Reinsurance transfers part of an insurer’s claim liability to another insurer. If aggregate claims are , a retained payout with leaves the reinsurer with . Quota share reinsurance retains a fixed fraction, whereas aggregate stop loss reinsurance retains losses only up to a fixed aggregate threshold.
With aggregate retention , the direct insurer pays and the reinsurer pays , where denotes the positive part. The threshold applies to the whole annual loss; applying a threshold to individual claims is a different contract.
If has exponential distribution with expected value , the tail integral formula for moments gives and , where . At matching retained expected value, the excess variance under quota share reinsurance is .
Among retained payouts with and the same expected value as , aggregate stop loss reinsurance minimizes the variance. Pointwise , and subtracting the identical squared distance of their common expected value from proves the claim. Equality requires equal payouts almost surely.
Articles by others on the same topic
Reinsurance is a financial arrangement in which an insurance company (the "ceding company") transfers a portion of its risk to another insurance company (the "reinsurer"). The primary purpose of reinsurance is to reduce the risk exposure of the ceding company by spreading risk among multiple parties, thereby enhancing the stability of the insurance market and ensuring that insurers can meet their financial obligations to policyholders.