Present value of a series of payments.
PV = PMT × [1 − (1 + r)^(−n)] / r FV = PMT × [(1 + r)^n − 1] / r
There are two core annuity calculations:
1. Present Value (PV) of an Ordinary Annuity — the lump-sum value today of all future equal payments, discounted at a given interest rate.
2. Future Value (FV) of an Ordinary Annuity — the total accumulated value of all payments at the end of the annuity term, assuming each payment earns interest.
Both assume payments occur at the end of each period (ordinary annuity). For an annuity-due (payments at the beginning), multiply the result by (1 + r).
PV = 300 × [1 − (1 + 0.005)^(−60)] / 0.005 = 300 × [1 − (1.005)^(−60)] / 0.005 = 300 × [1 − 0.74137] / 0.005 = 300 × [0.25863] / 0.005 = 300 × 51.7256 = $15,517.68
Result: Present Value ≈ **$15,517.68**
This result means that a series of $300 monthly payments over 5 years at a 6% annual interest rate is worth approximately $15,517.68 today. In other words, if you deposited $15,517.68 in an account earning 6% annually (compounded monthly) right now, you could withdraw exactly $300 every month for 60 months and the account would reach exactly $0 at the end. This is a core concept in loan pricing, structured settlements, and retirement income planning.
Annuity math is grounded in the time value of money: a dollar today is worth more than a dollar tomorrow because money can earn interest. Discounting (for PV) and compounding (for FV) are two sides of the same coin.
PV = 1000 × [1 − (1.05)^(−10)] / 0.05 = 1000 × 7.7217 ≈ **$7,721.73**.
FV = 200 × [(1 + 0.07/12)^(240) − 1] / (0.07/12) ≈ 200 × 520.93 ≈ **$104,185** (approximately).
For an investor *receiving* payments, a higher PV is better — it means the income stream is worth more today. For someone *making* payments (like a borrower), a lower PV is preferable, as it means a smaller debt obligation.
An annuity has a finite number of payments (n periods), while a perpetuity pays forever. A perpetuity's present value is simply PMT ÷ r, the limiting case of the annuity formula as n approaches infinity.
Standard annuity formulas use a nominal interest rate and assume constant payment amounts. To account for inflation, use a real interest rate (nominal rate minus inflation rate) or model a growing annuity where PMT increases by the inflation rate each period.
An ordinary annuity makes payments at the **end** of each period (most loans and investments), while an annuity-due makes payments at the **beginning** of each period (rent, insurance). The annuity-due values are always slightly higher — multiply ordinary annuity PV or FV by (1 + r) to convert.
Set the PV equal to your loan amount, enter the monthly interest rate (annual rate ÷ 12) and the number of monthly periods, then solve for PMT. For example, a $20,000 car loan at 5% annual for 48 months gives a monthly payment of about $460.59.
Yes. Use the Future Value mode: enter your planned monthly contribution as PMT, your expected annual return as the interest rate, and the number of months until retirement as n. The FV shows your projected nest egg, assuming a constant rate.
If r = 0, the formula breaks down mathematically (division by zero). In that case, PV = FV = PMT × n, because no interest is earned or discounted — you simply sum all equal payments.
Yes. Simply match your rate and period to the payment frequency. For quarterly payments, use r = annual rate ÷ 4 and n = number of quarters. The formula is identical; only the units change.
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