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Reinsurance is the practice whereby insurers transfer portions of their risk portfolios to other parties by some form of agreement to reduce the likelihood of paying a large obligation resulting from an insurance claim. (Investopedia)

The company issuing the reinsurance policy is referred simply as the "reinsurer". (Wikipedia)

The principal parties to any reinsurance contract are the reinsurer, who assumes the subject risk in return for premium, and the ‘reinsured’, who may also be referred to as the ‘reassured’, or the ‘cedant’. (LexisNexis)

The company that purchases the reinsurance policy is called a "ceding company" or "cedent" or "cedant" under most arrangements. (Wikipedia)