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Compound interest calculator and formula

Compound interest is interest that goes on to earn interest. A balance grows to A = P(1 + r/n) raised to nt. At 6 percent compounded monthly, $5,000 plus $200 a month becomes $41,872.85 after 10 years, and $12,872.85 of that is interest.

Balance after 10 years

$41,872.85

$12,872.85 of that is interest you did not pay in.

BalanceMoney you paid inYear 0 to 10, up to $41,873
You put in
$29,000.00
Interest earned
$12,872.85
Ending balance
$41,872.85
$
$
%
yr

How often interest is added to the balance.

The formula

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

AA is the ending balance, PP the starting amount, rr the annual rate as a decimal, nn how many times a year interest is added, and tt the number of years.

What this calculator works out

Enter a starting amount, anything you add each month, an annual rate and a number of years, and the calculator returns the balance at the end. It splits that balance into the money you paid in and the interest you did not, because the second number is the one people underestimate.

The chart above the results plots both lines. The solid line is the balance and the dashed line is what you paid in. The gap between them is the compounding, and it widens slowly at first and then quickly, which is the whole behaviour worth understanding.

The compound interest formula

For a single amount left alone, the balance after tt years is:

A=P(1+rn)ntA = P\left(1 + \frac{r}{n}\right)^{nt}

The rate rr goes in as a decimal, so 6 percent is 0.06. The pair nn and tt do the work that trips people up: r/nr/n is the rate for one period and ntnt is the number of periods. At 6 percent compounded monthly, the period rate is 0.005 and there are 12 periods a year, not 0.06 and 1.

Simple interest, by contrast, pays only on the original amount, so it grows by the same amount every year. Compound interest pays on the balance, and the balance keeps getting bigger.

Adding money every month

Most people are not leaving one amount alone. They are paying in every month, which adds a second term to the formula. A level payment made at the end of each period grows to:

FV=PMT×(1+rn)nt−1rnFV = PMT \times \frac{\left(1 + \frac{r}{n}\right)^{nt} - 1}{\frac{r}{n}}

The total is the starting amount grown by the first formula plus the deposits grown by the second. This calculator treats deposits as arriving at the end of each period, which is what a payroll deduction or a standing order actually does. Deposits made at the start of each period earn one extra period of interest, so they end up slightly higher.

The two terms are worth seeing separately, and the first worked example below splits them.

Why compounding frequency changes the total

The same annual rate produces different totals depending on how often interest is added, because each addition starts earning immediately. Over 10 years at 6 percent on $5,000 with nothing added, monthly compounding reaches $9,096.98 and annual compounding reaches $8,954.24.

The gap is real but smaller than most people expect, and it shrinks as the frequency rises. Going from monthly to daily adds far less than going from annual to monthly did. The APR against APY calculator puts a single number on that difference, which is what a bank is quoting when it advertises a yield rather than a rate.

This matters in the other direction too. The same mechanism runs on money you owe, which is what the loan payment calculator works through.

Worked examples

A starting amount plus \$200 a month for 10 years

You start with $5,000, add $200 at the end of every month, and earn 6 percent compounded monthly for 10 years. What is the balance?

  1. Find the period rate: r/n=0.06/12=0.005r/n = 0.06/12 = 0.005, so 0.5 percent a month.
  2. Count the periods: nt=12×10=120nt = 12 \times 10 = 120.
  3. Grow the starting amount: 5000×1.005120=5000×1.8193975000 \times 1.005^{120} = 5000 \times 1.819397, which is $9,096.98.
  4. Grow the deposits: 200×1.005120−10.005=200×163.8793200 \times \frac{1.005^{120} - 1}{0.005} = 200 \times 163.8793, which is $32,775.87.
  5. Add the two parts: $9,096.98 + $32,775.87 = $41,872.85.
  6. Check what you paid in: 5000+200×120=5000 + 200 \times 120 = $29,000.

The balance is $41,872.85. You paid in $29,000, so $12,872.85 of it is interest.

The same 10 years with nothing added

Same $5,000 and the same 6 percent compounded monthly, but no monthly deposits. What does it reach?

  1. Only the first term applies: A=5000×1.005120A = 5000 \times 1.005^{120}.
  2. 1.005120=1.8193971.005^{120} = 1.819397, so A=A = $9,096.98.
  3. Subtract what you started with: $9,096.98 minus $5,000.

It reaches $9,096.98, of which $4,096.98 is interest. The starting amount grows by about 82 percent over the 10 years, which shows how much of the first example's total came from the monthly deposits rather than from the opening balance.

Monthly compounding against annual

Take the same $5,000 at 6 percent for 10 years, but add the interest once a year instead of every month.

  1. Now n=1n = 1, so the period rate is the full 0.06 and there are 10 periods.
  2. A=5000×1.0610=5000×1.790848A = 5000 \times 1.06^{10} = 5000 \times 1.790848.
  3. That comes to $8,954.24.

Annual compounding reaches $8,954.24, so the interest is $3,954.24 against $4,096.98 for monthly. Same rate, same money, different total, because monthly compounding starts paying interest on interest eleven months sooner.

The mistake that costs the most

Putting the annual rate into a monthly calculation without dividing it first.

If you compound 6 percent for 120 periods instead of 0.5 percent, you are modelling a 6 percent monthly return, and the answer comes out about 600 times too big. It looks like a spectacular investment rather than an arithmetic error, which is exactly why it survives a glance.

Two habits catch it. Write the period rate down as its own number before you use it, and check the answer against the rule of 72: at 6 percent a year, money takes about 12 years to double, so a balance that has multiplied many times over in 10 years is wrong.

Common questions

Does the calculator assume deposits at the start or the end of the month?

The end, which matches a payroll deduction or a standing order. Deposits made at the start of each period earn one extra period of interest, so a start-of-period version of the same numbers comes out slightly higher.

What rate should I put in?

Use the rate you are actually quoted, and make sure you know whether it is a nominal annual rate or an annual yield. If a bank advertises an APY, it has already accounted for compounding, so enter it with annual compounding rather than compounding it again.

Is this before or after inflation and tax?

Before both. The output is what the account statement will say. To see what it buys, subtract inflation from the rate before you enter it, and remember that interest is often taxable in the year it is earned rather than when you withdraw.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.