3
$\begingroup$

I have the following risk adjusted portfolio which I optimise,

enter image description here

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

$\endgroup$
12
  • $\begingroup$ Do you want to maximize or minimize portfolio skewness? $\endgroup$
    – develarist
    Oct 15, 2020 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, 2020 at 9:46
  • $\begingroup$ The link requires login. Edit your question describing the scenario adjustment to $C$ $\endgroup$
    – develarist
    Oct 15, 2020 at 9:48
  • 1
    $\begingroup$ 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, 2020 at 10:21
  • 1
    $\begingroup$ A positive skewness induces more frequent 'positive extremes' (compared to a symmetric distribution.) $\endgroup$ Oct 15, 2020 at 11:08

2 Answers 2

4
$\begingroup$

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.

$\endgroup$
7
  • $\begingroup$ 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, 2020 at 12:16
  • 1
    $\begingroup$ 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. $\endgroup$ Oct 15, 2020 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, 2020 at 12:24
  • 3
    $\begingroup$ 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 $O(n^3)$. $\endgroup$ Oct 15, 2020 at 12:42
  • 1
    $\begingroup$ ... 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 :) $\endgroup$ Oct 15, 2020 at 12:44
2
$\begingroup$

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.

$\endgroup$
3
  • $\begingroup$ 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... :( $\endgroup$ Oct 15, 2020 at 19:24
  • $\begingroup$ should the $\frac{1}{3}$ coefficient be changed to $\frac{1}{6}$? $\endgroup$
    – develarist
    Oct 15, 2020 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, 2020 at 7:09

Your Answer

By clicking β€œPost Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.