Given an insurance or annuity contract and a duration *k*, the *reserve* at time *k* is defined exactly as in Definition 2.7. It is the amount that the insurer needs at time *k* in order to ensure that obligations under the contract can be met. Calculating reserves for each policy is an important responsibility of the actuary, known as *valuation*. The insurer wants to be confident that funds on hand, together with future premiums and investment earnings, are sufficient to pay the promised future benefits. It is important to thoroughly master the concept of insurance and annuity reserves in order to properly understand and analyze the nature of these contracts.

Throughout this chapter we will deal with the following model. As usual we start with a fixed investment discount function *v* and a life table. We have a contract issued on (*x*) with death benefit vector **b**, annuity benefit vector **c** and premium vector π. (For simplicity we will omit the possibility of guaranteed payments in our discussion, but this feature can easily be incorporated if desired.) Recall from Section 5.5 that we can view the death benefits as a vector of annuity benefits **b*****w**_{x}, where (**w**_{x})_{k} = *v*(*k*, *k* + 1)*q*_{x + k}. We can then form the *net cash flow* vector

(6.1)

which indeed represents the net cash flow on the contract from the insurer’s viewpoint. The insurer ...

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