Beta measures how sensitive a return series is to movements in a benchmark.

Beta measures how sensitive a return series is to movements in a benchmark. In portfolio risk work, it is not something to assume by inspection. It is estimated from data.

The standard approach is a regression of excess returns. Excess return means the return of the strategy minus the risk free rate for the same period. The benchmark is treated the same way. In that setup, the strategy’s excess returns are the dependent variable and the benchmark’s excess returns are the explanatory variable. The slope coefficient from that regression is beta.

Written conceptually, the model is:

strategy excess return = alpha + beta times benchmark excess return + residual

This construction matters because it separates three things. Alpha is the intercept. Beta is the estimated sensitivity to the benchmark. The residual is the part not explained by benchmark movements. The result is a statistically derived beta rather than an asserted exposure.

Estimating beta this way requires matched observations through time. A practical sample is a history of daily or other period returns for the strategy and the corresponding benchmark returns over the same dates. Those returns are converted to excess returns by subtracting the risk free rate from each series before running the regression.

The regression output provides the beta estimate and the statistics needed to evaluate it. The slope coefficient is the point estimate of beta. Its standard error shows how precisely beta is estimated from the sample. The t statistic and p value test whether the estimated sensitivity is distinguishable from zero under the model assumptions. A confidence interval places a range around the estimate and makes the estimation uncertainty explicit.

Goodness of fit metrics add another layer of interpretation. R squared shows how much of the variation in the strategy’s excess returns is explained by the benchmark’s excess returns in the regression. A higher R squared means benchmark variation explains more of the observed return variation. A lower R squared means more of the variation remains in the residual.

This is why beta should be described as an empirical estimate. It depends on the sample period, the benchmark definition, the return frequency, and the excess return convention. Regression makes those choices transparent and gives a repeatable method for quantifying benchmark sensitivity within a risk model.

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