[Fama-French Three Factor Model]
Fama-French Three Factor Model은 CAPM (단일팩터모델)이 개별자산의 움직임을 완벽히 설명하지 못한다는 한계로부터 나온 Multi-Factor Model이다. Fama-French의 등장배경은 아래 내가 직접 작성한 보고서를 참고하면 된다.
작성한 보고서의 일부분을 가져와보면 아래와 같다.
As smart beta investment has been promoted in the asset management industry, using Factor models has been more important. in that regard, the analysis of the Fama-French Factor model that becomes the foundation of smart beta investment is important. Fama-French three- Factor model is a very popular model for modern investment. In Fama and French's research that was published in 1992[1] and 1993[2], they argued that the three factors – market factor (market excess return), size factor and BE/ME factor (BE/ME[3]) – have better power for explaining asset or portfolio return in the stock market than the beta of the CAPM model in cross-sectional variation analysis [4].
According to the efficient market hypothesis, factors such as profits, dividends, cash flows, etc are already reflected in stock prices, so the way to make higher returns is by taking higher risks [5]. Here, the risk is beta and it implies a correlation between the return on assets and the return from the market. This is the conclusion of CAPM, developed by William Sharpe [6].
As the study progressed further, it was revealed that the beta of CAPM would not properly explain the difference in the returns of stocks. From CAPM, the only factor that affects the asset return is the market factor. However, empirically, factors, other than the market factor, are also reflected in calculating the return on assets[7]. Fama and French in their paper “The Cross-Section of Expected Stock Returns” in 1992 explained that market capitalization (size) and price-to-book ratio (PBR) are much more important than the beta of CAPM in determining the return of stocks. Fama and French in 1993 research "Common risk factors in the returns on stocks and bonds” introduces five factors that explain the average return on stocks and bonds better than the beta of CAPM. Three factors are a market factor (risk premium), size factor (SMB), and BE/ME factor (HML) for stock return, and two factors are maturity risk factor and default risk factor for bond return. Fama-French three- Factor model is the model that focuses on these three factors (risk premium, SMB, HML) for the stock market.
Let's take a look at the size factor first. The size factor is about small and large capitalization stocks. In fact, defects in CAPM began to emerge even before Fama and French's studies. According to Banz who was a Chicago graduate student, in 1981, he analyzed the stock returns data with a conclusion that small-cap stocks’ returns were systematically higher than large-cap stocks’ returns [8]. He examined the empirical relationship between the return and the total market value of NYSE common stocks and found that smaller size firms had higher risk-adjusted returns on average than larger ones. It examined that market value (ME, which is calculated as stock price multiply the number of issued shares) increases the explanatory power for the cross-sectional average return provided by the market beta. Additionally, the average return on small-cap stocks was too high and the average return on large-cap stocks was too low.
[1] Fama, E. F., & French, K. R. (1992). The cross‐section of expected stock returns. the Journal of Finance, 47(2), 427-465.
[2] Fama, E. F., & French, K. R. (1993). Common risk factors in the returns on stocks and bonds. Journal of financial economics, 33(1), 3-56.
[3] BE/ME: The value of an entity's book value divided by market capitalization, defined by dividing the future profit predicted in the market by the information inherent in the book value, means that BE/ME > 1 is undervalued (inverse of PBR)
[4] Time-series regression analysis estimates factor exposure using asset returns and factor returns, whereas cross-sectional analysis estimates factor returns through asset returns and factor exposures.
[5] Fama, E. F. (1965). The behavior of stock-market prices. The Journal of Business, 38(1), 34-105.
[6] Sharpe, W. F. (1964). Capital asset prices: A theory of market equilibrium under conditions of risk. The journal of finance, 19(3), 425-442.
[7] 곽승주. (2021). 파이썬으로 배우는 포트폴리오. 길벗.
[8] Banz, R. W. (1981). The relationship between return and market value of common stocks. Journal of financial economics, 9(1), 3-18.
In Fama and French's research that was published in 1992[1] and 1993[2], they argued that the three factors – market factor (market excess return), size factor and BE/ME factor (BE/ME[3]) – have better power for explaining asset or portfolio return in the stock market than the beta of the CAPM model in cross-sectional variation analysis [4].
FF Three factor model의 경우 특정 자산이나 포토폴리오에 대하여 CAPM (market single factor) 보다 더 나은 explanatory power를 가진다.
However, empirically, factors, other than the market factor, are also reflected in calculating the return on assets[7].
경험적으로, market factor 외에 다양한 factor들을 통해 자산의 움직임을 설명할 시, 더 좋은 설명력을 가졌다.
Fama and French in their paper “The Cross-Section of Expected Stock Returns” in 1992 explained that market capitalization (size) and price-to-book ratio (PBR) are much more important than the beta of CAPM in determining the return of stocks. Fama and French in 1993 research "Common risk factors in the returns on stocks and bonds” introduces five factors that explain the average return on stocks and bonds better than the beta of CAPM.
Fama와 French는 1992년 three factor, 1993년 five factor 모델을 소개했으며, 이는 beta of CAPM보다 더 나은 설명력을 가짐을 보여주었다.
Three factors are a market factor (risk premium), size factor (SMB), and BE/ME factor (HML) for stock return, and two factors are maturity risk factor and default risk factor for bond return. Fama-French three- Factor model is the model that focuses on these three factors (risk premium, SMB, HML) for the stock market.
- Three factors: market factor, size factor, BE/ME factor
- Two factors (Extra): maturity risk factor, default risk factor for bond return
위에 그림을 보면, 종속변수인 개별자산의 초과기대수익률을, 3가지 factor 독립변수들로 설명하며, 계수들 (베타들)을 통해 각 factor 기대수익률 대비 개별자산 기대수익률의 움직임 민감도를 알아보게 된다. 첫번째 factor로는 CAPM이 market factor를 그대로 가져오며, 두번째 factor로 SMB (Small Minus Big; Size premium), 세번째 factor로는 HML (High Minus Low; Value long, Growth short로 구성된다.
각 Factor들은 어떻게 계산할까?
- Market Factor : 시장 벤치마크 수익률 - 무위험이자율
- SMB(Small Minus Big) : 1/3 * Small (value + natural + growth) - 1/3 * Big (value + natural + grwoth)
- HML (High Minus Low) : 1/2 * Value (small + big) - 1/2 Growth (small + big)
실무에서는?
기대수익률로 진행, 이에 잔차의 기대값은 = 0
아래 코드에서 실제 factor model을 어떤식으로 만드는지 볼 수 있다.
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