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).

## Contents

* 1 Formula
* 2 Downside beta vs. beta
* 3 Use in Equilibrium Models of Risk-Reward
* 4 References
* 5 External links

## Formula

It is common to measure ri and rm as the excess returns to security i and the market m, um as the average market excess return, and Cov and Var as the covariance and variance operators, Downside beta is

β−=Cov⁡(ri,rm∣rm<um)Var⁡(rm∣rm<um),

while upside beta is given by this expression with the direction of the inequalities reversed. Therefore, β− 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

1. ↑ Roy, A. D. (1952). "Safety First and the Holding of Assets". Econometrica 20 (3): 431–449. doi:10.2307/1907413. ISSN 0012-9682. http://www.jstor.org/stable/1907413.
2. ↑ Ang, Andrew; Chen, Joseph; Xing, Yuhang (2006-12-01). "Downside Risk" (in en). The Review of Financial Studies 19 (4): 1191–1239. doi:10.1093/rfs/hhj035. ISSN 0893-9454. https://academic.oup.com/rfs/article/19/4/1191/1580531.
3. ↑ Lettau, Martin; Maggiori, Matteo; Weber, Michael (2014-11-01). "Conditional risk premia in currency markets and other asset classes" (in en). Journal of Financial Economics 114 (2): 197–225. doi:10.1016/j.jfineco.2014.07.001. ISSN 0304-405X. http://www.sciencedirect.com/science/article/pii/S0304405X14001378.
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.

## External links

* An Investigation of Beta and Downside Beta Based CAPM - Case Study of Karachi Stock Exchange, SSRN. M. Tahir, Q. Abbas, S.M. Sargana, U. Ayub and S.K. Saeed; February 2013
* Estrada (2007). "Mean-semivariance behavior: Downside risk and capital asset pricing". International Review of Economics and Finance 16 (2): 169. doi:10.1016/j.iref.2005.03.003.

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