TVM Calculator

What a time value of money calculator does

The time value of money (TVM) says a dollar today is worth more than a dollar later, because today's dollar can earn interest. Every loan, savings plan, and annuity comes down to five numbers: present value (PV), payment (PMT), future value (FV), interest rate, and number of periods (N). Enter any four and this TVM calculator solves for the fifth, the same job as the TVM keys on a financial calculator such as the TI BA II Plus or HP 12C.

Time value of money formula

PV × (1 + i)^N + PMT × (1 + i × k) × [(1 + i)^N − 1] ÷ i + FV = 0

• i is the rate per period. For a nominal rate (APR), i = annual rate ÷ periods per year. For an effective rate (APY), i = (1 + annual rate)^{1 ÷ periods per year} − 1.
• k is 0 when payments come at the end of each period (an ordinary annuity) and 1 when they come at the beginning (an annuity due).
• PV, PMT, and FV use signs for direction: money you receive is positive and money you pay out is negative.

PV, PMT, FV, and N have closed-form solutions. The interest rate does not, so the calculator finds it numerically, to well beyond the three decimal places it displays.

Using it as a future value calculator, present value calculator, or annuity calculator

Future value: $1,000 invested today plus $100 at the end of every year for 10 years at 5% compounded annually. Enter PV = −1,000, PMT = −100, N = 10, and a 5% rate, then solve for FV: $2,886.68.

Present value: to reach $10,000 in 5 years while saving $100 a month at 5% compounded monthly, enter PMT = −100, FV = 10,000, N = 60, and solve for PV: you need to deposit $2,492.98 today.

Loan payment: a $250,000 mortgage at 6.5% for 30 years with monthly payments is PV = 250,000, FV = 0, N = 360. Solving for PMT gives −$1,580.17 a month, and $318,861.22 of total interest.

Frequently asked questions

Why do I need negative numbers in a TVM solver?

The sign shows which way money moves. When you deposit money, PV is negative because it leaves your pocket, and FV is positive because you get it back. If every value has the same sign, no interest rate or number of periods can balance the equation, and the calculator says so.

What is the difference between nominal and effective rates?

A nominal rate (APR) is divided evenly across the periods in a year. An effective rate (APY) already includes compounding. A 6% nominal rate compounded monthly is 0.5% a month and an effective rate of about 6.168%; a 6% effective rate is about 0.487% a month.

When should I choose beginning of period?

Choose beginning when each payment is made at the start of the period, such as rent, a lease, or savings deposited on the first of each month. Each payment then earns one more period of interest. Most loans are paid at the end of each period.

Can I use it as an annuity calculator?

Yes. An annuity is a series of equal payments, which is exactly what PMT models. For the present value of an annuity paying $500 a month for 10 years at 5% compounded monthly, enter PMT = 500, FV = 0, N = 120, and a 5% rate, then solve for PV: −$47,140.68, the lump sum you would pay today for that income. Choose beginning of period for an annuity due.

How long does it take to double my money?

Enter PV = −1,000, FV = 2,000, PMT = 0, and a rate, then solve for periods. At 7% compounded annually it takes about 10.24 years. Solving for the rate instead, doubling in exactly 10 years needs about 7.177% a year.

Limitations

The calculator assumes one interest rate for the whole term, equal payments, and payments made as often as interest compounds. Daily uses 365 periods a year. Real loans can differ because of fees, rounding of each payment, or a different day-count rule. It is not financial advice.