Compound Interest Explained: Formula, Examples and the Rule of 72

Compound interest is interest calculated on both the original amount and the interest already added to it. Because each period's interest joins the balance, the next period earns slightly more, and the growth curve bends upwards. Over a few years the difference from simple interest is modest; over a few decades it is the whole story.

Simple interest versus compound interest

Simple interest is paid on the original sum only. Put 10,000 in an account paying 6% simple interest and you receive 600 a year, every year. After ten years you have 16,000. Nothing changes from one year to the next.

With compounding, the 600 earned in year one is added to the balance, so year two pays interest on 10,600. Year three pays interest on an even larger figure, and so on. The same 10,000 at 6% compounded once a year reaches 17,908.48 after ten years. Compounded monthly it reaches 18,193.97. The extra 2,000 or so comes from interest earning interest, and nothing else.

The same mechanism works against you when you borrow. Credit cards, overdrafts and some personal loans compound daily or monthly, which is why an unpaid balance grows faster than the headline rate suggests. The credit card payoff calculator turns that into a timetable: enter the balance, the APR and a monthly payment, and it shows how many months the balance takes to clear and the total interest paid. It charges interest monthly at one twelfth of the APR, whereas real cards often charge daily, so treat its answer as an estimate.

One condition has to hold for any of this to happen: the interest must stay in the account. If you withdraw the interest as it is paid, or an account pays it into a separate current account, you are back to simple interest no matter what the product is called. Reinvesting is what turns a steady income into a curve. To check a simple-interest figure for any principal, rate and period (in years, months or days), the simple interest calculator gives the interest and the total, and shows beside them what the same deposit would reach with yearly compounding.

The compound interest formula

The standard formula for a single lump sum is:

A = P × (1 + r/n)^(n × t)

• A is the final amount.
• P is the principal, the sum you start with.
• r is the annual interest rate as a decimal (6% is 0.06).
• n is the number of compounding periods per year (12 for monthly).
• t is the number of years.

The formula divides the annual rate across the periods, then multiplies the growth factor by itself once for every period. The interest earned is simply A minus P.

A worked example

You can see the acceleration if you check the balance at the end of each year. After year one it is 10,616.78, a gain of 616.78. After year five it is 13,488.50. In year ten alone the balance grows from 17,136.99 to 18,193.97, a gain of 1,056.98, nearly twice what the first year produced. The rate never changed; the base did.

Notice the first month: 10,000 × 0.005 = 50.00 of interest. The second month earns 10,050 × 0.005 = 50.25. Those extra pennies look trivial, and for the first few months they are. Compounding is a slow start followed by a long run.

Does compounding frequency matter?

Banks advertise daily, monthly, quarterly or annual compounding, and it sounds as though daily must be far better. It is better, but by less than most people expect. Here is the same 10,000 at 6% for 10 years under each schedule.

Compounding	Periods per year	Rate per period	Balance after 10 years
Annually	1	6%	17,908.48
Half-yearly	2	3%	18,061.11
Quarterly	4	1.5%	18,140.18
Monthly	12	0.5%	18,193.97
Daily	365	0.0164%	18,220.29
Continuously	infinite	-	18,221.19

Moving from annual to monthly compounding adds 285.49 over ten years. Moving from monthly to daily adds only 26.32 more. There is a ceiling, called continuous compounding, and the daily figure is already within a pound of it. The lesson is that a slightly higher rate beats a more frequent schedule every time. A 6.25% rate compounded annually outperforms 6% compounded daily.

Adding regular deposits

Most people do not invest one lump sum and walk away. They add a fixed amount every month. The lump sum formula still applies to the starting balance, and a second formula handles the stream of deposits. For payments made at the end of each period:

FV = PMT × ((1 + i)^N − 1) ÷ i

Here PMT is the regular payment, i is the rate per period and N is the total number of periods. The two results are added together.

The compound interest calculator runs this exact calculation and shows the balance year by year, so you can see when the interest line overtakes the deposit line. If you make deposits at the start of each month instead of the end, each payment earns one extra period of interest and the deposit total rises slightly, to 16,469.87.

The rule of 72

You do not always need a calculator. The rule of 72 estimates how long money takes to double: divide 72 by the annual rate in percent. At 6% that is 72 ÷ 6 = 12 years. The exact figure, from ln(2) ÷ ln(1.06), is 11.90 years, so the shortcut is close enough for planning.

Annual rate	Rule of 72 estimate	Exact doubling time
2%	36 years	35.0 years
3%	24 years	23.4 years
6%	12 years	11.9 years
8%	9 years	9.0 years
12%	6 years	6.1 years

The rule is most accurate around 8% and drifts a little at very low or very high rates. It works in reverse too: if you know a sum doubled in 9 years, the rate was about 8%. And it is a sobering way to look at debt. A credit card at 24% doubles what you owe in about three years if nothing is paid off.

Doubling times chain together, which makes the rule useful for longer horizons. Money at 6% doubles roughly every 12 years, so over 36 years it doubles three times: 10,000 becomes 20,000, then 40,000, then 80,000. The exact figure is 81,473, close enough that you can do the sum in your head while someone is still describing the product to you.

Why starting early matters more than how much you invest

Because growth accelerates, the years at the end of a long run do most of the work, and you only get those years by starting. Suppose four people each invest 200 a month at 7% compounded monthly and stop at 65. The only difference is the age at which they begin.

Starting age	Months of deposits	Total paid in	Value at 65	Growth
25	480	96,000	524,963	428,963
35	360	72,000	243,994	171,994
45	240	48,000	104,185	56,185
55	120	24,000	34,617	10,617

The person who starts at 25 pays in 24,000 more than the one who starts at 35, but finishes with 280,969 more. Those extra ten years of deposits are not what makes the difference; it is that every deposit made at 25 has forty years to compound instead of thirty. Doubling the monthly amount at 35 (400 a month) would reach roughly 487,988, still short of the 25-year-old investing half as much.

The practical version of this: a small amount started now beats a larger amount started later, and the gap widens the longer you wait.

Fees and inflation: the two quiet subtractions

Compounding is symmetrical. Anything that shaves a little off the rate every year compounds too, in the wrong direction.

Fees

A fund charging 1% a year turns a 7% gross return into a 6% net return. That sounds like a small haircut. Over 30 years on a 10,000 lump sum, 7% grows to 76,122.55 while 6% grows to 57,434.91. The fee has consumed 18,687.64, nearly a quarter of the final balance, and almost twice the original investment. With 200 a month over the same period, the gap is 243,994 against 200,903, a difference of over 43,000.

Fees are often quoted as an ongoing charge figure or expense ratio, and they are taken whether or not the investment goes up. The test is always the same: subtract the fee from the expected return and run the calculation on the net rate, because that is the rate you actually compound at. If you already know what went in and what came out, the ROI calculator works the other way round: enter the amount invested, any extra costs and fees, the final value and the years held, and it returns the total return and the annualised rate, which you can set against the figures above.

Inflation

A 6% return during 3% inflation is not a 3% real return, though that is a fair approximation. The exact real rate is (1.06 ÷ 1.03) − 1 = 2.91%. Use the real rate when you want to know what your savings will actually buy. At 3% inflation, 10,000 of purchasing power today is worth about 5,537 of today's money in 20 years. A percentage calculator is handy for turning these rate differences into real figures before you commit to a plan.

Savings goal maths: working backwards

Most planning questions run the formula in reverse. You know the target and the time; you want the monthly deposit. Rearranging the regular deposit formula gives:

PMT = FV × i ÷ ((1 + i)^N − 1)

Two other versions of the same question come up often. How much would you need as a lump sum today to hit 20,000 in five years at 4%? Divide the target by the growth factor: 20,000 ÷ 1.003333^60 = 16,380.06. And if you can only manage 300 a month, how long does it take? Just over 61 months, a little past the five-year mark. The savings goal calculator answers all three forms without any rearranging on your part.

A useful habit is to run the goal at two or three rates. If the plan only works at an optimistic rate, it is not a plan. If it works at a cautious rate, anything better is a bonus. It is also worth noticing how little the rate matters on short goals. Over five years, the difference between 2% and 4% on the 20,000 target is about 15 a month in required deposits. Over thirty years the same two rates produce wildly different outcomes. For short goals, focus on the deposit; for long ones, focus on the rate and the fees.

Where to go next

Try the worked examples above in the compound interest calculator and change one input at a time to see what moves the result most. For a target figure and a deadline, the savings goal calculator works backwards to a monthly amount. Both run entirely in your browser and nothing you type is uploaded.

Frequently asked questions

What is the difference between simple and compound interest?

Simple interest is paid only on the original sum, so the amount earned each year never changes. Compound interest is paid on the original sum plus any interest already added, so each year earns a little more than the last. On 10,000 at 6% for ten years, simple interest gives 16,000 while monthly compounding gives 18,193.97.

How often should interest compound to make a real difference?

Less often than you might think. On 10,000 at 6% for ten years, annual compounding gives 17,908.48, monthly gives 18,193.97 and daily gives 18,220.29. The jump from annual to monthly is worth about 285, but monthly to daily adds only 26. A slightly higher rate matters far more than a more frequent schedule, which is why comparing by AER or APY is the fair test.

How accurate is the rule of 72?

It is accurate enough for planning across the rates most people encounter. At 6% the rule says 12 years and the exact answer is 11.9 years. At 8% both give 9 years. The rule drifts at the extremes, overestimating slightly at very low rates and underestimating at very high ones, but for anything between 4% and 12% the error is well under a year.

Does compound interest work the same way on debt?

Yes, and that is why unpaid balances grow so quickly. A credit card compounding monthly at 24% adds interest to interest in exactly the way a savings account does, just at a far higher rate. By the rule of 72, a balance left untouched at 24% roughly doubles in three years. Paying more than the minimum early on breaks the cycle before it accelerates.

How much do I need to save each month to reach a goal?

Rearrange the deposit formula: monthly payment equals the target multiplied by the monthly rate, divided by the growth factor minus one. For 20,000 in five years at 4% compounded monthly, that works out to 301.66 a month. Without any interest you would need 333.33. A savings goal calculator does the rearranging for you and also shows how long a fixed monthly amount would take.