Portfolio Directional Distance Function Models with a Stochastic Jump Process

Abstract

Portfolio optimization problems include the selection of different assets to invest in order to maximize the return and minimize the risk. In practice, the models account for asset returns that skewness, Kurtosis, and heavy tails characterize. For this purpose, we describe the dynamics of assets’ returns with the Variance Gamma (VG) process from Lévy processes by considering the skewness and Kurtosis of the assets' return rate. We employ Data Envelopment Analysis (DEA) methodology alongside Directional Distance Function (DDF), which they able to evaluate different assets' performance by VG process through its constraints, and they identify inefficiencies within asset markets. We introduce two models that seek to simultaneously minimize the risk measure as the input and maximize the mean return as the output of the given asset using the pre-specified direction vector. In the first model, stricter assessments arise from directions emphasizing maximum conditional risk and return. In the second model, mean return-risk values remain constant, suggesting that asset inefficiency is unaffected by changing directions. This unchanging pattern may reflect similar impacts across scenarios or limitations in the mean return-risk metric’s ability to detect directional variations. Instead, asset inefficiency appears to be driven by intrinsic distributional properties, notably skewness and Kurtosis, rather than scenario-specific influences. An empirical example in the Iranian stock market of seven companies is used to validate the models.