How do I calculate the skewness of a portfolio of assets?

Question

I need to calculate the skewness of a portfolio consisting of 6 assets. I know that for that I would need the co-skewness matrix between the assets. Does anybody know the formula for co-skewness or any simple software to calculate a co-skewness matrix?

Any useful information would be highly appreciated.

Answer 1

Actually, co-skewness is represented by a rank 3 tensor, rather than a matrix.

I'm going to reproduce the formulation from Bhandari and Das, Options on portfolios with higher-order moments, but I'll add and omit some details.

The co-skewness tensor is $$ S_{ijk} = E \left[ r_i \times r_j \times r_k \right] = \frac{1}{T} \sum_{t=1}^T r_i(t) \times r_j(t) \times r_k(t) $$ where $r$ are asset returns over $T$ time periods.

Then, given portfolio weights $w$, mean asset returns $\mu$, covariance matrix $\Sigma$, and portfolio variance $\sigma_p^2(w) = w\prime \Sigma w$, we calculate moments: $$\begin{eqnarray*} m_1 & = & w\prime \mu \\ m_2 & = & \sigma_p^2 + m_1^2 \\ m_3 & = & \sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N w_i w_j w_k S_{ijk} \end{eqnarray*}$$

The portfolio skewness is then $$ S_p = \frac{1}{\sigma_p^3} \left[ m_3 - 3m_2 m_1 + 2m_1^3 \right] $$

In the case of a 6-asset portfolio, the co-skewness tensor will contain 216 components; however, due to symmetry, it only contains 56 independent components.

Therefore, it can be helpful to reformulate the portfolio skewness equation for computational efficiency. To do this, we can start with the definition of skewness for portfolio returns, $$ S_p = \frac{1}{\sigma_p^3} E [ \left( \sum_{i=1}^N w_i r_i \right)^3 ] \quad , $$

and then apply the multinomial theorem to obtain the portfolio skewness in terms of only the independent components.

Update

• Especially for longer time series, the return moments should be centered on the means, i.e., $r_i = R_i - \bar{R}_i$
• In the case of daily returns, $R_i(t) = \frac{P(t) - P(t-1)}{P(t-1)}$, where $P(t)$ is the closing price at time $t$.
• Be sure the prices for returns are comparable from period to period. For example, stock prices may need adjustments to account for dividend payments. See this Q & A on return measurement for more discussion.

note: I edited the equation for the co-skewness tensor above.

Answer 2

Have a look at PortfolioAnalytics in R.

> library(PerformanceAnalytics)
> data(managers)
> CoSkewness(managers, managers)

Answer 3

Maybe you like working with coskewness. But it is not needed if you just want to estimate the skewness of the portfolio. If you have retunr times serise $(r^i_t)_{t=1}^T$ for each asset $i$ and the weights $w_i$ that these assets have in your portfolio then you can form $$ r_t = \sum_{i=1}^6 w_i r^i_t \quad \text{for each } t, $$ and you simple estimate all moments on $(r_t)_{t_1}^T$. The solution will be the same as any matrix operation. Eg the variance is either $$ VAR(r_t) $$ or equivalently (!) $$ w^T \Sigma w $$ where $\Sigma$ is the covariance matrix of those 6 assets. The numbers will be the same. If you need comoments then you do so by using expressions of the form $$ E[r_t^k*(r_t^i)^m], $$ for some integer $k$ and $m$.

Answer 4

Happened to get stuck in this problem years ago. Don't need to use co-skewness or any co-moments to calculate the variance, skewness, and kurtosis of a portfolio. Just use the raw returns of the assets and the weights. This optimization can be implemented in Excel, R (using nonlinear) and Python (nonlinear or minimize with scipy).