Present Value of an Annuity in 9 Minutes
An annuity is a series of equal payments made at regular intervals for a set number of periods, like loan payments or rent. Instead of discounting each payment one by one, the present value of annuity formula does it all in one step.
Annuity present value explained simply
An annuity is a series of equal payments for a set number of periods. Its present value is PV = C × [1 − (1 + r)⁻ⁿ] ÷ r. In the example, $1,000 a year for 5 years at 6% is worth about $4,212.36 today.
What this annuity lesson covers
- What an ordinary annuity is
- The present value of annuity formula and the annuity factor
- How to value loan payments, leases and other equal cash flows
- How annuities relate to perpetuities
Annuity formula, explained
- C
- equal payment at the end of each period
- r
- discount rate per period
- n
- number of payments
How to calculate the present value of an annuity step by step (worked example)
You will receive $1,000 at the end of each year for 5 years. The discount rate is 6%. What is the stream worth today?
- Inputs: C = $1,000, r = 0.06, n = 5.
- Discount the last payment's factor: (1.06)⁻⁵ ≈ 0.747258.
- Annuity factor: (1 − 0.747258) ÷ 0.06 ≈ 4.212364.
- PV = $1,000 × 4.212364 ≈ $4,212.36.
Answer: About $4,212.36 today (rounded to the cent).
Ordinary annuity vs annuity due
| Ordinary annuity | Annuity due | |
|---|---|---|
| Payments arrive | End of each period | Start of each period |
| Examples | Loan repayments, bond coupons | Rent, leases, insurance premiums |
| $1,000 a year, 5 years, 6% | $4,212.36 | $4,465.11 |
Harder annuity questions
What yearly payment repays a $10,000 loan over 5 years at 6%?
Turn the formula around: C = 10,000 ÷ annuity factor = 10,000 ÷ 4.212364 ≈ $2,373.96 a year.
What if the payments last 10 years instead of 5?
$1,000 × [1 − 1.06⁻¹⁰] ÷ 0.06 ≈ $7,360.09. Doubling the length doesn't double the value, because later payments are discounted more.
Common annuity questions
What is the formula for the present value of an annuity?
PV = C × [1 − (1 + r)⁻ⁿ] ÷ r, where C is the payment per period, r is the rate per period and n is the number of payments. This version assumes payments come at the end of each period.
What is an annuity factor?
The annuity factor is the [1 − (1 + r)⁻ⁿ] ÷ r part of the formula. It tells you what $1 a period for n periods is worth today. At 6% for 5 years it is about 4.2124.
What is the difference between an annuity and a perpetuity?
An annuity pays for a fixed number of periods, while a perpetuity pays forever. In fact, an annuity's value equals a perpetuity with its first payment in period 1 minus a perpetuity with its first payment in period n + 1.
How do you calculate the present value of an annuity on a financial calculator?
Enter N as the number of payments, I/Y as the rate per period, PMT as the payment and FV as 0, then compute PV. The answer shows as negative on many calculators because it is treated as a cash outflow.