Range Value-at-Risk and Its Optimization in Vehicle Insurance

Bony Parulian Josaphat; Moch Fandi Ansori

doi:10.22342/jims.v32i2.1773

Bony Parulian Josaphat ⁽¹⁾ , Moch Fandi Ansori ⁽²⁾

(1) Department of Statistical Computing, Politeknik Statistika STIS, Indonesia,

(2) Department of Mathematics, Universitas Diponegoro, Indonesia

https://doi.org/10.22342/jims.v32i2.1773

Issue
Vol. 32 No. 2 (2026): JUNE

Submitted
August 9, 2024

Accepted
November 9, 2025

Published
April 23, 2026

Keywords:

Insurance claim, Newton method, RCTV, RVaR, Spiral optimization

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Abstract

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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References

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Authors

Bony Parulian Josaphat

Department of Statistical Computing, Politeknik Statistika STIS

bonyp@stis.ac.id (Primary Contact)

Moch Fandi Ansori

Department of Mathematics, Universitas Diponegoro

Josaphat, B. P., & Ansori, M. F. (2026). Range Value-at-Risk and Its Optimization in Vehicle Insurance. Journal of the Indonesian Mathematical Society, 32(2), 1773. https://doi.org/10.22342/jims.v32i2.1773

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1. INTRODUCTION

There have been numerous risk measures proposed in the subject of quantitative risk management, the two most well-known of which are the value-at-risk (VaR) and the tail value-at-risk. VaR is the greatest loss in a period for a given confidence level, whereas TVaR is the mean loss after VaR is exceeded. As VaR has been approved by the Basel Committee[1], it has emerged as the standard measurement of risk in the financial markets. Despite its simplicity and ease of application, VaR lacks coherence as a risk measure in the sense that[2]suggests. This is because VaR does not meet subadditivity, which means that the VaR of an aggregated risk is not necessarily less than the total of the VaRs of the individual risks that make up the aggregate, contrary to the principle of diversification. Furthermore, the VaR ignores the possibility of losses outside the considered quantile.

On the other hand, the TVaR, which accounts for losses beyond the VaR, is the most popular competitor for VaR. TVaR was proposed properly as a solution to VaR’s limitations in terms of risk sensitivity. Furthermore, at some confidence level, TVaR equals the conditional expected value of losses that are greater than or equal to VaR. Since TVaR is a coherent measure of risk[2], it has been considerably implemented by financial institutions. However, TVaR has some theoretical hurdles, such as not existing for distributions with a very heavy tail or infinite mean[3], i.e., it relies heavily on the extreme tail of the return distribution, potentially overestimating the original risk[4]. Nonetheless, regulatory agencies apply both risk measures; VaR more in banking, while TVaR more in insurance[1]. Furthermore, Liet al.[5]argued that RVaR may be considered as a trestle between VaR and TVaR and is robust in the sense that it is continuous in relation to the weak convergence of random variables, whereas neither TVaR nor VaR is continuous generally.

In particular, Jianget al.[6]implemented RVaR to illustrate the applicability of Pareto-optimal reinsurance policies that incorporate both the insurer’s and reinsurer’s risks and returns. Wang and Wei[7]identified and characterized Pareto-optimal allocations in quantile-based risk-sharing problems within the RVaR family. Liet al.[5]developed worst-case values and scenarios for RVaR under partial information assumption, for both single and aggregate risk models. Moreover, Fissler and Ziegel[8]statistically assessed, compared, and ranked the predictive performance of diferent RVaR models.

To overcome those limitations of VaR and TVaR simultaneously, we propose investigating the role of RVaR by finding a technique to optimize RVaR. Moreover, the existence of the contraction parameter \(a > 0\) gives RVaR some flexibility. Instead of infinite, this parameter can be applied to provide an upper bound on the risk for values that exceed VaR. As a result, when we construct the RVaR forecast, it reduces the size of the loss beyond VaR, making it smaller than the TVaR forecast. This is a useful characteristic in risk modeling, particularly when dealing with high-variability portfolio returns or data with outliers. Nevertheless, Jadhavet al.[4]only set the values of a while computing RVaR rather than optimizing RVaR.

Furthermore, none of the previously listed authors ([5],[6],[8],[7], and[9]) ofered a numerical optimization technique to ascertain parameter \(a\) directly or indirectly.

Besides TVaR, risk variability in the tail region of the distribution to the right of VaR was also noted by certain investigators. Essentially, the idea is that, regardless of the intended features and usefulness, TVaR only considers the mean losses on the tail and ignores variability, therefore it makes sense to include a second central moment or variance of the tail distribution. Thus, tail variance, or CTV, is a variability measure introduced by Furmanet al.[10]. Following[10], we then suggest a variability measure surrounding RVaR that is subject to the same restriction. We refer to this measure as range conditional tail variance (RCTV). We consider this measure because we believe that RCTV provides a better comprehension of the behavior of a risk throughout the tail of its distribution.

The goal of this study is to present an optimization technique for indirectly estimating the value of parameter \(a\) that maximizes RVaR while being constrained by RCTV and CTV. The Lagrangian approach is used to convert the constrained maximization problem into a problem of solving a system of nonlinear equations. The transformed problem is now a problem of unconstrained minimization. In this study, two methods are used to solve the RVaR optimization problem: Newton and spiral optimization (SpO) methods. Recently, the Newton method as well as its variants have been extensively used in a variety of optimization problems [11,12,13,14]. We choose Newton method because it can achieve convergence quickly. Of course, the method requires the selection of appropriate initial conditions as well as the calculation of the Jacobian matrix. Many people avoid Newton’s method due to the dificulties of deriving the Jacobian matrix. Moreover, they do not want to face the Jacobian matrix becoming singular for some points and at some iteration. In our unconstrained problem, we can still derive the Jacobian matrix although it has a long expression. The elements of the Jacobian matrix are given in Appendix. On the other hand,

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