Abstract
The so-called Problem of Optimal Premium Calculation deals with the selection of the appropriate premiums to be paid by the insurance policies. At first sight, this seems to be a statistical estimation problem: we should estimate the mean claim amount, which in actuarial terms is known as the net premium. Nevertheless, several extensions of this problem are clearly multi-objective decision problems. For example, when we allow the company to modify the premiums paid by the policyholders according to their past claim experience, there are several ways of designing the resulting Bonus-Malus System (BMS), and they usually involve several different objectives.
Optimal BMS design can thus be considered as a multi-objective problem, since it involves three conflicting objectives, which we have called Fairness, Toughness and Equilibrium (or Disequilibrium). Other researchers do not consider the multi-objective nature of this problem, since they always deal with a single objective, taking one of the objectives (Fairness) as the most important. In this chapter we apply a multi-objective approach. We represent in mathematical terms the three objectives, and we formulate the corresponding multi-objective program. Then we discuss several possible ways for solving the problem, and we apply the methodology to the improvement of a real BMS example.
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Notes
- 1.
For example, Norberg (1976) proposes the following quadratic distance function: \( {\displaystyle {\int}_0^{\infty }{\displaystyle \sum_{j=1}^n{\left(\lambda -{P}_j\right)}^2{\pi}_j\left(\lambda \right)dU\left(\lambda \right)}} \), which gives rise to the so-called Bayes Scale of Premiums. Verico (2002) proposes \( {\displaystyle {\int}_0^{\infty }{\left({\displaystyle \sum_{i=1}^n{P}_i{\pi}_i\left(\lambda \right)-\lambda}\right)}^2}dU\left(\lambda \right) \). The other references also consider quadratic functions for measuring the degree of fairness, with the only exception of the linear measure in Heras et al. (2002, 2004).
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Heras, A., Balbás, A., Balbás, B., Balbás, R. (2015). The Multiobjective Nature of Bonus-Malus Systems in Insurance Companies. In: Al-Shammari, M., Masri, H. (eds) Multiple Criteria Decision Making in Finance, Insurance and Investment. Multiple Criteria Decision Making. Springer, Cham. https://doi.org/10.1007/978-3-319-21158-9_7
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