BU.232.620 Empirical Project 1 (Linear Econometrics for Finance)
Introduction
In this empirical work we aim to explore the factors that drive the returns on stock portfolios, according to the seminal work of Fama and French ( 1993). Same as the Fama and French ( 1993), combing with the asset-pricing model, time-series regression is applied. Monthly returns on stocks and bonds are regressed on the returns to a market portfolio of stocks and mimicking portfolios for size, book-to-market equity (BE/ME), and term-structure risk factors in returns. Our findings reveal that market excess return plays a substantial role in explaining the variations in returns, although the explanatory power varies among different portfolios.
Data Description
In this work, data including monthly return series of the 25 stock portfolios, as well as the market excess return factor, are all from Ken French’s Data Library. Specifically, Monthly market excess return (Mkt-RF), size factor (SMB), and book-to-market factor (HML), Risk-Free Rate (RF) is defined as the Monthly returns of a one-month Treasury bill. Moreover, Returns on 25 portfolios formed on the intersections of five size groups (ME1 to ME5) and five book-to-market equity groups (BM1 to BM5).
Following tables present the Mean, Standard Deviation and T-Statistics of the portfolio intersections, which is applied to further discussion.
Table 1 ：Mean of Portfolio Intersections
Table 2: Standard Deviation of Portfolio Intersections
Table 3: T-Statistics of Portfolio Intersections
Figure 1: OLS Regression Results
In this case, because the lowest T-statistics (2.38) is larger than the 2.064, which is the critical value at 95% significance level and 24 degree of freedom, the null hypothesis: the means equal to zero will be rejected at 95% significnace level. And accoding to the OLS Regression Model, the skewness equals to 0.329 and kurtosis equals to 13.884, which shows these returns do not follow a normal distribution.
Models and Findings
To analyze the impact of market factors on the 25 portfolios, we employed a time-series regression model:
Rt−Rf= α+β(Mkt-RF)+ϵt
Where Rt is the return of the portfolio at time t, Rf is the risk-free rate at time t, Mkt-RF is the market excess return, α is the intercept, β is the coefficient of the portfolio to the market excess return.
Following tables present the result from regression about these parameters:
Table 4:β the coefficient
All 25 portfolios have positive Beta coefficients, indicating strong and positive sensitivity to market excess returns.
Table 5: T-Statitics on β
Because all the T-statistics on β is larger than the 2.064, which is the critical value at 95% significance level and 24 degree of freedom, the null hypothesis: The loadings of the 25 portfolios on the market portfolio significantly equal to zero should be rejected at 95% significnace level.
Table6: R² Goodness offit
The R² values suggest that the market factor explains a large portion of their return variation. Most of the intersections ’ R² value is larger than 0.7, and all intersections ’ R² is larger than 0.5, showing a great explanation power of our regression model. R² indicates how much of the time series variation in the return of each portfolio is explained by the market excess return (MKT - RF), for example, Portfolio ME1 BM1 has an R² of 0.511, meaning that 51.1% of its return variation is accounted for by the market excess return.
Conclusion
According to the empirical study especially from the time-series regression, it is comfirmed that the market excess return explains a significant proportion of the variation in returns for many portfolios. However, evethough all portfolios have over 0.5 goodness of fits, some of them with ralatively R² values indicate the potential for additional explanatory variables. After reading the Fama and French ( 1993), factors such as size (SMB) and book-to-market (HML) are expected to incorprate into the regression model to better capture the variations in returns.
In summary, this project indicated the importance of market factors in explaining portfolio returns while also emphasizing the limitations of using the single-factor model, as only market excess return (MKT - RF) is considered . Further enhancements could improve the explanatory power, by adding more varaibles in the model to better explain the returns ’ variations.