Novak (2015) stated that the CAPM is based on the study of ‘Harry Markowitz (1952) which explains how selected portfolio using the ‘mean-variance rule (E- rule) and latter paper of Tobin (1958) introduced ‘separation theorem’ which integrated the idea of linear efficient set which implies two steps for the selection of described portfolio.

Sharpe (1964) and Linter (1965) published studies concerned prices of assets under risk conditions. The studies answer the question that how to the relationship between higher risk stock (variance) and expected return is evaluated and how to differentiate the part of the risk that is market values. In addition, Sharpe observed that some of the risk inherent in assets can be avoided through diversification and total risk is not relevant influence on its price (Hens and Rieger, 2015).

This essay is structured into three sections. In the first section, it discusses the CAPM and its methodology. The second section encloses two support studies (Fama and MacBeth and Black, Jensen and Scholes) for CAPM which discuss the validity of CAPM results. Finally, in the third section, the challenge study of CAPM (Fama and French) is discussed.

Sharpe (1964) and Lintner (1965) CAPM is widely used ‘asset pricing model’ and related to expected return in excess of the risk-free rate on asset ‘*I*’ in relation to market beta. For CAPM, the commonly used formula is . Therefore, it states that expected return on any stock ‘*i’* is equal to risk-free interest rate and risk premium ‘*Rf*’. The risk premium is equal to per unit risk premium, also called ‘market risk premium’, and multiplied by Beta (β) which measure how risky the stock is.

The Beta (β) is calculated because it is not observable from the market. Thus, the testing of the CAPM is usually done in two steps which are estimating the stock beta and testing the model (Levy, 2011).

A stock beta is calculated as the covariance between the excess return on the market portfolio and the stock excess return divided by the variance of excess returns for the selected portfolio. The formula for covariance is,, where ‘e’ denotes the excess return. The second approach to calculate the beta is using the time-series regression for the excess stock return on the excess return of portfolio for each stock separately and slope estimate the beta as, .

The second stage involves calculating the single cross-sectional regression of the average stock return on the beta and constant, . In the equation, ‘*i’* is the average time period and is the return of the stock (Sigman, 2005).

According to CAPM, stocks with higher beta enclosed more risk and i.e. should deliver above average return for the investors. In order CAPM is valid; the two key predictions which can be tested using second stage regression are (1) In case that returns are systematically higher for small capitalisation stocks than CAPM would predict through augmented second stage regression, . In the equation, is the ratio of book value in relation to equity market value and represent market capitalisation for stock (Hens and Rieger, 2010).

The early test of CAPM was presented by Douglas (1969) and the study regressed the returns on the large cross-sectional sample of common stock in term of variance and covariance’s and derived same results as Linter. The study analysed data from 1926 to 1960 seven separate five-year periods and determined that average realized return is positively associated to the variance of stock over given period of time but no associated with covariance as predicted by the model.

Millers and Scholes review the work of Lintner and Douglas and concluded that results are varied because of omission of variable bias, errors in variable and Heteroscedasticity. Thus, the two major limitations of earlier models are ‘Heteroscedasticity’ and ‘measurement errors’ because betas are used as explanatory variables (Unger, 2014).

Black, Jensen and Scholes (1972) conducted additional tests of Sharpe-Lintner (SL) for ‘asset pricing model’ and during testing original SL model, and concluded that risk-free return does not hold to empirical findings. The study concluded that ‘two-factor model’ is good to explain the risk-return relationship. The use of zero beta portfolios which is . In such case, the CAPM depends upon two-factors which are zero and non-zero beta portfolios and two-factor CAPM is represented as (in excessive return form)

The zero-model specifies the equilibrium return on the asset as a function of the market factor. This is defined by the return on market portfolio and factors are clear using zero-beta portfolio which represents the minimum portfolio variance and uncorrelated with market portfolio. Thus, the zero-beta portfolio is similar to the risk-free rate of return (Sharpe-Linter model).

Gibbons (1982) and Shanken (1985) tested the CAPM model and assumed that market model is true. The results show that ‘ith’ asset has the linear function for the market portfolio proxy and calculated as . (Dash, 2016)

The two-factor model requires the intercept term ‘E’ which should be same for all assets. In addition, Gibbons (1982) stated that ‘two-factor model CAPM requires the constraints on the intercept for market model for the assets with the similar time interval. When these conditions are violated the CAPM is rejected. Stambaugh (1982) estimated that market model using ‘Language multiplier test’ and support the findings of Black, Jensen and Scholes (1972). Gibbons (1982) used likelihood ratio test (LRT) and MacBeth (1975) used the ‘Hotelling’ to study the validity of CAPM (Pignataro, 2013).

Fama and MacBeth (1973) tested the CAPM through two-stage approach but using time series of cross-sections. Rather using single time-series analysis for each stock and performing cross-sectional, the estimations are conducted through rolling window.

In the study, to estimate the CAPM, the five years observations are used and these are used as explanatory variables for cross-sectional regressions for each month in subsequent four years. Afterwards, estimations are rolled forward for four years and the process continues until a sample is reached. Since there is one estimation for lambdas for each period, then ‘t-ratio’ is used for average over ‘t’ divided by its standard deviation (it is standard deviation over time divided by the square root for the estimates of time-series of the lambdas) (Bereau, 2016)

The formula is useful to calculate each lambdas over ‘t’ period as .In the equation, is the number of cross-sectional regression which used in the test second stage. The four different parameters represented by ‘j’ are the coefficient on beta and the intercept etc and standard deviation is calculated as . The test statistics are represented through which represent asymptotically standard normal and ‘t-distribution’ highlight the finite sample degree of freedom (Cuthbertson and Nitzsche, 2004).

Fama and French (1992) ‘methodology’ is based on the notion that market risk is not sufficient to explain the cross-section of stock return and seek to measure abnormal returns for the impact of the characteristics of portfolio or firm under consideration. It is widely believed that value stock, small stocks and momentum stock outperform the market.

Therefore, in order to evaluate the performance, it is important to consider the characteristics of firm or portfolio to avoid incorrect labelling. Fama-French is also based on time-series of cross-sections model and cross-section regressions are performed through (Fama and French, 1992)

In the equation,

Fama and French show that book-to-market and size are highly significant in relation to returns and explanatory variables in the regression are characteristics of firm.

Fama and French used the ‘factor based model’ in the context of time-series’ regression using separate run for each portfolio . In the equation, R represents the return on portfolio or stock ‘*i*’ for time at time t. HML, RMRF and HML are the factors mimicking the portfolio returns for market excessive return, value and size of firm respectively (Unger, 2014).

The good reason for durability of CAPM is that it explains the return common variability in the context of the single factor which generates the return for the individual asset using the linear functional relationship. The important point is that multifactor model and Sharpe-Lintner-black CAPM are not mutually exclusive.

On the other hand, Fama and French undermined the inability of CAPM to observe the true market index can be rejected, otherwise, it is impossible to reject the theoretical CAPM. The model predictions about cross-section have empirical evidence and usability of model depend upon purpose in hand.

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