Using the annuity method, the reserve for year n is given by which expression?

• A. nV = 1 - a_x+n / a_x
• B. nV = a_x+n / a_x - 1
• C. nV = a_x / a_x+n
• D. nV = 1 / (1 + i)^n

Answer: (A)

The idea being tested is how reserves are measured under the annuity method: the reserve is the portion of the original premium that still has to be funded to cover future payments, expressed per unit premium. If the annual premium is taken as 1, the present value of all future premiums at time 0 is a_x. After year n, the remaining value of future premiums (in present-value terms, starting from age x+n) is a_{x+n}. The fraction of the original premium that is still needed to fund the remaining payments is 1 minus the ratio of the remaining premium value to the original premium value, i.e., 1 − a_{x+n}/a_x. Expressing this as a reserve per unit premium gives nV = (1 − a_{x+n})/a_x. This matches the given result. The other forms don’t align with how the annuity method allocates the remaining funded portion of the premium to future obligations.