Optimal Insurance Design Under a Value-at-Risk Framework

Abstract

This study designs an optimal insurance policy form endogenously, assuming the objective of the insured is to maximize expected final wealth under the Value-at-Risk (VaR) constraint. The optimal insurance policy can be replicated using three options, including a long call option with a small strike price, a short call option with a large strike price, and a short cash-or-nothing call option. Additionally, this study also calculates the optimal insurance levels for these models when we restrict the indemnity to be one of three common forms: a deductible policy, an upper-limit policy, or a policy with proportional coinsurance. Copyright Springer Science + Business Media, Inc. 2005value at risk, optimal insurance, deductible, policy limit, coinsurance,

Risk allocation in a public–private catastrophe insurance system: an actuarial analysis of deductibles, stop‐loss, and premiums

“…We follow the approach, with a major modification, taken by Wang et al . (), who model the consumer decision about how much insurance coverage to purchase as a minimisation of insurance costs, subject to a constraint that the consumer's own risk does not exceed a certain level . The insured aims to minimise the amount of coverage, which automatically minimises the total premium to be paid, under a desirable level of the deductible that meets the 99.9% TVaR damage constraint.…”

Optimal Reinsurance Arrangements Under Tail Risk Measures

“…Therefore, Proposition 1.1 also solves the dual problem by a truncated deductible contract. A variant of this optimal reinsurance problem has recently been studied by Wang, Shyu, and Huang (2005), in which the optimal reinsurance contract is the one that maximizes the expected wealth subject to the probability constraint , where the premium is . Problem 1.1 is significantly different from the problem in Wang, Shyu, and Huang in several aspects.…”

“…A variant of this optimal reinsurance problem has recently been studied by Wang, Shyu, and Huang (2005), in which the optimal reinsurance contract is the one that maximizes the expected wealth subject to the probability constraint , where the premium is . Problem 1.1 is significantly different from the problem in Wang, Shyu, and Huang in several aspects. First, Wang, Shyu, and Huang consider the deviation from the mean, , whereas a conventional VaR concept is about the quantile of W − W 0 .…”

“…First, Wang, Shyu, and Huang consider the deviation from the mean, , whereas a conventional VaR concept is about the quantile of W − W 0 . Second, which is independent of the loading factor ρ. Consequently, the minimal insolvency probability considered in Wang, Shyu, and Huang is independent of the loading factor. Hence, Proposition 1.1 can not derived from Wang, Shyu, and Huang 10…”

“…10 10 Moreover, some technical points have been overlooked by Wang, Shyu, and Huang (2005) such as the fact that the VaR constraint is binding when X has a continuous distribution, which is not true as proved by Figure 2. …”