1Department of Applied Mathematics, Islamic Azad University of Central Tehran Branch, Tehran, Iran.
2Department of Mathematics, Faculty of Mathematics and Computer Science, Allameh Tabataba'i University, Tehran Iran.
3Department of Applied Mathematics, Islamic Azad University of Central Tehran Branch, Tehran, Iran.
In this paper, linear Data Envelopment Analysis models are used to estimate Markowitz efficient frontier. Conventional DEA models assume non-negative values for inputs and outputs. however, variance is the only variable in these models that takes non-negative values. Therefore, negative data models which the risk of the assets had been used as an input and expected return was the output are utilized . At the beginning variance was considered as a risk measure. However, both theories and practices indicate that variance is not a good measure of risk. Then value at risk is introduced as new risk measure. In this paper,we should prove that with increasing sample size, the frontiers of the linear models with both variance and value at risk , as risk measure, gradually approximate the frontiers of the mean-variance and mean-value at risk models and non-linear model with negative data. Finally, we present a numerical example with variance and value at risk that obtained via historical simulation and variance-covariance method as risk measures to demonstrate the usefulness and effectiveness of our claim.