<![CDATA[A popular risk measure is tail value-at-risk (TVaR), which is the mean of a random risk's losses above the value-at-risk (VaR). Moreover, TVaR is the most popular competitor of VaR. However, TVaR has some theoretical obstacles; for example, it does not exist when the risk distribution has a very heavy tail or has an infinite mean. In practice, this may compel financial institutions or insurance companies to deposit additional funds to fulfill requisites specified by regulators. Many authors suggested using a generalization of TVaR known as range value-at-risk (RVaR), which measures the actual risk of an aggregated risk. Furthermore, we suggest the range conditional tail variance (RCTV), a second conditional moment of the tail distribution with the RVaR at its center. RVaR and RCTV are significantly more flexible than TVaR and conditional tail variance (CTV) since they both have a contraction parameter. We also provide analytical formulations for the RVaR and RCTV of exponentially distributed risk. This article \textcolor{black}{proposes} an optimization method for the RVaR by applying the Newton method and a metaheuristic algorithm, spiral optimization (SpO). \textcolor{black}{We use} the Newton technique and SpO with RCTV and CTV to find the contraction parameter that optimizes RVaR. This study shows the use of RVaR optimization to forecast the RVaR of vehicle insurance claim amounts in Australia. We find that the SpO method produces the estimation result quite well as noticed by the quadratic form of objective function converging to zero. On the other hand, the Newton method produces not only similar results for the estimation but also has less RVaR at the same probability levels, which means better in lowering the magnitude of TVaR. However, the empirical results show that, compared with Newton's method, the SpO method captures the RVaR more successfully.]]>
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