Why Aggregated Fund Portfolios Have Higher Factor-Model R-Squared
Summary
The discussion asks why an equally weighted portfolio of mutual funds can have a much higher four-factor model R-squared than the average R-squared from separate regressions for each fund. The explanation is that fund-specific return noise tends to cancel when many funds are combined, leaving the aggregate portfolio more closely related to common market factors.
A stylized stock-market example illustrates the point: individual stocks can have modest market-regression fit, while an aggregate portfolio may track the market much more closely. The comparison clarifies that the portfolio regression and the average of individual regression statistics answer different questions. The example is intentionally simplified, and the response provides intuition rather than a formal derivation or empirical analysis of the mutual fund data; aggregation does not guarantee a particular R-squared in every setting.
Key ideas
- Individual fund returns contain idiosyncratic noise that can obscure factor relationships.
- Combining funds can diversify away some fund-specific variation.
- A portfolio-level R-squared is not the same statistic as the average of fund-level R-squared values.
- The market-portfolio illustration gives intuition, not a general guarantee about the fit of every aggregate portfolio.
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Full text
# The R-squared of the four factor model. # The R-squared of the four factor model. Why does papers such as Fama and French (2010) and Barras et al. (2010) construct equal weighted portfolio of all funds when they analyse the aggregate performance of mutual funds? They both report an R-squared of 0.98 by weighting funds equally. When I do 2000 time series regression for all funds, and calculate the average R-squared obtained from all 2000 regressions, I get an average of 0.81. But, when I construct an equally weighted portfolio of all 2000 funds (similar to the papers mentioned above), I obtain an R-squared of 0.97. My main question is why does R-squared increase significantly when I construct an equally weighted portfolio of funds compared to taking the average R-squared of 2000 time series regressions. Appreciate any help you can provide. ## Answer by phdstudent (score 1) https://quant.stackexchange.com/a/35659 The reason is noise. There is much more noise in individual stock returns (or for that matter individual fund returns) than for an overall fund portfolio. In fact an overall fund portfolio should be quite close to the market. Think about it this way (very stylized example): - Imagine the market is the S&P500 and you are running CAPM regressions; - Run each of the S&P500 stock returns of the S&P500 on the market and you get what? Probably an $R^2$ around 0.2-0.5 for each stock? Take the average of those and you get a similar figure; - Now, value-weight those stocks and run them on the market factor. Which $R^2$ you get? You get 100%. It's just about the noise.
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