Portfolio Skewness and Kurtosis Require Higher-Order Optimization
Summary
The document explains why portfolio skewness and kurtosis objectives do not fit standard quadratic programming. Variance is quadratic in portfolio weights, but skewness and kurtosis involve third- and fourth-order terms, respectively, so flattening tensors into matrices does not make those objectives quadratic. A quadratic program can handle a quadratic objective with linear constraints; these higher-order moment objectives fall outside that form.
It mentions polynomial goal programming as one approach, allowing arbitrary weights on moments, and a utility-expansion method associated with Jondeau and Rockinger. The latter is grounded in utility theory. These are presented as alternatives rather than fully specified procedures: the text gives no implementation details, comparative evidence, or guidance on constraints and numerical behavior. The question of how to choose an optimizer therefore depends on the exact objective and modeling assumptions.
Key ideas
- Portfolio variance is quadratic in portfolio weights and can be expressed as a quadratic programming objective.
- Skewness and kurtosis introduce cubic and quartic terms, which standard quadratic programming does not represent.
- Flattening higher-order moment tensors does not change the polynomial order of the optimization objective.
- Polynomial goal programming can assign arbitrary weights to portfolio moments.
- A utility-expansion method offers a theory-based alternative, though the document does not detail its implementation.
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# Is quadratic programming used to maximize portfolio skewness and kurtosis?
# 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:
- how to transform a cubic optimisation problem into a quadratic
## Answer by Kermittfrog (score 3, accepted)
https://quant.stackexchange.com/a/58789
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.)Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.