Can the Normality of the Semi Variance Be Improved? Evidence from Financial Stock Indexes with Hourly, Daily, Quarterly and Annual Data of DJIA and SP500

Abstract

This study examines the financial and statistical properties of the variance and semi variance (SV). Since the mean-variance approach and its extended mean-semi variance approach assume normality of returns, it has been observed that practical and computational problems emerged in the cases of portfolio optimization and estimation risk. The reliability of the semi variance has to be re-examined. This paper shows that the variance and its partial domain (semi variance) produce non normal estimates when the mean returns are normally distributed. Accordingly, a new proposed measure of risk, Mean Semi Deviations (MSD), is introduced which focuses on the measurement of the percentage returns lost from the average. The financial and statistical properties of the three measures of risk are tested and examined taking into account the risk-return theoretical relationship using data from index returns (DJIA and S&P500). The data patterns used are hourly, daily, quarterly and annual data. The financial results of the paper show that the MSD outperforms the variance and the SV in terms of its association to mean returns. The statistical properties show that the MSD produces estimates that are normally distributed and less volatile for all patterns of data (except for daily data) which outperforms the variance and the SV. The contribution of the paper is that it shows a prerequisite approach to be followed for testing the normality and volatility of any downside risk measure before using it for portfolio optimization, selection and estimation risk.

Keywords: Risk measures, Variance, Semi Variance, Mean Semi Variance, DJIA, S&P500

JEL Classification: B23, O16