Downside beta

In investing, downside beta is the beta that measures a stock's association with the overall stock market (risk) only on days when the market’s return is negative. Downside beta was first proposed by Roy 1952 and then popularized in an investment book by Markowitz (1959).

Formula
It is common to measure $$r_i$$ and $$r_m$$ as the excess returns to security $$i$$ and the market $$m$$, $$u_m$$ as the average market excess return, and Cov and Var as the covariance and variance operators, Downside beta is


 * $$\beta^-=\frac{\operatorname{Cov}(r_i,r_m \mid r_m<u_m)}{\operatorname{Var}(r_m \mid r_m<u_m)},$$

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

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

Use in Equilibrium Models of Risk-Reward
The Capital asset pricing model (CAPM) can be modified to work with dual betas. Other researchers have attempted to use semi-variance instead of standard deviation to measure risk.