Including Portfolio Skewness in Mean-Variance Optimization

Summary

The note explains how to incorporate skewness into portfolio optimization when a standard mean-variance objective does not capture the third central moment. It first expresses each portfolio return as a weighted sum of asset returns, then estimates portfolio skewness from the sample’s centered cubed returns, scaled by the portfolio standard deviation cubed. Portfolio mean and variance can likewise be computed from asset weights, expected returns, and the covariance matrix.

One proposed formulation maximizes expected return minus a variance penalty while requiring skewness to exceed a chosen threshold. The response also suggests constraining a Cornish-Fisher value-at-risk measure instead. These are formulations to consider rather than a complete optimization procedure: the note gives no data, numerical example, or guidance on selecting the threshold or penalty, and the estimates depend on the observed return sample.

Key ideas

• Portfolio returns are weighted sums of constituent asset returns.
• Sample skewness uses the third centered moment divided by standard deviation cubed.
• Portfolio mean and variance can be expressed using asset expectations and the covariance matrix.
• A skewness threshold can be added as a constraint to a return-and-variance objective.
• A Cornish-Fisher value-at-risk measure is offered as an alternative constraint.

Tags

# How to optimize a portfolio using skewness?


# How to optimize a portfolio using skewness?












I am trying to do portfolio optimization for 5 stocks taking into account skewness of the portfolio but I am unable to incorporate skewness to the mean variance model.

Can anyone please help on how to go about it citing the formulae used for portfolio optimation including the objective function?

## Answer by Richi Wa (score 1)

https://quant.stackexchange.com/a/21467

You can not account or skewness in the mean-variance framework as skewness is the third central moment. Thus what I would do is

- formulate the skewness in terms of the asset returns. I.e. for each time-step you have $$ r_t = \sum_{i=1}^5 w_i r^i_t, $$ where $r_t^i$ is the return of asset $i$ at time $t$, $w_i$ is the weight and $r_t$ the portfolio return at $t$.

Then you can use the empirical estimator of skewness: $$ skew = \frac{ 1/T \sum_{t=1}^T (r_t-\mu)^3}{ \sigma^3}, $$ where you need the portfolio variance $$ \sigma^2 = w \Sigma w $$ and the expected value $$ \mu = 1/T \sum_{t=1}^T r_t, $$ where the above is the sample estimator and $$ \mu = \sum_{i=1}^5 w_i \mu_i $$ is the expression in terms of individual expectations. Then you can use this skewness above, $\sigma$ and $\mu$ to define the problem. E.g. $$ \mu - \lambda \sigma^2 \rightarrow Max $$ under the constraint $skew \ge x$ for some desired level $x$. Or you use the definition of Cornish-Fisher-VaR in the constraint.

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