ABSTRACT We propose the novel class of nonlinearly transformed risk measures (NTRMs) and apply them to the standard reinsurance problem in which an insurant seeks the risk-minimizing reinsurance contract. NTRMs transform both the outcomes and the probabilities of the insurant's uncertain final wealth by means of nonlinear functions. We prove relevant properties of NTRMs and show that they include popular Conditional Value-at-Risk (CVaR), Weighted Expected Shortfall (WES), Tail Nonlinearly Transformed Risk Measure (TNT), and Disutility Based Risk Measure (DBRM) as special cases. Regarding the reinsurance problem, we show that, under NTRMs, the optimal contract is of stop-loss type. We determine the optimal deductibles and provide comparative statics with respect to the insurant's risk aversion and initial wealth. Regarding comparative risk aversion, we show that the recently proposed WES, TNT, and DBRM help to overcome the restrictive all-or-nothing reinsurance decisions that prevail under CVaR. We further address the comparative statics with respect to initial wealth and show that under CVaR as well as WES and TNT, initial wealth is irrelevant: increasing initial wealth does not alter the optimal deductible. We show that NTRMs are able to overcome this shortcoming when the nonlinear transformation function of the outcomes is appropriately chosen.

中文翻译：

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非线性转换风险测度：优化再保险的属性和应用

摘要 我们提出了一类新的非线性转换风险度量 (NTRM)，并将它们应用于标准再保险问题，在该问题中，被保险人寻求风险最小化再保险合同。NTRM 通过非线性函数转换了被保险人不确定的最终财富的结果和概率。我们证明了 NTRM 的相关属性，并表明它们包括流行的条件风险价值 (CVaR)、加权预期短缺 (WES)、尾部非线性转换风险度量 (TNT) 和基于损失的风险度量 (DBRM) 作为特殊情况。关于再保险问题，我们表明，在 NTRM 下，最优合约是止损类型。我们确定最佳免赔额，并提供关于被保险人的风险规避和初始财富的比较静态。关于比较风险规避，我们表明最近提出的 WES、TNT 和 DBRM 有助于克服 CVaR 下普遍存在的限制性的全有或全无再保险决策。我们进一步解决了初始财富的比较静态问题，并表明在 CVaR 以及 WES 和 TNT 下，初始财富无关紧要：增加初始财富不会改变最佳免赔额。We show that NTRMs are able to overcome this shortcoming when the nonlinear transformation function of the outcomes is appropriately chosen. 增加初始财富不会改变最佳免赔额。We show that NTRMs are able to overcome this shortcoming when the nonlinear transformation function of the outcomes is appropriately chosen. 增加初始财富不会改变最佳免赔额。We show that NTRMs are able to overcome this shortcoming when the nonlinear transformation function of the outcomes is appropriately chosen.