Downside beta

It is common to measure ${\displaystyle r_{i}}$ and ${\displaystyle r_{m}}$ as the excess returns to security ${\displaystyle i}$ and the market ${\displaystyle m}$, ${\displaystyle u_{m}}$ as the average market excess return, and Cov and Var as the covariance and variance operators, Downside beta is

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

while upside beta is given by this expression with the direction of the inequalities reversed. Therefore, ${\displaystyle \beta ^{-}}$ can be estimated with a regression of the excess return of security ${\displaystyle i}$ on the excess return of the market, conditional on (excess) market return being negative.

Downside beta was once hypothesized to have greater explanatory power than standard beta in bearish markets.^[1][2] As such, it would have been a better measure of risk than ordinary beta.

The Capital asset pricing model (CAPM) can be modified to work with dual betas.^[3] Other researchers have attempted to use semi-variance instead of standard deviation to measure risk.^[4]