Market efficiency measures

Arguably the most commonly used asset pricing model used to measure the market ef-ficiency is the Capital Asset Pricing Model (CAPM), which was developed by Sharpe (1964), Lintner (1965) and Mossin (1966). CAPM is based on the principles of the port-folio selection theory by Harry Markowitz (1952), which is further discussed in this sub-section. Like the MPT, CAPM assumes that the investor is rational and risk-averse and will select only efficient mean-variance portfolios. This leads to investor selecting port-folios that maximize the return with the given variance and minimize the variance of portfolio return with the given expected return. (Markowitz, 1952; Fama & French, 2004)

Figure 3 below illustrates the CAPM portfolio opportunities. The horizontal axis displays the portfolio’s risk, which is measured by the standard deviation (σ) of the return. The vertical axis displays the portfolio’s expected return. The minimum variance frontier for risky assets is illustrated by the curve abc. It illustrates the relationship between the re-turn and the risk of the portfolio. Furthermore, the mean-variance efficient frontier with a riskless asset and the minimum variance frontier for risky assets meet at the point T, in which point the investor can gain moderate return with relatively low volatility: (Fama &

French, 2004)

Figure 3. Investment opportunities illustrated by CAPM (Fama & French, 2004).

Furthermore, the CAPM illustrates the relationship between the expected return of an individual asset or portfolio and systematic risk: as the systematic risk increases, inves-tors demand a higher return for the individual asset or portfolio. Thus, the expected rate of return can be derived as follows: (Fama & French, 2004)

(1) 𝐸(𝑅_𝑖) = 𝑅_𝑓+ [𝐸(𝑅_𝑀) − 𝑅_𝑓]𝛽_𝑖𝑀

where: 𝐸(𝑅_𝑖) = Expected return on asset or portfolio 𝑖 𝑅_𝑓= Risk − free rate of return

𝐸(𝑅_𝑀) = Expected market return 𝛽_𝑖𝑀= Market beta of asset or portfolio 𝑖

Also, Bodie et al. (2018) present a list of assumptions on which CAPM is based on:

1) Investors are rational and optimize the mean-variance relationship

2) Investors commonly plan for a single period 3) All the investors have homogeneous expectations

4) All the assets trade on public exchanges and are publicly held 5) Investors can borrow or lend at a risk-free rate and sell short assets 6) No transaction costs or taxes

A portfolio performance measure often related to CAPM is the Sharpe ratio. First intro-duced as reward-to-variability ratio by Sharpe (1966), it measures the ratio between ex-cess return over the risk-free rate of return and the standard deviation of these exex-cess returns. It enables the comparison between the returns of portfolios or assets by looking at the returns and considering the amount of risk these returns have generated. Thus, as the ratio increases, the reward increases relatively to risk. The ratio can be derived as follows: (Sharpe, 1966)

(2)

𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 =

𝜎_𝑖

where: 𝑅_𝑖 = Return on portfolio or asset 𝑖 𝑅_𝑓= Risk − free rate of return

𝜎_𝑖 = Standard deviation of the excess return

Even though CAPM is still widely used in finance due to its comprehensive applications, for example Reinganum (1981) and Lakonishok & Shapiro (1986) started to question the explanatory power of the CAPM on portfolio returns. Fama & French (1992, 1993) also note that the explanatory power of CAPM was sufficient in 1926-1968, but it had started to diminish over the years. Furthermore, they presented their famous 3-factor model, which had two additional risk factors: a size factor and a value factor. These additional factors were added to have a model with better explaining power on portfolio returns.

The size factor captures the difference between the returns of separate diversified port-folios of small market capitalization stocks and large market capitalization stocks. The

value factor captures the difference between the returns of separate diversified portfo-lios of high book-to-market value stocks low book-to-market value stocks. The Fama-French 3-factor model can be presented as: (Fama & Fama-French, 1992; 1993)

(3) 𝑅_𝑖𝑡− 𝑅_𝑓𝑡 = 𝛼_𝑖𝑡+ 𝛽_1,𝑖(𝑅_𝑀𝑡− 𝑅_𝑓𝑡) + 𝛽_2,𝑖𝑆𝑀𝐵_𝑡+ 𝛽_3,𝑖𝐻𝑀𝐿_𝑡+ 𝜀_𝑖𝑡

where: 𝑅_𝑖𝑡 = Return on asset or portfolio 𝑖 for time 𝑡 𝑅_𝑓𝑡 = Risk − free rate of return for time 𝑡 𝛼_𝑖𝑡 = Alpha for asset or portfolio 𝑖 for time 𝑡

𝛽_1,2,3 = Factor coefficients for asset or portfolio 𝑖 for time 𝑡 𝑅_𝑀𝑡 = Return on market portfolio for time 𝑡

𝑆𝑀𝐵_𝑡= Size factor 𝐻𝑀𝐿_𝑡= Value factor 𝜀_𝑖𝑡 = Error term

Inspired by the findings of Jegadeesh & Titman (1993) and the arising excitement of mo-mentum investing, Carhart (1997) expands the Fama-French 3-factor model to increase its explanatory power by adding a momentum factor. The momentum factor captures the difference between returns of momentum versus contrarian portfolio in the past year. Furthermore, the Carhart 4-factor model can be presented as: (Carhart, 1997)

(4) 𝑅_𝑖𝑡− 𝑅_𝑓𝑡 = 𝛼_𝑖𝑡+ 𝛽_1,𝑖(𝑅_𝑀𝑡− 𝑅_𝑓𝑡) + 𝛽_2,𝑖𝑆𝑀𝐵_𝑡+ 𝛽_3,𝑖𝐻𝑀𝐿_𝑡+ 𝛽_4,𝑖𝑈𝑀𝐷_𝑡+ 𝜀_𝑖𝑡

where: 𝑆𝑒𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 (3)

𝛽_4,𝑖 = Factor coefficient for asset or portfolio 𝑖 for time 𝑡 𝑈𝑀𝐷_𝑡 = 𝑀𝑜𝑚𝑒𝑛𝑡𝑢𝑚 𝑓𝑎𝑐𝑡𝑜𝑟

Fama and French (2015) also expand their own 3-factor model due to some critique (Tit-man, Wei & Xie, 2004; Novy-Marx, 2013) by adding two new risk factors: a profitability factor and an investment factor. The profitability factor captures the difference between the returns of separate diversified portfolios of robust profitability and weak profitability.

The investment factor captures the difference between the returns of separate diversi-fied portfolios of low investment firms and high investment firms. According to their study, up to 94% of the cross-section variance of the observed returns of the portfolios can be explained by the Fama-French 5-factor model. It can be presented as follows:

(Fama & French, 2015)

(5) 𝑅_𝑖𝑡− 𝑅_𝑓𝑡 = 𝛼_𝑖𝑡+ 𝛽_1,𝑖(𝑅_𝑀𝑡− 𝑅_𝑓𝑡) + 𝛽_2,𝑖𝑆𝑀𝐵_𝑡+ 𝛽_3,𝑖𝐻𝑀𝐿_𝑡+ 𝛽_4,𝑖𝑅𝑀𝑊_𝑡+ 𝛽_5,𝑖𝐶𝑀𝐴_𝑡+ 𝜀_𝑖𝑡

where: 𝑆𝑒𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 3)

𝛽_4,5 = Factor coefficients for asset or portfolio 𝑖 for time 𝑡 𝑅𝑀𝑊_𝑡 = Profitability factor

𝐶𝑀𝐴_𝑡 = Investment factor

Finally, Fama and French (2018) add a sixth, the momentum factor UMD into their 5-factor model to construct the Fama-French 6-5-factor model, which can explain the returns even better: (Fama & French, 2018)

(6) 𝑅_𝑖𝑡− 𝑅_𝑓𝑡 = 𝛼_𝑖𝑡+ 𝛽_1,𝑖(𝑅_𝑀𝑡− 𝑅_𝑓𝑡) + 𝛽_2,𝑖𝑆𝑀𝐵_𝑡+ 𝛽_3,𝑖𝐻𝑀𝐿_𝑡+ 𝛽_4,𝑖𝑅𝑀𝑊_𝑡+ 𝛽_5,𝑖𝐶𝑀𝐴_𝑡+ 𝛽_6,𝑖𝑈𝑀𝐷_𝑡+ 𝜀_𝑖𝑡

where: 𝑆𝑒𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 5)

𝛽₆ = Factor coefficient for asset or portfolio 𝑖 for time 𝑡 𝑈𝑀𝐷_𝑡 = Momentum factor