Risk Measure Analysis

Assessing the financial risks is an essential component of portfolio management. This assessment involves the employment of one or more risk measures, i.e., quantitative coefficients employed to capture how risky a portfolio is. As the risk measures are important both to create and evaluate portfolios, it is imperative to understand their characteristics and possible shortcomings. This study analyzes six risk measures: variance, value at risk (VaR), conditional value at risk (CVaR), expectile-based value at risk (EVaR), omega ratio, and Sortino ratios. As a first step, closed form solutions were calculated for all measures, as function of the variance and mean of a simple distribution. In a second moment, the risk measures behavior was studied when the mean and variance are kept constant and only the worst scenarios are displaced. Their behavior was then studied numerically for two more examples: the Johnson’
$S_{U}$
distribution and the Brazilian energy market. Among the risk measures analyzed, CVaR and EVaR are the only ones that are invariant to mean and with proper sensitivity to variance and displacement of worst scenarios.