Downside beta

In investing, downside beta is the element of beta that investors associate with risk in the sense of the uncertain potential for loss. It is defined to be the scaled amount by which an asset tends to move compared to a benchmark, calculated only on days when the benchmark’s return is negative.^[1]

Formula

Downside beta measures downside risk. The Capital asset pricing model (CAPM) can be modified to use semi-variance instead of standard deviation to measure risk.^[2]

Where $r_{i}$ and $r_m$ are the excess returns to security $i$ and market $m$, and $u_{m}$ is the average market excess return, and Cov and Var are the covariance and variance operators, the CAPM can be modified to incorporate downside (or upside) beta as follows.^[3]

${\displaystyle \beta ^{-}={\frac {\operatorname {Cov} (r_{i},r_{m}\mid r_{m}<u_{m})}{\operatorname {Var} (r_{m}\mid r_{m}<u_{m})}}}$

Therefore, $\beta ^{-}$ and $\beta ^{+}$ can be estimated with a regression of excess return of security $i$ on excess return of the market, conditional on excess market return being below the mean for downside beta (or above the mean for upside beta).^[1] Downside beta is calculated from data points of the asset or portfolio return using only those days when the benchmark return is negative. Downside beta and upside beta are also differentiated in the dual-beta model.

Downside beta vs. beta

Downside beta has greater explanatory power than standard beta in bearish markets.^[1][4] Portfolios that are constructed by minimizing downside beta may be able to maintain more of their value during times of market decline.

References

1. Kaplanski, Guy. "Traditional beta, downside risk beta and market risk premiums". The Quarterly Review of Economics and Finance. 44 (5): 636–653. doi:10.1016/j.qref.2004.05.008.
2. 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.
3. Bawa, V.; Lindenberg, E. (1977). "Capital market equilibrium in a mean-lower partial moment framework". Journal of Financial Economics. 5: 189–200. doi:10.1016/0304-405x(77)90017-4.
4. Estrada (2007). "Mean-semivariance behavior: Downside risk and capital asset pricing". International Review of Economics and Finance. 16: 169. doi:10.1016/j.iref.2005.03.003. Estrada computed standard betas and downside betas for stocks across 23 developed markets and 27 emerging markets. This research showed that downside beta did a better job of explaining variations of cross-section returns in both types of market than did standard beta. In emerging markets, downside beta explained 55% of variations in mean returns.

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