What a Market-Model Regression Separates
A stock return can move with the market and also differ from the return predicted by that market movement. The market-model regression describes these two features separately: beta measures the fitted sensitivity to market returns, while the intercept and residuals describe two distinct forms of return not captured by that sensitivity. Keeping those quantities separate matters when interpreting a regression: a high or low beta does not, by itself, describe how much residual variation the stock has.
Suppose an analyst observes an asset return and a market return for each of \(n\) matching periods. Use the same return units and period for both series. The model below is written for ordinary, or raw, returns; an intercept from a regression of excess returns has a different reference point and should not be casually interchanged with this one.
Before using the model as a statistical description, the observations should be matched over comparable periods and the relationship should be reasonably represented by a straight line. Beta estimates how much the fitted asset return changes, on average in the fitted line, for a one-unit change in market return. Alpha is the fitted asset return at a market return of zero. A residual is the observed return minus the fitted return for that particular period.
For population interpretation, a common assumption is \(E[\epsilon_{i,t}\mid R_{M,t}]=0\): after accounting for the market return, the residual has conditional mean zero. This assumption supports interpreting beta as the conditional linear response and alpha as the conditional mean at a zero market return. It does not guarantee a causal effect, and the model may be inadequate if the relationship is nonlinear or changes over time. Additional assumptions, such as independent errors with constant variance, are needed for standard regression inference, not merely to calculate the fitted line.
Estimating the Slope, Intercept, and Residuals
For ordinary least squares (OLS) with an intercept, the fitted line is chosen to minimize the sum of squared residuals. Provided the observed market returns are not all identical, its slope and intercept can be calculated from the paired observations:
The slope is the sample covariance of asset and market returns divided by the sample variance of market returns. The common covariance and variance denominator cancels in this ratio. The intercept is then set so the fitted line passes through the point of sample means \((\bar{R}_M,\bar{R}_i)\). These formulas describe an OLS fit; a line fitted by another method need not have these properties.
Worked Example: Fitting Beta and Alpha to Four Returns
A hypothetical asset and market have these four period returns, all measured in percentage points. Fit the OLS market model, calculate every residual, and check the fitted residual conditions.
Method. First calculate the two means, then the centered cross-product and market sum of squares. This avoids mistaking an individual period's return for the average relationship.
The market returns are \(-2,-1,1,2\), so \(\bar{R}_M=0\). The asset returns are \(-1,-1.5,1.5,5\), so their sum is \(4\) and \(\bar{R}_i=1\).
The centered cross-product is \((-2)(-1)+(-1)(-1.5)+(1)(1.5)+(2)(5)=15\). The market sum of squares is \((-2)^2+(-1)^2+1^2+2^2=10\). Therefore \(\hat{\beta}_i=15/10=1.5\), and \(\hat{\alpha}_i=1-(1.5)(0)=1\).
The fitted return is \(1+1.5R_M\). In the four periods, fitted returns are \(-2, -0.5, 2.5,\) and \(4\). Subtracting these from the observed asset returns gives residuals \(1,-1,-1,1\), respectively, all in percentage points.
The residuals sum to \(1-1-1+1=0\). Their cross-product with market returns is \((-2)(1)+(-1)(-1)+(1)(-1)+(2)(1)=0\). Both checks pass, as they should for an OLS fit with an intercept. The positive 1% intercept is not the mean residual: the fitted residual average here is zero.
Why OLS Residuals Have These Properties
The residual checks follow directly from the least-squares objective. For a proposed intercept \(a\) and slope \(b\), let \(e_t=R_{i,t}-a-bR_{M,t}\) and minimize \(Q(a,b)=\sum_t e_t^2\). At an interior minimum, the partial derivatives are zero:
Thus the fitted residuals sum to zero and have zero sample cross-product with the market returns. Since \(\sum_t e_t=0\), subtracting the market mean inside that cross-product also gives \(\sum_t(R_{M,t}-\bar{R}_M)e_t=0\). These are algebraic properties of OLS with an intercept; they do not prove that the model is a good description of future returns or that residuals are economically irrelevant.
The zero residual average is especially important to distinguish from alpha. Alpha is the fitted value at a market return of zero. The residual records a period-specific difference from the fitted line. Averaging the fitted line across the observed periods gives \(\hat{\alpha}_i+\hat{\beta}_i\bar{R}_M=\bar{R}_i\), not \(\hat{\alpha}_i\) unless the market sample mean is zero.
Worked Example: Interpreting an Intercept When the Market Mean Is Not Zero
For a hypothetical asset, the four market returns are \(0,1,2,3\) percentage points, and asset returns are \(0.7,1.3,2.5,4.3\) percentage points. Fit the regression and explain why the asset's average return is not its alpha.
Method. Center both series to calculate the slope, then use the sample means to calculate the intercept. The market mean is not zero, so using the asset mean as alpha would be an error.
The market mean is \((0+1+2+3)/4=1.5\), and the asset mean is \((0.7+1.3+2.5+4.3)/4=2.2\). The centered market values are \(-1.5,-0.5,0.5,1.5\); the centered asset values are \(-1.5,-0.9,0.3,2.1\).
The centered cross-product is \(2.25+0.45+0.15+3.15=6\), and the market sum of squares is \(2.25+0.25+0.25+2.25=5\). Therefore \(\hat{\beta}_i=6/5=1.2\). The intercept is \(\hat{\alpha}_i=2.2-(1.2)(1.5)=0.4\) percentage points.
The fitted returns are \(0.4,1.6,2.8,4.0\), giving residuals \(0.3,-0.3,-0.3,0.3\). Their sum is zero, and their cross-product with the market returns is \(0+(-0.3)+(-0.6)+0.9=0\). The sample average asset return is \(2.2\), while alpha is \(0.4\): the fitted line's value at the sample's average market return is \(0.4+(1.2)(1.5)=2.2\).
Residual Variation and the Risk Decomposition
Beta describes the market-linked component of the fitted return, but it does not measure the size of the residuals. To describe risk, distinguish the variance associated with market movements from the variance left in the residuals. In the population model, if \(E[\epsilon_i\mid R_M]=0\), then \(\operatorname{Cov}(R_M,\epsilon_i)=0\). Since the intercept is constant, taking the variance of \(R_i=\alpha_i+\beta_iR_M+\epsilon_i\) gives:
The final equality depends on the zero-covariance condition; without it, the covariance term must remain. The first term is market-related variance in this one-factor model. The second is residual, or firm-specific, variance relative to the model. Residual variation can reflect many influences, including firm-specific news, measurement noise, and omitted common factors. It is not automatically a pure measure of risk unique to the firm.
Worked Example: Same Beta, Different Residual Risk
Two hypothetical stocks have been fitted against the same four market returns, \(-2,-1,1,2\) percentage points. Stock B has \(\hat{\alpha}=0.5\), \(\hat{\beta}=0.8\), and residuals \(0.4,-0.4,-0.4,0.4\). Stock C has \(\hat{\alpha}=0\), \(\hat{\beta}=0.8\), and residuals \(-1,1,1,-1\). Compare their descriptive return variances by treating the four observations as equally weighted outcomes and dividing sums of squared deviations by four.
Method. Use the model decomposition for each stock. The market mean is zero, the residual means are zero, and each residual series has zero cross-product with the market series, so the sample outcomes satisfy the variance decomposition when using the same divisor of four throughout.
The market variance under the stated convention is \((4+1+1+4)/4=2.5\) squared percentage points. The common market-related variance for either stock is \((0.8)^2(2.5)=1.6\).
For B, the residual variance is \((0.16+0.16+0.16+0.16)/4=0.16\). Its total variance is \(1.6+0.16=1.76\) squared percentage points, and its standard deviation is \(\sqrt{1.76}=1.3266\) percentage points.
For C, the residual variance is \((1+1+1+1)/4=1\). Its total variance is \(1.6+1=2.6\) squared percentage points, and its standard deviation is \(\sqrt{2.6}=1.6125\) percentage points. Both stocks have the same fitted beta, but C has greater residual variation and greater total variation in these observations. The standard deviations are rounded to four decimal places.
This is a descriptive calculation with divisor four, not the usual regression estimate of residual variance using residual degrees of freedom. Changing the variance convention changes the numerical estimates; it does not change the distinction between beta and residual variation.
Common Interpretive Traps
A regression output can tempt an analyst to give the intercept more meaning than the model supports. Alpha is an estimated intercept, not proof that an asset will outperform, nor a guaranteed abnormal return. It is estimated from a particular sample and model specification. Likewise, a large positive residual is a positive unexplained return for one observation, not a higher beta. A large residual variance indicates considerable variation around the fitted line, not necessarily a large average residual.
A common wrong calculation is to set alpha equal to the sample mean asset return. In the previous example, this gives the incorrect value \(2.2\) percentage points. It amounts to evaluating the fitted line at a market return of zero while ignoring that the sample mean market return is \(1.5\). The correction is:
Another trap is to assume beta alone determines total risk. In the variance decomposition, the market component depends on beta and market variance, while residual variance is a separate term. Two assets with identical betas can therefore have quite different total variances. This distinction is useful when evaluating whether an asset's variation is closely associated with market movements or is mostly left unexplained by the chosen model.
Check Your Understanding
For each problem, treat returns and residuals as percentage points. Use an OLS regression with an intercept where a fit is requested.
- Market returns are \(-1,0,1\), and asset returns are \(-1.1,0.4,1.9\). Find the slope, intercept, fitted return when the market return is \(2\), and the three residuals. (Answer: \(\hat{\beta}=1.5,\ \hat{\alpha}=0.4\); fitted return \(3.4\); residuals \(0,0,0\).)
- Market returns are \(-2,-1,1,2\), and asset returns are \(-1.2,-1.45,0.05,1.8\). Calculate the fitted slope and intercept, then the residual sum of squares and the usual regression residual variance estimate using \(n-2\) degrees of freedom. (Answer: \(\hat{\beta}=0.75,\ \hat{\alpha}=-0.2\); residuals \(0.5,-0.5,-0.5,0.5\); sum of squares \(1\); residual variance estimate \(0.5\) squared percentage points.)
- A fitted model has \(\hat{\alpha}=0.3\), \(\hat{\beta}=1.1\), and the sample mean market return is \(2\). What is the fitted mean asset return? A colleague instead reports the fitted mean as alpha; identify the error. (Answer: \(2.5\) percentage points; alpha is the fitted return at a zero market return, not at the sample mean market return.)
- Assume market and residual returns have zero covariance. If beta is \(1.1\), market variance is \(4\), and residual variance is \(1.5\), calculate market-related variance, total variance, and total standard deviation. (Answer: market-related variance \(4.84\), total variance \(6.34\) squared percentage points, total standard deviation \(2.518\) percentage points.)
- For market returns \(-1,0,1\), an asset's returns are \(-2,1,2\). Calculate the OLS slope, intercept, and residuals. Does the residual average equal alpha? (Answer: \(\hat{\beta}=2,\ \hat{\alpha}=1/3\); residuals \(-1/3,2/3,-1/3\); no, the residual average is \(0\), not \(1/3\).)