# Upside beta

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.

## Formula

Upside beta measures this upside risk. Defining ${\displaystyle r_{i}}$  and ${\displaystyle r_{m}}$  as the excess returns to security ${\displaystyle i}$  and market ${\displaystyle m}$ , ${\displaystyle u_{m}}$  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.[2]

${\displaystyle \beta ^{+}={\frac {\operatorname {Cov} (r_{i},r_{m}\mid r_{m}>u_{m})}{\operatorname {Var} (r_{m}\mid r_{m}>u_{m})}},}$

with downside beta ${\displaystyle \beta ^{-}}$  defined with the inequality directions reversed. Therefore, ${\displaystyle \beta ^{-}}$  and ${\displaystyle \beta ^{+}}$  can be estimated with a regression of excess return of security ${\displaystyle i}$  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. Upside beta and downside beta are also differentiated in the dual-beta model.