How to add the effect of skewness in the portfolio optimisation objective function?

Question

I have the following risk adjusted portfolio which I optimise,

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 $𝑔𝑛𝑠=𝜇𝑛𝑠−𝜈𝑛$. Where $𝜇𝑛𝑠$ are the returns of the assets and $𝜈𝑛$ 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 $𝑑𝑛𝑛=Σ^S_s 𝑝𝑠*(𝜎𝑛𝑠)^2$. Where $𝜎𝑛𝑠$ 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

Answer 1

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.

Answer 2

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.