Compound interest explained with one example
Compound interest is interest that is added to a balance and then earns interest of its own. Simple interest is paid only on the original amount. Over a few years the gap is small, and over decades it becomes most of the balance. The compound interest calculator runs every figure in this guide, which follows one deposit of USD 10,000 at 6% a year for 10 years.
Simple and compound interest on the same deposit
Every amount below is in USD. Picture two savers who each put 10,000.00 into an account paying 6% a year. The first account pays simple interest. The second adds the interest to the balance once a year, and from then on that interest earns 6% too.
With simple interest the yearly payment never changes. It is 6% of 10,000.00, which is 600.00, every year. With compound interest the payment grows, because each year starts from a larger balance. Year 1 pays 600.00, year 2 pays 636.00, and year 10 pays 1,013.69.
| Year | Simple interest balance | Compound balance (yearly) | Gap |
|---|---|---|---|
| 1 | 10,600.00 | 10,600.00 | 0.00 |
| 2 | 11,200.00 | 11,236.00 | 36.00 |
| 3 | 11,800.00 | 11,910.16 | 110.16 |
| 4 | 12,400.00 | 12,624.77 | 224.77 |
| 5 | 13,000.00 | 13,382.26 | 382.26 |
| 6 | 13,600.00 | 14,185.19 | 585.19 |
| 7 | 14,200.00 | 15,036.30 | 836.30 |
| 8 | 14,800.00 | 15,938.48 | 1,138.48 |
| 9 | 15,400.00 | 16,894.79 | 1,494.79 |
| 10 | 16,000.00 | 17,908.48 | 1,908.48 |
After 10 years, simple interest has earned 6,000.00 and compound interest has earned 7,908.48. The gap in the last column widens faster each year. That widening is what people mean when they say compound interest snowballs.
The compound interest formula
For a single deposit, the balance after a number of years comes from one line. Each symbol has a plain meaning.
- A = P x (1 + r / n)^(n x t)
- A is the balance at the end.
- P is the amount deposited at the start.
- r is the yearly rate as a decimal, so 6% is 0.06.
- n is how many times a year interest is added, such as 1, 4, or 12.
- t is the number of years.
With yearly compounding, n is 1, so the sum is 10,000 x 1.06 to the power of 10. That is 17,908.48, the last row of the table. Interest earned is always the end balance minus everything paid in.
How often interest is added
Banks add interest yearly, quarterly, monthly, or daily. More frequent compounding means interest starts earning interest sooner. The rate stays at 6%, but the balance after 10 years moves a little with each step.
| Compounding | Balance after 10 years | Interest earned | Effective annual rate |
|---|---|---|---|
| Yearly | 17,908.48 | 7,908.48 | 6.00% |
| Quarterly | 18,140.18 | 8,140.18 | 6.14% |
| Monthly | 18,193.97 | 8,193.97 | 6.17% |
| Daily | 18,220.29 | 8,220.29 | 6.18% |
Moving from yearly to monthly compounding adds 285.49 over 10 years on this deposit. Moving from monthly to daily adds only 26.32 more. The effective annual rate puts all four on one scale. It is the yearly growth once compounding is counted, so two accounts with different schedules can be compared by that single figure.
What regular deposits add
Most savings plans add money along the way. Suppose the same saver also pays in 200.00 at the end of every month, with interest added monthly at 6%. After 120 months the balance reaches 50,969.84.
| Part of the balance | Amount |
|---|---|
| The first 10,000.00, grown for 10 years | 18,193.97 |
| 120 monthly deposits of 200.00, with their interest | 32,775.87 |
| Final balance | 50,969.84 |
| Total paid in, 10,000 + (200 x 120) | 34,000.00 |
| Interest earned | 16,969.84 |
Each deposit compounds for a shorter time than the one before it. The first 200.00 grows for almost 10 years, and the last one earns nothing at all. The deposits still end up as the larger part of the balance, because 24,000.00 went in through them.
The rule of 72
The rule of 72 is a mental shortcut for how long money takes to double. The working is 72 divided by the yearly rate as a whole number. At 6%, 72 / 6 gives 12 years.
The shortcut is close but not exact. With yearly compounding, the 10,000.00 deposit is worth 18,982.99 after 11 years and 20,121.96 after 12, so it doubles during year 12. The rule is an estimate, and it drifts further from the true figure at very low or very high rates.
Why time matters more than the rate
Compounding rewards years more than percentage points. The table compares a monthly deposit of 200.00 under different rates and terms, with interest added monthly.
| Monthly deposit of 200.00 | Paid in | Final balance |
|---|---|---|
| 10 years at 6% | 24,000.00 | 32,775.87 |
| 10 years at 8% | 24,000.00 | 36,589.21 |
| 20 years at 6% | 48,000.00 | 92,408.18 |
| 30 years at 6% | 72,000.00 | 200,903.01 |
Two extra percentage points over 10 years add 3,813.34. Ten extra years at the lower rate double the amount paid in and almost triple the balance. In the 30 year row, interest makes up 128,903.01 of the total, well over half.
What the calculator leaves out
Every figure here assumes a rate that never changes. Real accounts change their rates, and investments go up and down. Tax on interest, account fees, and inflation are also left out. Each of them lowers the real result, and inflation reduces what the final balance can buy. The figures show how compounding works, not what any account will pay.
Compounding also works against borrowers. Loan interest is charged on a balance that falls each month, and the guide on how EMI is calculated shows that schedule. The loan and EMI calculator runs it for any loan.
Compound interest questions
Is compound interest the same as APY? Not quite. APY, the annual percentage yield, is a name used for the effective annual rate on savings in some countries. It is a result of compounding rather than the process itself.
Does compounding help with a lump sum and with monthly deposits? Yes. A lump sum compounds for the whole term. Each deposit compounds from the day it lands, so earlier deposits grow more than later ones.
How is the time to double worked out exactly? Divide the natural log of 2 by the natural log of 1 plus the yearly rate. At 6% with yearly compounding that is about 11.9 years, close to the 12 years the rule of 72 gives.