# Is quadratic programming used to maximize portfolio skewness and kurtosis?

Quadratic programming, a type of convex optimization, is used to solve the minimum variance portfolio weights $$w = \arg \min_w \sigma_P^2 = w^\top \Sigma w$$

because the objective function coincides with quadratic programming, which takes the form: $$x = \arg \min_x x^\top A x$$

The maximum skewness and maximum kurtosis portfolios, on the other hand, are tensors that look like they would require a type of optimization of higher order (order-3 and order-4) than quadratic programming (which is order-2):

$$\arg \max_w \enspace s_P = w M_3 (w^\top\otimes w^\top)$$ $$\arg \max_w \enspace k_P = w M_4 (w^\top\otimes w^\top \otimes w^\top)$$ where $$M_3$$ and $$M_4$$ are the co-skewness and co-kurtosis matrices respectively. Would these two objective functions comply with the quadratic programming formula (second from the top)? If not, what is an appropriate optimizer? Or would quadratic programming work as long as the tensors $$s_P$$ and $$k_P$$ are flattened into 2-dimensional matrices?

Someone followed up the answers to this question with:

• I think that you cannot make us of QP here as the problem itself is not quadratic but cubic / quartic. Oct 19, 2020 at 6:56
• I also wonder whether this function (with skewness and/or kurtosis) has only one minumum/maximum, or if the extension to higher orders makes appearing also local minima/maxima.. Oct 19, 2020 at 9:29

The quadratic programming approach is used to solve problems of the form

$$\sum_i\beta_ix_i+\sum_i\sum_j \gamma_{ij}x_ix_j \quad s.t.\quad Ax\leq a\quad \mathrm{and}\quad Bx=b.$$

A portfolio optimisation that involves decisions over skew and kurtosis introduces terms in $$\sum_i\sum_j\sum_k\kappa_{ijk} x_ix_jx_k$$ and $$\sum_i\sum_j\sum_k\sum_l\theta_{ijkl} x_ix_jx_kx_l$$ $$-$$ the problem is thus not solvable using a QP.

A couple of older papers went with the polynomial goal programming (PGP) approach; I found one comprehensible example here. Another, supposedly faster approach is the utility-expansion method given in Jondeau/Rockinger. The PGP approach provides arbitrary weights for the moments whereas the ansatz of Jondeau/Rockinger is footed in utility theory (see my other post on this where I offered a cursory description of this.)

• could you add a link to your post on the utility-expansion method? Oct 19, 2020 at 14:11
• in your formula for quadratic problems, shouldn't the dual sum of the second term exclude the case where $i=j$? and just to check, when applied to the covariance matrix, $\beta_i$ becomes $\sigma_i^2$, right, and $\gamma_{i\neq j}$ is covariance $\sigma_{i\neq j}$? Oct 19, 2020 at 15:00
• I don't think that these methods are too complicated (I have never implemented them, though). IMHO, the idea never really took off in the academic world due to unwieldy data sets, complex estimation, etc. For toy problems with, say, up to ten or twenty assets you could easily give any good solver a try I'd say... Oct 21, 2020 at 11:40
• If we neglect non-negativity conditions, you could write something along the lines of multivariate newton raphson: fourier.eng.hmc.edu/e176/lectures/NM/node21.html Oct 21, 2020 at 14:13
• What I meant: formulate the FOC of your optimization (this is a vector) and derive the Jacobian of that. Then apply the NR-method. If you only add skew (for starters), you simply add $/gamma^2w^TM_i$ to each line $i$ of your FOCs where M is the skewness tensor. Oct 21, 2020 at 14:21