Nonlinearly transformed risk measures: properties and application to optimal reinsurance

被引:1
作者
Brandtner, Mario [1 ]
Kuersten, Wolfgang [1 ]
Rischau, Robert [1 ]
机构
[1] Friedrich Schiller Univ Jena, Finance Banking & Risk Management, Carl Zeiss Str 3, D-07743 Jena, Germany
关键词
Nonlinearly transformed risk measures; weighted expected shortfall; tail nonlinearly transformed risk measure; disutility based risk measure; optimal reinsurance; VALUE-AT-RISK; PORTFOLIO SELECTION; VULNERABILITY;
D O I
10.1080/03461238.2019.1670249
中图分类号
O1 [数学];
学科分类号
0701 ; 070101 ;
摘要
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.
引用
收藏
页码:376 / 395
页数:20
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