What is the present value?
Enter a future amount, a payment each period, or both, with the discount rate and the number of periods. See what they are worth today.
- Present value
- $6,139.13
At 5% a year, $10,000.00 after 10 periods plus $0.00 each period is worth $6,139.13 today.
- Rate per period
- 5%
- Effective yearly rate
- 5%
- Total of payments
- $0.00
- Interest (the discount)
- $3,860.87
Present value: $6,139.13. At 5% a year, $10,000.00 after 10 periods plus $0.00 each period is worth $6,139.13 today.
The results are estimates for information only. They are not financial, tax, or legal advice. Check the numbers with your lender or a qualified professional before you decide. Terms of use
How to calculate
Computes the present value of a future amount, of level payments (an annuity), or of both, at a yearly discount rate, with payments at the end or the start of each period.
Example with the default inputs (Find the Present, Number of periods 10, Discount rate (yearly) 5%, Payment each period $0.00, Future value $10,000.00, Periods a year 1, Compounding a year 1, Payments at the End (END)): At 5% a year, $10,000.00 after 10 periods plus $0.00 each period is worth $6,139.13 today.
Method: Present value = (future value + payment × (1 + i × t) × ((1 + i)^N − 1) ÷ i) ÷ (1 + i)^N, with i the rate per period and t = 1 for payments at the start of each period, 0 at the end.
- The interest rate is the same in every period, and every payment is the same size.
- The yearly rate is nominal, compounded as many times a year as set; payments are made as many times a year as set.
- The payments and the future amount are money received, and the present value is what they are worth today.
- No fees or taxes are included. This is an estimate for planning, not financial advice.
Worked examples
Each example is checked against the calculator on every build.
- Find the Present, Number of periods 10, Discount rate (yearly) 5%, Payment each period $0.00, Future value $10,000.00, Periods a year 1, Compounding a year 1, Payments at the End (END) gives Present value $6,139.13, Interest (the discount) $3,860.87.Source: The compound interest formula A = P(1 + r)^n solved for P (SEC Investor.gov, https://www.investor.gov/financial-tools-calculators/calculators/compound-interest-calculator)
- Find the Present, Number of periods 240, Discount rate (yearly) 8%, Payment each period $500.00, Future value $0.00, Periods a year 12, Compounding a year 12, Payments at the End (END) gives Present value $59,777.15, Total of payments $120,000.00.Source: Microsoft Excel PV function example: ($59,777.15) (https://support.microsoft.com/en-us/office/pv-function-23879d31-0e02-4321-be01-da16e8168cbd)
- Find the Present, Number of periods 5, Discount rate (yearly) 6%, Payment each period $1,000.00, Future value $0.00, Periods a year 1, Compounding a year 1, Payments at the Start (BGN) gives Present value $4,465.11.Source: Present value of an annuity due, Broverman (2017), Mathematics of Investment and Credit, section 2.1
- Find the Present, Number of periods 20, Discount rate (yearly) 4%, Payment each period $50.00, Future value $1,000.00, Periods a year 2, Compounding a year 2, Payments at the End (END) gives Present value $1,490.54.Source: Present value of an annuity plus a lump sum, Broverman (2017), Mathematics of Investment and Credit, chapter 2
How the present value is worked out
Inputs. The number of periods N, the yearly discount rate I/Y in percent, a payment received each period (PMT), a future amount received after the last period (FV), the periods a year P/Y and the compounding times a year C/Y (whole numbers from 1 to 365), and whether payments come at the end or the start of each period. All amounts are positive. An empty payment or future value counts as 0. Find the picks the value to work out: usually the present value, but any one of the five can be found from the others.
Rate per period. i = (1 + I/Y ÷ (100 × C/Y))^(C/Y ÷ P/Y) − 1. With P/Y = C/Y = 1 (the default) this is I/Y ÷ 100 and N is in years.
Present value. With g = (1 + i)^N, t = 0 for payments at the end and 1 at the start, and a = (1 + i × t) × (g − 1) ÷ i (a = N when i = 0):
PV = (FV + PMT × a) ÷ g
This is the same equation as a financial calculator's, PV × (1 + i)^N = FV + PMT × a, with the present value as money paid and the payments and future value as money received.
The other values. From the same equation: FV = PV × g − PMT × a; PMT = (PV × g − FV) ÷ a; N = ln(g) ÷ ln(1 + i) with g = (PMT × (1 + i × t) − FV × i) ÷ (PMT × (1 + i × t) − PV × i) (at 0%, N = (PV − FV) ÷ PMT); and the rate is every yearly rate above −100% and at most 1,000% where PV × g − PMT × a − FV changes sign, found on a grid (every 0.01% from −20% to 50%, every 0.1% from −99.9% to −20% and from 50% to 1,000%, and −99.95% to −99.999999% by factors of ten) and refined by bisection.
Other results. Total of payments = N × PMT. Interest (the discount) = N × PMT + FV − PV. Rate per period = 100 × i. Effective yearly rate = 100 × ((1 + i)^(P/Y) − 1).
Precision and overflow. The rate search reads the sign of the equation from the form above when (1 + i)^N is at most 1, and from the same equation divided by (1 + i)^N when it is above 1, so no step overflows; a rate at which (1 + i)^N is not a finite number above 0 in 64-bit floating point is dropped. With the present value, the payment and the future value all 0 (or empty) the equation holds at every rate, so there is no rate. The other computed values use the same divided equation when (1 + i)^N is above 1, so an answer inside the limits never fails on a larger number on the way. The total of payments and the interest (the discount) are exact decimal arithmetic on the typed amounts and the computed value, each read as the decimal it prints as, rounded once at the end. The computed value, the rates and the powers are 64-bit floating point, so a value that is exactly half a cent in decimals can land just below it and show rounded down.
Limits and no answer. N is above 0 and at most 12,000; the rate is above −100% and at most 1,000%; amounts are from $0 to $1 trillion. A computed amount below $0 or over $1 trillion, or a computed N of 0 or less or over 12,000, is no answer. So is a (1 + i)^N that is not a finite number above 0 in 64-bit floating point, N at 0% with no payment, N when g ≤ 0 or g = 1, and a rate when none fits.
Display. Money shows to the cent, rounded half up from the value worked out; periods at most 2 decimals; rates at most 4 (the rate per period 6).
Assumptions
- The discount rate is the same in every period and every payment is the same size.
- The payments and the future amount are certain; no taxes or fees are included.
- This is an estimate for planning, not financial advice.
Worked examples by hand
A single future amount. FV = 10,000, N = 10, 5% a year, yearly. g = 1.05^10 = 1.628895. PV = 10,000 ÷ 1.628895 = $6,139.13. The discount is 10,000 − 6,139.13 = $3,860.87.
Excel's PV example. PMT = 500 a month for N = 240 months at 8% a year (P/Y = C/Y = 12), END. i = 0.08 ÷ 12 = 0.0066667; g = 1.0066667^240 = 4.926803; a = (g − 1) ÷ i = 589.0204. PV = 500 × 589.0204 ÷ 4.926803 = $59,777.15. The payments total $120,000.
An annuity due. PMT = 1,000 at the start of each year for 5 years at 6%. g = 1.06^5 = 1.338226; a = 1.06 × 0.338226 ÷ 0.06 = 5.975319. PV = 1,000 × 5.975319 ÷ 1.338226 = $4,465.11. At the end of each year it would be 4,465.11 ÷ 1.06 = $4,212.36.
Payments and a future amount. PMT = 50 twice a year for N = 20 half-years, FV = 1,000, 4% a year compounded twice a year (P/Y = C/Y = 2). i = 0.02; g = 1.02^20 = 1.485947; a = 24.297370. PV = (1,000 + 50 × 24.297370) ÷ 1.485947 = $1,490.54.
Other questions people ask
What is present value?
Present value (PV) is what money you will receive later is worth today, given a rate of interest you could earn in the meantime. $10,000 received in 10 years is worth $6,139.13 today at 5% a year, because $6,139.13 invested at 5% grows to $10,000 in 10 years.
What is the present value formula?
For a single future amount: PV = FV ÷ (1 + i)^N, where i is the rate per period and N the number of periods. For a level payment PMT at the end of each period: PV = PMT × (1 − (1 + i)^−N) ÷ i. For payments at the start of each period, multiply the payment part by (1 + i). This page adds the two parts.
What discount rate should I use?
Use the return you could earn elsewhere on money of the same risk: a savings or CD rate for money that is sure to arrive, a higher rate for payments that might not. A higher discount rate gives a lower present value. This page does not look up any rate; you type it.
What is the present value of an annuity?
It is today’s value of a series of equal payments. $500 a month for 20 years at 8% a year is worth $59,777.15 today, even though the payments add up to $120,000. Set the future value to 0 for payments only.
What is the difference between an ordinary annuity and an annuity due?
In an ordinary annuity each payment comes at the end of its period; in an annuity due it comes at the start. Each payment of an annuity due arrives one period sooner, so its present value is (1 + i) times larger: $1,000 a year for 5 years at 6% is worth $4,212.36 at the end of each year and $4,465.11 at the start.
How is present value used to price a bond?
A bond’s price is the present value of its coupons and its face value. A 10-year bond paying $50 twice a year plus $1,000 at the end, at 4% a year compounded twice a year, is worth $1,490.54 on a coupon date. For prices between coupon dates, with accrued interest and a day count, use the bond calculator.