Annuities Immediate Due and Perpetuities: Notes and FAQs

An annuity is a series of equal payments made at regular intervals, such as monthly loan instalments or yearly pension payouts. An annuity-immediate (ordinary annuity) pays at the end of each period, while an annuity-due pays at the beginning of each period. A perpetuity is an annuity that continues forever.

The timing distinction matters because money has a time value: a payment received at the start of a period can be invested one period longer than the same payment received at the end. Annuity formulas price mortgages, leases, insurance premiums, and retirement plans, so getting the timing right changes real money.

Key Concepts

• Payment (P): the level amount paid each period. Term (n): the number of payments. Rate (r): the interest rate per payment period.
• Annuity-immediate: first payment one period from now; standard for loans repaid in arrears. Annuity-due: first payment immediately; standard for rent and lease payments.
• Actuarial symbols: a-angle-n for the present value of an immediate annuity, s-angle-n for its accumulated (future) value; ä-angle-n and s-double-dot-angle-n for the due versions.
• Perpetuity-immediate: payments of P at the end of every period forever. Perpetuity-due: first payment now, then forever.

Formulas

Present value of an annuity-immediate: a-angle-n = [1 − (1 + r)^(−n)] / r, so PV = P × a-angle-n. Present value of an annuity-due: ä-angle-n = a-angle-n × (1 + r), because every payment arrives one period earlier. Accumulated value of an annuity-immediate: s-angle-n = [(1 + r)^n − 1] / r, so FV = P × s-angle-n; the due version multiplies by (1 + r). Present value of a perpetuity-immediate: PV = P / r; perpetuity-due: PV = P(1 + r) / r = P + P/r. All assume a constant rate, equal payments, and no missed payments.

Worked Example

A lottery offers ₹5,000 per year for 10 years at a 7% discount rate. As an annuity-immediate, the present value factor is a-angle-10 = [1 − (1.07)^(−10)] / 0.07. Compute (1.07)^10 = 1.96715136, so (1.07)^(−10) = 1 / 1.96715136 = 0.50834829. Then 1 − 0.50834829 = 0.49165171, and dividing by 0.07 gives 7.02358153. The present value is 5,000 × 7.02358153 = ₹35,117.91. If the first ₹5,000 arrives immediately (annuity-due), multiply by 1.07: 35,117.91 × 1.07 = ₹37,575.77. The ₹2,457.86 difference is exactly one period of discounting on every payment. A perpetuity paying ₹5,000 per year forever at 7% would be worth 5,000 / 0.07 = ₹71,428.57 as an immediate perpetuity.

Common Misconceptions

• Due and immediate annuities are nearly the same. They differ by a full factor of (1 + r); at 7% that is 7%, or thousands of rupees on a large loan or pension valuation.
• A perpetuity must be worth an infinite amount. Because distant payments are discounted so heavily, the infinite sum converges to the finite value P / r.
• Annuity formulas work with any payment schedule. They require level payments at regular intervals; irregular cash flows must be discounted individually and summed.

Real-World Applications

Annuity-immediate formulas amortize home and car loans and value bonds; annuity-due formulas price rents, leases, and insurance premiums paid in advance; perpetuities value preference shares, endowment funds, and consols, the perpetual bonds once issued by governments.

Annuities package a stream of equal payments into one present or future value; the only things to get right are whether the first payment is now or one period away, and whether the stream ever ends.