Mean-Variance Efficiency, Higher Moments, and Investor Preferences
Summary
This discussion distinguishes mean-variance efficiency from preferences over skewness and kurtosis. Mean-variance analysis focuses on expected return and variance, so it does not directly describe whether returns are positively or negatively skewed or how heavy their tails are. A risk-adjusted measure such as the Sharpe ratio likewise does not capture those distribution features, which motivates considering alternatives such as the Omega ratio when higher moments matter.
The answers emphasize that there is no universal trade-off between average return and a preference for positively skewed outcomes. Some investors may accept smaller typical returns in exchange for a small chance of a large gain; others may favor steadier profits while accepting a small chance of a severe loss, as in selling far out-of-the-money puts. Loss aversion can also lead investors to avoid potential losses even when that means giving up possible gains. These are descriptions of differing preferences, not a quantitative model or empirical comparison; the right choice depends on the risks an investor is willing to bear.
Key ideas
- Mean-variance efficiency evaluates expected return and variance but omits skewness and kurtosis.
- The Sharpe ratio does not represent the full shape of a return distribution.
- Some investors may prefer small frequent returns with a low probability of a large payoff.
- Other investors may prefer steady gains while bearing a small probability of large losses.
- Loss aversion can make investors willing to forgo potential gains to reduce exposure to losses.
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# Relationship mean variance efficiency and skewness of the return distribution? # Relationship mean variance efficiency and skewness of the return distribution? I am wondering what the relationship is between skewness, kurtosis and mean variance efficiency is. Is it correct that particular investors are willing to give up mean variance efficiency in return for greater probability of positive returns. Does this mean that a investor trades a lower average return for bearing more exposure towards positively skewed return distributions and a higher kurtosis? Similar to a lottery type pay-off function? Or do they accept a lower risk adjusted return instead of average return? Holding other factors constant such as risk aversion. ## Answer by Chris (score 1) https://quant.stackexchange.com/a/49118 Mean-variance efficiency generally only considers the first two moments. Skew and kurtosis are outside of it, and Sharpe is commonly criticized for not accounting for skew and kurtosis (leading to metrics like Omega as a replacement). Regarding your specific question, it depends entirely on what risks a given investor/trader wants to take. One person may prefer a return stream with smaller average returns but a small chance of a large payoff as a result of skew and/or kurtosis. Another may prefer small steady profits with a small risk of large losses (eg, selling deep OTM puts). All depends on which risks you're willing to take on. ## Answer by Charles Fox (score 0) https://quant.stackexchange.com/a/49100 Preferences vary by person. Some people have loss aversion. Their satisfaction from gaining \$1 is significantly smaller than their pain from losing \$1. People with this condition may give up potential gains to avoid small potential losses.
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