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^[1] 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.^[2]^[3] 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.^[4] Other researchers have attempted to use semi-variance instead of standard deviation to measure risk.^[5]

References

^ Roy, A. D. (1952). "Safety First and the Holding of Assets". Econometrica. 20 (3): 431–449. doi:10.2307/1907413. ISSN 0012-9682. JSTOR 1907413.
^ Ang, Andrew; Chen, Joseph; Xing, Yuhang (2006-12-01). "Downside Risk". The Review of Financial Studies. 19 (4): 1191–1239. doi:10.1093/rfs/hhj035. ISSN 0893-9454.
^ Lettau, Martin; Maggiori, Matteo; Weber, Michael (2014-11-01). "Conditional risk premia in currency markets and other asset classes". Journal of Financial Economics. 114 (2): 197–225. doi:10.1016/j.jfineco.2014.07.001. ISSN 0304-405X.
^ Bawa, V.; Lindenberg, E. (1977). "Capital market equilibrium in a mean-lower partial moment framework". Journal of Financial Economics. 5 (2): 189–200. doi:10.1016/0304-405x(77)90017-4.
^ Hogan, W.W.; Warren, J.M. (1977). "Toward the development of an equilibrium capital-market model based on semi-variance". Journal of Financial and Quantitative Analysis. 9 (1): 1–11. doi:10.2307/2329964. JSTOR 2329964. S2CID 153337865.

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