercept Slope (B Elzab)

153 Table 39 Microsoft Excel output for the multiple regression analysis of Elzab two factor model Multiple R R square Adjusted R square Standard error Observations 235 Regression Residual Total Intercept Slope 1 (B Elzab) Slope 2 (B Elzab_info) IB System Figure 24 Graphic presentation of IB System single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

154 Table 40 Microsoft Excel output for the single regression model of IB System Multiple R R square Adjusted R square Standard error Observations 212 Regression Residual Total Intercept Slope (B IBsystem) InternetGroup Figure 25 Graphic presentation of InternetGroup single regression model R-rf y = 0.702x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

155 Table 41 Microsoft Excel output for the single regression model of InternetGroup Multiple R R square Adjusted R square Standard error Observations 232 Regression E-06 Residual Total Intercept Slope (B InternetGroup) E Table 42 Microsoft Excel output for the multiple regression analysis of InternetGroup two factor model Multiple R R square Adjusted R square Standard error Observations 232 Regression E-05 Residual Total Intercept Slope 1 (B InternetGroup) E Slope 2 (B InternetGroup_info)

156 Macrosoft Figure 26 Graphic presentation of Macrosoft single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 43 Microsoft Excel output for the single regression model of Macrosoft Multiple R R square Adjusted R square Standard error Observations 232 Regression E-15 Residual Total Intercept Slope (B Macrosoft) E

157 Table 44 Microsoft Excel output for the multiple regression analysis of Macrosoft two factor model Multiple R R square Adjusted R square Standard error Observations 232 Regression E-16 Residual Total Intercept Slope 1 (B Macrosoft) E Slope 2 (B Macrosoft_info) Prokom Figure 27 Graphic presentation of Prokom single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

158 Table 45 Microsoft Excel output for the single regression model of Prokom Multiple R R square Adjusted R square Standard error Observations 240 Regression E-34 Residual Total Intercept Slope (B Prokom) E Table 46 Microsoft Excel output for the multiple regression analysis of Prokom two factor model Multiple R R square Adjusted R square Standard error Observations 240 Regression E-108 Residual Total Intercept Slope 1 (B Pro) E Slope 2 (B Pro_info) E

159 Softbank Figure 28 Graphic presentation of Softbank single regression model R-rf y = x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 47 Microsoft Excel output for the single regression model of Softbank Multiple R R square Adjusted R square Standard error Observations 234 Regression E-22 Residual Total Intercept Slope (B Softbank) E

160 Table 48 Microsoft Excel output for the multiple regression analysis of Softbank two factor model Multiple R R square Adjusted R square Standard error Observations 234 Regression E-36 Residual Total Intercept Slope 1 (B Softbank) E Slope 2 (B Softbank_info) E Telecommunication industry Elektrim Figure 29 Graphic presentation of Elektrim single regression model R-rf y = 0.841x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

161 Table 49 Microsoft Excel output for the single regression model of Elektrim Multiple R R square Adjusted R square Standard error Observations 221 Regression E-06 Residual Total Intercept Slope (B Elektrim) E Table 50 Microsoft Excel output for the multiple regression analysis of Elektrim two factor model Multiple R R square Adjusted R square Standard error Observations 221 Regression E-10 Residual Total Intercept Slope 1 (B Elektrim) E Slope 2 (B Elektrim_telekom) E

162 Netia Figure 30 Graphic presentation of Netia single regression model R-rf y = 1.209x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors Table 51 Microsoft Excel output for the single regression model of Netia Multiple R R square Adjusted R square Standard error Observations 226 Regression E-18 Residual Total Intercept Slope (B Netia) E

163 Table 52 Microsoft Excel output for the multiple regression analysis of Netia two factor model Multiple R R square Adjusted R square Standard error Observations 226 Regression E-19 Residual Total Intercept Slope 1 (B Netia) E Slope 2 (B Netia_telekom) TPSA Figure 31 Graphic presentation of TPSA single regression model. 0.2 R-rf y = 1.359x R 2 = Rm(WIG)-rf Source: The analysis of the thesis s authors

164 Table 53 Microsoft Excel output for the single regression model of TPSA Multiple R R square Adjusted R square Standard error Observations 241 Regression E-50 Residual Total Intercept Slope (B TPSA) E Table 54 Microsoft Excel output for the multiple regression analysis of TPSA two factor model Multiple R R square Adjusted R square Standard error Observations 241 Regression E-95 Residual Total Intercept Slope 1 (B TPSA) E Slope 2 (B TPSA_teleko E

165 Food industry Beefsan Figure 32 Graphic presentation of Beefsan single regression model R-Rf y = 0.38x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 55 Microsoft Excel output for the single regression model of Beefsan Multiple R R square Adjusted R square Standard error Observations 237 Regression Residual Total Intercept (a) Slope (B Beefsan)

166 Table 56 Microsoft Excel output for the multiple regression analysis of Beefsan two factor model Multiple R R square Adjusted R square Standard error Observations 237 Regression Residual Total Intercept (a) Slope 1 (B Beefsan) Slope 2 (B Beefsan_food) Ekodrob Figure 33 Graphic presentation of Ekodrob single regression model R-Rf y = 0.28x R 2 = Rm-Rf Source: The analysis of the thesis s authors

167 Table 57 Microsoft Excel output for the single regression model of Ekodrob Multiple R R square Adjusted R square Standard error Observations 234 Regression Residual Total Intercept (a) Slope (B Ekodrob) Table 58 Microsoft Excel output for the multiple regression analysis of Ekodrob two factor model Multiple R R square Adjusted R square Standard error Observations 234 Regression Residual Total Intercept (a) Slope 1 (B Ekodrob) Slope 2 (B Ekodrob_food)

168 Indykpol Figure 34 Graphic presentation of Indykpol single regression model R-Rf y = 0.27x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 59 Microsoft Excel output for the single regression model of Indykpol Multiple R R square Adjusted R square Standard error Observations 254 Regression Residual Total Intercept (a) Slope (B Indykpol)

169 Table 60 Microsoft Excel output for the multiple regression analysis of Indykpol two factor model Multiple R R square Adjusted R square Standard error Observations 254 Regression Residual Total Intercept (a) Slope 1 (B Indykpol) Slope 2 (B Indykpol_food) Jutrzenka Figure 35 Graphic presentation of Jutrzenka single regression model R-Rf y = 0.81x R 2 = Rm-Rf Source: The analysis of the thesis s authors

170 Table 61 Microsoft Excel output for the single regression model of Jutrzenka Multiple R R square Adjusted R square Standard error Observations 254 Regression E-16 Residual Total Intercept (a) Slope (B Jutrzenka) E Table 62 Microsoft Excel output for the multiple regression analysis of Jutrzenka two factor model Multiple R R square Adjusted R square Standard error Observations 254 Regression E-16 Residual Total Intercept (a) Slope 1 (B Jutrzenka) E Slope 2 (B Jutrzenka_food)

171 Kruszwica Figure 36 Graphic presentation of Kruszwica single regression model R-Rf y = 0.62x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 63 Microsoft Excel output for the single regression model of Kruszwica Multiple R R square Adjusted R square Standard error Observations 254 Regression E-08 Residual Total Intercept (a) Slope (B Kruszwica E

172 Table 64 Microsoft Excel output for the multiple regression analysis of Kruszwica two factor model Multiple R R square Adjusted R square Standard error Observations 254 Regression E-08 Residual Total Intercept (a) Slope 1 (B Kruszwica) E Slope 2 (B Kruszwica_food) Mieszko Figure 37 Graphic presentation of Mieszko single regression model R-Rf y = 0.15x R 2 = Rm-Rf Source: The analysis of the thesis s authors

173 Table 65 Microsoft Excel output for the single regression model of Mieszko Multiple R R square Adjusted R square Standard error Observations 254 Regression Residual Total Intercept (a) Slope (B Mieszko) Table 66 Microsoft Excel output for the multiple regression analysis of Mieszko two factor model Multiple R R square Adjusted R square Standard error Observations 254 Regression Residual Total Intercept (a) Slope 1 (B Mieszko) Slope 2 (B Mieszko_food)

174 Pepees Figure 38 Graphic presentation of Pepees single regression model R-Rf y = 0.58x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 67 Microsoft Excel output for the single regression model of Pepees Multiple R R square Adjusted R square Standard error Observations 254 Regression E-07 Residual Total Intercept (a) Slope (B Pepees) E

175 Table 68 Microsoft Excel output for the multiple regression analysis of Pepees two factor model Multiple R R square Adjusted R square Standard error Observations 254 df SS MS F Istotność F Regression E-07 Residual Total Intercept (a) Slope 1 (B Pepees) E Slope 2 (B Pepees_food) Rolimpex Figure 39 Graphic presentation of Rolimpex single regression model R-Rf y = 0.97x R 2 = Rm-Rf Source: The analysis of the thesis s authors

176 Table 69 Microsoft Excel output for the single regression model of Rolimpex Multiple R R square Adjusted R square Standard error Observations 259 Regression E-17 Residual Total Intercept (a) Slope (B Rolimpex) E Table 70 Microsoft Excel output for the multiple regression analysis of Rolimpex two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression E-22 Residual Total Intercept (a) Slope 1 (B Rolimpex) E Slope 2 (B Rolimpex_food) E

177 Sokolow Figure 40 Graphic presentation of Sokolow single regression model R-Rf y = 0.85x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 71 Microsoft Excel output for the single regression model of Sokolow Multiple R R square Adjusted R square Standard error Observations 259 Regression E-18 Residual Total Intercept (a) Slope (B Sokolow) E

178 Table 72 Microsoft Excel output for the multiple regression analysis of Sokolow two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression E-29 Residual Total Intercept (a) Slope 1 (B Sokolow) E Slope 2 (B Sokolow_food) E Strzelec Figure 41 Graphic presentation of Strzelec single regression model R-Rf y = 0.28x R 2 = Rm-Rf Source: The analysis of the thesis s authors

179 Table 73 Microsoft Excel output for the single regression model of Strzelec Multiple R R square Adjusted R square Standard error Observations 258 Regression Residual Total Intercept (a) Slope (B Strzelec) Table 74 Microsoft Excel output for the multiple regression analysis of Strzelec two factor model Multiple R R square Adjusted R square Standard error Observations 258 Regression Residual Total Intercept (a) Slope 1 (B Strzelec) Slope 2 (B Strzelec_food)

180 Wawel Figure 42 Graphic presentation of Wawel single regression model R-Rf y = 0.20x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 75 Microsoft Excel output for the single regression model of Wawel Multiple R R square Adjusted R square Standard error Observations 259 Regression Residual Total Intercept (a) Slope (B Wawel)

181 Table 76 Microsoft Excel output for the multiple regression analysis of Wawel two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression Residual Total Intercept (a) Slope 1 (B Wawel) Slope 2 (B Wawel_food) Wilbo Figure 43 Graphic presentation of Wilbo single regression model R-Rf y = 0.68x R 2 = Rm-Rf Source: The analysis of the thesis s authors

182 Table 77 Microsoft Excel output for the single regression model of Wilbo Multiple R R square Adjusted R square Standard error Observations 260 Regression E-11 Residual Total Intercept (a) Slope (B Wilbo) E Table 78 Microsoft Excel output for the multiple regression analysis of Wilbo two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-11 Residual Total Intercept (a) Slope 1 (B Wilbo) E Slope 2 (B Wilbo_food)

183 Zywiec Figure 44 Graphic presentation of Zywiec single regression model. R-Rf y = 0.44x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 79 Microsoft Excel output for the single regression model of Zywiec Multiple R R square Adjusted R square Standard error Observations 260 Regression E-11 Residual Total Intercept (a) Slope (B Zywiec) E

184 Table 80 Microsoft Excel output for the multiple regression analysis of Zywiec two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-43 Residual Total Intercept (a) Slope 1 (B Zywiec) E Slope 2 (B Zywiec_food) E Banking BOS Figure 45 Graphic presentation of BOS single regression model R-Rf y = 0.09x R 2 = Rm-Rf Source: The analysis of the thesis s authors

185 Table 81 Microsoft Excel output for the single regression model of BOS Multiple R R square Adjusted R square E-05 Standard error Observations 260 Regression Residual Total Intercept (a) Slope (B BOS) Table 82 Microsoft Excel output for the multiple regression analysis of BOS two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression Residual Total Intercept (a) Slope 1 (B BOS) Slope 2 (B BOS_banking)

186 BPH Figure 46 Graphic presentation of BPH single regression model R-Rf y = 1.05x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 83 Microsoft Excel output for the single regression model of BPH Multiple R R square Adjusted R square Standard error Observations 260 Regression E-36 Residual Total Intercept (a) Slope (B BPH) E

187 Table 84 Microsoft Excel output for the multiple regression analysis of BPH two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-55 Residual Total Intercept (a) Slope 1 (B BPH) E Slope 2 (B BPH_banking) E BRE Figure 47 Graphic presentation of BRE single regression model R-Rf y = 0.97x R 2 = Rm-Rf Source: The analysis of the thesis s authors

188 Table 85 Microsoft Excel output for the single regression model of BRE Multiple R R square Adjusted R square Standard error Observations 259 Regression E-25 Residual Total Intercept (a) E Slope (B BRE) E Table 86 Microsoft Excel output for the multiple regression analysis of BRE two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression E-27 Residual Total Intercept (a) E Slope 1 (B BRE) E Slope 2 (B BRE_banking)

189 BZ WBK Figure 48 Graphic presentation of BZ WBK single regression model R-Rf y = 1.06x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 87 Microsoft Excel output for the single regression model of BZ WBK Multiple R R square Adjusted R square Standard error Observations 260 Regression E-32 Residual Total Intercept (a) Slope (B BZ WBK) E

190 Table 88 Microsoft Excel output for the multiple regression analysis of BZ WBK two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-39 Residual Total Intercept (a) Slope 1 (B BZ WBK) E Slope 2 (B BZ WBK_banking) E DZ Polska Figure 49 Graphic presentation of DZ Polska single regression model R-Rf y = 0.17x R 2 = Rm-Rf Source: The analysis of the thesis s authors

191 Table 89 Microsoft Excel output for the single regression model of DZ Polska Multiple R R square Adjusted R square Standard error Observations 255 Regression Residual Total Intercept (a) Slope (B DZ Polska) Table 90 Microsoft Excel output for the multiple regression analysis of DZ Polska two factor model Multiple R R square Adjusted R square Standard error Observations 255 Regression Residual Total Intercept (a) Slope 1 (B DZ Polska) Slope 2 (B DZ Polska_banking)

192 Fortis Figure 50 Graphic presentation of Fortis single regression model R-Rf y = 0.39x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 91 Microsoft Excel output for the single regression model of Fortis Multiple R R square Adjusted R square Standard error Observations 259 Regression Residual Total Intercept (a) Slope (B Fortis)

193 Table 92 Microsoft Excel output for the multiple regression analysis of Fortis two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression Residual Total Intercept (a) Slope 1 (B Fortis) Slope 2 (B Fortis_banking) Handlowy Figure 51 Graphic presentation of Handlowy single regression model R-Rf y = 0.51x R 2 = Rm-Rf Source: The analysis of the thesis s authors

194 Table 93 Microsoft Excel output for the single regression model of Handlowy Multiple R R square Adjusted R square Standard error Observations 260 Regression E-14 Residual Total Intercept (a) Slope (B Handlowy) E Table 94 Microsoft Excel output for the multiple regression analysis of Handlowy two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-13 Residual Total Intercept (a) Slope 1 (B Handlowy) E Slope 2 (B Handlowy_bankin

195 ING Figure 52 Graphic presentation of ING single regression model R-Rf y = 0.51x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 95 Microsoft Excel output for the single regression model of ING Multiple R R square Adjusted R square Standard error Observations 260 Regression E-13 Residual Total Intercept (a) Slope (B ING) E

196 Table 96 Microsoft Excel output for the multiple regression analysis of ING two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-17 Residual Total Coefficients Standard error t Stat P-value Lower 95% Upper 95 Intercept (a) Slope 1 (B ING) E Slope 2 (B ING_banking) E Kredyt Bank Figure 53 Graphic presentation of Kredyt Bank single regression model R-Rf 0.10 y = 0.48x R 2 = Rm-Rf Source: The analysis of the thesis s authors

197 Table 97 Microsoft Excel output for the single regression model of Kredyt Bank Multiple R R square Adjusted R square Standard error Observations 259 Regression E-10 Residual Total Intercept (a) Slope (B Kredyt Bank) E Table 98 Microsoft Excel output for the multiple regression analysis of Kredyt Bank two factor model Multiple R R square Adjusted R square Standard error Observations 259 Regression E-12 Residual Total Intercept (a) Slope 1 (B Kredyt Bank) E Slope 2 (B Kredyt Bank_ban

198 Nordea Figure 54 Graphic presentation of Nordea single regression model R-Rf y = 0.36x R 2 = Rm-Rf Source: The analysis of the thesis s authors Table 99 Microsoft Excel output for the single regression model of Nordea Multiple R R square Adjusted R square Standard error Observations 256 Regression Residual Total Intercept (a) Slope (B Nordea)

199 Table 100 Microsoft Excel output for the multiple regression analysis of Nordea two factor model Multiple R R square Adjusted R square Standard error Observations 256 Regression Residual Total Intercept (a) Slope 1 (B Nordea) Slope 2 (B Nordea_banking) Pekao Figure 55 Graphic presentation of Pekao single regression model R-Rf y = 1.08x R 2 = Rm-Rf Source: The analysis of the thesis s authors

200 Table 101 Microsoft Excel output for the single regression model of Pekao Multiple R R square Adjusted R square Standard error Observations 260 Regression E-42 Residual Total Intercept (a) Slope (B Pekao) E Table 102 Microsoft Excel output for the multiple regression analysis of Pekao two factor model Multiple R R square Adjusted R square Standard error Observations 260 Regression E-67 Residual Total Intercept (a) E Slope 1 (B Pekao) E Slope 2 (B Pekao_banking) E