Deriving Portfolio Skewness and Kurtosis from Asset Returns
Summary
The document discusses how portfolio skewness and kurtosis relate to the higher-order co-moment tensors of asset returns. Its main practical point is that the tensor formulas can be understood as contracting each tensor with the portfolio weights until a scalar statistic remains. It outlines a computational procedure for skewness: take slices of the three-dimensional co-skewness tensor, evaluate each slice as a quadratic form in the weights, then combine the resulting vector with the weights. The same sequence of reductions applies to the four-dimensional co-kurtosis tensor.
The discussion also suggests a direct alternative when asset return observations are available: calculate portfolio returns period by period, then estimate skewness and kurtosis from that portfolio return series. It does not provide a full derivation from expectation operators or identify who first published the formulas. The tensor notation is explained mainly through dimensions and computation, so readers seeking a probability-theory derivation will need another source.
Key ideas
- Portfolio higher moments can be calculated from co-skewness and co-kurtosis tensors and portfolio weights.
- Tensor contractions successively reduce the co-moment arrays to scalar portfolio statistics.
- For skewness, contract tensor slices with the portfolio weights, then combine the resulting vector with those weights.
- When asset return observations are available, calculate portfolio returns first and apply sample skewness or kurtosis estimators.
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# Derivation of portfolio skewness and portfolio kurtosis
# Derivation of portfolio skewness and portfolio kurtosis
Where can I find derivation of formula for portfolio skewness and kurtosis? I can find formulas everywhere, but not their derivations?
For example, the portfolio variance formula, $\sigma_P = w^\top \Sigma w$ is well known, where $\Sigma$ is the covariance matrix, and I can find the derivation of that formula in a lot of books, but I can't find anything on the formuals for:
- portfolio skewness, $s_P = w^\top M_3 (w\otimes w)$, and
- portfolio kurtosis, $k_P = w^\top M_4 (w\otimes w\otimes w)$,
where $M_3$ is the co-skewness matrix and $M_4$ is the co-kurtosis matrix.
They are just given the way they are. I'm not strong enough at probability theory to use it to derive the formulas from the expectations operator. Who was the first person to derive them? Where were they first published?
## Answer by Richi Wa (score 1)
https://quant.stackexchange.com/a/3662
What is the data basis that you start from? If you just have the covariance matrix, then you can only calculate portfolio variance or volatility by $$ w^T \Sigma w$$ where $w$ are the portfolio weights and $\Sigma$ is the covariance matrix. If you have the individual asset continuously compounded returns $r^j_t$ where $j$ indexes assets, $j=1,\ldots,N$, and $t$ stands for time, $t=1,\ldots,T$, then you can also calculate the portfolio returns for each points in time $$r_t = \sum_{j=1}^N w_j r^j_t$$ and then apply the standard variance estimator on $(r_t)_{t=1}^T$. Coming back to your question, having $(r_t)_{t=1}^T$ you can calculate skewness and kurtosis on this sample. You find the formulas on wikipedia.
## Answer by Kermittfrog (score 0)
https://quant.stackexchange.com/a/58733
The key here is in finding that $-$ for our application in finance $-$, the Kronecker product notation is 1) a way to shorten notation and 2) a function that is well represented in mathematical toolboxes such as `Matlab`or `R`.
Assume there are $N=3$ random returns ($x_1,x_2,x_3$) and some weight vector $w$ of dimension $N\times 1$. The vector of expected returns with typical entry $E(x_i)$ has the dimension $\mu$ is $N\times1$, and $\mu^T\times w$ is a scalar ($1\times1$). The covariance matrix $\Sigma$ with typical entry $E((x_i-\mu_i)(x_j-\mu_j))$ has the dimension $N\times N$, and $w^T\times \Sigma \times w$ is again a scalar ($1\times1$). The co-skewness tensor $M_3$ with typical entry $E((x_i-\mu_i)(x_j-\mu_j)(x_k-\mu_k))$ has the dimension $N\times N\times N$, and again $w^TM_3\left(w\otimes w\right)$ is $1\times 1$. The same then holds for the kurtosis with $M_4$ of size $N\times N\times N \times N$.
Effectively, each statistics is simply an $K$-dimensional object, with $K=1,2,3,4$...
Now to the question: "How to compute the portfolio skewness?" (or kurtosis)
Assume your co-skewness matrix $M_3$ is stored as a three-dimensional array. You now right multiply the 3d-array ($N \times N \times N$) with a $N\times 1$ vector of portfolio weights $w$ $-$: the result is an $N\times N$ matrix! The next step is then the typical left/right multiplication with $w^T$ and $w$ et voilà: You have a scalar.
The algorithm (for the skewness) is:
- Build your $3$-dimensional skewness tensor $M_3$ with typical element as above
- For each entry $i$ in the last dimension, calculate $q_i=w^TM_3(.,.,i)w$. This yields a $N\times 1$ vector (by construction). Finally, $w^Tq$ gives you the portfolio skewness.
The same approach works for the portfolio kurtosis, i.e. reduce a 4d-array to 3d to 2d to 1d to scalar.
HTH?Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.