A compact mean-variance-skewness model for large-scale portfolio optimization and its application to the NYSE market

Abstract

This paper develops a portfolio optimization model that uses the first three moments of the distribution of the rate of return on investment in selecting portfolios. An alternative measure of skewness is designed for the purpose, and, in the grand scheme of compact factorization, the proposed model is transformed to an equivalent quadratic program with a quadratic constraint with 2 T nonlinear variables and terms, where usually T⩽50. Extensive computational results are obtained on a real-world dataset of the returns of about 3500 stocks that were traded in the NYSE from 3 January to 17 September 2002. In summary, the portfolios built by the proposed model gave the average return on investment of 66.85% over the course of 150 trading days, a period in time when US economy and stock markets suffered tremendously after the tragic events of September 2001.

Appendix: Proofs

Proof of Proposition 1

Consider vectors z₁=(z₁⁺, z₁⁻) and z₂=(z₂⁺, z₂⁻) with g(z₁)⩾g(z₂). Then,

Similarly, g(z₁)⩽g(z₂)⇔s(z₁)⩽s(z₂).□

Proof of Theorem 1

It suffices to show that

for is nonconvex.

Consider a vector z₁=(z₁⁺,z₁⁻) with s(z₁)=κ. Note that z₁⁻≠0. W.l.o.g., assume that z₁ satisfies z_1j⁺⩾z_1,j+1⁺ for j=1, …, T−1 and that there are l and k (l<k) such that z_1l⁺>z_1k⁺>0.

Consider a vector z₂ obtained from z₁ by exchanging z_1l⁺ and z_1k⁺ and their counterparts z_1l⁻ and z_1k⁻ which are equal to 0. Note that s(z₂)=κ as well. Now, choose a vector that is a convex combination of z₁ and z₂, say, z₃=(z₃⁺,z₃⁻) ≔ 0.5z₁⁺+0.5z₂⁺=(0.5z₁⁺+0.5z₂⁺,z₁⁻). We have

In the above, the strict inequality is obtained by the well-known Arithmetic–Geometric Inequality and by the fact that z_1l⁺≠z_1k⁺. This proves that S _z is nonconvex and completes the proof.□