Compound interest from scratch: why time matters more than the amount
Ania paid in PLN 60,000. Bartek paid in PLN 180,000 - three times as much. At retirement it is Ania who has PLN 92,435 more. There is no trick here: just a formula, which we are about to take apart piece by piece.
What compound interest actually is
Compound interest is when interest is added to the capital, and in the next period the interest itself starts earning. Interest on interest. It sounds unremarkable - and for the first few years it really is. After twenty years the effect can be hard to believe.
The easiest way to understand it is by contrast with simple interest, where interest is always calculated on the initial amount and then set aside next to the capital, where it just sits and does nothing.
Simple interest: the interest does not work
You put in PLN 10,000 at 7% a year. Every year you get the same PLN 700 - in the first year and in the tenth. After 10 years you have PLN 10,000 + 10 × PLN 700 = PLN 17,000.
Compound interest: interest earns interest
The same, but we add the interest to the capital. In the second year the 7% is calculated not on PLN 10,000 but on PLN 10,700. In the third - on PLN 11,449. The base on which you earn interest keeps growing:
| Year | Capital at start | Interest (7%) | Capital at end |
|---|---|---|---|
| 1 | PLN 10,000.00 | PLN 700.00 | PLN 10,700.00 |
| 2 | PLN 10,700.00 | PLN 749.00 | PLN 11,449.00 |
| 3 | PLN 11,449.00 | PLN 801.43 | PLN 12,250.43 |
| 4 | PLN 12,250.43 | PLN 857.53 | PLN 13,107.96 |
| 5 | PLN 13,107.96 | PLN 917.56 | PLN 14,025.52 |
| 6 | PLN 14,025.52 | PLN 981.79 | PLN 15,007.30 |
| 7 | PLN 15,007.30 | PLN 1,050.51 | PLN 16,057.81 |
| 8 | PLN 16,057.81 | PLN 1,124.05 | PLN 17,181.86 |
| 9 | PLN 17,181.86 | PLN 1,202.73 | PLN 18,384.59 |
| 10 | PLN 18,384.59 | PLN 1,286.92 | PLN 19,671.51 |
Look at the interest column. In the first year - PLN 700. In the tenth - PLN 1,286.92. You did not add a single zloty. The whole difference is interest that earned its own interest.
10 years · simple interest: PLN 17,000.00 · compound: PLN 19,671.51 · gain from the mechanism alone: PLN 2,671.51
30 years · simple interest: PLN 31,000.00 · compound: PLN 76,122.55 · gain from the mechanism alone: PLN 45,122.55
We are still talking about the same PLN 10,000 initial deposit and the same 7% rate. Tripling the time raised the advantage of compound interest almost seventeen times (16.9×). That is the whole lesson of this article, just in numbers.
The formula
Initial capital growing over the years - that is the whole equation:
P - initial capital
r - annual rate of return (7% = 0.07)
m - number of compounding periods per year (monthly = 12)
t - number of years
All the power sits in the exponent m·t. Notice that time (t) is in the exponent, while capital (P) is just an ordinary multiplier at the front. Doubling the deposit doubles the result. Doubling the time squares the result. This is not a metaphor - it is literally the same line of the formula.
And when you add money every month, a second term comes in - the formula for the future value of an annuity:
i - the rate per period (r/m)
n - total number of payments (m · t)
Our calculator works out both terms at once and adds the results - which is why you can enter both a starting capital and a monthly payment.
Compounding: why 12% is not always 12%
Compounding is the moment interest is added to the capital. The more often it happens, the sooner the interest starts working. Take PLN 10,000 at 12% for one year:
| Compounding | Capital after a year | Effective rate |
|---|---|---|
| Annual | PLN 11,200.00 | 12.00% |
| Semi-annual | PLN 11,236.00 | 12.36% |
| Quarterly | PLN 11,255.09 | 12.55% |
| Monthly | PLN 11,268.25 | 12.68% |
| Daily | PLN 11,274.75 | 12.75% |
The same "12" in the advert, and a different result every time. 12% compounded monthly is in reality 12.68% a year. That is why financial institutions are required to state the effective rate - the only number you can compare between offers. It is a close cousin of the APR (RRSO in Poland) on the loan side: both numbers exist so that cost cannot be hidden in the way it is calculated.
What you really earn - or really pay - when interest is added more often than once a year.
The rule of 72 - do it in your head
Want to know when your capital will double? Divide 72 by the interest rate:
| Annual rate | Rule of 72 | Exact result | Approximation error |
|---|---|---|---|
| 4% | 18.0 years | 17.7 years | 0.3 years |
| 5% | 14.4 years | 14.2 years | 0.2 years |
| 7% | 10.3 years | 10.2 years | 0.1 years |
| 8% | 9.0 years | 9.0 years | 0.0 |
| 10% | 7.2 years | 7.3 years | 0.1 years |
| 12% | 6.0 years | 6.1 years | 0.1 years |
| 18% | 4.0 years | 4.2 years | 0.2 years |
At realistic rates (4-12%) the approximation is off by less than half a year - and you can do it in your head. And the most important consequence: if at 8% doubling takes 9 years, then over 36 years your capital doubles four times, which means it grows sixteenfold. Not by 400%. Sixteen times.
Ania and Bartek - the most important table in this article
Two people, the same rate of return of 7% a year, the same payment of PLN 500 a month. The only difference between them is when they start. We calculate the result up to age 65.
Starts at age 25
Puts aside PLN 500 a month for 10 years, until age 35. Then she stops paying in anything at all and leaves the capital alone for the next 30 years.
Starts at age 35
Puts it off for ten years, but then pays in PLN 500 a month consistently for 30 years, right up to retirement. He never misses a single month.
Ania paid PLN 60,000 out of her own pocket. Bartek - PLN 180,000, or three times as much. And in the end it is Ania who has more: PLN 702,421.17 versus PLN 609,985.50.
Her advantage is not higher payments or a better rate of return. It is ten extra years in which the interest had time to grow and start earning interest of its own.
Let's look at it from another angle - how much each zloty paid in earned:
| Ania | Bartek | |
|---|---|---|
| Starting age | 25 | 35 |
| Years of paying in | 10 | 30 |
| Total paid in | PLN 60,000.00 | PLN 180,000.00 |
| Capital at age 35 | PLN 86,542.40 | PLN 0.00 |
| Earned by interest | PLN 642,421.17 | PLN 429,985.50 |
| Capital at age 65 | PLN 702,421.17 | PLN 609,985.50 |
| What each zloty paid in became | PLN 11.71 | PLN 3.39 |
Each of Ania's zloty became PLN 11.71. Each of Bartek's - PLN 3.39. The same rate of return, the same currency, the same market. The difference: time in the account.
Not "start when you earn more". Not "wait for a good moment in the market". Start with an amount you can realistically keep up every month, and start now. Starting ten years earlier turned out to be worth more than paying in three times as much - and no market analysis will change that.
Plug your own numbers into this model in the compound interest calculator - you will see your age, your payment and your result.
The other side of the coin: when compound interest works against you
The same mechanism drives debt. Credit cards, payday loans, an overdraft - in all of them interest is added to the debt, and the next month you are charged interest on that too. The curve is identical. Only this time you are on the side that pays.
| Credit card: PLN 5,000 of debt, 18% a year | Debt |
|---|---|
| Today | PLN 5,000.00 |
| After a year | PLN 5,900.00 |
| After 2 years | PLN 6,962.00 |
| After 3 years | PLN 8,215.16 |
| After 4 years | PLN 9,693.89 - the debt has almost doubled |
| After 5 years | PLN 11,438.79 |
The rule of 72 works both ways: 72 ÷ 18 = 4 years to double (4.2 years exactly). And because cards charge interest monthly, the effective interest rate on such a card is in reality 19.56% a year - and it doubles the debt in 3.9 years.
The practical conclusion, awkward for the whole investment industry: paying off expensive debt is the best "investment" on the market. By paying off a card at 18%, you get a guaranteed 19.56% return - with no risk and no tax. We go into this in the article "Overpay your loan or invest?".
Two brakes everyone forgets about
1. The Belka tax
The taxman takes 19% of capital gains (in Poland this is known as the Belka tax). Earned PLN 100,000 in interest? You keep PLN 81,000. The exception is the IKE and IKZE retirement accounts - there the gain is exempt from this tax, which over a 30-year horizon makes a huge difference.
There is one detail in your favour: with shares and ETFs bought and held, you pay tax only when you sell. So the gross amount works for the whole period. With a deposit it is the other way round - the Belka tax is taken at every maturity, so only the reduced amount keeps earning.
2. Inflation - compound interest in reverse
Prices grow exponentially too. With 5% inflation, PLN 100,000 in 20 years' time will have the purchasing power of PLN 37,688.95 today. With 3.5% inflation - PLN 50,256.59. Your future fortune loses value through exactly the same mechanism by which it grows.
That is why only the real rate counts:
With an 8% return and 6% inflation you really earn 1.89%, not 2%.
A deposit at 5% with 7% inflation is a real loss, even though your statement shows nothing but pluses. Work it out in the inflation calculator.
Three things to remember
1. Time beats the amount. Time sits in the exponent, the amount is just a multiplier. Starting ten years earlier beat payments three times as high.
2. Regularity beats timing. A fixed payment every month works better than waiting for "the right moment" - because you only ever recognise that moment in hindsight.
3. What counts is the real rate, after tax and after inflation. The rest is an illusion on your bank statement.
Work it out yourself
Compound interest calculator
All the numbers in this article were worked out by the same engine that powers our calculator - you can reproduce them to the grosz. Enter your capital, your monthly payment and your time horizon. You will see a chart, a year-by-year table and the result after the Belka tax and after inflation. Try the Ania and Bartek scenario with your own age, too.
Open the compound interest calculatorInflation calculator
What your future money will really be worth. Compound interest working against your savings.
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Have an expensive loan? See how much interest you cut by paying it off faster - a guaranteed return, with no risk.
Calculate your overpaymentRead on
- Overpay your loan or invest? - since compound interest works both ways, let's work out which side to stop it on first.
- APR or interest rate? - the same compounding mechanism, only on the loan side. A payday loan at "10% a month" has an APR of 219%.
- How a bank calculates your borrowing capacity - before compound interest starts working for you, make sure it is not working against you in your card limit.
Calculation assumptions: the "simple versus compound interest", "compounding" and "rule of 72" tables use annual compounding (unless the table says otherwise). Ania and Bartek scenario: 7% a year, monthly compounding, payments of PLN 500 at the end of each month, ignoring tax and inflation - exactly as our calculator does it, so you can reproduce the results yourself. The 7% rate is a historical assumption, not a forecast; the real result may be significantly lower, including a loss of capital. This text is educational and does not constitute investment advice.