y 2 del ángulo de la articulación: Señala al ECM el ángulo del bastidor trasero con

Historically, small firms have exhibited greater risk and experienced higher returns than large capitalization firms38. 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 size39.

The main result of the study carried out by Fama and French in 199240 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, 2nd edition, p. 122 39

Banz R. W., The relationship between return and market value of common stocks, Journal of Financial Economics 9, 1981, p. 3-18

40

Fama E. F., French K. R., The cross-section of expected stock returns, Journal of Finance 47, 1992, p. 427-465

prospects and lower Book to Market values, higher prices and lower average stock returns41. Fama and French found that the size effect appeared to be weaker in the subperiod 1977-1990.

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

[

m f

]

i i i i f i R b E R R s E SMB hE HML R E( )− = ( )− + ( )+ ( )+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 cross- section 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 earnings42. 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. 446

42

Fama E. F., French K., Size and Book-Market Factors in Earnings and Returns, Journal of Finance 50, 1995, p. 131-55

with low book to market value (or high P/BV ratio) are typical of companies that enjoy high return on equity capital highly profitable43. 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 + 0.52 R2 = 0.74 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 0% 5% 10% 15% 20% 25% ROE P /B V Getin BZ WBK PKO BP Pekao BPH BRE Kredyt Bank ING Millennium Handlowy BOS

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. 233- 255

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 Studies44. 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.90

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

)

(

_,2004 2 t t

MCAP

R

=α

+β

×

Where: 2 t

R

- 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

2004 , t

MCAP

- 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

)

ln(

_,2004 2 t t

MCAP

R

=α

+β

×

Where: 2 t

R

,

α

,

β

- as previously

)

ln(MCAP

_t,2004 - natural log of the market capitalization of security t as at the end of 2004

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 2 versus the market capitalization) is presented in the figure below.

Figure 31 Graphic presentation of regression (the coefficient of determination, R2, versus the market capitalization)

y = 0.00x + 0.14 R2 = 0.45 0% 10% 20% 30% 40% 50% 60% 70% 0 5 10 15 20 25 Market Capitalization (PLNbn) R2

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, R2, versus the natural log of market capitalization)

y = 0.05x - 0.78 R2 = 0.49 0% 10% 20% 30% 40% 50% 60% 70% 15.0 16.0 17.0 18.0 19.0 20.0 21.0 22.0 23.0 24.0 25.0

Natural log of Market Capitalization R2

Source: The analysis of the thesis’s authors

The fit of the line appears to be superior for the regression given in this way.

Table 33 Microsoft Excel output for the regression analysis (R2 versus

) ln(MCAP_t,2004 Regression statistics Multiple R 0.70094746 R square 0.491327342 Adjusted R square 0.478284454 Standard error 0.115163433 Observations 41 ANOVA df SS MS F Significance F Regression 1 0.499604513 0.499605 37.67013238 3.33511E-07 Residual 39 0.517242038 0.013263 Total 40 1.016846551

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept (a) -0.77889126 0.158738135 -4.906768 1.6801E-05 -1.099969126 -0.45781339

Slope (b) 0.049309412 0.008033989 6.1376 3.33511E-07 0.033059151 0.065559674

The analysis of variance shows that the variation of the independent variable (ln(MCAP_t,2004)) explains as much as 49% of the variance of the dependent 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 6.138. At the 0.05 significance level and for 39 degrees of freedom, the

023 . 2 =

critical

t . The absolute value of the t-statistic of the slope coefficient is greater than 2.023, we reject the hypothesis H₀:β =0. Also, since |F-stat| = 37.67 is greater than the F_critical=4.091 (at 0.05 significance level, df₁=1, and

2

df =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 H₀ :α =0, as the absolute value of t-stat for the intercept (4.907) is also larger than t_critical =2.026. This can also be seen by examining the P values for 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 0.05.

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 R and that relation is linear. The larger the company, the more of the 2 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

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.

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Appendix 1

The appendix includes the MS Excel outcomes (ANOVA 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 ANOVA table for the two-factor regression model.

Construction industry

Budimex

Figure 1 Graphic presentation of Budimex single regression model.

y = 0.6946x + 0.0007 R2 _{= 0.2091} -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 y=R-rf x=R(WIG)-rf

Table 1 Microsoft Excel output for the single regression model of Budimex Regression statistics Multiple R 0.45731001 R square 0.20913244 Adjusted R square 0.20583716 Standard error 0.03911449 Observations 242 ANOVA df SS MS F Significance F Regression 1 0.097096658 0.09709666 63.46421223 6.56267E-14 Residual 240 0.367186436 0.00152994 Total 241 0.464283094

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept 0.00074465 0.002515669 0.29600517 0.767482094 -0.004210958 0.00570026

Slope (B Budimex) 0.69461719 0.087192892 7.96644288 6.56267E-14 0.522856148 0.86637823

Source: The analysis of thesis’s authors

Table 2 Microsoft Excel output for the multiple regression analysis of Budimex two factor model Regression statistics Multiple R 0.74702406 R square 0.55804495 Adjusted R square 0.55434658 Standard error 0.02930094 Observations 242 ANOVA df SS MS F Significance F Regression 2 0.259090835 0.12954542 150.8894881 4.19453E-43 Residual 239 0.205192258 0.00085855 Total 241 0.464283094

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept 0.00145381 0.001885212 0.77116524 0.441370724 -0.002259941 0.00516756

Slope 1 (B Bud) 0.69461719 0.065316803 10.6345864 7.04608E-22 0.565947123 0.82328726

Slope 2 (B Bud_bud) 1.15930338 0.084397374 13.7362494 4.86419E-32 0.993045766 1.325561

Budopol

Figure 2 Graphic presentation of Budopol single regression model.

y = 0.417x - 0.0049 R2 = 0.0255 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 R-rf Rm(WIG)-rf

Source: The analysis of the thesis’s authors

Table 3 Microsoft Excel output for the single regression model of Budopol

Regression statistics Multiple R 0.061589121 R square 0.00379322 Adjusted R square -0.000519364 Standard error 0.069638233 Observations 233 ANOVA df SS MS F Significance F Regression 1 0.004265461 0.004265 0.879570197 0.349298952 Residual 231 1.120230685 0.004849 Total 232 1.124496146

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept -0.004888535 0.004563347 -1.471657 0.142474433 -0.015706793 0.00227543

Slope (B Budopol) 0.416987322 0.160042276 0.937854 0.349298952 -0.165233144 0.46542573

Echo

Figure 3 Graphic presentation of Echo single regression model.

y = 0.6084x + 0.0022 R2 = 0.1436 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 R-rf Rm(WIG)-rf

Source: The analysis of the thesis’s authors

Table 4 Microsoft Excel output for the single regression model of Echo

Regression statistics Multiple R 0.378919872 R square 0.14358027 Adjusted R square 0.139996924 Standard error 0.043045091 Observations 241 ANOVA df SS MS F Significance F Regression 1 0.074242616 0.074243 40.06876915 1.20126E-09 Residual 239 0.442838289 0.001853 Total 240 0.517080905

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept 0.00216568 0.002774555 0.78055 0.435839444 -0.003300023 0.00763138

Slope (B Echo) 0.608417027 0.096116591 6.32999 1.20126E-09 0.419073279 0.79776078

Table 5 Microsoft Excel output for the multiple regression analysis of Echo two factor model Regression statistics Multiple R 0.551165196 R square 0.303783073 Adjusted R square 0.297932511 Standard error 0.038892249 Observations 241 ANOVA df SS MS F Significance F Regression 2 0.157080427 0.07854 51.92373865 1.93466E-19 Residual 238 0.360000478 0.001513 Total 240 0.517080905

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept 0.002582371 0.002507508 1.029856 0.304123205 -0.002357379 0.00752212

Slope 1 (B Echo) 0.611203455 0.08684441 7.037914 2.06897E-11 0.440121381 0.78228553

Slope 2 (B Echo_bud) 0.831327942 0.112336563 7.400333 2.32571E-12 0.610026742 1.05262914

Source: The analysis of thesis’s authors

Elbudowa

Figure 4 Graphic presentation of Elbudowa single regression model.

y = 0.59x - 0.002 R2 _{= 0.1333} -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 -0.1 -0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08 0.1 0.12 R-rf Rm(WIG)-rf

Table 6 Microsoft Excel output for the single regression model of Elbudowa Regression statistics Multiple R 0.365117006 R square 0.133310428 Adjusted R square 0.129684112 Standard error 0.043461022 Observations 241 ANOVA df SS MS F Significance F Regression 1 0.069438178 0.06943818 36.76194259 5.17049E-09 Residual 239 0.451437638 0.00188886 Total 240 0.520875816

Coefficients Standard error t Stat P-value Lower 95% Upper 95%

Intercept -0.002048199 0.002800526 -0.73136233 0.465274316 -0.007565063 0.00346866

Slope (B Elbudowa) 0.589991147 0.09730749 6.06316275 5.17049E-09 0.398301402 0.78168089

Source: The analysis of thesis’s authors

Table 7 Microsoft Excel output for the multiple regression analysis of Elbudowa two factor model Regression statistics Multiple R 0.389356744 R square 0.151598674 Adjusted R square 0.144469251 Standard error 0.043090277 Observations 241 ANOVA df SS MS F Significance F

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