A symmetric LPM model for heuristic mean–semivariance analysis

While the semivariance (lower partial moment degree 2) has been variously described as being more in line with investors’ attitude towards risk, implementation in a forecasting portfolio management role has been hampered by computational problems. The original formulation by Markowitz (1959) requires a laborious iterative process because the cosemivariance matrix is endogenous and a closed form solution does not exist. There have been attempts at optimizing an exogenous asymmetric cosemivariance matrix. However, this approach does not always provide a positive semi-definite matrix for which a closed form solution exists. We provide a proof that converts the exogenous asymmetric matrix to a symmetric matrix for which a closed form solution does exist. This approach allows the mean–semivariance formulation to be solved using Markowitz's critical line algorithm. Empirical results compare the cosemivariance algorithm to the covariance algorithm which is currently the best optimization proxy for the cosemivariance. We also compare our formulation to Estrada's (2008) cosemivariance formulation. The results demonstrate that the cosemivariance algorithm is robust to a 45 security universe and is still effective at increasing portfolio skewness at a 150 security universe. There are four major benefits to a usable mean–semivariance formulation: (1) managers may engineer skewness into the portfolio without resorting to option strategies, (2) managers will be able to evaluate the skewness effect of option strategies within their portfolio, (3) a workable mean–semivariance algorithm leads to a workable n-degree lower partial moment (LPM) algorithms which provides managers access to a wider variety of investor utility functions including risk averse, risk neutral, and risk seeking utility functions, and (4) a workable LPM algorithm leads to a workable UPM/LPM (upper partial moment/lower partial moment) algorithm.

Research highlights
► We clarify the problem of endogenous and exogenous cosemivariance matrices.
► We compare the resulting E–S formulation to the Markowitz E–V formulation and Estrada's (2008) cosemivariance heuristic formulation.
► Empirical tests were run for 150 securities from 2001 to 2009 both with and without the financial crisis of 2008.
► Tests indicate that this E–S formulation delivers statistically superior results to the E–V and Estrada E–S.
► This E–S formulation provides an algorithm to solve upper partial moment/lower partial moment (UPM/LPM) formulations.