Distinguish the beta anomaly from the volatility anomaly, and examine what the data show for both.
The idea
The beta anomaly is the finding that low-beta stocks have earned higher risk-adjusted returns, measured as CAPM alpha, than high-beta stocks. The empirical security market line is therefore flatter than the CAPM predicts and can even slope downward. The volatility anomaly is the separate finding that stocks with low total volatility, or low idiosyncratic volatility measured against a factor model, have earned higher returns than high-volatility stocks. Let σM be market volatility and σE a stock's idiosyncratic volatility. Total variance then splits into a market part and a stock-specific part, as in this identity. Variance = β² × σM² + σE². A stock can therefore have a low beta but high total volatility, so the two sorts rank stocks differently. In the data the highest-beta and highest-volatility portfolios have earned lower returns than the low-risk ones. To build a beta-neutral long-short portfolio, short beta-adjusted amounts, not equal dollars. The usual error is treating beta and volatility as interchangeable.
Formulas
Total variance split into systematic and idiosyncratic parts
total variance = β² × σm² + σε²
- β
- the asset's beta to the market
- σm
- the volatility of the market's return
- σε
- the idiosyncratic (residual) volatility
Square each part before adding. A low beta does not mean low total volatility, which is why beta and volatility sorts differ.
Short position for a beta-neutral long-short
short amount = long amount × βL ÷ βH
- long amount
- the money invested in the low-beta portfolio
- βL
- the beta of the low-beta portfolio
- βH
- the beta of the high-beta portfolio
Equal dollars on each side would leave a net short beta; the high-beta leg must be smaller.
Excess return of a betting-against-beta portfolio
BAB = rL ÷ βL − rH ÷ βH
- rL
- the excess return of the low-beta portfolio
- βL
- the beta of the low-beta portfolio
- rH
- the excess return of the high-beta portfolio
- βH
- the beta of the high-beta portfolio
Each leg is scaled to a beta of one, levering the low-beta side and de-levering the high-beta side, so net beta is zero.
A practice question
Which statement correctly distinguishes the beta anomaly from the volatility anomaly?
- The beta anomaly concerns stocks sorted by total volatility, while the volatility anomaly concerns stocks sorted by market beta.Why not: Swaps the two sorting variables.
- The beta anomaly is found only in bond markets, while the volatility anomaly is found only in equity markets.Why not: Both are documented chiefly in equities; neither is confined to one asset class in this way.
- The two are the same finding, because a stock's beta determines its total volatility exactly.Why not: Total variance also contains idiosyncratic variance, so beta and volatility rank stocks differently.
- The beta anomaly concerns returns of stocks sorted by market beta, while the volatility anomaly concerns returns of stocks sorted by total or idiosyncratic volatility. - correct
The beta anomaly sorts on systematic risk (beta) and finds a flat or inverted security market line. The volatility anomaly sorts on total or idiosyncratic volatility and finds the most volatile stocks earn the lowest returns.
A harder one
A long-short strategy goes long USD 1 of a low-beta portfolio with a beta of 0.7 and shorts a high-beta portfolio with a beta of 1.4. What amount of the high-beta portfolio must be shorted for the position to be beta-neutral?
- USD 1.00Why not: Uses a dollar-neutral position; this leaves a net beta of 0.7 - 1.4 = -0.7, not zero.
- USD 0.50 - correct
- USD 2.00Why not: Inverts the ratio (1.4/0.7), which would short twice the amount needed and leave a net beta of -2.1.
- USD 0.70Why not: Shorts an amount equal to the low-beta portfolio's beta, which does not equate the two exposures.
Beta-neutral requires 1 x 0.7 = x x 1.4, so x = 0.7/1.4 = 0.50. The short leg is smaller because each dollar of the high-beta portfolio carries more market risk.
The hardest tier
Stock A has a beta of 0.8 and idiosyncratic volatility of 30% a year. The market's volatility is 16% a year. Which is closest to Stock A's total volatility?
- 42.8%Why not: Adds systematic and idiosyncratic volatilities directly instead of adding variances.
- 12.8%Why not: Includes only the systematic volatility (beta x market volatility) and omits the idiosyncratic part.
- 30.0%Why not: Uses only the idiosyncratic volatility and omits the systematic component.
- 32.6% - correct
Systematic volatility = 0.8 x 16 = 12.8%. Total variance = 12.8² + 30² = 163.8 + 900 = 1063.8, so total volatility = 32.6%. A low-beta stock can still be highly volatile.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
Start free →More in Alpha (and the Low-Risk Anomaly)
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- Account for how the choice of benchmark shifts measured alpha and set out what makes a benchmark good for measuring it.
- Set out Grinold's fundamental law of active management with its assumptions and limits, and use it to work out the highest information ratio that can be attained.
- Use a factor regression to build a multifactor benchmark, gauge how sensitive a portfolio is to those factors, and measure alpha relative to the benchmark.
- Show how style analysis can be used to cope with factor exposures that change over time.
- Outline the problems that come up when alphas are measured for strategies with nonlinear payoffs.
- Lay out the possible explanations offered for the risk anomaly.