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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling · Annuity and accumulation functions

Which expression equals the present value of an n-year annuity-due of 1 per year at effective annual rate i, with v = 1/(1+i) and d = i/(1+i)?

The present value of an n-year annuity-due of 1 is (1 − v^n) divided by d, the sum of the geometric series 1 + v + ... + v^(n−1), whose common ratio v gives denominator 1 − v = d.

  1. A(1 − v^n)/i
  2. B(1 − v^n)/dCorrect
  3. C(1 − v^n)/(i·d)
  4. D(v^n − 1)/d
  5. (1 + i^n)/d

Explanation

Annuity-due PV = 1 + v + ... + v^(n−1) = (1 − v^n)/(1 − v) = (1 − v^n)/d. Dividing by i gives the annuity-immediate. The other options are not geometric-series results.

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