Compound interest
The eighth wonder of the world - as Einstein supposedly said (the quote is probably made up, but the maths is very real). Interest added to the capital starts earning on its own. The effect is barely noticeable in the first few years and staggering after twenty.
| Year | Paid in this year | Interest this year | Total paid in | Capital at year end |
|---|
How does it actually work?
Simple interest: the interest doesn't work
You deposit PLN 10,000 at 7% a year. With simple interest, interest is always calculated on the starting amount. Every year you get the same PLN 700 - in the first year and in the twentieth. The interest sits next to the capital and just lies there.
After 10 years you have PLN 10,000 + 10 × PLN 700 = PLN 17,000.
Compound interest: interest earns interest
Now the same thing, but we add the interest to the capital. In the second year, 7% is calculated not on PLN 10,000 but on PLN 10,700. In the third - on PLN 11,449. The capital on which interest is charged 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 |
| … | … | … | … |
| 10 | PLN 18,384.59 | PLN 1,286.92 | PLN 19,671.51 |
Notice: in the first year interest is PLN 700, in the tenth - already PLN 1,286.92. You didn't add a single zloty. The difference is purely interest that earned its own interest.
Simple interest: PLN 17,000 · Compound interest: PLN 19,671.51 · Gain from the mechanism alone: PLN 2,671.51. Over 30 years the same difference grows to PLN 45,122.55 - and we are still talking about the same PLN 10,000 initial deposit. Compound interest doesn't grow in a straight line over time. Over time it accelerates.
The formula
Starting capital growing over the years - that is the whole equation:
P - starting 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. Growth is not a straight line - it is a curve that bends upwards more steeply every year. That is why the chart above looks like a ski jump, not a gentle slope.
And when you add money every month, a second term comes in - the formula for the future value of an annuity:
i - rate per period (r/m)
n - total number of contributions (m · t)
The calculator above computes both terms at once and adds the results - that is why you can enter both a starting capital and a monthly contribution.
Compounding: why 12% isn't 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% |
| 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, but a different result. That is why banks are required to state the effective rate - the only number you should compare between offers.
What you really earn 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:
It is an approximation, but at realistic rates (4-12%) it is off by less than half a year. Perfect for mental arithmetic: 8% → 72/8 = 9 years. And since doubling takes 9 years, over 36 years the capital will double four times - in other words it will grow sixteenfold.
The most important lesson: time beats the size of your contributions
This is the moment when most people's jaws drop. Two people, the same rate of return of 7% a year, result calculated up to age 65:
There is no trick here. Ania gave her money more time to work - and those extra years turned out to be worth more than Bartek's three times larger contributions. That is why the best time to start was ten years ago, and the second best is today.
The other side of the coin: when compound interest works against you
The same mechanism drives debt. Credit cards, payday loans, overdrafts - in all of them interest is added to the debt, and the next month you are charged interest on it too.
If you only pay the minimum, the debt doubles in ~4 years (rule of 72: 72/18 = 4). The same exponential curve, only this time you are on the side that pays.
The practical takeaway: paying off expensive debt is the best "investment" on the market. By overpaying a card at 18%, you get a guaranteed 18% return - with no risk and no tax. No savings deposit will beat that.
You can work this out in the overpayment calculator - it will show how much you really save by paying off a loan faster.
Two brakes everyone forgets about
1. The Belka tax. The Polish tax office takes 19% of capital gains. 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.
2. Inflation. Compound interest works on prices too - only in the opposite direction. With 5% inflation, PLN 100,000 in 20 years will have the purchasing power of PLN 37,689 today. That is why the only thing that counts is the real rate:
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 gains. Work it out in the inflation calculator.
1. Time matters more than the amount - starting ten years earlier
beats contributions three times as large.
2. Regularity beats timing - a fixed contribution every month
works better than waiting for "the right moment".
3. What counts is the real rate, after tax and after inflation -
the rest is an illusion.
Got an expensive loan? See how much you gain by overpaying
Compound interest works both ways. Before you start investing, check whether paying off your loan would give you a better return - with no risk.