A conditional covariance slope
Keep rows where benchmark return bᵢ is strictly below target τ. Within that subset, recompute the strategy mean and benchmark mean. Downside beta equals sample covariance(strategy, benchmark) divided by the sample variance of the benchmark. Both covariance and variance use the same retained rows and their own conditional means. This is a conditional regression slope; it is distinct from downside-beta definitions based on clipping returns or calculating cosemivariance over the whole sample.
Work through a declining benchmark subset
Suppose paired benchmark returns are −3%, −2%, −1%, and 2%, while strategy returns are −4%, −2%, 0%, and 1%. With a zero threshold, the first three rows qualify. Their benchmark mean is −2% and strategy mean is −2%. Sample covariance is 2 in percentage-point-squared units and sample benchmark variance is 1, giving downside beta of 2. The positive benchmark row has no effect on this calculation.
Small subsets can produce fragile slopes
At least two qualifying observations and nonzero conditional benchmark variance are needed for a finite slope. Identical benchmark losses cannot identify sensitivity. A beta of two does not imply that every future benchmark decline produces exactly twice the strategy loss; the regression also has a level and residual variation. Align return timestamps, avoid mixing currencies, and compare the retained sample count when changing the threshold. Filtering a very short history can make a striking slope depend on just two dates.