Question

How do you compare the Capital asset pricing model to the 3 factors fama french model and 5 factor fama french model?

Answer 1

2. Literature review

What kinds of factors determine the price of an asset? Since Markowitz formulated a model of asset pricing [1], the debate on this question continues. The main determinants of asset prices and risk factors that affect the demand for assets and asset prices have been an important issue in finance theory and practice. One can find enormous number of studies on this issue. Earlier studies in this area are by Markowitz, Sharpe, Ross, Fama and French [1, 2, 3, 4].

Since the literature on asset pricing model (APM) is very well known and can be reached easily in finance textbooks, I do not go into a detailed explanation of evolution of APM. However, I would like to briefly state that all the asset pricing models developed so far have included risk as the most important determinant. For example, [1] defines the expected return and variance of returns on a portfolio as the basic criteria for portfolio selection.

Markowitz’s model requires large data inputs. Because of input drawback, new models have been developed to simplify the inputs to portfolio analysis. William Sharpe’s market model [2] is as follows:

Rit=αi+βi∗RMt+eit,Rit=αi+βi∗RMt+eit, E1

where Rit is the return of stock i in period t, αi is the unique expected return of security i, βi is the sensitivity of stock i to market movements, Rmt is the return on the market in period t and eit is the unique risky return of security i in period t and has a mean of zero and finite variance σ ²ei , uncorrelated with the market return, pairwise and serially uncorrelated. This equation explains the return on asset i by the return on a stock market index. β in Eq. (1) is a risk measure arising from the relationship between the return on a stock and the market’s return.

Later on, the equilibrium models have been developed. The difference between the market model and the equilibrium model was that asset returns are related to excess market return rather than market return. The first and basic form of the general equilibrium model, which was developed by Sharpe, Lintner and Mossin [2, 4, 5] called capital asset pricing model (CAPM), is given in Eq. (2):

Rit=Rf+β∗i (RMt−Rf)+eitRit=Rf+βi* (RMt−Rf)+eit E2

where Rit is the return of stock i in period t, Rf is risk free rate, βi is the sensitivity of stock i to excess return on a market portfolio, Rmt is the return on the market in period t and eit is the unique risky return of security i in period t and has a mean of zero and variance σ ²ei .

Black et al. [6] derived a new model of the CAPM by relaxing the assumption of risk-free lending and borrowing. Basu [7] considers a different time series model, which is written in terms of returns in excess of the risk-free rate Rf and shows that returns are positively and linearly related to β, as follows:

Rit−Rft=αi+βi(RMt−Rft)+eitRit−Rft=αi+βiRMt−Rft+eit E3

While the CAPM is still the most widely accepted description for security pricing, empirical studies found contradicting evidence (see [7, 8, 9, 10, 11, 12, 13, 14]). Therefore, researchers concentrated on finding better models for the behaviour of stock returns and added more explanatory variables into CAPM.

In the early 1990s, one of the most influential researches was by Fama and French [15, 16]. Fama and French [15] reject the market beta associated with the CAPM and instead find that stock size and book-to-market (B/M) ratio better capture the cross-sectional variation in average stock returns. One year later the same researchers proposed that a 3F-FF asset pricing model augmenting the CAPM with size and book-to-market proxies for risk might be a superior description of average returns [16]. After these two influential studies, along with earlier evidence against the CAPM drove the finance community into investigating the reasons behind the anomalies found in [10, 11, 12, 13, 14].

Recent studies have found additional factors that seem to exhibit a strong relationship with average returns. Novy-Marx [17] finds that firms with high profitability generate significantly higher returns than unprofitable firms. Aharoni et al. [18] find that a statistically significant relation exists between an investment proxy and average returns. In the wake of these findings, Fama and French [19] expanded the 3F-FF model with profitability and investment. They reveal that the 5F-FF model performs better than the 3F-FF model in explaining average returns for their sample. The same model was tested using international data [20, 21], and they have found similar results.

This study adds to research conducted on CAPM, three-factor model and the new 5F-FF model by testing all these models on the Turkish stock market.

3. Data selection and issues

3.1. The sample

The data sample used in the analysis consists of monthly price, total return and accounting data downloaded from ‘Finnet Data Yayıncılık’. The data set contains nonfinancial 263 firms listed in BIST (Borsa Istanbul or Istanbul stock exchange) for the period between 31.12.1999 and 30.05.2017. The collected accounting data includes total assets, total liabilities, outstanding shares, owner’s equity and operating income, where operating income is defined as ‘net sales minus operating expenses’ and operating expenses is defined as the ‘sum of all expenses related to operations’. Data was collected for all available active and dead stocks in Istanbul stock exchange totalling 204 observations. The data was quoted in the Turkish Lira (TRY hereafter).

The downloaded sample included a large amount of stocks, which were already dead at the beginning of the research period, as well as some missing data types and data errors, which ought to be removed.

3.2. Variable definitions

This subsection defines the variables needed in the factor creation process. Market capitalisation or market cap was used as a measure of size for each stock and was calculated by multiplying the price (P) at the 31st of December each year with outstanding shares at the 31st of December for the same year. The price data was obtained from FDY. Book equity was calculated as yearly total assets minus total liabilities from FDY. Book-to-market ratio (B/M) was calculated from the previous two variables by dividing book equity by market cap. Operating profitability (OP) was calculated by dividing operating income by book equity following [22]. Finally, investment (Inv) was calculated as in Eq. (4):

Total Assetst−1−Total Assetst−2Total Assetst−2Total Assetst−1−Total Assetst−2Total Assetst−2 E4

TRY 3-month Libor rate is used as a proxy of risk-free rate (Rf ), while market return (Rm ) is approximated by natural log difference of BIST-100 Index of the Istanbul stock exchange. It consists of 100 stocks, which are selected among the stocks of companies listed on the national market (excluding list C companies). Monthly returns for stocks are all calculated as natural log difference of monthly stock data.

3.3. The return period

Fama and French use 6-month gap between the ends of the fiscal year and the portfolio formation date can be considered as convenient and conservative. Since all the accounting data in BIST is available by the end of May of each year, I use 5-month gap. Hence, to ensure that all accounting variables are known by investors, I assume that all accounting information is made public by the end of May, and I use monthly returns from the beginning of June to the end of the following year in May. And, each year at the end of May, I sort the portfolios.

Fama and French [15, 16] used value-weighted returns in their study; however, they also stressed that equal-weighted returns do a better job than value-weighted returns in explaining returns by 3F-FF model. Lakonishok et al. [23] also suggest to use equal-weighted portfolios to investigate the relationship between risk factors and stock returns. Hence, the equal-weighted monthly returns on each portfolio were used in this study.

3.4. Filtering data

At the end of each year, I eliminated firms that have the following specifics: (1) negative book-to-market values were removed, and (2) the companies with yearly increase in their investment, as defined in Eq. (4), which is either less than −50% or higher than 100% in a certain year were eliminated. This would imply that the company in question lost half of its assets, or more than doubled its assets in the given year, which seems very unlikely during normal recurring circumstances.

4. Methodology

4.1. Model definitions

In this study, I test three models, namely, CAPM, 3F-FF model and 5F-FF model for BIST. The model definitions are given below:

Rit−Rft=ai+(RMt−Rft)+eit−CAPMRit−Rft=ai+(RMt−Rft)+eit−CAPM E5

Rit−Rft=ai+(RMt−Rft)+siSMBt+hiHMLt+eit(3F,‐,FF model)Rit−Rft=ai+RMt−Rft+siSMBt+hiHMLt+eit3F‐FF model E6

Rit−Rft=ai+(RMt−Rft)+siSMBt+hiHMLt+riRMWt+ciCMAt+eit(5F,‐,FF model)Rit−Rft=ai+RMt−Rft+siSMBt+hiHMLt+riRMWt+ciCMAt+eit5F‐FF modelE7

where R _it is the return of portfolio i at time t (portfolio creation procedure is given in the following topic); R _ft is the risk-free rate approximated by 3-month TRY Libor rate at time t; R _it −R _ft is the excess return of portfolio i at time t; R _Mt is the monthly market return approximated by natural log difference of BIST-100 Index at time t; SMBt , HMLt , RMWt and CMAt are defined in details in the following topic; ai is the intercept; βi is the coefficient of R _Mt −R _ft for portfolio i; si is the coefficient of SMBt for portfolio i; hi is the coefficient of HMLtfor portfolio i; ri is the coefficient of RMWt for portfolio i; and ci is the coefficient of RMWt for portfolio i.

4.2. Construction of Fama-French factors

The next step in the analysis is to create sorted portfolios from which the Fama and French factor return series could be calculated. The factors used in the analysis were constructed in a manner similar to the process described in [19], relying solely on 2 × 3 sorts for creating the factors. Other sorting choices might have been used; however, [19] finds no differences in model performance when testing differing sorting methods.

My first data point is at the 31st of December 1999, and the investment variable is calculated as in Eq. (4); the first available year of accounting data used in the sorting process is at the end of fiscal year 2000. The portfolios were sorted at the end of May each year, and therefore the first available return observation in the final analysis is the return of June 2000, sorted according to accounting data at the end of fiscal year 1999. Thus, the time period for the actual analysis is June 2000 to May 2017 or 204 months of return data.

The sorting process is as follows:

(1) First of all, stocks are sorted according to their market cap to define small-sized and big-sized stocks. Fifty percent of the market cap was used as the breakpoint for size. (2) For all other factors, yearly sample 30th and 70th percentiles were used as breakpoints in the sorting method. (3) After the determination of the breakpoints, the stocks in the sample were independently distributed for every year into six size-B/M (where B/M is book-to-market ratio) portfolios, six size-OP (where OP is operational profits divided by book equity showing profitability) portfolios and six size-Inv (where Inv is yearly increase in total assets) portfolios created from the intersections of the yearly breakpoints. (4) All portfolios are value-weighted according to their market cap. (5) Monthly returns were calculated for each of the 18 portfolios. (6) After calculating the sorted portfolio returns, the actual factor returns were calculated.

There are two size groups and three book-to-market (B/M), three operating profitability (OP) and three investment (Inv) groups. The resulting groups are labelled with two letters. The first letter describes the size group, small (S) or big (B). The second letter describes the B/M group, high (H), neutral (N) or low (L); the OP group, robust (R), neutral (N) or weak (W); or the Inv group, conservative (C), neutral (N) or aggressive (A). Stocks in each component are value-weighted to calculate the component’s monthly returns

. Regression portfolio statistics

In this section, I give descriptive statistics for the regression portfolios and explanatory factors used in the regressions.

The main aim of this research is to see if well-targeted regression models can explain average monthly excess returns on portfolios with large differences in constituent size, B/M, profitability and investment. In Table 3 , the standard deviations of monthly excess return of portfolios seem to be very high. One of the explanations is that portfolio groups include small numbers of stocks. The second explanation could be the economic crisis experienced in 2001 in Turkey. This crisis created very high volatility in the financial markets, and the daily change in stock market index (viz. BIST-100) reached to 30%. When I exclude data covering the years from 2000 to 2003, standard deviations decrease by 35% on average. On the other hand, it should also be noted that global crisis in 2008 and Greece’s haircut in 2010–2011 created very high volatility in many stock markets. In normal circumstances, I would expect the standard deviations to be half of the

Factor spanning regressions

Factor spanning regressions are a means to test if an explanatory factor can be explained by a combination of other explanatory factors. Spanning tests are performed by regressing returns of one factor against the returns of all other factors and analysing the intercepts from that regression.

Table 4 shows regressions for the 5F-FF model’s explanatory variables, where four factors explain returns on the fifth. In the RM-Rf regressions, the intercept is not statistically significant (t = −0.55). Regressions to explain HML, RMW and CMA factors are strongly positive. However, regressions to explain SMB show insignificant intercept, with intercept of −0.27% (t = −1.34). These results suggest that removing either the RM-Rf or SMB factor would not hurt the mean-variance-efficient tangency portfolio produced by combining the remaining four factors.

Hypothesis tests of joint significance of the regressions’ alphas and regressions

Having presented the methodology and statistical results, in this part, I present an answer to important question if the estimated models can completely capture expected returns. To obtain a more absolute answer to this question GRS f-tests were conducted on results obtained from the first hypothesis’ portfolio regressions. The GRS statistic is used to test if the alpha values from regressions are jointly indistinguishable from zero.

If a model completely captures expected returns, the intercept should be indistinguishable from zero. Hence, the first hypotheses are:

H₀: αˆα̂ 1 = αˆα̂ 2 = … = αˆα̂ 16 = 0 (the regression alpha is not significantly different from zero).

H₁: αˆα̂ 1, αˆα̂ 2, …, αˆα̂ 16 ≠ 0 (the regression alpha is significantly different from zero).

The second hypotheses for all equations are:

H₀: αˆα̂ 1 = αˆα̂ 2 = … = αˆα̂ 48 = 0 (the regression alphas are jointly indistinguishable from zero).

H₁: αˆα̂ 1, αˆα̂ 2,…, αˆα̂ 48 ≠ 0 (the regression alphas are jointly distinguishable from zero).

Finally, average individual regression alphas and joint GRS regression f-values were used together in order to compare the performance between the tested models.

7.1. The GRS regression equation

The GRS test was developed by Gibbons et al. [24] and serves as a test of mean-variance efficiency between a left-hand-side collection of assets or portfolios and a right-hand-side model or portfolio. The following regression defines the GRS test:

fGRS=TN×T−N−LT−L−1×αˆ′×Σˆ−1×αˆ1+μ¯′×Ωˆ−1×μ¯∼F(N,T−N−L)fGRS=TN×T−N−LT−L−1×α̂′×Σ̂−1×α̂1+μ¯′×Ω̂−1×μ¯∼FNT−N−L E15

where αˆα̂ is a N×1 vector of estimated intercepts, ΣˆΣ̂ an unbiased estimate of the residual covariance matrix, μ¯¯μ¯ a L×1 vector of the factor portfolios’ sample means and ΩˆΩ̂ an unbiased estimate of the factor portfolios’ covariance matrix.

The GRS test is used in this study to determine whether the alpha values from individual model regressions are jointly non-significant and hence to find out if a model completely captures the sample return variation. As intercepts from individual regressions approach zero, the GRS statistic will also approach zero. However, since the GRS statistic derives its results from comparing the optimal LHS and RHS portfolios, the resulting statistic is not strictly comparable between models.

7.2. Model performance

A set of several summary metrics were deployed in order to compare the performance of the asset pricing models. GRS statistics and average alpha values were used as the main two statistics in order to determine how good the different asset pricing models performed in explaining portfolio returns. In addition to these statistics, average absolute alpha spread was added for a more complete picture of the alpha results. Furthermore, different models’ explanatory power was measured using R² values.

If a capital asset pricing model (CAPM, three-factor model or five-factor model) completely captures expected returns, the intercept (alphas) is indistinguishable from zero in a regression of an asset’s excess returns on the model’s factor returns.

Table 5 shows the GRS statistics of [24] that tests this hypothesis for combinations of LHS portfolios and factors. The GRS test easily rejects all models considered for all LHS portfolios and RHS factors. The probability, or p-value, of getting a GRS statistic larger than the one observed if the true intercepts are all zero is shown in column ‘pGRS’. One can see from Table 5 that except CAPM in Panel A and Panel C, sets of left-hand-side returns, the p-values round to zero to at least three decimals. Only five-factor model in Panel C has a p-value of 0.30, and it is still significant at the 5% level.