Outline the problems that come up when alphas are measured for strategies with nonlinear payoffs.
The idea
Standard alpha and Sharpe measures assume that a portfolio's return is linearly related to the benchmark or factor returns and is roughly symmetric. Strategies with nonlinear payoffs, such as selling options, buying options or dynamic trading that replicates them, break that assumption. A put-writing strategy collects a steady premium in calm markets, which shows up as a high alpha and Sharpe ratio. Yet it carries rare large losses, with negative skewness and fat tails, that a sample without a crash never reveals. A convex strategy whose beta rises in up markets and falls in down markets yields a positive intercept in a linear regression even with no skill. Remedies include adding option-like factors, such as (market return)², using long samples that contain stress periods, and examining skewness and drawdowns as well as mean and volatility. Exam point: a high alpha from an option-based strategy may be pay for tail risk, not skill.
Formulas
True expected return of a put-writing strategy
E(R) = (1 − p) × G − p × L
- p
- the probability of a loss month
- G
- the monthly gain from premium in calm months
- L
- the size of the loss in a crash month
A sample with no crash shows only G and overstates the expected return and the Sharpe ratio.
A practice question
Which strategy has a clearly nonlinear payoff relative to the equity index, making a linear alpha regression unreliable?
- Holding a diversified stock portfolio with a beta of 1.0.Why not: A beta-one portfolio has a linear payoff relative to the index.
- Holding a diversified portfolio with a beta of 1.5, financed by borrowing.Why not: Leverage scales the exposure but keeps the payoff linear.
- Writing out-of-the-money index put options each month and holding the premium in cash. - correct
- Holding a long position in equity index futures.Why not: Index futures have a linear payoff in the index level.
Option positions have kinked, nonlinear payoffs, so returns are not a straight-line function of index returns. Linear regression then gives misleading alphas and betas.
A harder one
A fund writes put options. Each month it earns +1.2% with probability 0.97 and loses 12% with probability 0.03. A 24-month track record happens to contain no loss. Which is closest to the strategy's true expected monthly return?
- 0.80% - correct
- 1.20%Why not: Reports the observed average from the sample without a loss, ignoring the 3% chance of a 12% loss.
- 1.16%Why not: Includes the probability-weighted gain but omits the probability-weighted loss of 0.36%.
- -5.40%Why not: Weights the gain and the loss equally instead of by their probabilities 0.97 and 0.03.
Expected return = 0.97 x 1.2 + 0.03 x (-12) = 1.164 - 0.36 = 0.80% a month. The sample average of 1.2% overstates it because the loss never appeared.
The hardest tier
A strategy has a market beta of 1.5 when the market rises and 0.5 when it falls, with no skill involved. In a sample with two equally likely outcomes, the market excess return is +8% in the up state and -4% in the down state, so the strategy returns +12% and -2%. What alpha does a linear regression of the strategy's returns on the market's return give?
- 0.00%Why not: Assumes that because the strategy has no skill the regression alpha must be zero, ignoring that the payoff is convex.
- 3.00%Why not: Uses the average beta of 1.0 instead of the fitted slope of 1.17.
- 2.67% - correct
- 5.00%Why not: Uses the strategy's mean return as alpha without removing any market exposure.
Mean market = 2.0%, mean strategy = 5.0%. Slope = Cov/Var = 42/36 = 1.167. Alpha = 5.0 - 1.167 x 2.0 = 2.67%. A convex payoff produces positive alpha in a linear regression with no skill at all.
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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- Lay out the possible explanations offered for the risk anomaly.