What future value means
Future value is what a sum of money today, plus any equal payments added along the way, grows to after a fixed number of periods at a fixed rate per period. It is the same idea whether the periods are months on a savings account, quarters on a bond, or years on a business plan; the formula only cares about the rate and the count of periods matching each other, not what a period is actually called. This tool keeps that framing deliberately loose, working in generic periods (n) rather than assuming years, because the underlying time-value-of-money formula taught in finance courses is written that way and applies equally to all of them.
To keep the numbers simple, every amount here is entered as a positive figure: present value, payment, and the future value they produce. Some textbooks instead enter an outflow like a deposit as a negative number so the formula balances to zero; this tool skips that convention and treats every figure as money moving in the same direction, growing over time.
The formula and its two special cases
The general future value formula, combining a present value with equal payments each period, is one line:
The first term is the present value compounding on its own, exactly like a lump sum earning interest. The second term is the future value of the run of payments, each one compounding for however many periods it has left once it lands. Add them together and you have the total. Two situations break out of that general case and are worth naming separately.
Annuity due. If each payment lands at the start of its period rather than the end, it gets one extra period to compound, so the whole payment term is multiplied by (1 + r):
Zero rate. When r is exactly 0 the formula above divides by zero, so it collapses to plain addition instead, with no growth term at all and payment timing making no difference:
The calculator switches to this second line automatically whenever the rate per period works out to zero, rather than trying to force the general formula through an undefined division.
Turning an annual rate into a per-period rate
The formula only ever wants r as a rate per period, but rates are usually quoted per year. The "derive from annual rate" option divides a stated nominal annual rate by however many periods sit inside a year: a 12% annual rate over 12 periods a year gives 12% ÷ 12 = 1% per period, and a 6% annual rate paid quarterly gives 6% ÷ 4 = 1.5% per period. That plain division is how a stated, or nominal, annual rate is meant to be read, and it is the convention this calculator uses throughout. It is a different figure from an effective annual rate, which asks what a per-period rate compounds up to over a full year and is very slightly higher once you reverse the division; this tool sticks to the simpler, more common textbook direction, annual rate down to a per-period one, since that is the direction the FV formula itself needs.
Why ordinary annuity and annuity due give different answers
Take the calculator's own default: a present value of 0, a payment of 100 a period, a rate of 5% a period, over 10 periods. As an ordinary annuity, each payment sits idle for the rest of the run after it lands, and the total comes to 1,257.79. As an annuity due, every one of those same ten payments has already had one period to compound by the time the run ends, and the total rises to 1,320.68, a difference of 62.89, which is exactly 5% of the ordinary total. That is not a coincidence: multiplying the whole payment term by (1 + r) always adds exactly r times itself, whatever the rate happens to be. Rent paid in advance, insurance premiums and lease payments are commonly annuity due in the real world; loan repayments and most savings contributions are commonly ordinary annuities, paid at the end of the period they cover.
Reading the contributions, the chart and the tables
The headline splits the future value into what was actually paid in, the present value plus every payment added along the way, and what growth from the rate contributed on top. The chart draws the same split period by period: the dashed line is pure addition, present value plus payments with nothing compounding, while the solid line is the actual future value, and the widening gap between them is growth doing its work. The period-by-period table underneath gives every row of that same data; past 120 periods it thins itself out to a readable set of rows, always keeping the final period, with a note above the table saying so.
The smaller table asks one more question: how much does the future value move if the rate were a single point higher or lower, everything else held exactly as entered? It is a quick sensitivity check, not a forecast, useful for seeing how exposed a plan is to a rate that turns out different from assumed.
Which calculator actually fits your question
This page is the formula tool: periods instead of years, a rate per period instead of an annual percentage, and an explicit ordinary annuity or annuity due choice, the shape a finance course or a textbook problem usually arrives in. The Compound Interest Calculator answers a related but different everyday question, working in years and a monthly contribution with a chosen compounding frequency, aimed at an actual savings account rather than a generic formula. If there is no rate at all involved, only a flat percentage on a fixed sum over a fixed time, the Simple Interest Calculator handles that plainer arithmetic instead. If the question runs the other way, starting from a target amount and asking when you will reach it, the Savings Calculator is built around that goal-first framing.
Questions people ask
What is the difference between future value and present value?
Future value asks what a sum today, plus any regular payments, grows to after a number of periods at a given rate. Present value asks the reverse question: what a future sum is worth today once that same rate and time are stripped back out. The two use the same variables and the same rate; only the direction of the question changes.
What is the difference between an ordinary annuity and an annuity due?
An ordinary annuity pays at the end of each period; an annuity due pays at the start. Because an annuity-due payment gets one extra period to compound, its future value is always the ordinary annuity's future value multiplied by (1 + r), never less, for any rate above zero.
How do I get a per-period rate from an annual rate?
Divide the stated annual rate by the number of periods in a year: a nominal 12% a year over 12 monthly periods is 1% a period; the same 12% over 4 quarterly periods is 3% a period. The "derive from annual rate" option on this calculator does that division for you and shows the result.
What happens if the payment (PMT) is zero?
With no payment, the formula reduces to the future value of a single lump sum, FV = PV × (1 + r)n, and the ordinary annuity and annuity due toggle stops making any difference, since there is no payment left for timing to affect.
What happens if the rate is zero?
The formula switches to plain addition, FV = PV + PMT × n, since there is no growth term to divide by a zero rate. Timing no longer matters here either: with nothing compounding, it makes no difference whether a payment is dated to the start or the end of its period.
Does a period have to be a year?
No. A period is whatever r is quoted for, a month, a quarter, a year, even a week, as long as n counts periods of that same length and r is the rate for one of them. This tool works in generic periods rather than years for exactly that reason.
Why is present value entered as a positive number here?
Some textbooks enter an amount paid in, such as a present value or a payment, as a negative number so the whole equation balances to zero. This tool keeps every figure positive and simply adds the growth on top, which gives the same future value without the sign bookkeeping; check which convention your own course or textbook expects before comparing figures directly.
This calculator illustrates the standard future value formula for study and planning purposes. It is not financial advice. Real accounts, loans and investments may compound unevenly, charge fees, or apply a day-count or sign convention different from the one used here.