Convex Optimization of Portfolio Kurtosis

This work presents five convex reformulations of portfolio kurtosis that allows us to pose kurtosis as a parametric convex risk measure. These new reformulations are based on new formulas for estimation of portfolio moments and co-moments matrices, second order cone and semidefinite programming. The advantages of these new reformulations are that allow us to optimize portfolio kurtosis using disciplined convex programming rules; can be solved using state of art solvers that supports second order cone and semidefinite programming; are suitable for a medium size problems; and have the flexibility to model several objective functions like risk adjusted return ratio maximization, kurtosis minimization, kurtosis constraints and risk parity based on kurtosis. Also, we present the heuristic of lower semi-kurtosis, a parametric optimization model based on lower semi-co-kurtosis matrix that allows us to minimize the left tail of portfolio returns