# 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, $$w^\top \Sigma w$$ is well known 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 and portfolio kurtosis. 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?

• none of the answers so far show a derivation of portfolio skewness and portfolio kurtosis. a source would be good. Apparently, while the covariance matrix used in portfolio variance is $N\times N$, the coskewness matrix used in portfolio skewness is $N\times N^2$, while the cokurtosis matrix used in portfolio kurtosis is $N\times N^3$ quantatrisk.com/2013/01/20/coskewness-and-cokurtosis – develarist Sep 9 at 0:00

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.
• she's looking for the $w^\top \Sigma w$ equivalent of portfolio skewness and portfolio kurtosis, derived from the top. The definitions of portfolio skewness and portfolio kurtosis are given at the following but it has no derivations either quantatrisk.com/2013/01/20/coskewness-and-cokurtosis – develarist Sep 9 at 0:08