# derivation of formula for portfolio skewness and 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 is well known and I can find the derivation of that formula in a lot of books, but I can't find anything on portfolio skewness formula and 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?

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## 1 Answer

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.

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i know how to calculate it when i have the formulae, but the derivation of the given formulas is what i need. you just can't say this is the formula as it is without no proof or derivation. that is what i need – mary Jun 23 '12 at 18:38
@mary When you go to the level of portfolio returns then the standard estimators for skewness and kurtotis apply. So any derivation of these estimators would hold for a general prove. The first formula above e.g. is just algebra and the standard estimator. In the case that we assume iid returns (w.r.t. time) we just work with a sample and the portfolio case is not different from the standard case. – Richard Aug 20 '12 at 20:11