Using Fama–French Regressions to Compare Portfolio Returns
Article Quant Q&A · Author: deb
Summary
The document uses monthly regressions on a benchmark market return and style factors to interpret two portfolios' return patterns. It reads coefficient t-statistics as evidence of factor exposures: both funds show market sensitivity, while the second has statistically significant size, profitability, quality, and momentum terms. The reported fit statistics show that the chosen factors explain a larger share of the second fund's observed returns. The response also notes residual distribution diagnostics and the lack of notable autocorrelation in the reported Durbin–Watson statistics.
For inference, it flags heteroscedasticity as a possible problem and suggests weighted least squares, while proposing rolling regressions to track changing exposures over time. These are starting points rather than a complete comparison framework. The answer does not establish that weighting is appropriate without a variance model, nor does it specify rolling window selection or out-of-sample performance measures. Factor significance and in-sample fit alone do not determine which fund is preferable; the investor's objective and risk constraints matter.
Key ideas
- Factor regression coefficients and their t-statistics can help describe portfolio return exposures.
- A higher in-sample R-squared means the included factors explain more observed return variation, not necessarily better future performance.
- Heteroscedasticity can make conventional standard errors unreliable, so inference may need robust treatment or a justified variance model.
- Rolling regressions can reveal how estimated factor exposures change over time.
- Fund selection requires an objective beyond factor significance or regression fit.
Tags
Full text
# Analyzing portfolio returns using Fama-French Factors
# 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.
```
- What do I need to pay attention to when I am comparing them?
- Do one care about hetroscedasticity? If yes, what do one do about that?
- 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.
## Answer by vaibhav (score 2)
https://quant.stackexchange.com/a/78718
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
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.