# Higher Moments and Minimum Variance

Several subsequent posts will analyze the relevance and comparative performance of higher and mixed moments to minimum variance portfolios (along with CVaR / ES and lower partial moments), extending previous US sector and global rotation models. To help motivate this, consider the following quote from Amaya and Vasquez [2010], which conjectures that *compensation for volatility depends on skewness level* (p. 16):

Stocks with negative skewness are compensated as suggested by the mean-volatility model: more volatility translates into more returns. However, as skewness increases and becomes positive, the positive relation between volatility and returns turns into a negative relation. For stocks with positive skewness, higher volatility means lower returns. Even more, the lowest returns are earned by the portfolio with stocks that have high positive skewness and high volatility.

This is an interesting extension of minimum variance, nuancing the role of volatility consistent with previously conjectured lottery-seeking investor behavior and thus further affirmation of portfolio theory being dead (ibid):

Stocks with low skewness are compensated with high returns when volatility increases, but stocks with high skewness are compensated with high returns when volatility decreases. This compensation for volatility presents somewhat of a puzzle given that, under the mean-variance model, investors always prefer low volatility and not high volatility as implied by our result. Hence, we argue that investors accept low returns and high volatility only because they are more attracted to high positive skewness; they are skewness lovers or lotto investors.

Although this return-variance-skew relationship is intuitively obvious to traders, this article nicely extends the intuition motivating minimum variance optimization into higher moments.

There is also an argument to be made with respect to kurtosis and skewness. In the literature it says that positive skewness is good but kurtosis is bad. But when you think it through, kurtosis is only bad when you have negative skewness. If you have a greater likelihood of large gains (skewness), then you may prefer positive kurtosis in order for those gains to be as large as possible. Mean variance is a very limited way to look at the world, but it does simplify all the not so obvious things. There is more in heaven and earth than are dreamt ofin our philosphy.

@Gary: agreed; absolutely correct. An upcoming post will analyze both (co-) skew and (co-) kurtosis for the rotation models, along with modifying the weight optimizations. As will be shown, skewness preference is significantly more subtle than positive/negative.