R-squared in mutual funds
R-squared (\(R^2\)), or the coefficient of determination, measures the proportion of a mutual fund’s return variance that is explained by variance in its benchmark index. It ranges from 0 to 1 (or 0 to 100 per cent). An R-squared of 1.0 (100 per cent) means all return variation is explained by the benchmark; an R-squared of 0 means the fund’s returns have no statistical relationship with the benchmark.
R-squared is an essential companion to beta because beta is only a reliable predictor of fund behaviour when R-squared is high. A fund with a calculated beta of 1.20 but an R-squared of 0.40 cannot be relied upon to behave in line with that beta, much of its movement is independent of the benchmark.
Formula
R-squared is the square of the Pearson correlation coefficient \(r\) between fund and benchmark returns:
\[ R^2 = r^2 = \left[\frac{\text{Cov}(R_p, R_m)}{\sigma_p \times \sigma_m}\right]^2 \]
Equivalently, from the CAPM regression \(R_p = \alpha + \beta R_m + \epsilon\):
\[ R^2 = 1 - \frac{\text{SS}{\text{residual}}}{\text{SS}{\text{total}}} \]
Where \(\text{SS}{\text{residual}}\) is the unexplained variation and \(\text{SS}{\text{total}}\) is the total variation in fund returns.
A high R-squared means little residual variation (the regression line fits the fund returns well). A low R-squared means large residuals (the benchmark explains little about the fund’s day-to-day return fluctuations).
Interpretation
| R-squared | Interpretation |
|---|---|
| 85–100 | High correlation; fund closely follows the benchmark |
| 70–85 | Moderate-high; some idiosyncratic exposure |
| 50–70 | Moderate; significant non-benchmark risk |
| < 50 | Low; fund behaviour largely independent of benchmark |
CAPM reliability threshold: Most practitioners treat R² below 0.70 as a threshold below which CAPM-derived statistics (beta, alpha) are unreliable. A beta computed with R² = 0.40 tells you little about how the fund will behave relative to the index.
Typical R-squared values in Indian mutual funds
| Category | Typical R-squared |
|---|---|
| Nifty 50 index fund | 0.98–1.00 |
| Large-cap equity (actively managed) | 0.85–0.97 |
| Flexi-cap equity | 0.80–0.95 |
| Mid-cap equity | 0.75–0.92 |
| Small-cap equity | 0.65–0.88 |
| Sector fund (thematic) | 0.40–0.80 (depends on sector/benchmark match) |
| Aggressive hybrid | 0.70–0.90 |
| Balanced advantage | 0.50–0.75 |
| Arbitrage fund | Near 0 (when benchmarked to equity; ~0.90 when benchmarked to liquid fund index) |
Index funds have R-squared near 1.0 by construction. Sector funds can have very low R-squared against a broad market index because their returns are dominated by sector-specific factors.
R-squared and the closet indexing problem
R-squared is used to identify closet indexers, actively managed funds whose portfolio is so similar to the benchmark that their R-squared approaches the level of a true index fund (above 0.95–0.98), yet they charge active management fees. A closet indexer has:
- High R-squared: ~0.97
- Low tracking error: <2%
- Low or near-zero alpha
- Information ratio near zero
These funds deliver index-like returns at active fund costs. The direct vs regular plan TER differential makes closet indexing in the regular plan particularly value-destructive.
R-squared for benchmark selection
R-squared is also a diagnostic for whether the right benchmark is being used. A fund with very low R-squared against its stated benchmark may be drifting from its mandate. For example:
- A large-cap equity fund showing R² = 0.65 against Nifty 100 TRI should probably be investigated, it may have significant mid-cap or small-cap exposure.
- An aggressive hybrid fund showing R² = 0.90 against a pure equity index may actually be holding very little fixed income despite its hybrid mandate.
SEBI’s category-and-benchmark rationalisation (October 2017, circular SEBI/HO/IMD/DF3/CIR/P/2017/114) was partly motivated by inconsistencies in fund behaviour relative to stated benchmarks, situations where R-squared analysis would have flagged the problem.
R-squared, beta, and systematic vs unsystematic risk
The decomposition of total risk (variance) into systematic and unsystematic components uses R-squared:
\[ \sigma_p^2 = R^2 \times \sigma_p^2 + (1 - R^2) \times \sigma_p^2 \]
More precisely:
\[ \sigma_p^2 = \beta_p^2 \times \sigma_m^2 + \sigma_\epsilon^2 \]
Where:
- \(\beta_p^2 \times \sigma_m^2\) = systematic risk (explained by the market)
- \(\sigma_\epsilon^2\) = unsystematic (idiosyncratic) risk
- \(R^2 = \frac{\beta_p^2 \times \sigma_m^2}{\sigma_p^2}\) is the proportion of total risk attributable to the market
For a fund with R² = 0.90, 90 per cent of its variance is systematic (market-driven) and 10 per cent is fund-specific. For a thematic fund with R² = 0.45, only 45 per cent is market-driven.
Limitations
- R-squared is sensitive to benchmark choice. The same fund will show different R-squared values against different benchmarks. Always verify that the benchmark used is the fund’s stated regulatory benchmark.
- Period sensitivity. Rolling R-squared can shift significantly, a small-cap fund may show R² = 0.75 in a normal market and R² = 0.90 during a market panic when all stocks become highly correlated.
- R-squared does not indicate performance quality. A high R-squared is not inherently good, it simply means the fund is benchmark-like. An index fund investor wants high R-squared; an active investor who is paying for stock-picking skill may expect lower R-squared.
See also
- Beta in mutual funds
- Alpha (Jensen’s alpha) in mutual funds
- Tracking error in index funds
- Information ratio
- Standard deviation as a mutual fund risk metric
- Total return index benchmarking
- Mutual fund
References
- Bodie, Z., Kane, A., and Marcus, A. J., Investments, 12th edition, McGraw-Hill.
- AMFI, Risk statistics methodology, amfiindia.com.
- SEBI circular SEBI/HO/IMD/DF3/CIR/P/2017/114 dated 6 October 2017, category and benchmark rationalisation.
- Morningstar India, R-squared data for Indian mutual funds, morningstar.in.
- Value Research, Fund analytics platform, valueresearchonline.com.