R squared

What Is R-squared?

R-squared, also known as the coefficient of determination ((R^2)), is a statistical measure that represents the proportion of the variance for a dependent variable that can be explained by one or more independent variables in a regression analysis. In the context of portfolio theory and investment analysis, R-squared indicates the percentage of a portfolio's or security's price movements that can be explained by movements in a benchmark index. A higher R-squared value suggests that the model better fits the observed data, implying a stronger relationship between the variables.

History and Origin

The concept of R-squared is intrinsically linked to the development of regression analysis, a statistical technique that gained prominence in the late 19th and early 20th centuries. While the precise origin of the term "R-squared" itself is less documented, the underlying statistical principles of analyzing variance and fitting models to data have evolved significantly over time. It is a fundamental concept in econometrics and statistics, used to evaluate the goodness of fit of a model, as recognized by institutions that standardize statistical methodologies.²³ Its application in finance became widespread with the advent of modern portfolio theory, allowing investors and analysts to quantify how well an asset or portfolio's performance correlates with broad market movements.

Key Takeaways

• R-squared is a statistical measure (coefficient of determination) indicating how much of a dependent variable's variance is explained by an independent variable or variables in a regression model.
• In finance, it quantifies the degree to which an asset's or fund's movements can be attributed to its benchmark index.
• Values range from 0 to 1 (or 0% to 100%), with higher values indicating a stronger correlation to the benchmark.
• A high R-squared (e.g., 85% to 100%) suggests that movements in the benchmark explain most of the fund's performance.
• A low R-squared (e.g., below 70%) implies that other factors, beyond the benchmark's influence, are significant drivers of the fund's performance.

Formula and Calculation

The R-squared value is derived from a statistical procedure known as regression analysis. It is calculated as the ratio of the explained sum of squares (SSR) to the total sum of squares (SST). Alternatively, it can be expressed as 1 minus the ratio of the sum of squared errors (SSE) to the total sum of squares (SST).

The formula for R-squared is:

Where:

• (SSE) (Sum of Squared Errors or Unexplained Variation) measures the aggregate squared difference between the actual data points and the values predicted by the regression model.
• (SST) (Total Sum of Squares or Total Variation) measures the total variation in the dependent variable.

This formula essentially quantifies how much better the regression line explains the data compared to simply using the mean of the dependent variable.²²

Interpreting the R-squared

Interpreting R-squared involves understanding what the resulting percentage signifies about the relationship between the variables. In investing, an R-squared of 100 (or 100%) means that all movements of a fund or security can be completely explained by movements in its designated benchmark index.²¹ For example, an R-squared of 95% for a mutual fund against the S&P 500 Index suggests that 95% of the fund's price fluctuations are attributable to the S&P 500's movements, with the remaining 5% due to other factors such as the fund manager's stock selection.²⁰

Generally, a high R-squared (typically above 70%) indicates a strong correlation between the fund's returns and the benchmark's returns.¹⁹ This is often seen in index funds designed to mimic a specific market index. Conversely, a low R-squared (below 40%) suggests a low correlation, implying that the fund's performance is driven more by its own specific characteristics or the decisions of its portfolio managers, rather than the broad market benchmark.¹⁸ Such a low R-squared may be desirable for investors seeking true diversification from a particular index, or for actively managed funds aiming to generate returns independent of the benchmark.

Hypothetical Example

Consider an investor analyzing a technology sector mutual fund. The investor wants to understand how much of the fund's performance is tied to the broader technology market. They choose the NASDAQ-100 Index as the benchmark.

After performing a regression analysis of the mutual fund's monthly returns against the NASDAQ-100's monthly returns over the past three years, the R-squared value is calculated to be 0.88, or 88%.

This R-squared of 88% signifies that 88% of the technology fund's price movements can be explained by the movements of the NASDAQ-100 Index. The remaining 12% of the fund's performance is due to factors unique to the fund, such as the specific stock holdings, the skill of the fund's portfolio managers, or other idiosyncratic elements. For an investor looking for a fund that closely tracks the tech sector, an 88% R-squared suggests a strong alignment.

Practical Applications

R-squared is a widely used metric in financial analysis and investment management. It helps investors and analysts understand the relationship between a fund or security and its benchmark.

• Fund Evaluation: Investors use R-squared to assess how closely a mutual fund or exchange-traded fund (ETF) tracks its stated benchmark. A high R-squared for an index fund confirms its objective of mirroring the index. For actively managed funds, a moderate R-squared might be desired, indicating some correlation but also the potential for the manager's strategy to contribute unique returns.¹⁷
• Risk Assessment: R-squared, in conjunction with beta, helps in understanding systematic risk. A high R-squared makes a fund's beta a more reliable indicator of its sensitivity to market movements.¹⁶
• Portfolio Diversification: When constructing a portfolio, R-squared can help investors select assets that have low correlation to existing holdings or overall market benchmarks, thus contributing to better asset allocation.
• Economic Modeling: Beyond finance, R-squared is a general statistical measure used in various economic models to evaluate how well independent variables explain the variance in dependent variables. The Federal Reserve System, for instance, utilizes such statistical tools in its economic research and analysis to understand market behavior and policy impacts.¹⁵,¹⁴

Limitations and Criticisms

Despite its widespread use, R-squared has several limitations and criticisms that investors and analysts should consider:

• Not a Measure of Causation: A high R-squared indicates correlation, not causation. It does not imply that changes in the independent variable cause changes in the dependent variable, only that they move together.¹³
• Inflation with More Variables: R-squared tends to increase as more independent variables are added to a regression model, even if those variables are not truly relevant.¹² This can lead to overfitting, where the model performs well on historical data but poorly on new, unseen data.¹¹ Adjusted R-squared attempts to correct for this by penalizing the addition of irrelevant variables.¹⁰
• Does Not Indicate Model Quality: A high R-squared does not automatically mean a model is "good" or appropriate for a task.⁹ It doesn't tell us if the model's predictions are biased or if it meets the underlying assumptions of the regression.⁸ A low R-squared doesn't necessarily mean a model is bad; it might still provide valuable insights, particularly in fields with inherent high variability.⁷
• Sensitivity to Outliers: Extreme data points, or outliers, can significantly influence the R-squared value, potentially leading to a misleading impression of the model's fit.
• Inapplicability to Non-Linear Relationships: R-squared is best suited for linear relationships. It may not adequately capture the nuances of non-linear interactions between variables.⁶
• Doesn't Evaluate Predictive Power: While R-squared measures how well a model explains past data, it doesn't directly assess its predictive power for future outcomes. A model with high R-squared might still have large prediction errors in practical application.⁵ Researchers and investment firms like Research Affiliates emphasize that no investment strategy can guarantee returns, and past performance, even when well-explained by a model, is not indicative of future results.⁴

R-squared vs. Beta

R-squared and beta are both important metrics in portfolio analysis, but they measure different aspects of a fund's relationship with its benchmark.

While R-squared tells you how closely a fund's performance aligns with its benchmark, beta tells you how much that fund's performance tends to move when the benchmark moves. If a fund has a low R-squared, its beta may not be a statistically significant or reliable measure, as the benchmark doesn't sufficiently explain the fund's movements.³ Therefore, these two metrics are often considered together to provide a more complete picture of a fund's risk and return characteristics.

FAQs

What is a good R-squared value for a mutual fund?

For a mutual fund, a "good" R-squared depends on the fund's objective. If a fund aims to track a benchmark index (like an index fund), an R-squared of 90% or higher is generally considered good, indicating it closely mirrors the index's performance.² For an actively managed fund that seeks to outperform a benchmark through unique strategies, a moderate R-squared (e.g., 40-70%) might be acceptable, suggesting that the fund's returns are influenced by the benchmark but also by other factors.¹ A very low R-squared means the fund's performance is largely unrelated to the benchmark.

Can R-squared be negative?

No, R-squared cannot be negative. By definition, R-squared ranges from 0 to 1 (or 0% to 100%). It represents the proportion of variance explained, and variance is a non-negative value. While a regression model might perform worse than a simple average (leading to a higher sum of squared errors than total sum of squares), in such cases, R-squared is conventionally reported as 0.

Does a high R-squared mean a good investment?

Not necessarily. A high R-squared simply means that a large portion of an investment's movements can be explained by the movements of its benchmark index. It does not, however, imply that the investment is "good" or will generate high returns. For example, a fund tracking a declining market could have a high R-squared, but still lose money. R-squared should be evaluated alongside other metrics like alpha, beta, and the overall investment objectives and risk tolerance.