Keywords

1 Introduction

Capital asset pricing model (CAPM) was developed independently by William Sharpe [1], John Lintner [2] and Jan Mossin [3] twelve years after Harry Markowitz had laid down the foundation of modern portfolio management [4]. This model became a useful instrument to estimate the systematic risk of individual financial instruments or portfolio of securities, and its parameter—beta became one of the most frequently used risk measures. CAPM is also the capital cost model, which is most often used in practice, thanks to its simple form and easy application as well as clear interpretation (refer to [5,6,7,8,9]).

It should be noted that the classic capital asset pricing model has had both supporters and opponents since it was developed. Among the former, one should mention [10,11,12]. The latter have criticized CAPM mostly because of unrealistic assumptions that underlie the model and because it is a single-factor model. Roll’s critique [13] refers to the model construction itself and concerns the lack of possibility of its empirical verification since it is impossible to create or observe a truly diversified market portfolio. While Fama and French in series of their works [14,15,16] prove that CAPM model fails to apply in capital estimation because beta coefficient does not provide a correct explanation for the expected return rates. At the same, they proved that good results are given by supplementing model with additional factors, i.e., price to book value ratio and measures of company size (see Fama and French [17]). In [18], it was found that most errors in cost of stock capital estimation using CAPM model are also caused by wrong assessment of risk premiums, while Pástor and Stambaugh in [19] presented that capital estimations may be corrected by Bayesian approach. However, Gibbson [20] who proposed a methodology to avoid errors in the measurement of variables and increase of appropriateness of estimates of model parameters proposes to reject CAPM. Whereas authors of [21] were not able to come to clear conclusions regarding appropriateness of CAPM and beta coefficient estimates. As they found by themselves, decades may pass by until additionally collected data allow for rejection of CAPM for a reasonable significance level.

Despite the criticism, the classic capital asset pricing model is still widely used amongst practitioners. According to a study on European firms (discussed in [22]), about 45% of chief financial officers (CFOs) rely on the CAPM and 73.5% of interviewed US CFOs “always or almost always” use the CAPM to estimate the cost of equity (see Graham and Harvey [5]).

Here, a question arises why Fama–French three-factor model, which seems to be the improved version of the capital asset pricing model, has not been used in practice and research as often as CAPM. In fact, the research on application three-factor model to the Polish capital market has been provided rather rarely. Therefore, the study aims to find out which model is more appropriate to explain rates of return of portfolios containing selected companies listed on the Warsaw Stock Exchange. Investigation is provided for the years 2007–2017.

2 Fama–French Three-Factor Model

The three-factor model was developed as a response to the poor performance of the CAPM in explaining realized returns. Its authors claim in [17] that anomalies relating to the CAPM are captured by the three-factor asset pricing model. These three risk factors defined in the Fama–French model are:

  1. 1.

    risk premium, i.e., excess market portfolio return;

  2. 2.

    the difference between the excess return on a portfolio of stocks with small capitalization and the excess return on a portfolio of stocks with big capitalization;

  3. 3.

    the difference between the excess return on a portfolio of stocks which is characterized by the high value of the book-to-market value (BV/MV) and the excess return on a portfolio of stocks which is characterized by low book-to-market value.

The three-factor model is formulated as following [15]:

$$R_{i} - R_{f} = \alpha + \beta_{M} \left( {R_{M} - R_{f} } \right) + \beta_{\text{SMB}} {\text{SMB}} + \beta_{\text{HML}} {\text{HML}} + \varepsilon$$
(16.1)

where, \(R_{i}\)—rate of return from the portfolio, \(R_{f}\)—risk-free rate of return; \(R_{M}\)—rate of return from the market portfolio; \((R_{M} - R_{f} )\)—risk premium; SMB—the difference between the excess return on a portfolio of small stocks and the excess return on a portfolio of big stocks (SMB, small minus big); HML—the difference between the excess return on a portfolio of high-book-to-market stocks and the excess return on a portfolio of low-book-to-market stocks (HML, high minus low); \(\alpha\)—an intercept; \(\beta_{M}\), \(\beta_{\text{SMB}} , \beta_{\text{HML}}\)—model parameters describing the effect of each risk factor; \(\varepsilon\)—residual.

One should note that the classical capital asset pricing model—CAPM, as a single-factor model, is a simplified version of the Fama–French model, i.e., it can be formulated as (16.1) for \(\beta_{\text{SMB}} = \beta_{\text{HML}} = 0\).

Empirical tests of the Fama–French three-factor model have been provided for different markets and time spans. For instance, in the studies [15, 17, 23] Fama–French model is applied to the US market, in [24]—to seven developed markets (the US, Canadian, Australian, Japanese, German, British and French), in [25]—to Australian market. The author of [26] considers the Brazilian market and in [27] five Asian markets are considered. The authors of above-mentioned research confirm good performance of the Fama–French model. Although Blanco [23] and Lam [28] for the US market point out that the results depend on the portfolio construction. Also, the research provided for Australian in [29], British [30] and Turkish [31] markets shows that the three-factor model has limited abilities to explain rates of return, and it is hard to conclude that it performs better than CAPM.

There are also some studies on single-factor and three-factor capital asset pricing models dedicated for the Polish capital market, which were carried out for different stocks, periods and return intervals. Application of CAPM to evaluate returns from portfolios and companies listed on Warsaw Stock Exchange is described in [9, 32,33,34,35,36,37], to mention some of the research. One should notice that the Fama–French model is not used too often. Although, research presented in [38,39,40,41,42] confirm better performance of the Fama–French model in comparison to CAPM. However, [43] prove that the Fama–French model is adequate to describe bull market, but it is useless in explanation of returns in the bear market. Also [44] reports wrong valuation of stock returns.

3 Assumptions and Data

Capital asset pricing model seems to be simple and elegant; however, there are some issues which must be determined before the model is estimated and applied. For instance, it is necessary to define [37, p. 45]: interval range for rates of return, market portfolio, risk-free instrument, estimation period and methods of model estimation.

In our study, the investigation is provided using daily logarithmic rates of return evaluated for closing prices. This approach is similar to [45] although in the above-mentioned literature monthly returns are usually used. The market portfolio is represented by WIG—the main index of Warsaw Stock Exchange and WIBOR 1Y—Warsaw Interbank Offered Rate (reference rate set for deposits for the 12 months maturities) is used as a proxy of the risk-free rate of return. All data concerning quotations, book values and capitalization of stocks are available on www.stooq.pl.

The time span of analysis is from October 15, 2007, to March 31, 2017. This period is characterized by the changing situation on the Polish capital market. Therefore, seven sub-periods of increasing, decreasing or stable tendency of the Warsaw Stock Exchange index are distinguished (see Fig. 16.1; Table 16.1). Selection of sub-periods results from the instability of beta which should be estimated for shorter periods, especially when less developed capital markets are concerned what is pointed out in [37 pp. 50–51, 81, 137–176, 45]. Also [43] prove different performance of the Fama–French model in bull and bear market.

Fig. 16.1
figure 1

Source Own elaboration on the basis of [46, p. 49]

Warsaw stock exchange index WIG quotations in the years 2007–2017.

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Table 16.1 Selected periods of investigation
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In the study, two criteria of stocks selection are determined. Firstly, the company has been quoted incessantly in the whole horizon of investigation. Secondly, the company has been comprised in the portfolio of a one among three Warsaw Stock Exchange indexes: WIG20, mWIG40 or sWIG80. These indexes base on the value of portfolios which are composed from shares of 20 big and most liquid companies, 40 medium size companies and 80 smaller companies listed at WSE Main List, respectively.

It is worth mentioning that in the years 2006–2017 only seven companies were incessantly comprised in the index WI20, six in mWIG40, and none in sWIG80. However, 35 companies belonged to the portfolio of any index during the period of analysis what is connected with periodical revisions of the index portfolio.Footnote 1

Table 16.2 contains the list of companies selected for further investigation with information about the index portfolio which they belong to. These companies are used to construct four portfolios denoted as:

PB:

portfolio containing ten big companies belonging to the index WIG20,

PM:

portfolio containing ten medium size companies belonging to the index mWIG40,

PS:

portfolio containing ten small size companies belonging to the index sWIG80,

PT:

portfolio containing all (i.e. 30) selected companies.

The composition of these four portfolios is stable for the whole analyzed time span and all distinguished sub-periods.

Table 16.2 List of selected companies
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4 Analysis of Rates of Return and Risk from Portfolios

In our study, we consider rates of return evaluated for all portfolios constructed from selected shares of companies which are classified as big (PB), medium (PM) and small (PS), together with a portfolio comprised from all 30 distinguished equities (PT). Analysis consists of verification of following hypotheses concerning returns and risk.

$$H_{0} :E\left( {R_{\text{PB}} } \right) = 0;H_{0} :E\left( {R_{\text{PM}} } \right) = 0\;{\text{and}}\;H_{0} :E\left( {R_{\text{PS}} } \right) = 0,E\left( {R_{\text{PT}} } \right) = 0$$
(16.2)

using normally distributed test statistics:

$$u = \frac{{R_{k} }}{{S_{k} }}\sqrt {T_{k} }$$
(16.3)
$$\begin{aligned} & H_{0} :E(R_{\text{PB}} ) = E(R_{\text{PM}} );H_{0} :E(R_{\text{PM}} ) = E(R_{\text{PS}} ); \\ & H_{0} :E(R_{\text{PS}} ) = E(R_{\text{PB}} ),{\text{and for each portfolio}} \\ & H_{0} :E(R_{t1} ) = E(R_{t2} ), \\ \end{aligned}$$
(16.4)

applying Cochran-Cox test:

$$u = \frac{{R_{1} - R_{2} }}{{\sqrt {\frac{{S_{1}^{2} }}{{T_{1} }} + \frac{{S_{2}^{2} }}{{T_{2} }}} }}$$
(16.5)
$$\begin{aligned} & H_{0} :D^{2} \left( {R_{\text{PB}} } \right) = D^{2} \left( {R_{\text{PM}} } \right);H_{0} :D^{2} \left( {R_{\text{PM}} } \right) = D^{2} \left( {R_{\text{PS}} } \right); \\ & H_{0} :D^{2} \left( {R_{\text{PS}} } \right) = D^{2} \left( {R_{\text{PB}} } \right),{\text{and for each portfolio}} \\ & H_{0} :D^{2} \left( {R_{t1} } \right) = D^{2} \left( {R_{t2} } \right) \\ \end{aligned}$$
(16.6)

using Fisher statistics:

$$F = \frac{{S_{\hbox{max} }^{2} }}{{S_{\hbox{min} }^{2} }}$$
(16.7)

where E(R)—expected returns, D2(R)—variance of returns, RPB, RPM, RPS and RPT—returns from the portfolio: PB, PM, PS and PT, respectively, \(R_{k}\)—the average rate of return observed in the k-th sample (i.e., the k-th period or portfolio), \(S_{k}^{2}\)—the variance of rates of return from the k-th sample, Tk—count of observations in the k-th sample, \(S_{\hbox{max} }^{2} = \hbox{max} \left\{ {S_{1}^{2} , S_{2}^{2} } \right\}\), \(S_{\hbox{min} }^{2} = \hbox{min} \left\{ {S_{1}^{2} , S_{2}^{2} } \right\}\), t1, t2—two neighboring periods of analysis.

The results of rates of return analysis are presented in Tables 16.3, 16.4, 16.5, 16.6 and 16.7. Null hypotheses (16.2), (16.4) and (16.6) are rejected for the significance level 0.05 what is marked in Tables by bold letters. In Tables 16.3, 16.4 and 16.6, values of test statistics (16.3) and (16.5) are positive for bigger values of returns from the portfolio defined on the left-hand side of hypotheses (16.2) and (16.4). In Tables 16.5 and 16.7, cells containing statistics (16.7) are italicized when the bigger variance concerns portfolio from the right-hand side of hypotheses presented in (16.6).

Table 16.3 Values of test statistics (16.3) and standard deviations of constructed portfolios
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Table 16.4 Values of test statistics (16.5) comparing returns from constructed portfolios
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Table 16.5 Values of test statistics (16.7) comparing the risk of portfolios
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Table 16.6 Values of test statistics (16.5) comparing returns from portfolios in distinguished periods
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Table 16.7 Values of test statistics (16.7) comparing the risk of portfolios in distinguished periods
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Table 16.3 contains information about rates of returns and risk generated by each portfolio in the distinguished sub-periods. It is visible that the performance of all portfolios in all periods is poor although significantly negative returns are observed only in the first bear market period for all portfolios except the one comprising small companies. Risk, measured by standard deviation, seems to be of the same range in all portfolios and time spans.

Comparing returns generated by pairs of portfolios in different sub-periods, no significant differences are observed (Table 16.4). Whereas the risk of the portfolio PM (Table 16.5) is significantly bigger than the risk of portfolios

  • PB in all periods except the ones denoted as the first bear market and the last bull market periods and

  • PS when the market was stable and during two first bull market periods.

Taking into consideration distinguished periods of analysis, one may notice that returns do not differ significantly except the first bear market period (Table 16.6). Although risk (Table 16.7) is significantly bigger in the first bear market period and the first stable period in comparison to the two first bull market periods for all portfolios. Also, the risk of each portfolio in the second bull market period is bigger than the one in the second stable market period. Whereas risk in the first stable market period (denoted by stable1) is significantly higher than the one in the first bear market period for the portfolio of medium size companies and significantly smaller for the portfolio of small size companies. The last-mentioned portfolio additionally generates a bigger risk in the second bear market period than in the second stable market period.

To sum up this stage of investigation, we notice that the performance of constructed portfolios does not differ significantly between each other and in distinguished periods (except the first bear market period when returns were significantly smaller) than in the neighbouring time span. However, significant differences are observed for risk generated by considered portfolios (when risk is measured by standard deviation).

5 Fama–French Three-Factor Model Construction

To construct the Fama–French model, it is necessary to distinguish (additionally in comparison to CAPM) risk factors SMB and HML. The former (small minus big) is calculated as the difference in returns from a portfolio of stocks characterized by small market value (MV) and from a portfolio of stocks with big MV.Footnote 2 The latter (high minus low) represents the difference in returns on a portfolio of high book-to-market value (BV/MV) stocks and on a portfolio of low BV/MV stocks. Therefore, companies are classified according to their capitalization, i.e., big or small value of MV and high, medium and low value of BV/MV rate. Split into small and big companies is made using a median of capitalization. Classification into three groups let us distinguish 30% of companies with the highest and the lowest book-to-market value ratios, and 40% of the rest companies belong to the group of medium BV/MV rate values.

In other words, six portfolios combining both risk factors are defined for each year of analysis. Table 16.8 contains information about composition of each portfolio, denoted as: SL (small and low), SM (small and medium), SH (small and high), BL (big and low), BM (big and medium) and BH (big and high), in every year of analysis.

Table 16.8 Number of companies which are comprised in each portfolio
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Daily logarithmic rates of return are calculated for each portfolio assuming an equal share of each company in the portfolio. Then values of risk factors SMB and HML are evaluated, according to formulas:

$${\text{SMB}} = \frac{{R_{\text{SL}} + R_{\text{SM}} + R_{\text{SH}} }}{3} - \frac{{R_{\text{BL}} + R_{\text{BM}} + R_{\text{BH}} }}{3}$$
(16.8)
$${\text{HML}} = \frac{{R_{\text{SH}} + R_{\text{BH}} }}{2} - \frac{{R_{\text{SL}} + R_{\text{BL}} }}{2}$$
(16.9)

where RSL, RSM, RSH, RBL, RBM, RBH—rates of returns from the portfolios: SL, SM, SH, BL, BM, BH, respectively.

Arithmetic means of all three risk factors evaluated in each period of analysis are presented in Table 16.9. It is visible that HML average is always negative. It means that in average, returns from the companies, characterized by high values of BV/MV rate, are lower than returns generated by low book-to-market stocks. This conclusion is in line with Fama and French [15] findings. It is also visible that average risk premium in the bull market periods is positive, whereas excess market portfolio return in the bear market periods is negative.

Table 16.9 Table captions should be placed above the tables
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6 Estimation Results

CAPM and Fama–French three-factor models are estimated using the OLS method. Parameter estimates and determination coefficients of the single-factor and three-factor capital asset pricing models obtained for four constructed portfolios and all distinguished sub-periods are presented in Tables 16.10 and 16.11.

Table 16.10 Results of Fama–French three-factor model estimation
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Table 16.11 Results of CAPM estimation
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It is worth mentioning that the risk premium is statistically significant in all estimated models. It is also visible that portfolio PB, containing the biggest and the most liquid companies, is an aggressive one in the majority of periods. While the reaction of returns to risk premium for portfolios PM and PS is similar, and the parameter estimates are below one. All risk factors present in Fama–French models are statistically significant in ten models (among 21)—in four models estimated for the portfolio PS and in three models estimated for big companies PB and medium size companies PM. The three-factor models explain returns better than the single-factor models although the differences in determination coefficients are not very big, except the models estimated for the portfolio PS in the last four sub-periods. In both types of models, the best performance (measured by R2) is observed for portfolio PB in two first sub-periods.

7 Conclusion

The results of the presented research let us formulate the following conclusions concerning the application of the Fama–French three-factor model to the constructed portfolios.

  1. 1.

    Risk premium is the most important factor affecting rates of return, and it is statistically significant in all estimated capital asset pricing models. Whereas two other risk factors (which appear in the three-factor models only) are significant in about half of models.

  2. 2.

    Presence of risk factors concerning capitalization and book-to-market value rate of companies improve the performance of the capital asset pricing models. However, this improvement is visible mostly for portfolios constructed from small and medium size companies.

Answering the question asked in introduction, why CAPM has been still in use much more often than Fama–French model, one should realize that the latter is much more complicated than the former. Evaluation of two additional (in three-factor in comparison to single-factor model) risk factors requires much more data and work connecting with data processing. Whereas the improvement of performance seems not to be essential. Therefore, results obtained from the three-factor model might not cover “the costs” of its construction and estimation.