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Pension buy‐ins and buy‐outs have become an important aspect of managing pension risk in recent years. As a step toward understanding these pension de‐risking instruments, we develop models for pricing investment risk and longevity risk embedded in pension buy‐ins and buy‐outs. We also bring a contingent‐claims framework to price credit risk of buy‐in bulk annuities. Overall, our model can be used to assess the pricing of investment, longevity, and credit risks being transferred in pension buy‐in and buy‐out transactions.

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The finite-time ruin problem, which implicitly involves the inversion of the Laplace transform of the time to ruin, has been a long-standing research problem in risk theory. Existing results in the Sparre Andersen risk models are mainly based on an exponential assumption either on the interclaim times or on the claim sizes. In this paper, we utilize the multivariate version of Lagrange expansion theorem to obtain a series expansion for the density of the time to ruin under a more general distribution assumption, namely the combination of n exponentials. A remark is further made to emphasize that this technique can also be applied to other areas of applied probability. For instance, the proposed methodology can be used to obtain the distribution of some first passage times for particular stochastic processes. As an illustration, the duration of a busy period in a queueing risk model will be examined. KEYWORDSTime to ruin, finite-time ruin probability, multivariate Lagrange expansion theorem, duration of a busy period, combination of n exponentials, K m /G/1 queueing model, number of claims until ruin.

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