How to use the time value of money solver
- Pick the value you want to find in Solve for: the payment, the future value, the present value, the number of periods or the annual rate.
- Fill in the other four values. Enter money you receive as a positive number and money you pay as a negative number, so a loan has a positive present value and negative payments.
- Set Periods per year to match how often payments happen (12 for monthly, 1 for yearly) and choose whether payments fall at the end or the start of each period.
- Read the answer, then the balance table, which replays the money period by period so you can see the result is not just a formula output.
Formula
PV x (1 + i)^n + PMT x (1 + i x t) x ((1 + i)^n - 1) / i + FV = 0
i = annual rate / 100 / periods per year
t = 0 for payments at the end of a period, 1 for the start
When i = 0: PV + PMT x n + FV = 0
Every solved value comes from rearranging this one equation. The number of periods uses a logarithm, and the rate has no closed form, so it is found by scanning for a sign change and then bisecting.
A 30 year loan, and how long money takes to double
Borrow $200,000 over 360 monthly payments at 6% a year with payments at the end of each month. Here i is 0.5%, the present value is 200,000 and the future value is 0. Solving for the payment gives ($1,199.10), shown in parentheses because it is money you pay. Replaying the loan month by month leaves a balance of zero after payment 360, and the payments add up to $431,676.38, which is $231,676.38 of interest.
To find how long $1,000 takes to become $2,000 at 6% a year with no payments, set the present value to -1,000, the future value to 2,000, the payment to 0 and periods per year to 1. The number of periods is ln(2) / ln(1.06) = 11.8957, so a little under 12 years. A savings goal of $50,000 in 120 months at 5% a year needs a payment of ($321.99) a month.
Why one solver covers many questions
Loan payments, savings plans, annuities, leases and bond prices all balance the same flows of money: what you start with, what you add or take out each period, and what is left at the end. When four of the five quantities are known, the fifth is fixed. That is why a financial calculator has five keys for N, rate, present value, payment and future value.
This page keeps that structure but shows its work. For single-purpose versions see the Annuity Calculator, the Present Value Calculator and the Interest Rate Calculator. For loans quoted with a flat rate, the Add-On Interest Loan Calculator finds the true annual rate.
Signs, timing and when there is no answer
Signs carry the meaning. If you borrow and also enter a positive payment, no rate or number of periods can make the equation balance, and the tool says so instead of showing a made-up number. Payments at the start of a period earn one more period of interest, so the same payment grows to a larger future value.
Solving for the rate can have no answer or more than one when the cash flows change sign several times. If more than one rate fits, the page lists them and shows the first one at or above zero. If the number of periods is not whole, the page tells you how many payments you would need and that the last one is smaller.
Assumptions and limits
- The rate is constant and compounds once per payment period. A nominal annual rate is divided by the periods per year to get the rate per period.
- Payments are equal and happen at the same point of every period. Fees, taxes, skipped payments and changing rates are not modeled.
- The rate search covers rates from -95% to 500% per period. Rates outside that range are reported as not found.
- Everything is calculated in your browser and nothing you type is stored or sent anywhere.
Frequently asked questions
What does the sign of each value mean?
A positive number is money you receive and a negative number is money you pay. A loan has a positive present value and negative payments. A savings plan has negative payments and a positive future value to collect.
Why do payments at the start of the period give a different answer?
Each payment earns interest for one extra period when it is made at the start. The tool multiplies the payment stream by (1 + i) in that case, which lowers the payment needed to reach the same goal.
What if I get a message that no rate fits?
It means the equation has no solution between -95% and 500% per period for the numbers you entered. The usual cause is two values with the same sign when one should be negative, such as a loan amount and a payment that are both positive.
How is the balance table a check on the answer?
It starts from the present value and applies interest and the payment for every period, without using the closed form. The last balance should match the negative of the future value, and the page shows the gap.