The Effects of Largest Claim and Excess of Loss Reinsurance on a Company’s Ruin Time and Valuation

Abstract

Abstract:We compare two types of reinsurance: excess of loss (EOL) and largest claim reinsurance (LCR), each of which transfers the payment of part, or all, of one or more large claims from the primary insurance company (the cedant) to a reinsurer. The primary insurer's point of view is documented in terms of assessment of risk and payment of reinsurance premium. A utility indifference rationale based on the expected future dividend stream is used to value the company with and without reinsurance. Assuming the classical compound Poisson risk model with choices of claim size distributions (classified as heavy, medium and light-tailed cases), simulations are used to illustrate the impact of the EOL and LCR treaties on the company's ruin probability, ruin time and value as determined by the dividend discounting model. We find that LCR is at least as effective as EOL in averting ruin in comparable finite time horizon settings. In instances where the ruin probability for LCR is smaller than for EOL, the dividend discount model shows that the cedant is able to pay a larger portion of the dividend for LCR reinsurance than for EOL while still maintaining company value. Both methods reduce risk considerably as compared with no reinsurance, in a variety of situations, as measured by the standard deviation of the company value. A further interesting finding is that heaviness of tails alone is not necessarily the decisive factor in the possible ruin of a company; small and moderate sized claims can also play a significant role in this.

The effect of reinsurance treaties on the cedent loss reserving

“…The following recall definition of an LCR treaty. We refer interested readers to study Jiang and Tang (2008), Ladoucette and Teugels (2006) and Fan et al (2017), among others.…”

“…Riegle (2015) used the chain ladder (CL) method to predict price uncertainty under a long-tail QS reinsurance treaty. The impact of reinsurance treaties on the insurer's lifetime has been investigated by Fan et al (2017). As far as we know, a few amounts of literature have studied the impact of the reinsurance treaty on the loss reserves.…”

Functional laws for trimmed Lévy processes

“…The idea is that the largest claim up to a specified time incurred by an insurance company (the "cedant") is referred to a higher level insurer (the "reinsurer"). See Fan et al [5] for details and references to the applications literature. This is known as the largest claim reinsurance (LCR) treaty: having set a fixed follow-up time t, we delete from the process the largest claim occurring up to and including that time.…”

“…Let C(r n ) denote the set of accumulation points of a sequence (r n ) ⊆ R as n → ∞. Recall that L τ and R τ , the last and first modulus record time processes, are defined in ( 2) and ( 6); R τ , the first positive record time process, is defined in (5).…”

Special Issue “Actuarial and Financial Risks in Life Insurance, Pensions and Household Finance”

“…Yuguang Fan, Philip S. Griffin, Ross Maller, Alexander Szimayer and Tiandong Wang (Fan et al 2017) carry out an extensive simulation-based study of the effects of two reinsurance policies, namely the Largest Claim and the Excess of Loss, on the ruin probability, ruin time and value of an insurance company under the classical compound Poisson risk model. Their exercise should help guide insurers in the design of their most appropriate reinsurance strategy.…”