# Analyzing portfolio returns using Fama-French Factors

Here is my problem - I have monthly returns from few portfolios. I also have monthly return from benchmark portfolio. I downloaded F-F 5 factor daily data. Also downloaded Momentum data. Converted them to monthly data. Replaced Mkt-RF by Benchmark-RF. Also downloaded Low Beta(BaB) from AQR.

I ran single regression for each portfolio in Python. Here are the summary stats for two portfolios -

==============================================================================
Dep. Variable:                 Fund 1   R-squared:                       0.688
Model:                            OLS   Adj. R-squared:                  0.678
Method:                 Least Squares   F-statistic:                     69.63
Date:                Mon, 18 Mar 2024   Prob (F-statistic):           1.99e-52
Time:                        17:56:29   Log-Likelihood:                 483.88
No. Observations:                 229   AIC:                            -951.8
Df Residuals:                     221   BIC:                            -924.3
Df Model:                           7
Covariance Type:            nonrobust
=================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0058      0.002      2.642      0.009       0.001       0.010
Mkt-RF            0.9959      0.054     18.550      0.000       0.890       1.102
Size             -0.0349      0.145     -0.242      0.809      -0.320       0.250
Value             0.0412      0.147      0.280      0.779      -0.248       0.331
Profitability     0.1389      0.195      0.711      0.478      -0.246       0.524
Quality          -0.1735      0.199     -0.874      0.383      -0.565       0.218
Momentum          0.2036      0.077      2.630      0.009       0.051       0.356
Low Beta         -0.0132      0.106     -0.125      0.901      -0.222       0.195
==============================================================================
Omnibus:                       10.453   Durbin-Watson:                   1.979
Prob(Omnibus):                  0.005   Jarque-Bera (JB):               15.139
Skew:                          -0.297   Prob(JB):                     0.000516
Kurtosis:                       4.111   Cond. No.                         121.
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.
-------------------------------------------------------------------------------------------
-------------------------------------------------------------------------------------------
OLS Regression Results
==============================================================================
Dep. Variable:                 Fund 2   R-squared:                       0.749
Model:                            OLS   Adj. R-squared:                  0.741
Method:                 Least Squares   F-statistic:                     94.19
Date:                Mon, 18 Mar 2024   Prob (F-statistic):           9.22e-63
Time:                        17:56:29   Log-Likelihood:                 501.01
No. Observations:                 229   AIC:                            -986.0
Df Residuals:                     221   BIC:                            -958.6
Df Model:                           7
Covariance Type:            nonrobust
=================================================================================
coef    std err          t      P>|t|      [0.025      0.975]
---------------------------------------------------------------------------------
const             0.0078      0.002      3.792      0.000       0.004       0.012
Mkt-RF            0.7943      0.050     15.945      0.000       0.696       0.893
Size              0.5272      0.134      3.932      0.000       0.263       0.791
Value             0.0498      0.136      0.365      0.715      -0.219       0.319
Profitability    -0.4035      0.181     -2.226      0.027      -0.761      -0.046
Quality          -0.3824      0.184     -2.075      0.039      -0.746      -0.019
Momentum         -0.2807      0.072     -3.908      0.000      -0.422      -0.139
Low Beta          0.0761      0.098      0.775      0.439      -0.117       0.269
==============================================================================
Omnibus:                        1.413   Durbin-Watson:                   1.990
Prob(Omnibus):                  0.493   Jarque-Bera (JB):                1.156
Skew:                          -0.164   Prob(JB):                        0.561
Kurtosis:                       3.114   Cond. No.                         121.
==============================================================================

Notes:
[1] Standard Errors assume that the covariance matrix of the errors is correctly specified.


1. What do I need to pay attention to when I am comparing them?
2. Do one care about hetroscedasticity? If yes, what do one do about that?
3. I ran single regression? Would a rolling regression be more appropriate? How to compare performance in such situation? How to choose between the funds?

This question may have been answered before. If yes, please point me to the relevant discussion.

It is very important to understand your end goal. FF regressions are used to understand return of portfolio which can be attributed to FF style factors. In this analysis I am assuming that you are trying to understand returns of the funds.

I would look at the t-statistics of various factors. Many insights can be extracted, but I will point out few(please note that I am using t-statistics of co-efficients to come to this conclusion): Fund 1's returns seem to be primarily influenced by the market risk premium, while the additional factors (size, value, profitability, quality, low beta) do not appear to have a significant impact. Fund 2's returns are influenced by the market risk premium, size, profitability, quality, and momentum factors. The size factor seems to have a particularly significant impact. Both funds exhibit some evidence of departures from normality in the residuals, but there is no significant autocorrelation. Fund 2 generally has a higher R-squared value, indicating that the factors included in the model explain a larger proportion of its returns compared to Fund 1.

Fund 2 demonstrates significant exposures(Calculate the exposure by summing the absolute values of the significant coefficients) to profitability, quality, and momentum factors, indicating a more active approach compared to Fund 1.

For your second question:

Yes, heteroscedasticity can be a concern in regression analysis, including Fama-French regressions. Heteroscedasticity occurs when the variance of the residuals (errors) is not constant across all levels of the independent variables. This violates one of the assumptions of classical linear regression, which assumes homoscedasticity (constant variance of residuals).

In the context of Fama-French regressions, heteroscedasticity can lead to inefficient parameter estimates and biased standard errors, which can affect the accuracy of statistical inferences and hypothesis testing.

To address heteroscedasticity, I would recommend changing OLS to WLS

1. When dealing with time series data, in my opinion, a rolling regression can be more appropriate than a single regression. A rolling regression allows you to assess how the relationship between variables evolves over time by estimating the regression coefficients over rolling windows of data.