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?

• 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 '20 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.

• 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. – Ric Aug 20 '12 at 20:11
• 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 '20 at 0:08
• @develarist I see what you mean. For applications like portfolio optimization the matrix formulation might be extra useful. If one just wants to calculate these quantities for given weights, then it is sometimes forgotten that one can first look at the portfolio pnl and then calculate statistics from there. – Ric Sep 10 '20 at 6:26

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 Matlabor 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?

• The question is how to derive portfolio skewness, not how to calculate it – develarist Oct 16 '20 at 23:23
• What do you mean by derive? It is E((x-mx)^3), no? Simply insert x=w^tr and solve the third order expectation - which is a bit tedious, I admit. But nevertheless, it is straightforward. If you want, you can open another question on this (or point me to one) and we can get to the bottom of this. – Kermittfrog Oct 17 '20 at 10:07