Lay out the possible explanations offered for the risk anomaly.
The idea
Explanations offered for the risk anomaly fall into three groups. The first is leverage constraints. Investors who cannot or will not borrow reach for high-beta stocks to raise expected return, bidding their prices up and their future returns down, while low-beta stocks are neglected. The second is institutional and agency explanations. Managers benchmarked to an index with a tracking-error budget find low-beta stocks costly to hold, since a beta below one creates tracking error, so they have little incentive to exploit the anomaly. The third is behavioural. Investors overpay for lottery-like, positively skewed, high-volatility stocks and are overconfident in their forecasts. The risk-based explanation, that low-risk stocks are riskier in bad times, has little empirical support. Exam point: match each explanation to its mechanism, which is a constraint, benchmarking or preference, and recall that the rational risk story is the weakest.
Formulas
Tracking error from a beta gap
TE = |1 − β| × σm
- β
- the portfolio's beta to the benchmark
- σm
- the benchmark's volatility
Counts only the beta shortfall, with no other active positions. It shows why a tracking-error budget keeps benchmarked managers out of low-beta stocks.
A practice question
Which explanation of the risk anomaly rests on investors who cannot or will not borrow buying high-beta stocks to raise their expected returns?
- Leverage constraints. - correct
- Lottery-like payoff preferences.Why not: A behavioural story about skewness, which works through high-volatility stocks, not through borrowing limits.
- Benchmarking and tracking-error limits for institutional managers.Why not: An institutional story about managers' mandates, not about investors who cannot borrow.
- Compensation for greater exposure to bad economic states.Why not: A risk-based story that says low-risk stocks must be riskier, which is not a borrowing-constraint argument.
With leverage constraints, investors seeking higher expected returns overweight high-beta stocks instead of borrowing, bidding up their prices and lowering future returns, while low-beta stocks are left cheap.
A harder one
The risk-free rate is 2% and the market risk premium is 6%. An investor wants an expected return of 11% but is not allowed to borrow, so her fully invested portfolio must have a beta high enough to reach it. Which beta is needed?
- 1.83Why not: Divides the target return by the premium, forgetting to subtract the risk-free rate.
- 0.67Why not: Inverts the ratio (premium divided by required excess return).
- 1.50 - correct
- 1.12Why not: Divides the required excess return by the market's total expected return of 8% instead of its 6% premium.
CAPM: 11 = 2 + beta x 6, so beta = (11 - 2)/6 = 1.50. Investors who cannot lever must reach for high-beta stocks, which under the leverage-constraint explanation bids their prices up and lowers their future returns.
The hardest tier
Researchers observe that the highest-volatility stocks have earned the lowest average returns, have positively skewed returns, and are held disproportionately by individual retail investors. Which explanation fits best?
- Investors overpay for lottery-like, positively skewed stocks because of a preference for such payoffs. - correct
- Leverage constraints, because investors who cannot borrow overweight the highest-volatility stocks.Why not: Leverage constraints work mainly through high-beta stocks and do not account for skewness or retail ownership.
- Benchmarking, because institutional managers must hold high-volatility stocks to match an index.Why not: Institutional benchmarking does not explain heavy retail holdings or the preference for positive skewness.
- Compensation for systematic risk, because high-volatility stocks do worst in recessions and so must offer higher returns.Why not: A risk story predicts higher returns for riskier stocks, contrary to the lowest returns observed.
Skewness plus retail ownership plus low returns point to a preference for lottery-like payoffs: investors bid up the stocks that offer a small chance of a large gain, depressing their average returns.
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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