Two Regression Statistics, Two Questions
A market regression can summarize two distinct features of an asset’s returns. Beta describes how sensitive the asset’s return is to market returns: it is the estimated slope of the regression. R-squared describes how much of the asset’s observed return variation is accounted for by that fitted linear relationship. A large value of one does not automatically imply a large value of the other.
This distinction matters when interpreting regression output. A security can have a large beta, meaning its fitted return changes substantially with the market, but a low R-squared if substantial variation remains unrelated to the market. Conversely, a small beta can be estimated in a regression that explains a large share of the asset’s variation. These statistics answer different questions; neither alone describes every dimension of risk.
The Regression Model and the Measures
Let \(R_i\) denote an asset’s return for an observation period and \(R_M\) the market proxy’s return for the same period. With matched observations and a fitted intercept, the regression model is
Here, \(\alpha_i\) is the intercept, \(\beta_i\) is the slope, and \(\varepsilon_i\) is the residual: the part of the observed return not captured by the fitted line. Beta is estimated from the paired returns. With a nonzero market variance, its ordinary least squares estimate is the sample covariance of asset and market returns divided by the sample variance of market returns.
In this tutorial, “variance explained” refers to the variation captured by the fitted linear component, not to a claim that the market causes each observed return. For a regression with an intercept, R-squared can also be written as the squared correlation between the asset and market returns. It ranges from 0 to 1 in the usual in-sample ordinary least squares setting.
These calculations are descriptive properties of a fitted regression. They do not require returns to follow a particular probability distribution. The observations must, however, refer to the same periods and use consistent return units. For example, paired monthly returns should not be mixed with quarterly returns. The standard R-squared definition above assumes the regression includes an intercept; without one, the usual centered-variation interpretation and identities may not hold.
Why Beta and R-Squared Differ
The distinction follows directly from the variance decomposition in a regression with an intercept. In ordinary least squares, residuals are orthogonal to the fitted values and to the centered market returns. Thus the asset’s total centered variation separates into the variation in its fitted values and the remaining residual variation. The following argument gives the connection.
After centering the returns, the fitted component is \(\widehat{\beta}_i(R_M-\overline{R}_M)\). Its variance is \(\widehat{\beta}_i^2\operatorname{Var}(R_M)\), because multiplying a variable by a constant multiplies its variance by the constant squared.
With an intercept, the ordinary least squares residuals have zero sample covariance with fitted values. Therefore, \(\operatorname{Var}(R_i)=\widehat{\beta}_i^2\operatorname{Var}(R_M)+\operatorname{Var}(\widehat{\varepsilon}_i)\), using the same variance convention for each term.
R-squared is the fitted component’s share of total asset-return variation. Hence \(R^2=\widehat{\beta}_i^2\operatorname{Var}(R_M)/\operatorname{Var}(R_i)\). The slope appears squared and is scaled by market variance; R-squared is not the slope itself.
This identity explains why beta alone cannot determine R-squared. Even with beta held fixed, the residual variation can change the total variation and therefore the explained share. Similarly, two assets can have the same R-squared but different slopes if their market-related variation and residual variation differ in proportion.
Worked Example: A Large Beta With Low R-Squared
Consider hypothetical paired monthly returns, all measured in percentage points, from a regression with an intercept. The summary statistics are \(\operatorname{Var}(R_M)=4\), \(\operatorname{Cov}(R_i,R_M)=6\), total asset-return variance \(100\), and residual variance \(91\), all in squared percentage-point units. Calculate beta and R-squared, then interpret them. Assume the reported values are mutually consistent ordinary least squares summaries.
Method. Calculate beta from covariance divided by market variance. Then calculate the fitted variance and divide by total asset variance to obtain R-squared.
Step 1: Calculate beta.
The estimated slope is 1.5. In the fitted line, a one-percentage-point increase in the market return corresponds to a 1.5-percentage-point increase in the asset’s fitted return.
Step 2: Calculate the variance in the fitted component.
Step 3: Calculate R-squared.
The residual variance also checks the decomposition: \(9+91=100\). The asset has a relatively large positive beta, but the market regression explains only 9% of its observed variation. The other 91% is residual variation in this regression. The mistake to avoid is reading beta \(1.5\) as “150% of variation explained”; beta is a slope, while R-squared is a proportion of variation.
Worked Example: The Same Beta With a Different R-Squared
A second hypothetical asset is regressed on the same market proxy, using matched monthly returns in percentage points and an intercept. Its market variance is again 4 and its covariance with the market is again 6. Its residual variance is 9. Find its beta and R-squared, and compare the fit with the preceding example.
Method. Use covariance divided by market variance for beta. Use the fitted-plus-residual variance decomposition for R-squared.
Step 1: Calculate beta.
The slope matches the first asset’s slope.
Step 2: Calculate fitted variance and total variance.
Step 3: Calculate R-squared.
The residual variance check is \(18-9=9\), as stated. Both assets have beta 1.5, so their estimated market sensitivities are the same. Their R-squared values differ substantially: 9% for the first and 50% for the second, because the second asset has much less residual variation relative to its fitted variation. The common trap is to assume equal betas imply equally strong regression fit.
Worked Example: The Same R-Squared With Different Betas
Compare two other hypothetical assets, each observed over the same monthly periods as a market proxy with variance 4, in squared percentage-point units. For Asset A, the market covariance is 2 and residual variance is 1. For Asset B, the market covariance is 6 and residual variance is 9. Both regressions include an intercept and use the same variance convention. Calculate each beta and R-squared.
Method. Calculate each slope from its covariance, then calculate fitted variance and divide it by the sum of fitted and residual variances.
Asset A, Step 1: Beta.
Asset A, Step 2: Fitted variance, total variance, and R-squared.
Asset B, Step 1: Beta.
Asset B, Step 2: Fitted variance, total variance, and R-squared.
Both assets have R-squared of 50%, but their betas differ: 0.5 versus 1.5. In each case, residual variance equals fitted variance, so exactly half the total variation is explained by the fitted market relationship. That shared proportion does not make their market sensitivities equal.
Interpreting Both Measures Together
A useful reading of regression output keeps the two interpretations separate. Beta answers, “How much does the fitted asset return move for a given market-return change?” R-squared answers, “What fraction of the asset’s centered return variation does this regression fit?” Always interpret beta in the return units and period used in the regression, and interpret R-squared as a proportion rather than as a return or a slope.
| Statistic | What it measures | What it does not establish by itself |
|---|---|---|
| Beta | The fitted slope: asset-return sensitivity to market returns | How much of the asset’s total variation the regression explains |
| R-squared | The share of centered asset-return variation explained by the fitted regression | The size or direction of the slope, or a causal effect |
R-squared is not the percentage of an asset’s return “caused by” the market, nor is it the percentage of its risk that can be eliminated by an investment decision. It is a measure of fit for a particular regression, using a particular set of observations and a particular market proxy. A low value means that this fitted market relationship accounts for a relatively small share of observed variation; it does not mean beta must be small.
Likewise, a high R-squared does not mean beta is high. It means the observed returns lie relatively close to the fitted line around the sample mean. A nearly flat line can fit tightly if the asset’s returns have little variation beyond that small fitted relationship. A steep line can fit poorly if large residual movements remain.
Both measures also depend on the regression setup. Changing the market proxy, the return frequency, or the observation window can change the estimated slope and fit. This is one reason to read the regression specification along with its statistics rather than treating a reported beta or R-squared as a context-free characteristic.
Common Exam Trap
The most frequent error is treating beta as a percentage of variation explained. For example, if beta is 1.5, an incorrect interpretation is “the market explains 150% of the asset’s return variation.” That is wrong for two reasons: beta is a slope, and R-squared is the variation share. In the first worked example, beta was 1.5 while R-squared was only 9%.
Another error is to infer low beta from low R-squared, or high beta from high R-squared. Neither conclusion follows. First calculate or identify the slope to discuss sensitivity. Separately calculate or identify R-squared to discuss the fraction of variation explained. In a simple regression with an intercept, checking that fitted variance plus residual variance equals total variance is a practical safeguard against mixing up the measures.
Check Your Understanding
For each problem, use an ordinary least squares regression with an intercept and consistent return periods. Variances and covariances are expressed in squared percentage-point units.
- A hypothetical fund has market variance 9, covariance with the market 9, and total return variance 45. Calculate beta and R-squared. (Answer: \(\widehat{\beta}=1.0\); \(R^2=20\%\).)
- A hypothetical asset has beta 0.8, market variance 25, and residual variance 84. Find its fitted variance, total variance, and R-squared. (Answer: Fitted variance \(=16\); total variance \(=100\); \(R^2=16\%\).)
- A regression has market variance 4, asset-market covariance \(-2\), and total asset variance 5. Find beta, fitted variance, and R-squared, and state the direction of the fitted relationship. (Answer: \(\widehat{\beta}=-0.5\); fitted variance \(=1\); \(R^2=20\%\); the fitted relationship is negative.)
- Two assets have market variance 4. Asset A has beta 1.2 and residual variance 16; Asset B has beta 0.6 and residual variance 1.44. Calculate each R-squared and compare their sensitivities. (Answer: Asset A: fitted variance \(=5.76\), total variance \(=21.76\), \(R^2\approx26.47\%\). Asset B: fitted variance \(=1.44\), total variance \(=2.88\), \(R^2=50\%\). Asset A has the larger beta.)
- A regression reports beta 0.8 and R-squared 64%. The market-return standard deviation is 5 percentage points. Find the fitted variance, total asset-return variance, residual variance, and residual standard deviation. (Answer: Fitted variance \(=(0.8)^2(5^2)=16\); total variance \(=16/0.64=25\); residual variance \(=25-16=9\); residual standard deviation \(=3\) percentage points.)