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

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

β − = Cov ⁡ ( r i , r m ∣ r m < u m ) Var ⁡ ( r m ∣ r m < u m ) , {\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,  β − {\displaystyle \beta ^{-}} can be estimated with a regression of the excess return of security  i {\displaystyle i} on the excess return of the market, conditional on (excess) market return being negative.

## Downside beta vs. beta

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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

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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

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1. ^ 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.
2. ^ 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.
3. ^ 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.
4. ^ 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.
5. ^ 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.