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Benchmark return OLS slope

For an equal-weight intercept fit, b=Sxy/Sxx when benchmark variation is positive.

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Define explanatory and response returns

Let X be the chosen benchmark and Y the asset return in matched periods. Use one compatible return basis. The least-squares line includes an intercept and minimises the sum of squared Y-direction residuals.

Use the centered sums

b=∑i(xi−xˉ)(yi−yˉ)∑i(xi−xˉ)2=SxySxxb=\frac{\sum_i(x_i-\bar{x})(y_i-\bar{y})}{\sum_i(x_i-\bar{x})^2}=\frac{S_{xy}}{S_{xx}}

Sxx must be positive; this is not a symmetric fit in X and Y.

Check the original record

Centered cross-product is 21 and benchmark squared sum 10 in percentage-point units, giving slope 2.1. Computing both covariance and benchmark variance with the same divisor gives the same ratio.

Retain the estimated-coefficient distinction

This is a sample fitting formula, not proof of a population coefficient, a causal exposure or future performance. Inference needs its own assumptions and calculation.

Further references