Skewness Preferences Under Log Utility and Target-Based Objectives
Summary
The answer distinguishes preferences for return skewness according to an investor’s objective. It says that maximizing expected log wealth can favor higher odd moments, such as mean return and skewness, while favoring lower even moments, such as volatility and kurtosis. The explanation is based on a Taylor expansion of the logarithm, though the document gives no derivation or qualifications for the approximation.
A different objective—maximizing the chance of exceeding a fixed return threshold—can imply a different preference. The cited discussion says the preferred skew may depend on the Sharpe ratio over the relevant horizon: investors with a higher Sharpe may favor positive skew, while those with a lower Sharpe may favor negative skew to improve the chance of reaching a near-term target. This is a brief account of an argument, not a settled general rule; conclusions depend on the objective and time horizon, and the document supplies no empirical test.
Key ideas
- Expected log wealth can imply preference for positive skewness and lower volatility, according to the answer's Taylor expansion argument.
- A target-exceedance objective can produce different skewness preferences from log-wealth maximization.
- The cited discussion links the preferred skew direction to the Sharpe ratio over the investment horizon.
- Investors with different horizons or objectives may rationally take opposite sides of skew exposure.
Tags
Full text
# Is positive skewness preferences rational or irrational? # Is positive skewness preferences rational or irrational? Is positive skewness preference rational or irrational? I have a great trouble understanding why investors should prefer positive skewness over negative one. Sometimes it is argued that preference for positive skewness is a behavioral and consequently irrational phenomenon as stated here. And sometimes it is argued it can be justified by assumption of concave utility and risk –averse investors as stated here (the second answer). It is clear that in this regard this preference is perfectly rational. Now which interpretation is correct? An intuitive response would be highly appreciated. ## Answer by steveo'america (score 3) https://quant.stackexchange.com/a/40635 I think the usual argument is that if an investor is maximizing expected log wealth, then this implies preference for higher odd order moments (mean return, skew, etc.) and for lower even order moments (volatility, kurtosis, etc.). This comes from the Taylor expansion of the log. However, if one wishes to maximize the probability that returns over a given period exceed some fixed amount, the story becomes more complicated. This paper on higher order moment preferences argues that if the Sharpe is larger than 1 (on the timescale in question) one would prefer positive skew, but otherwise would prefer negative skew. The idea seems to be that if your Sharpe is low, and you are at significant risk of not beating the threshold, you would "sell lottery tickets" and risk having a payout in order to boost returns in the short term. This would allow two traders with perfect knowledge of the moments of returns to trade skew with each other if they had different investment horizons (the long term investor buys skew, while the short term sells skew).
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