We studied the relationship between expected returns and thirteen risk measures, namely Expected Loss, Value at Risk, Expected Shortfall, Expectile Value at Risk, Entropic, Maximum Loss, Standard Deviation, Negative Semi-Deviation, Shortfall Deviation, Expected Loss Deviation, Shortfall Deviation Risk, Deviation Expectile Value at Risk, and Deviation Entropic. We consider measures that assess the loss, deviation, and both risk concepts simultaneously in a sample of United States (US) stock returns from January 1982 to January 2021. Analyzing the difference between the returns of the higher risk and lower risk portfolios, we report a significant difference for all risk measures tested, except for the Shortfall Deviation. We also perform time-series regressions using the returns of the Fama–French three factors and the Momentum factor as independent variables. When controlling factors are included in the model alphas are not significant, suggesting evidence of no risk premium.

我们研究了预期收益与十三个风险度量之间的关系，即预期损失、风险价值、预期短缺、预期风险价值、熵、最大损失、标准差、负半偏差、短缺偏差、预期损失偏差、短缺偏差风险、风险偏差期望值和偏差熵。我们考虑在 1982 年 1 月至 2021 年 1 月的美国股票回报样本中同时评估损失、偏差和两种风险概念的措施。通过分析高风险和低风险投资组合的回报之间的差异，我们报告除短缺偏差外，所有测试的风险指标均存在显着差异。我们还使用 Fama-French 三个因子的回报和动量因子作为自变量进行时间序列回归。当模型中包含控制因素时，α 值不显着，表明没有风险溢价。