I have the following risk adjusted portfolio which I optimise,

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where gamma is the risk return trade off, $r$ are the returns and $C$ is the covariance matrix which considers scenarios, so it is not defined as $r^\top r$, but as shown in the following Markowitz paper (page 3, $C = D + GPG'$): https://www.jstor.org/stable/2327552?seq=1

$P$ is a diagonal $SxS$ matrix with the probability

$G$ is an $NxS$ matrix whose entries are given by $𝑔𝑛𝑠=πœ‡π‘›π‘ βˆ’πœˆπ‘›$. Where $πœ‡π‘›π‘ $ are the returns of the assets and $πœˆπ‘›$ are the returns of the nth asset class weighted by the probabilities of the scenarios. $N$ total numer of assets

$D$ is a diagonal $NxN$ matrix whose entries are given by $𝑑𝑛𝑛=Ξ£^S_s 𝑝𝑠*(πœŽπ‘›π‘ )^2$. Where $πœŽπ‘›π‘ $ is the standard deviation of the nth asset for the sth scenario

Now I want to add also the third moment thus the skewness to this optimisation function, but I do not really know how, and if I have to include the scenarios in this skewness and how.

Can you guide me pls? Thanks

  • $\begingroup$ Do you want to maximize or minimize portfolio skewness? $\endgroup$ – develarist Oct 15 '20 at 9:41
  • $\begingroup$ this is a very good question..actually I do not really know..they are both acceptable I guess for returns. what is the typical way of thinking about it? many thanks for your answer $\endgroup$ – Luigi87 Oct 15 '20 at 9:46
  • $\begingroup$ The link requires login. Edit your question describing the scenario adjustment to $C$ $\endgroup$ – develarist Oct 15 '20 at 9:48
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    $\begingroup$ notwithstanding the scenario adjustment, become familiar with the co-skewness matrix used to calculate portfolio skewness quantatrisk.com/2013/01/20/coskewness-and-cokurtosis $\endgroup$ – develarist Oct 15 '20 at 10:21
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    $\begingroup$ A positive skewness induces more frequent 'positive extremes' (compared to a symmetric distribution.) $\endgroup$ – Kermittfrog Oct 15 '20 at 11:08

Let's derive a possible approach from utility theory.

Our investor is risk averse and exhibits CARA utility using an exponential utility function with risk aversion parameter $\gamma>0$ (risk averse agent):

$$u(x)=\frac{1-e^{-\gamma x}}{\gamma}$$

A 3rd order Taylor series expansion around $x=0$ yields

\begin{align} u(x)\approx& x - \frac{1}{2}\gamma x^2+\frac{1}{6}\gamma^2x^3 \end{align}

Thus, the expected utility (which is to be maximized) is \begin{align} E\left[u(x)\right]&\approx E(x)-\frac{1}{2}\gamma E(x^2)+\frac{1}{6}\gamma^2 E(x^3)\\ &=\mu_x-\frac{1}{2}\gamma\left(\sigma_x^2+\mu_x^2\right)+\frac{1}{6}\gamma^2\left(skew_x+3\mu_x\sigma_x^2+\mu_x^3\right) \end{align}

In a portfolio application, we can now make use of standard notation and the helpful hint from @develarist in the comments and maximize

$$ w^T\mu-\frac{1}{2}\gamma w^T\Sigma w+\frac{1}{6}\gamma^2 w^TM_3(w\otimes w) $$

subject to your investment restrictions.

Effectively, this approach is (only) a starting point for incorporating skewness in your optimisation. Here, the tradeoff is clearly between $-.5\gamma$ 'penalty' for variance and a 'reward' of $\frac{1}{6}\gamma^2$ for positive skewness. You can certainly disentangle the two and simply introduce two parameters of your choice, say $a$ and $b$ to penalize/reward portfolio variance and portfolio skewness.

  • $\begingroup$ thanks a lot. this is a great explanation. So actually increasing $Ξ³$ the risk is minimise and the positive skewness is increased...if $Ξ³$ is a vector of values from -1000 to +1000 if for each value I run my optimisation so the efficient frontier should become a surface with x volatility, y return, z skewness I could simply choose the point in the surface which I like the most (eg. max sharpe) and in turn I will have the gamma..does it sound a reasonably strategy? I do not want to impose a gamma but I want to choose the best point in the efficient frontier which will then have a certain gamma $\endgroup$ – Luigi87 Oct 15 '20 at 12:16
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    $\begingroup$ Hi Luigi, yes to all three. 1) Your resulting surface is threedimensional, 2) varying $\gamma$ will trace out various combinations, 3) You need a concept for "optimality", i.e. which parameter $\gamma$ is ultimately to be used. $\endgroup$ – Kermittfrog Oct 15 '20 at 12:19
  • $\begingroup$ thanks..just one last point..M3 is the coskewness matrix right? if I calculate it as $E[Ξ£*ΞΌ]$ where sigma is not a typical covariance but my scenario based covariance, do you think the conclusion as above would stil apply? $\endgroup$ – Luigi87 Oct 15 '20 at 12:24
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    $\begingroup$ Technically, M3 is not a matrix (anymore) but a 3D-tensor. Nevertheless, you can represent it as a (2D) matrix, of course. I am afraid that you cannot simply replace a (scenario based) covariance into the M3 matrix as it is an expectation, i.e. you must specify (scenario) expectations regarding 2nd and 3rd (co-)moments. Your scenario specification effort is hence $O(n^3)$. $\endgroup$ – Kermittfrog Oct 15 '20 at 12:42
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    $\begingroup$ ... by the way: The $O(n^m)$ behavior ($m$=number of moments to consider) is one of the reasons why the "moment-based-portfolio-optimization" approaches have never really taken off from academia :) $\endgroup$ – Kermittfrog Oct 15 '20 at 12:44

Instead of starting from a CARA utility function like how the other answer does, an alternative for incorporating portfolio skewness in the mean-variance model's objective function, without risk-aversion parameter $\gamma$ or going through a Taylor series expansion of some arbitrarily asserted utility function, could be

$$\arg \max_w \enspace w^T\mu-\frac{1}{2} \left( w^T\Sigma w \right) +\frac{1}{3} \left[ w^TM_3(w\otimes w )\right], \hspace{1cm} 1_N^\top w = 1$$

where $M_3$ is the co-skewness matrix. This formulation would be suitable if investors' preferences are unknown and we don't want to assert arbitrary assumptions for investor preferences.

  • $\begingroup$ Hi @develariat; this looks very ad hoc to me. But adding weights to vol/skew is of course a bit ad hoc as well. I’ll try to find some of the references this weekend. (No promises, though... :( $\endgroup$ – Kermittfrog Oct 15 '20 at 19:24
  • $\begingroup$ should the $\frac{1}{3}$ coefficient be changed to $\frac{1}{6}$? $\endgroup$ – develarist Oct 15 '20 at 23:26
  • $\begingroup$ @develarist, thanks for proposing an answer. I agree and happy with both the soluion even if the manager of the fund I am implementing this wants to have a risk aversion parameter, so i will stick to the Kermittfrog solution, however, the difficult part for me here is the definition of $M_3$ considering the fact that it has to be scenario based and should be defined in a similar manner as $C$...but so far I cannot find anything in literature :(..any idea would be very much appreciated $\endgroup$ – Luigi87 Oct 16 '20 at 7:09

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