Time Value of Money Calculator

What the time value of money means

A dollar today is worth more than a dollar promised next year, because the dollar you hold now can be saved or invested and earn a return. The time value of money (TVM) puts a number on that difference. It links five quantities: a starting amount (present value), an ending amount (future value), an interest rate, a number of periods and, optionally, a stream of equal payments.

Knowing any four of the five lets you solve for the fifth. That is the job of this calculator: pick the unknown with the radio buttons and the form shows only the fields that matter for it.

This tool gives mathematical estimates under simple assumptions. It is not investment, tax or financial advice. See the disclaimer.

How to use the calculator

• Future value: enter the present value, the interest rate, the number of periods and, if you add or withdraw money each period, the amount.
• Present value: enter the future value you want, the rate, the periods and any regular payment. You get the amount that would have to be on hand today.
• Repeating payment: enter present and future value, the rate and the periods. The result is the size of the equal payment made at the end of each period, labelled Add when money has to be paid in to reach the target and Withdraw when the balance would overshoot it and can be drawn down.
• Interest rate: enter present and future value and the periods, plus a payment if there is one. The result is the rate per period.
• Periods: enter present and future value and the rate, plus a payment if there is one. The result is how many periods it takes. At least two of present value, future value and payment must be non-zero. A future value of 0 is allowed when there is a payment, which tells you when withdrawals use a balance up or when deposits clear a debt entered as a negative present value.

Use the Cash flow selector to say whether the optional payment adds money to the account or takes it out. Leave the payment empty for a one-off lump sum. The interest rate is the rate for one period, so if periods are months, enter the monthly rate.

The formulas

Let PV be the present value, FV the future value, r the rate per period as a decimal, n the number of periods and C the payment made at the end of each period (positive for deposits, negative for withdrawals).

FV = PV × (1 + r)ⁿ + C × ((1 + r)ⁿ − 1) ÷ r

PV = FV ÷ (1 + r)ⁿ − C × ((1 + r)ⁿ − 1) ÷ (r × (1 + r)ⁿ)

C = (FV − PV × (1 + r)ⁿ) × r ÷ ((1 + r)ⁿ − 1)

n = ln((FV × r + C) ÷ (PV × r + C)) ÷ ln(1 + r)This is exact, not an approximation. If the ratio inside the logarithm is zero or negative, no number of periods reaches the target and the calculator says so.

without payments: r = (FV ÷ PV)^1/n − 1With payments there is no closed form, so the calculator finds the rate numerically.

Worked examples

Future value. You hold $5,000 and add $200 at the end of every period for 12 periods at 4% per period. The growth factor is 1.04¹² = 1.6010. The lump sum grows to 5,000 × 1.6010 = $8,005.16, and the deposits add 200 × (1.6010 − 1) ÷ 0.04 = $3,005.16. The future value is $11,010.32.

Present value. To have $50,000 in 10 periods at 6% with no extra payments, you need 50,000 ÷ 1.06¹⁰ = 50,000 ÷ 1.7908 = $27,919.74 today.

Repeating payment. You start with $2,000 and want $30,000 after 8 periods at 3.5%. Your $2,000 grows to 2,000 × 1.3168 = $2,633.62, leaving a gap of $27,366.38. Divide by the annuity factor (1.3168 − 1) ÷ 0.035 = 9.0517 to get $3,023.35 per period. The positive result is shown as Add: that much has to be paid in at the end of every period. If instead you start with $5,000 and want $1,000 after 10 periods at 5%, the balance would grow to 5,000 × 1.6289 = $8,144.47 on its own. The gap is negative, so the answer is a Withdraw of $568.02 per period.

Interest rate. Money that grows from $8,000 to $12,000 in 6 periods earned (12,000 ÷ 8,000)^1/6 − 1 = 6.99% per period.

Periods. At 5% per period, money doubles after ln 2 ÷ ln 1.05 = 14.21 periods. The familiar rule of 72 gives 72 ÷ 5 = 14.4, a close shortcut. With a payment, a $10,000 balance earning 5% and drawn down by $1,000 per period reaches zero after ln(1,000 ÷ 500) ÷ ln 1.05 = 14.21 periods as well. A balance that shrinks every period can never climb to a higher target, and the calculator reports that instead of a number.

Planning withdrawals

Choose Withdraw in the cash flow selector to model spending from a balance. Suppose a fund of $400,000 earns 5% a year and you take out $30,000 at the end of each year for 15 years. The balance grows to 400,000 × 2.0789 = $831,571.27 while the withdrawals, compounded, total 30,000 × 21.5786 = $647,356.91. What is left is $184,214.36.

To find the opening balance that would be used up exactly, solve for present value with no future value target. Here 30,000 × (1 − 1.05^-15) ÷ 0.05 = $311,389.74 pays out $30,000 a year for 15 years and leaves nothing. To see how a balance grows when nothing is withdrawn, use the future value calculator; to discount a stream of uneven cash flows, use the NPV calculator.

Assumptions and common mistakes

• Match the rate to the period. A 6% annual rate with monthly payments needs a monthly rate of 0.5% and a period count in months. Mixing yearly rates with monthly periods is the most frequent error.
• Payments come at the end of each period. Deposits made at the start of a period earn one more period of interest and would give a slightly higher result.
• The rate is constant. Real returns vary; the calculator assumes the same rate every period and no taxes, fees or inflation.
• A zero rate is not accepted. Without interest there is nothing to discount, and the formulas above divide by r. Enter a small positive rate instead.
• Check the direction of a repeating payment. If the target is higher than your starting balance would grow to by itself, you need to add money each period (shown as Add); if it is lower, the balance can fund withdrawals (shown as Withdraw).

Nominal and real values differ: a future sum is worth less in purchasing power if prices rise. To compare an investment with a required return over uneven cash flows, see the IRR calculator and the present value calculator.