Present Value of Annuity
Present value of a series of equal payments.
Formula
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Example
A $1000/year pension for 10 years at 5% is worth about $7722 today.
How it works
This formula tells you how much a fixed stream of future payments is worth in today’s money. Each payment is discounted back to the present by dividing by powers of 1+r, because money received later is worth less than money received now due to interest and opportunity cost; the formula is just the sum of that geometric series. It works for ordinary annuities, where payments occur at the end of each period and stay constant in size.
When to use it
Use this when valuing loans, leases, pensions, bond coupon streams, or any contract that pays the same amount at regular intervals. It is especially useful when you need the lump-sum price today of a predictable payment schedule.
Worked example, step by step
Suppose you will receive $2,500 at the end of each year for 8 years, and the discount rate is 6% per year. PV = P(1 - (1+r)^(-n)) / r PV = 2500(1 - (1.06)^(-8)) / 0.06 (1.06)^8 = 1.593848, so (1.06)^(-8) = 1 / 1.593848 ≈ 0.627412 PV = 2500(1 - 0.627412) / 0.06 PV = 2500(0.372588) / 0.06 PV = 931.47 / 0.06 PV ≈ 15,524.54 So the 8-year payment stream is worth about $15,525 today.
Common mistakes to avoid
- Using the formula for payments made at the beginning of each period without adjusting for an annuity due
- Putting the discount rate in as a percentage like 6 instead of a decimal like 0.06
- Forgetting that n counts the number of payment periods, not the number of calendar years if compounding and payments occur more frequently
Did You Know?
Lottery winners face this math when choosing a lump sum over annual payments — the lump is worth less than the total.
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