Annuity Future Value

Savings accumulation from repeated contributions.

Worked example — Annuity Future Value

Future value	6.1×10⁵
Total contributions	1.8×10⁵
Investment growth	4.3×10⁵
Growth share	70.491%

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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Lesson

The theory — Annuity Future Value

An annuity here is a stream of equal deposits at equal intervals, and its future value is what that stream is worth at the end: every deposit, plus all the growth each one earned in the time it had left. One warning before the algebra, because it catches almost everyone — the r and n in the formula below are not the numbers in the boxes above. They are the rate per period and the number of periods.

What each symbol means

d: the deposit made every period. Constant: this formula has no room for a payment that changes.
r: the rate per period, and this is the one that catches people out. The box above asks for an annual 7%, but the formula uses 0.0058333 — that is 0.07 / 12, because the deposits are monthly.
n: the number of periods, not years. Thirty years of monthly deposits is n = 360, which is the figure the working above substitutes.

Where the formula comes from

Line the deposits up by how long each one grows. The first has n − 1 periods left, the last has none, so the total is d·(1+r)ⁿ⁻¹ + d·(1+r)ⁿ⁻² + ⋯ + d.
That is a geometric series: each term is the next one multiplied by (1+r). Its closed form is derived in step 5 of Sequences & series, the first step on this learning path — with first term d, ratio (1+r) and n terms it sums to d·((1+r)ⁿ − 1) / ((1+r) − 1). If that formula is not already familiar, start there and come back.
The denominator (1+r) − 1 is simply r, which leaves FV = d·((1+r)ⁿ − 1) / r — the formula printed above. Substituting the per-period figures: 500·((1+0.0058333)³⁶⁰ − 1) / 0.0058333 = 6.1 × 10⁵, which is 360 separate deposits collapsed into one line.

Assumes: A constant rate and a constant deposit, with every deposit landing at the end of its period and compounding at exactly the same frequency. Real returns are not constant, and a fund that compounds daily while you pay monthly does not fit this formula.
Breaks when: Nothing here is adjusted for inflation, so the 6.1 × 10⁵ is in the money of the final year rather than today’s. It also assumes you never miss a deposit and never withdraw. The zero-rate case is worth trying, because the display and the answer part company: set the rate to 0 and the working prints FV = 500·((1+0)³⁶⁰ − 1) / 0, a zero denominator — yet the result is a correct 1.8 × 10⁵, because as r approaches 0 the whole fraction approaches n, leaving exactly d × n.

why the last 10 years do most of the work 🖖

Compound interest is exponential, not linear. In a 30-year retirement plan, roughly half the total balance accumulates in the final 10 years. This is why starting early matters so much more than contributing more later — the base on which interest compounds is much larger after 20 years of growth. Doubling your deposit at year 25 helps far less than starting 5 years earlier would have.

what an annuity really is 🖖

An annuity is simply a stream of equal deposits made at regular intervals — monthly, quarterly, or yearly. This tool adds them up while every past deposit keeps earning on its own, then splits the final balance into two parts: the money you put in (blue line) and the growth it generated (green shading). The takeaway: once the green balance line pulls clearly away from the blue contributions line, most new money is coming from returns, not from your wallet.

the same formula doses your medicine 🖖

The engine here is a geometric series: FV = d·((1+r)ⁿ − 1)/r. Swap the growth factor (1+r) for a shrinking one and you get the exact equation pharmacologists use for repeated drug doses. Because each deposit grows while each drug dose decays between intervals, an annuity climbs without limit while blood concentration levels off at a steady-state plateau — same equation, opposite destiny, decided only by whether the ratio sits above or below 1.

Practice

Check yourself

Predict the answer first, then use the controls above to find out. Reveal only after you have committed to a guess — that is what makes it practice.

The defaults save 500 a month for 30 years at 7% — 6,000 a year. Set Deposits per year to 1 and Deposit per period to 6000: identical money, paid in one annual lump. Predict what the future value does.

Show answer

It drops from 6.1 × 10⁵ to 5.668 × 10⁵, on total contributions that stay at 1.8 × 10⁵. The line underneath prices the difference: 429,985 of growth against 386,765, so roughly 43,000 for nothing but timing. Money paid in January compounds eleven months longer than money paid in December, and you have just done that thirty times over.
Same experiment, but watch the crossover point — the year growth overtakes what you have paid in. Try one deposit of 6000 a year, then 500 twelve times, then 16.438356 daily. Predict how the crossover moves.

Show answer

It does not move: Year 11 in all three. Frequency changes the destination — 5.668 × 10⁵ annually, 6.1 × 10⁵ monthly, 6.141 × 10⁵ daily — but the crossover is a race between a total growing exponentially and a total growing in a straight line, and shifting deposits a few weeks earlier barely touches when one passes the other. Notice how little that last step buys, too: monthly to daily adds about 4,000, where annual to monthly added 43,000.

Problem solved in full

The future value of 500 a month for 30 years at 7% 6 steps
500 a month for 30 years at 7%. Derive the future value from the series rather than a spreadsheet, then work out what it is actually worth.
1. Each deposit is a separate lump sum that compounds for however long it has left. Adding them up is a geometric series with ratio (1+i), and the closed form is the only formula in the whole of retirement arithmetic.
  
  $FV = k = 0 \sum n - 1 d (1 + i)^{k} = d \frac{( 1 + i ) ^{n} - 1}{i}$
2. Convert to the compounding period before anything else. Both the rate and the count are monthly — mixing an annual rate with a monthly count is the classic way to be out by a factor of twelve.
  
  $i = \frac{0.07}{12} = 0.0058333, n = 30 \times 12 = 360$
3. Substitute. Note where the leverage is: the numerator grew by a factor of 8.1 over the term, and dividing by a small i multiplies again.
  
  $(1 + i)^{360} = 8.1165 ⟹ FV = 500 \cdot \frac{8.1165 - 1}{i} = 609985$
4. Contributions are trivially deposit times count. Everything above that is compounding, and here it is 70% of the total — the account earns more than the saver pays in.
  
  $contributions = 500 \times 360 = 180000, growth = 429985 = 70.491%$
5. The deposits are not equal in effect. The first has 359 months to grow and the last has none.
  
  $first deposit \to 500 (1.0058333)^{359} = 4035; last \to 500$
6. Finally, deflate. The pot is nominal and the deposits were fixed in nominal terms, so the honest question is what 609,985 buys in thirty years.
  
  $\frac{609 985}{1.02 5 ^{30}} = \frac{609 985}{2.0976} = 290806$
Answer

609,985 nominal — and about 291,000 in today's money. Inflation at 2.5% divides the pot by 2.10 over thirty years, which turns a 430,000 gain into a real gain of roughly 111,000 on 180,000 paid in. Nothing on the panel is wrong; it simply answers in the currency of 2056. The other thing hiding in the geometric series is how unequal the deposits are: the first 500 arrives as 4,035 and the last arrives as 500, so the opening year matters eight times more than the closing one. Starting a decade earlier beats saving a third more.

$FV = d \frac{( 1 + i ) ^{n} - 1}{i} = 609985 nominal; 290806 in today’s money$

Learning path

Money over time

Leads to Loan payment the summed value of a stream of equal payments.

References (1)

Rate per period versus quoted annual rate — the distinction the lesson opens with: R. J. Wilders, "Measuring Interest," in Financial Mathematics for Actuarial Science: The Theory of Interest, 19–66. Chapman and Hall/CRC, 2020.

By the 42Math team · Updated 2026-09-03

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

Retirement Fund - Simulates saving $500/month for 30 years at a 7% annual rate. Demonstrates how a long timeline allows growth to compound until it accounts for over 60% of the final wealth.
College Fund - Simulates saving $300/month for 18 years at a 5% annual rate. Ideal for education savings from birth to graduation, illustrating lower-risk accumulation.
Quarterly Plan - Simulates saving $2,000/quarter for 12 years at a 6% annual rate. Demonstrates how quarterly compound interest accumulates on mid-term capital investments.