Effortless Math › Personal Finance › Compound Interest, Worked Out
Compound Interest, Worked Out
Compound interest is the only idea in personal finance that pays off for fifty years, and it is usually taught as a slogan instead of a calculation. Here it is as a calculation: the formula, the same money run through simple and compound side by side, and what changing the compounding frequency actually buys you.
Simple interest first
Simple interest is computed on the original principal, every period, forever. It never earns on itself.
Worked example — simple interest
$2,000.00 at 5% simple interest for 10 years.
Simple interest is rare in savings. You meet it in federal student loans, in some car title and personal loans, and in short-term notes.
The compound interest formula
Compound interest is computed on the principal plus everything it has already earned. That is the whole difference, and over a decade it is a large one.
| Symbol | Means | In the example below |
|---|---|---|
| A | Final amount | what we are solving for |
| P | Principal — the starting amount | $2,000.00 |
| r | Annual rate, as a decimal | 0.05 |
| n | Compounding periods per year | 1, 12, or 365 |
| t | Years | 10 |
Worked example — compound, annually
The same $2,000.00 at 5%, compounded once a year for 10 years.
How often it compounds
More frequent compounding earns more, but the returns diminish quickly. The same $2,000.00 at 5% for 10 years:
| Compounding | n | Final amount | Gained over annual |
|---|---|---|---|
| Annually | 1 | $3,257.79 | — |
| Monthly | 12 | $3,294.02 | $36.23 |
| Daily | 365 | $3,297.33 | $39.54 |
Going from annual to monthly is worth about $36.23. Going from monthly all the way to daily adds only $3.31 more. This is why banks advertise APY rather than the rate — APY already folds the compounding in, so two accounts can be compared honestly.
Compare APY, not the interest rate
A 4.9% rate compounded daily and a 5.0% rate compounded annually are close enough that the headline rate tells you nothing. The APY is the single number that accounts for both the rate and the frequency. It is on every account disclosure, and it is the only fair comparison.
Adding money every month
Most saving is not one lump sum. It is a fixed amount every month, which needs a different formula — the future value of a series:
where i is the monthly rate (annual rate ÷ 12) and N is the number of months.
Worked example — $200 a month at 7%
$200 every month, at a 7% annual return compounded monthly.
Why starting early beats saving more
In the formula, t sits in the exponent and everything else does not. That single structural fact is why ten years of delay costs more than most people expect.
Two savers, same $200 a month, same 7%
One starts at 22 and stops at 65. The other starts at 32 and stops at 65.
Ten years of $200 deposits — $24,000.00 — turns into $346,413.23 of final balance. That ratio is the entire argument for starting before you feel ready.
The same formula, pointed at you
Compound interest does not care which side of the transaction you are on. A credit card balance compounds against you at 20% to 30% a year, which is three to four times any return you are likely to earn. Paying off a card at 24% is mathematically identical to finding a guaranteed, tax-free 24% investment — which does not exist.
That is why carrying a balance while saving is usually backwards. See what paying the minimum actually costs.
Practice, with answers
1. $5,000 at 4% compounded annually for 8 years. Find the final amount.
Answer: $6,842.85
2. The same $5,000 at 4% for 8 years, compounded monthly. How much more does it earn?
Answer: $6,881.98, which is $39.13 more.
3. $150 a month at 6% for 25 years. What is the balance, and how much of it is growth?
Answer: $103,949.09, of which $58,949.09 is growth.
Check what you learned
Take the 10-question saving and compound interest quiz. It opens on the hub, and you will see your score, the correct answers, and explanations when you finish.
Keep going
- The Rule of 72 — check any of these answers in your head in about five seconds
- Credit card minimum payment math — the same arithmetic running the other way
- How to read a pay stub — where the money you are compounding comes from
Questions people ask
What is the difference between simple and compound interest?
Simple interest is calculated only on the original principal, so it earns the same amount every period. Compound interest is calculated on the principal plus all the interest already earned, so each period earns slightly more than the last. On $2,000 at 5% for 10 years, simple interest pays $1,000.00 and annual compounding pays $1,257.79.
Does compounding daily make a real difference?
Less than people expect. On $2,000.00 at 5% for 10 years, moving from annual to monthly compounding is worth $36.23. Moving from monthly to daily adds only $3.31 on top of that. The rate matters far more than the frequency, which is why you should compare APY rather than either one alone.
What rate should I assume for long-term investing?
Historical US stock market returns have averaged roughly 7% a year after inflation over long periods, which is why 7% is the conventional figure in classroom examples. It is an average across decades, not a promise about any particular year, and any single year can be sharply negative.
Is the Rule of 72 accurate?
It is a good estimate for rates between about 4% and 12%, usually within a few months of the exact answer. Outside that band it drifts. It is a mental check, not a substitute for the formula.
How do I calculate compound interest without a financial calculator?
Any calculator with an exponent key will do it. Compute 1 + r/n, raise it to the power nt, then multiply by the principal. A spreadsheet is easier still: =P*(1+r/n)^(n*t), and =FV(rate, nper, pmt) when you are adding money every month.
Part of the Personal Finance hub — ten units, a free pacing guide, and worked examples with answers.