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Characterise the low-risk anomaly in asset returns and weigh up the case for it.

FRM® Part II · Risk Management and Investment Management · Alpha (and the Low-Risk Anomaly) · FRM2.IM.3.a

The idea

The low-risk anomaly is the finding that stocks with low beta or low volatility have earned returns as high as, or higher than, high-risk stocks, so their risk-adjusted performance has been better. This contradicts the CAPM, which says expected return rises in proportion to beta. The security market line should therefore slope upward at the market risk premium, yet empirically the line has been much flatter. The case for the anomaly is that the evidence covers long samples, many countries and several asset classes, and survives controls for size, value and momentum, which argues against pure data mining. The case against is that part of the alpha can be other factor tilts, trading costs and limited capacity. Let IR be the information ratio and T the number of years. Then the t-statistic of an alpha is roughly t = IR × √T. Exam point: the anomaly says risk is not rewarded as the CAPM predicts, not that low-risk stocks win every year.

Formulas

t-statistic of an alpha

t = IR × √T

IR
the information ratio, alpha ÷ tracking error, annual
T
the number of independent years

Assumes independent annual observations. A significant historical alpha must still be weighed against data mining and costs.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which statement best characterises the low-risk anomaly in equity returns?

  1. Stocks with low beta or low volatility have earned returns similar to or higher than high-risk stocks, so their risk-adjusted performance beats what the CAPM predicts. - correct
  2. Stocks with high beta have earned the highest raw returns and the best Sharpe ratios, which is exactly what the CAPM predicts.Why not: Describes the CAPM's rewarded-risk prediction, which is the opposite of the anomaly.
  3. Stocks with the best returns over the past twelve months have continued to outperform stocks with the worst past returns.Why not: Describes momentum, an anomaly sorted on past returns, not on beta or volatility.
  4. Small-capitalisation stocks have earned higher returns than large-capitalisation stocks after adjusting for market beta.Why not: Describes the size effect; the low-risk anomaly sorts on beta or volatility, not on market capitalisation.

The low-risk anomaly is that low-beta and low-volatility stocks have not earned the lower returns the CAPM implies; their risk-adjusted returns have been higher. Momentum and size are different effects.

A harder one

The risk-free rate is 2% and the market risk premium is 6%. A low-beta portfolio (beta 0.6) returned 8.0% and a high-beta portfolio (beta 1.4) returned 9.0%. What is the CAPM alpha of the low-beta portfolio minus the CAPM alpha of the high-beta portfolio?

  1. -3.8%Why not: Subtracts in the wrong order (high minus low), which flips the sign of the spread.
  2. -1.0%Why not: Takes the raw return difference 8.0% - 9.0% and ignores the betas, so it measures no alpha at all.
  3. 3.8% - correct
  4. 2.4%Why not: Uses only the low-beta portfolio's alpha and forgets to subtract the high-beta portfolio's negative alpha.

Expected returns: low = 2 + 0.6 x 6 = 5.6%, so alpha = 8.0 - 5.6 = 2.4%. High = 2 + 1.4 x 6 = 10.4%, so alpha = 9.0 - 10.4 = -1.4%. Spread = 2.4 - (-1.4) = 3.8%, the pattern the low-risk anomaly describes.

The hardest tier

A low-volatility equity portfolio returned 7% with a standard deviation of 10% and a beta of 0.55. The risk-free rate is 2%, the market risk premium is 7% and the market's standard deviation is 16%. Which pair gives the portfolio's Sharpe ratio and CAPM alpha?

  1. Sharpe 0.50; alpha 1.15% - correct
  2. Sharpe 0.70; alpha 1.15%Why not: Leaves the risk-free rate out of the Sharpe ratio numerator (7/10 instead of (7-2)/10).
  3. Sharpe 0.50; alpha 3.15%Why not: Omits the risk-free rate from the CAPM expected return, using beta x premium alone.
  4. Sharpe 0.31; alpha 1.15%Why not: Divides by the market's standard deviation instead of the portfolio's own 10%.

Sharpe = (7 - 2)/10 = 0.50. CAPM expected return = 2 + 0.55 x 7 = 5.85%, so alpha = 7 - 5.85 = 1.15%. Sharpe uses the portfolio's own volatility, and alpha needs the risk-free rate added back.

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