Deferred Annuities and Perpetuities for Exam FM

Deferred Annuities: Core Concepts

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.

It is important to distinguish between the deferral period and the payment period. The deferral period is the number of periods before the first payment. For an annuity-immediate deferred k periods, the first payment occurs at time k+1. For an annuity-due deferred k periods, the first payment occurs at time k. This distinction matters on Exam FM because choosing the wrong timing shifts your answer by one discount factor.

The two equivalent formulas for the present value of a deferred annuity-immediate are worth memorizing. First, PV = v^k * a-angle-n, which discounts the annuity value back through the deferral period. Second, PV = a-angle-(k+n) - a-angle-k, which treats the deferred annuity as the difference between a longer annuity and the payments during the deferral period. Both formulas give identical results, so choose whichever is more convenient for the given problem.

Deferred Annuity Calculations

Example: What is the present value of $1000 per year for 10 years, with the first payment in 6 years, at i = 5%? The deferral period is k = 5 (the first payment is at time 6, which is one period after time 5). PV = 1000 * v^5 * a-angle-10 = 1000 * (1.05)^(-5) * 7.7217 = 1000 * 0.7835 * 7.7217 = $6049.77.

Another example: A retiree will receive $2000 per month for 20 years, with the first payment 10 years from now. At a monthly rate of 0.5%, the present value is 2000 * v^119 * a-angle-240 at 0.5%, where the deferral is 119 months (the first payment at month 120 is one period after month 119 for an annuity-immediate). Alternatively, if treated as a deferred annuity-due, the deferral is 120 months and PV = 2000 * v^120 * a-double-dot-angle-240.

For accumulated values of deferred annuities, the deferral period does not affect the accumulated value at the end of the payment period. The accumulated value at time k+n is simply s-angle-n, the same as a non-deferred annuity. The deferral only affects present values.

Perpetuities: Immediate and Due

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 formulas 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. To derive it, start from a-angle-n = (1 - v^n)/i and let n approach infinity. Since 0 < v < 1, v^n approaches 0, giving 1/i.

The relationship between perpetuity-immediate and perpetuity-due follows the same pattern as finite annuities: a-double-dot-angle-infinity = (1+i) * a-angle-infinity = (1+i)/i = 1/d. You can verify this using d = i/(1+i), so 1/d = (1+i)/i.

Deferred Perpetuities

Deferred perpetuities combine both concepts. The PV of a perpetuity-immediate with first payment at time k+1 is v^k / i. This formula appears frequently on Exam FM in comparison problems. For instance, if a scholarship fund must provide $5000 per year starting 4 years from now (first payment at time 4), the present value at i = 6% is v^3 * (5000/0.06) = (1.06)^(-3) * 83,333.33 = $69,954.67. The deferral is 3 periods because the perpetuity-immediate begins one period before the first payment.

A useful special case is the deferred perpetuity-due with first payment at time k. Its present value is v^k / d = v^k * (1+i) / i. Be careful to identify whether the problem specifies when the first payment occurs or when the deferral period ends.

Comparing Deferred and Immediate Annuities

A common Exam FM problem involves finding the deferral period that makes two options equivalent. For example, suppose Option A pays $500 per year for 20 years starting immediately, and Option B pays $800 per year for 20 years starting after k years of deferral. At i = 4%, find k such that both options have the same present value. Set 500 * a-angle-20 = 800 * v^k * a-angle-20. Simplifying, v^k = 500/800 = 0.625, so k = -ln(0.625)/ln(1.04) = 11.98, meaning approximately 12 years of deferral.

Another problem type involves finding the level payment for a deferred annuity that has the same present value as a given immediate annuity. These problems test your ability to set up equations of value at a common valuation date.

Continuously Payable Deferred Annuities

For continuous annuities (payments made continuously), the present value of a continuously payable annuity of 1 per year for n years, deferred k years, is v^k * a-bar-angle-n, where a-bar-angle-n = (1 - v^n)/delta and delta = ln(1+i) is the force of interest. The continuous perpetuity has present value 1/delta. These formulas appear occasionally on Exam FM and connect the discrete annuity results to continuous compounding.

The continuously payable deferred perpetuity starting at time k has present value v^k / delta. This is the continuous analogue of the deferred perpetuity-immediate.

Study Tips and Common Pitfalls

The most frequent mistake with deferred annuities is miscounting the deferral period. Always draw a timeline showing when each payment occurs. Label time 0, the deferral period, and the payment period clearly. Remember that for an annuity-immediate deferred k periods, the first payment is at time k+1, not time k.

Another common error involves perpetuities with growing payments. A growing perpetuity with first payment P at time 1, growing at rate g per period, has present value P/(i-g) when i > g. If deferred k periods, multiply by v^k. Make sure you identify the correct first payment amount after accounting for growth during the deferral period.

Practice at least 10 deferred annuity problems and 5 perpetuity problems before exam day. These topics often combine with other concepts like increasing annuities or loan refinancing, so familiarity with the basic formulas allows you to focus on the problem setup rather than formula recall.