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{{Cite journal|last=Roy|first=A. D.|date=1952|title=Safety First and the Holding of Assets|url=http://www.jstor.org/stable/1907413|journal=Econometrica|volume=20|issue=3|pages=431–449|doi=10.2307/1907413|jstor=1907413 |issn=0012-9682|url-access=subscription}} and then popularized in an investment book by [https://www.jstor.org/stable/j.ctt1bh4c8h Markowitz (1959)].
Formula
It is common to measure and as the excess returns to security and the market , as the average market excess return, and Cov and Var as the covariance and variance operators, Downside beta is
:
while upside beta is given by this expression with the direction of the inequalities reversed. Therefore, can be estimated with a regression of the excess return of security 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.{{Cite journal|last1=Ang|first1=Andrew|last2=Chen|first2=Joseph|last3=Xing|first3=Yuhang|date=2006-12-01|title=Downside Risk|url=https://academic.oup.com/rfs/article/19/4/1191/1580531|journal=The Review of Financial Studies|language=en|volume=19|issue=4|pages=1191–1239|doi=10.1093/rfs/hhj035|issn=0893-9454|url-access=subscription}}{{Cite journal|last1=Lettau|first1=Martin|last2=Maggiori|first2=Matteo|last3=Weber|first3=Michael|date=2014-11-01|title=Conditional risk premia in currency markets and other asset classes|url=http://www.sciencedirect.com/science/article/pii/S0304405X14001378|journal=Journal of Financial Economics|language=en|volume=114|issue=2|pages=197–225|doi=10.1016/j.jfineco.2014.07.001|issn=0304-405X}} 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.{{cite journal|last=Bawa|first=V.|author2=Lindenberg, E.|year=1977|title=Capital market equilibrium in a mean-lower partial moment framework|journal=Journal of Financial Economics|volume=5|issue=2|pages=189–200|doi=10.1016/0304-405x(77)90017-4}} Other researchers have attempted to use semi-variance instead of standard deviation to measure risk.{{cite journal|last=Hogan|first=W.W.|author2=Warren, J.M.|year=1977|title=Toward the development of an equilibrium capital-market model based on semi-variance|journal=Journal of Financial and Quantitative Analysis|volume=9|issue=1|pages=1–11|doi=10.2307/2329964|jstor=2329964|s2cid=153337865 }}