In investing, upside beta is the element of traditional beta that investors do not typically associate with the true meaning of risk.
[1] It is defined to be the scaled amount by which an asset tends to move compared to a benchmark, calculated only on days when the benchmark's return is positive.
as the excess returns to security
as the average market excess return, and Cov and Var as the covariance and variance operators, the CAPM can be modified to incorporate upside (or downside) beta as follows.
defined with the inequality directions reversed.
can be estimated with a regression of excess return of security
on excess return of the market, conditional on excess market return being below the mean (downside beta) and above the mean (upside beta).
"[3] Upside beta is calculated using asset returns only on those days when the benchmark returns are positive.