In Quantitative Finance 2016, Chen, Hu and Lin (CHL) claimed the following: ‘ … there is yet no coherent risk measure related to investment performance.’ (p. 682). Our paper suggests and analyzes four coherence axioms that portfolio performance ratios should satisfy.

Our Portfolio Riskless Translation Invariance axiom must be satisfied to assure separation of the objective decision to optimize a portfolio’s risky composition from the subjective decision to optimize the weight of the portfolio's level of risk-free asset. Performance ratios with fixed thresholds other than the risk-free rate do not satisfy this axiom, allowing portfolio managers to affect an ex-ante performance ratio merely by changing the proportion of the risk-free asset in the portfolio rather than by improving the composition of the portfolio’s risky components. The magnitude of this potential drawback is examined using S&P-500 stock index data.

Replacing the fixed threshold, T, with a threshold $T(\gamma ,\alpha )$ that equals γ times the portfolio’s risk premium plus (1-γ) times the risk-free rate, eliminates the above shortcoming for any selected γ. In addition, using performance ratios with threshold $T(\gamma ,\alpha )$ rather than fixed T, assures consistency of performance ratios of effective stochastic dominance and risk-free asset rules.

在 2016 年量化金融中，Chen、Hu 和 Lin (CHL) 声称：“……目前还没有与投资业绩相关的连贯风险衡量标准。” （第 682 页）。我们的论文提出并分析了投资组合绩效比率应该满足的四个连贯公理。

我们的投资组合无风险转换不变性公理必须得到满足，以确保将优化投资组合风险构成的客观决策与优化投资组合无风险资产水平权重的主观决策分离。具有固定阈值而不是无风险利率的绩效比率不满足这个公理，允许投资组合经理仅通过改变投资组合中无风险资产的比例而不是通过改善资产的构成来影响事前绩效比率投资组合的风险成分。使用 S&P-500 股票指数数据检查了这种潜在缺陷的严重程度。

更换固定阈值，Ť，与阈$吨(\gamma ,\alpha )$等于γ乘以投资组合的风险溢价加上 (1- γ ) 乘以无风险利率，消除了任何选定γ的上述缺点。此外，使用具有阈值的性能比$吨(\gamma ,\alpha )$而不是固定T，确保有效随机优势和无风险资产规则的性能比率的一致性。