Calculate investment returns with compound interest.
Guide to Compound Interest
What is compound interest?
Compound interest is one of the most powerful financial concepts, often called the "eighth wonder of the world" by Albert Einstein. Unlike simple interest where interest is calculated only on the initial principal, compound interest is calculated on the principal plus all previously accrued interest. This leads to exponential growth over time, significantly exceeding results from simple interest.
Compound interest formula
The compound interest formula is A = P(1 + r/n)^(nt), where P is the initial principal, r is the annual interest rate as a decimal, n is the number of compounding periods per year, and t is the number of years. For continuous compounding the formula simplifies to A = Pe^(rt). This formula helps calculate the future value of an investment at different compounding frequencies.
Rule of 72
The Rule of 72 is a quick method to estimate the time needed to double an investment at a given rate of return. Simply divide 72 by the annual interest rate. At 8% per year, the investment doubles in about 9 years (72 ÷ 8 = 9). This simple formula is extremely useful for quick financial planning and illustrating compound interest's power.
Practical applications
- Savings and bank deposits
- Stock investments (reinvested dividends)
- Pension and investment funds
- Loans and credit (interest on interest)
Compounding frequency affects growth
Compounding frequency significantly affects the final investment value. More frequent compounding yields greater compounding effect. The difference between annual and monthly compounding with 1000 PLN at 10% for 10 years is about 150 PLN. With longer periods and higher rates, the difference grows significantly. This should be considered when choosing financial products.
The last ten years earn more than the first thirty
Compounding is described as a snowball, which undersells it. Ten thousand at 7% grows by 66,123 across its first three decades and by 73,622 in the single decade that follows. The money does not grow faster later; there is simply far more of it doing the growing.
How it works
- Projects a balance forward from a starting sum, a rate and a number of years.
- Separates contributions from growth, since after long periods growth is nearly all of the balance.
- Handles different compounding frequencies, which matter less than most people expect.
FV = P × (1 + r)ⁿ with periodic compounding: FV = P × (1 + r ÷ m)^(m × n) Rule of 72: years to double ≈ 72 ÷ rate as a percentage
Worked example
Ten thousand invested at 7%, left alone for forty years.
- year 10: 19,672
- year 20: 38,697
- year 30: 76,123
- year 40: 149,745
The first thirty years add 66,123. The next ten add 73,622 — more than the previous three decades combined. Nothing about the rate changed; the balance being multiplied simply got large.
Reading the result
- This is the whole argument for starting early, and it is arithmetic rather than encouragement. Every year removed from the front of the period is removed from the end, where the doublings are largest — the ten years you skip at twenty-five cost you the 73,622 decade, not the 9,672 one.
- The Rule of 72 is a good mental shortcut. At 7% it predicts a doubling every 10.3 years against a true 10.24 — an error of 0.4%, close enough for any decision you would make in your head.
- Compounding frequency matters far less than the rate. Moving from annual to monthly compounding at 7% adds roughly a fifth of a percentage point of effective yield; moving from 7% to 8% is worth vastly more, and is where attention belongs.
- The projection assumes an undisturbed balance and a constant rate, and reality supplies neither. Treat it as a demonstration of the mechanism rather than a forecast — and subtract inflation if you want the answer in today's purchasing power.
Common questions
- Why does the balance seem to explode late on?
- Because each doubling adds as much as everything that came before it. Going from 76,123 to 149,745 adds more than the entire journey from 10,000 to 76,123, since a fixed percentage of a large number is a large number.
- Does compounding monthly rather than annually matter?
- Barely. At 7%, annual compounding gives 7.00% effective and monthly gives about 7.23%. It is real but small next to the rate itself — chasing frequency while ignoring the rate optimises the wrong variable.