On coverage limits and deductibles for SAI loss severities
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On coverage limits and deductibles for SAI loss severities Yinping You1 · Xiaohu Li2 · Rui Fang3 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper studies allocation of coverage limits and deductibles for dependent losses in the frame of utility theory. The optimal allocation of deductibles is derived for SAI losses without frequency impact, and the optimal allocation of coverage limits (deductibles) for SAI loss severities with RWSAI frequencies are proved to be arrayed in ascending (descending) order. Sufficient conditions to exclude the worst allocation of coverage limits are built for comonotonic loss severities with RWSAI frequencies. A real application in house property insurance is presented as well. Keywords Majorization · RWSAI · SAI · Stochastic orders · WSAI Mathematics Subject Classification Primary 91B16 · 91B30 · Secondary 60E15
1 Introduction In insurance practice, there are usually two commonly used forms of coverage: deductible and coverage limit for each one of multiple random risks/losses concerned with an insurance policy. Sometimes the insurer encourages the insured/policyholder to allocate the deductibles or the coverage limits of those risks according to their own will. In literature, except for some generic discussion on the impact of deductible and coverage limit, see for example Gaffney and Ben-Israel (2016), some research have been developed on the optimal allocation of coverage limits and deductibles. See, for example, Cheung (2007), Hua and Cheung (2008a, b), Zhuang et al. (2009), Lu and Meng (2011), Li and You (2012, 2015), You and Li (2014a, 2015), Cai and Wei (2015). The basic model involved in these studies are as follows: Covered by an insurance policy, the vector of random risks X = (X 1 , · · · , X n ) is independent of their corresponding occurrence times (T1 , · · · , Tn ). Given n the discount rate δ ≥ 0, the policyholder attains the total discounted loss Z · X = i=1 Z i X i , where the discounts Z = (Z 1 , · · · , Z n ) ≡ (e−δT1 , · · · , e−δTn ).
B
Yinping You [email protected]
1
Fujian Province University Key Laboratory of Computation Science and School of Mathematical Sciences, Huaqiao University, Quanzhou 362021, Fujian, China
2
Department of Mathematical Sciences, Stevens Institute of Technology, Hoboken, NJ 07030, USA
3
Department of Mathematics, Shantou University, Shantou 515063, Guangdong, China
123
Annals of Operations Research
In insurance practice the policyholder is granted a total of > 0 coverage limit, which can be allocated to these risks through an vector l = (l1 , · · · , ln ). Let A be the set of all n admissible allocation vectors such that i=1 li = and li ≥ 0 for all i = 1, · · · , n. Denote x ∧ l = min{x, l} and x+ = max{x, 0}. Then, for any l ∈ A , the policyholder gets the n potential discounted benefit i=1 e−δTi (X i ∧ li ) and hence bears the total retained loss n
Z i [X i − (X i ∧ li )] =
i=1
n
Z i (X i − li )+ = Z · (X − l)+ .
i=1
On the other hand, the policyholder c
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