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CalcMax

Savings Calculator

Range: 0 – 10,000,000

Range: 0 – 100,000

Range: 0 – 20

Range: 1 – 50

Result

80,749.18

Final balance

Total contributions
65,000.00
Investment earnings
15,749.18

A savings account quotes one number, and it is almost never the number people think it is. The rate on the advert is the annual percentage yield: the rate that already accounts for interest being credited and then earning interest of its own. The other number — the plain interest rate, the one you would divide by twelve to get a monthly rate — is a different quantity, and banks are required to define the two apart precisely because mixing them changes the answer. So this page takes the APY as its input, converts it to a periodic rate by taking its n-th root rather than dividing it, and reports what the balance becomes alongside the two figures that make it readable: what you put in, and what the balance earned. On a 5,000 balance with 500 going in every month at 4.00% APY, ten years gives 80,749.18 — of which 65,000 is your own money.

5,000 to start, 500 a month for 10 years, at seven APYs

APYBalance at 10 yearsYou put inInterest earned
065000650000
168583.58650003583.58
272393.38650007393.38
376443.586500011443.58
480749.186500015749.18
585326.056500020326.05
690190.966500025190.96

Only the APY moves in these rows; the starting balance, the deposit, and the term are all fixed. The second column is not a straight line, and the amount it bends is the whole point: at 0% the balance is the 65,000 you paid in, and at 6% it is 90,190.96, so the account added 25,190.96 on top of your money. Compare consecutive rows and the increments themselves grow — 3,583.58 from 0% to 1%, then 3,809.80, then 4,050.20, then 4,305.60 — which is compounding arriving on top of the deposits rather than instead of them. The third column never moves, and that is what makes it useful: it is the fixed reference that keeps the other two honest, and the difference between the second and third columns is the fourth.

The same headline rate, read two ways — 5,000 to start, 500 a month, 10 years

Rate on the advertBalance read as an APYBalance read as a nominal rateDifference
168583.5868600.5616.98
272393.3872465.8372.45
376443.5876617.48173.9
480749.1881079.07329.89
585326.0585876.19550.14
690190.9691036.66845.7

This is the table to look at if only one of these numbers is familiar. The middle column is what this page computes and what the account will really pay, because the rate on a savings advert is an APY. The right-hand column is what you get by feeding the same number into a calculator that expects a nominal rate — the future value of an annuity page, for instance — which divides it by twelve instead of taking its root. Reading across, the gap starts at 16.98 on a 1% advert and reaches 845.70 on a 6% one, so the error grows with the rate: at 1% the two conventions nearly agree, and at 6% they are 845.70 apart on a ten-year balance. Neither column contains an arithmetic mistake. They are two different quantities that share a name on the page, and the reason the regulation defines them separately is that the difference is easy to miss and always in the same direction.

Formula

Balance = starting balance × (1 + i)ⁿ + deposit × [(1 + i)ⁿ − 1] ÷ i where i = (1 + APY ÷ 100)^(1 ÷ periods per year) − 1 and n = years × periods per year

Starting balance
What is already in the account on day one, before anything is added — it is your money too, so it counts towards the total you paid in
Deposit
The amount added at the end of each period, at whatever frequency you choose; it can be zero, in which case the page becomes a plain compounding calculation on the starting balance
APY
The annual percentage yield the bank advertises — the rate that already includes the effect of compounding over a 365-day year. Because compounding is already inside it, the periodic rate is its n-th root, not the APY divided by n
i
The periodic rate — the APY converted down to one period by taking the n-th root, which is the inverse of how an APY is built up from a periodic rate
n
The total number of periods — the years times the number of deposits and compounding periods a year
Interest earned
The part of the final balance that is not your own deposits — what the account added on top

Use it when the rate you have in front of you is the one on a savings account advert or statement — an annual percentage yield, which is a compounded figure — and you want the balance it produces rather than a rate conversion. That covers a regular saver working out where a monthly habit lands, someone checking whether a quoted APY matches the balance their bank actually shows, and anyone comparing two accounts that quote rates on different compounding. It is also the fastest way to see why the two readings diverge: the same 4.00%, read as an APY, gives 80,749.18 over ten years, and read as a nominal rate compounded monthly it gives 81,079.07 — a 329.89 gap that is not a fee and not an error but the compounding counted twice. It is the wrong page for a market return, where the number is an expectation rather than a promise and can be negative; that is what the investing pages are for.

Worked examples

  1. The default case, and the reading it is not

    1. Periodic rate: (1 + 0.04)^(1 ÷ 12) − 1 = 0.3274% a month. This is a twelfth root, not 4% ÷ 12.
    2. Periods: 10 × 12 = 120.
    3. Starting balance grows to 5,000 × 1.003274¹²⁰ = 7,401.22.
    4. The deposits accumulate to 500 × [(1.003274¹²⁰ − 1) ÷ 0.003274] = 73,347.96.
    5. Final balance: 7,401.22 + 73,347.96 = 80,749.18. You paid in 5,000 + 500 × 120 = 65,000.

    Now read the same 4.00% the other way, as a nominal rate compounded monthly — 4% ÷ 12 = 0.3333% a month, which is what you would feed the future value of an annuity page — and the balance comes to 81,079.07. The 329.89 difference is the whole subject of this page: the APY already contains a year of compounding, so dividing it by twelve compounds a second time and pays you interest on interest the bank never promised. Neither figure is a mistake in arithmetic; they answer two different questions, and the one on the advert is the APY.

  2. A long horizon, where the starting balance matters less than the deposits

    1. Periodic rate: (1.05)^(1 ÷ 12) − 1 = 0.4074% a month. Periods: 360.
    2. Starting balance: 100,000 × 1.004074³⁶⁰ = 432,194.24.
    3. Deposits: 1,000 × [(1.004074³⁶⁰ − 1) ÷ 0.004074] = 815,375.90.
    4. Final balance: 1,247,570.14, against 460,000 paid in.

    Thirty years at a 5% APY turns 460,000 of deposits into 1,247,570 — and the quiet part is that the starting 100,000, which is twenty-two percent of the money paid in, ends up being about thirty-five percent of the final balance. Money already in the account has the whole term to work; money arriving in year twenty-nine has almost none. The same asymmetry is why the monthly deposit contributes less than its share of the total despite being the larger sum over the term.

  3. Depositing once a year instead of monthly

    1. Periodic rate: (1.04)^(1 ÷ 1) − 1 = 4% a year, which is the APY itself — with annual compounding the APY and the periodic rate are the same number.
    2. Starting balance: 5,000 × 1.04¹⁰ = 7,401.22.
    3. Deposits: 500 × [(1.04¹⁰ − 1) ÷ 0.04] = 6,003.05.
    4. Final balance: 13,404.27 on 10,000 paid in.

    The frequency here does double duty, and this is the case where the APY is easiest to see: with annual compounding the APY and the periodic rate are literally the same number, 4%, because there is only one period to compound over. Move to monthly and the periodic rate drops to 0.3274% — not 0.3333% — because twelve turns of it have to build back up to exactly 4% over the year. That is the constraint the root satisfies and the division does not.

  4. An APY of zero

    1. Zero APY gives a periodic rate of zero.
    2. Both terms collapse: the starting balance stays at 5,000 and the 120 deposits add to 60,000.
    3. Final balance 65,000, which is exactly what was paid in, so interest earned is 0.

    This is the one case where the three figures are forced into agreement, and it is worth pausing on because it is also the case that breaks a naive implementation: written as a closed form, the deposit factor is 0 ÷ 0 when the rate is zero, and a page without a branch for it prints NaN. Unlike the annuity pages there is no rounding drift here, because with no growth the arithmetic is exact to the cent.

  5. No deposits at all, only a balance left alone

    1. Periodic rate 0.3274% a month, 120 periods.
    2. 5,000 × 1.003274¹²⁰ = 7,401.22, and nothing is added along the way.
    3. Total paid in is the 5,000, so interest earned is 2,401.22.

    With the deposit set to zero the page reduces to plain compounding on a single balance — 5,000 becoming 7,401.22 over ten years at 4.00% APY — which is the same question the compound interest page answers, except that there the rate is read as a nominal one. That difference is small over ten years and grows with the term, so an account held for thirty years is where the two readings part company most.

Limitations

The APY is assumed to hold for the whole term. Real savings rates move, and they move with the central bank's policy rate rather than with anything the account holder does, so a ten-year projection is a statement about a rate that has not been promised for ten years. A variable-rate account will not match this page. The APY is a 365-day-year figure. In a leap year a bank is permitted to use a daily rate of 1/366 of the interest rate instead, so the realised total can differ from this projection by a fraction of a day's interest. No fees, no taxes, and no inflation. The balance is nominal, and no account minimum, withdrawal limit, or balance tier is modelled — many real accounts pay a lower rate, or none, below a minimum balance, or reduce the rate in a month with a withdrawal. Deposits are assumed to land at the end of each period. A deposit made at the start of a period earns one more period of interest, which at the default monthly rate adds about a third of a percent of that deposit. Federal deposit insurance limits are not modelled. The balances this page can produce, at the top of its ranges, are far above the amount any one account is insured for, and the projection treats the whole balance as equally safe. This is a deposit account, not an investment. There is no market risk in this model and the balance cannot fall — which is precisely what separates it from the investing pages, where the equivalent input is an expected return rather than a rate the bank pays. Reading a 7% market expectation into this page's APY field is the mistake it cannot catch for you.

Frequently asked questions

Should I enter the APY or the interest rate in this savings calculator?
The APY — the annual percentage yield, which is the number on the advert and on the statement. The two are defined as different quantities on purpose: the APY reflects the frequency of compounding, and the plain interest rate does not. Enter the interest rate here and you will get a balance slightly lower than the account will really produce, because the compounding the bank already did gets left out.
Why is the monthly rate the twelfth root of the APY and not the APY divided by twelve?
Because the APY is built by compounding, so undoing it means taking a root. Twelve periods each growing by 0.3274% multiply out to exactly 4% over the year; twelve periods each growing by 0.3333% multiply out to 4.074%, which is more than the bank promised. The root is what makes the twelve steps land back on the advertised figure, and it is the inverse of the calculation the regulation uses to define the APY in the first place.
How much does the difference between the two readings actually come to?
On the default case — 5,000 to start, 500 a month, ten years — the APY reading gives 80,749.18 and the nominal reading gives 81,079.07, a gap of 329.89. It is worth being clear about which way round that is: reading the APY as if it were a nominal rate pays you more, because it compounds a second time on top of compounding that is already inside the number. The gap grows with both the rate and the term, so it is small on a short low-rate account and material on a long one.
Does it matter how often I make deposits?
It changes the balance, and it also changes the periodic rate, because the compounding frequency and the deposit frequency are the same thing on this page. Depositing annually at 4.00% APY uses a periodic rate of exactly 4%; depositing monthly uses 0.3274% a month. Same annual figure, different steps to get there — which is why the two frequencies give different balances even though nothing about the advertised rate changed.
What happens if the APY is zero?
The balance is exactly the sum of what you paid in, and interest earned is zero — on the default inputs, 65,000 all three ways. A zero rate also breaks the closed form, because the deposit factor becomes 0 ÷ 0, so the kernel behind this page handles it as its own branch rather than as a very small rate.
Is this the same as an investment or annuity calculator?
It is the same arithmetic with one important substitution. The annuity and investment pages take a nominal annual rate and divide it by the number of periods; this page takes an APY, which already contains compounding, and takes its root instead. The inputs look alike and the formulas look alike, but the rate means something different, and a rate used with the wrong convention gives an answer that is close enough to look right — which is exactly why the two readings are shown side by side in the table below.

References

Related calculators

Future Value of Annuity CalculatorFuture value of a series of equal payments.Investment CalculatorGrowth of regular contributions over time.401(k) Calculator401(k) balance at retirement, with employer match.Compound Interest CalculatorSee how savings grow with compounding at any frequency.APY CalculatorAnnual percentage yield from a nominal rate.