Semi Beta (Downside Beta)
Semi betas decompose the standard CAPM beta into downside and upside components by conditioning on the direction of market returns. Downside beta measures an asset's sensitivity to the market specifically during falling markets, capturing the risk that investors care about most -- losses.
Overview
Standard CAPM beta treats upside and downside market movements symmetrically, implicitly assuming that investors are equally concerned about gains and losses of the same magnitude. Behavioral finance and prospect theory (Kahneman & Tversky, 1979) demonstrate that this is not the case: investors exhibit loss aversion, weighting losses roughly twice as heavily as equivalent gains.
Semi betas address this asymmetry by estimating separate betas for rising and falling markets. The downside beta () measures how much an asset falls when the market falls, while the upside beta () measures how much it rises when the market rises. An ideal investment has low downside beta (participates less in market declines) and high upside beta (captures more of market rallies).
Ang, Chen, and Xing (2006) showed that stocks with high downside beta earn higher average returns than predicted by CAPM, suggesting that downside risk is priced in the cross-section of expected returns.
Mathematical Formulation
Downside Beta
The downside beta is computed using only observations where the market return falls below a threshold (typically the mean market return or zero):
The conditional covariance and variance are computed using only the subset of observations where . This focuses the estimation entirely on the bear-market regime.
Upside Beta
Symmetrically, the upside beta uses only observations where the market return exceeds the threshold:
Regression Form
Equivalently, the downside beta can be obtained by running an OLS regression on the filtered sub-sample:
The upside regression is analogous but restricted to . The slope coefficients from these sub-sample regressions are the semi betas.
Relationship to Standard Beta
The standard CAPM beta can be expressed as a weighted average of the downside and upside semi betas:
where and are the number of down-market and up-market observations respectively.
Threshold Selection
The choice of threshold affects the results:
- : Separates positive and negative market returns. Simple and intuitive.
- :Uses the sample mean market return. Bawa & Lindenberg (1977) recommend this choice.
- : Uses the risk-free rate. Separates periods where the market earned a positive vs. negative excess return.
Advantages & Limitations
Advantages
- Asymmetric risk: Captures the fundamentally different behavior of assets in bull versus bear markets.
- Loss aversion alignment: Focuses on downside risk, which aligns with how investors actually experience and evaluate risk.
- Cross-sectional pricing: Downside beta has been shown to explain cross-sectional return variation beyond standard CAPM beta.
- Portfolio construction: Enables selection of assets with favorable asymmetry (low downside beta, high upside beta).
Limitations
- Reduced sample size: Each semi beta uses only half the data, increasing estimation uncertainty.
- Threshold sensitivity: Results depend on the choice of threshold, which is somewhat arbitrary.
- Non-stationarity: The relationship between asset and market returns in down markets may not be stable over time.
- Not widely reported: Unlike standard beta, semi betas are not routinely provided by financial data services.
References
- Bawa, V. S., & Lindenberg, E. B. (1977). "Capital Market Equilibrium in a Mean-Lower Partial Moment Framework." Journal of Financial Economics, 5(2), 189-200.
- Ang, A., Chen, J., & Xing, Y. (2006). "Downside Risk." Review of Financial Studies, 19(4), 1191-1239.
- Estrada, J. (2002). "Systematic Risk in Emerging Markets: The D-CAPM." Emerging Markets Review, 3(4), 365-379.