Deferred Annuities and Perpetuities for Exam FM
Study deferred annuities and perpetuities and their Exam FM applications.
Deferred Annuities
A deferred annuity has a waiting period before payments begin. An n-year annuity-immediate deferred k years has its first payment at time k+1 and last payment at time k+n. Its present value is v^k * a-angle-n = a-angle-(k+n) - a-angle-k. The deferral period simply multiplies the annuity value by the discount factor v^k.
Example: What is the present value of $1000 per year for 10 years, with the first payment in 6 years, at i = 5%? PV = 1000 * v^5 * a-angle-10 = 1000 * (1.05)^(-5) * 7.7217 = 1000 * 0.7835 * 7.7217 = $6049.77.
Perpetuities
A perpetuity is an annuity with infinite payments. The present value of a perpetuity-immediate paying 1 per period is a-angle-infinity = 1/i. For a perpetuity-due: a-double-dot-angle-infinity = 1/d. These are the limits of the finite annuity formulas as n approaches infinity, since v^n approaches 0.
A perpetuity has no accumulated value (it would be infinite). The present value formula 1/i is widely used as a building block in bond pricing and stock valuation.
Exam FM Applications
Deferred perpetuities combine both concepts: the PV of a perpetuity-immediate with first payment at time k+1 is v^k / i. A common Exam FM problem involves comparing the cost of a deferred annuity to an immediate annuity or finding the deferral period that makes two options equivalent. Another frequent problem type asks for the present value of increasing or level payments that start after a deferral period. Draw the timeline and identify the valuation point carefully.