Factor Regression

What it answers#

• The exposure chart shows each estimated beta and its 95% confidence interval. The table gives the exact estimates and test statistics.
• Alpha and residuals show the return that the selected factors do not explain.
• R-squared and residuals show how much of the observed variation the model explains.

Important modeling choices#

• The regression uses excess returns: portfolio returns minus the Fama-French risk-free series.
• Each factor model has a different history. Later-starting models can shorten the usable sample window, and the tool shows that adjustment.
• Monthly mode compounds daily observations and excludes incomplete first and last months. Annual mode compounds daily observations into calendar-year returns and drops incomplete edge years.
• The portfolio configuration reuses the same typed portfolio model as the backtester, but the tool excludes cash flows because they distort return attribution.

Inputs and workflow#

• The tool reads the portfolio rows of the analysis, the same rows as Portfolio Backtest. One Run gives one regression for each row, side by side in one table.
• This analysis does not support cash flows. A row with cash flows stays in the table with the reason. For a DCA or withdrawal plan, first use Portfolio Backtest. Then, analyze the static portfolio configuration.
• Custom factors live under More in the settings of this tool. Each custom factor is a long-short spread built from public ticker returns. Specify tickers and weights, which must sum to zero. The tool adds the custom factor return series as an additional regressor alongside the selected factor model.
• After the run, select a row to read its coefficients and diagnostics. Save and Share keep the selected row.

How to read the output#

Start with estimated betas and their 95% confidence intervals. Then read R-squared for the return variation explained by the model and annualized alpha for the modeled return it does not explain.

The coefficient table contains exact p-values and confidence bounds from the t-distribution. Use the F-statistic to evaluate joint significance. Use the fitted-versus-actual chart and residual series to identify persistent or regime-specific errors and errors that are mostly noise. The residual summary gives the mean, standard deviation, skewness, and excess kurtosis.

Negative SMB indicates large-cap exposure. Negative HML indicates growth exposure. Residuals clustered in one period may point to a missing driver.

Diagnostics reference#

• P-values use the exact t-distribution with n - k - 1 degrees of freedom. Here n is the observation count and k is the factor count. Significance markers are *** p < 0.001, ** p < 0.01, and * p < 0.05.
• Confidence intervals are 95% two-tailed, computed from the same t-distribution critical values.
• F-statistic tests joint significance of all factor loadings (excluding the intercept). A large F with a small p-value means the factors collectively explain meaningful variation.
• Intercept diagnostics report standard error, t-stat, p-value, and confidence interval for the alpha estimate itself.
• Residual summary reports mean (should be near zero for OLS), standard deviation, min, max, skewness, and excess kurtosis. Heavy tails or skew may indicate the factor model is missing an important driver.

Which model to use#

• CAPM is the simplest baseline when you only want market beta and alpha.
• Fama-French 3 factor is a strong default for broad equity portfolios with size and value tilts.
• Carhart 4 factor is useful when momentum may explain part of the return pattern.
• Fama-French 5 factor adds profitability and investment exposures to the 3-factor model.
• q-factor (Hou-Xue-Zhang) is an investment-based model with market, size, investment-to-assets, and return-on-equity factors. Use monthly frequency, because daily q-factor data is not publicly available.
• AQR multi-factor covers market, size, value, momentum, betting-against-beta, and quality.
• Fixed income factors provides term spread, default spread, and corporate bond excess return factors for fixed income attribution. Use this when the portfolio has significant bond exposure.

Factor coverage and limitations#

• CAPM and FF3 use data from July 1926 onward.
• Carhart 4 factor momentum data is available from November 1926 onward.
• Fama-French 5 factor daily data begins July 1963.
• q-factor monthly data is available from January 1967 onward.
• AQR multi-factor coverage varies by factor. Many US equity factor series start in the early 1970s. Check each factor's data window.
• Fixed income factors start in the mid-1960s for most series.
• When the requested date range extends beyond factor data coverage, the tool trims the regression window and surfaces a warning.