RatedrillPresent value of an annuity
The present value of an ordinary annuity is PV = PMT x (1 - (1 + r)^-n) / r: the lump sum today that is worth the same as n equal payments at rate r per period. It prices loans, leases, pensions and bond coupons, and solving it for PMT gives the loan payment formula.
Syntax
PV = PMT x (1 - (1 + r)^-n) / r| Part | What it is |
|---|---|
PMT | Payment per period, at the end of each period |
r | Discount rate per period |
n | Number of payments |
Examples
PMT = 100 per year for 3 years, r = 5%
PV = 100 x (1 - 1.05^-3) / 0.05Result: PV = 272.32
Excel: =PV(5%, 3, -100)
PMT = 500 per month for 5 years, 6% a year
PV = 500 x (1 - 1.005^-60) / 0.005Result: PV = 25,862.78
Excel: =PV(6%/12, 60, -500)
Pension of 1,000 per year for 20 years at 7%
PV = 1,000 x (1 - 1.07^-20) / 0.07Result: PV = 10,594.01
Common mistakes
- Forgetting the minus sign in the exponent: it is (1 + r)^-n, which is 1 / (1 + r)^n.
- Using this formula for payments that start immediately; multiply by (1 + r) for an annuity due.
- Getting a result larger than PMT x n: the present value must be smaller than the undiscounted sum (272.32 versus 300 above).
Practice Present value of an annuity
These drills check your answer on live data, free, no account needed.
- Present value of a payment streamA legal settlement pays $1,000 at the end of each year for 25 years. At a discount rate of 3.25% per year, what is the payment stream worth today?
- Deferred annuityA trust will pay $2,000 a year for 20 years, with the first payment 3 years from today. At 7.5% per year, what is the trust's payment stream worth today?
- Monthly income from savingsYou retire with $200,000 earning 3.25% APR, compounded monthly. What equal withdrawal at the end of each month uses up the balance in exactly 35 years?
Questions
What is the annuity factor?
The annuity factor is (1 - (1 + r)^-n) / r, the present value of 1 per period. At 5% for 3 years it is 2.7232; multiply any payment by it to get the present value.
How is this related to the loan payment formula?
A loan is an annuity: the amount borrowed is the present value of the payments. Rearranging gives PMT = PV x r / (1 - (1 + r)^-n), which is exactly the mortgage payment formula.
Related
- Annuity formula (future value)The future value of an ordinary annuity is PMT x ((1 + r)^n - 1) / r. Examples for yearly and monthly deposits, the Excel FV function and common errors.
- Mortgage payment formulaHow to calculate a monthly mortgage payment with M = P x r / (1 - (1 + r)^-n). Examples for 200,000 at 6% and 300,000 at 5%, Excel PMT and common mistakes.
- How much can I borrowThe loan you can afford for a given payment is the present value of an annuity: P = PMT x (1 - (1 + r)^-n) / r. Mortgage and car loan examples and Excel PV.
- PerpetuityA perpetuity pays a fixed amount forever and is worth PV = C / r. The growing perpetuity formula C1 / (r - g), examples for preferred stock and dividends.