💷 Money
How loans and compound interest actually work
A loan is a simple idea wrapped in a slightly intimidating formula. Once you have seen the formula once, you can tell where every monthly payment goes, why the balance falls so slowly at first, and what an extra hundred a month is really worth.
Simple interest and compound interest
Simple interest is charged on the original amount only. Borrow 1,000 at 5 percent for three years and you pay 50 a year, 150 in total, and the balance behaves predictably.
Compound interest is charged on the original amount plus any interest already added. The same 1,000 at 5 percent compounds to 1,157.63 after three years, not 1,150. The gap looks trivial at small sums and short periods. It is not trivial at mortgage sums or over decades, and it works in both directions: it is why savings grow faster than the headline rate suggests and why credit card balances are so hard to clear.
The formula is A = P × (1 + r)n, where P is the starting amount, r is the rate per period as a decimal and n is the number of periods. The rate and the period must match — 5 percent a year compounded monthly uses r = 0.05/12 and n = 12 per year.
Compounding frequency, and why the headline rate is not the real rate
Because compounding happens on a schedule, the frequency changes the effective cost. A nominal 5 percent compounded monthly is not 5 percent a year; it is 1.05/12 raised to the twelfth power, minus one, which is 5.116 percent. Compounded daily it is about 5.127 percent. The formula for the effective annual rate is (1 + r/n)n − 1.
| Compounding | Effective annual rate |
|---|---|
| Annually | 5.000% |
| Quarterly | 5.095% |
| Monthly | 5.116% |
| Daily | 5.127% |
| Continuously | 5.127% |
This is why regulators require lenders to quote an annualised effective figure — APR in the United Kingdom and the European Union, APR in the United States for consumer credit — rather than the nominal rate. Two loans advertised at "5 percent" can differ in cost if one compounds monthly and the other annually.
The amortisation formula
A repayment loan has a fixed payment that is calculated so the balance reaches exactly zero at the end of the term. That payment is:
M = P × r × (1 + r)n ÷ ((1 + r)n − 1)
where M is the monthly payment, P the principal, r the monthly interest rate as a decimal and n the number of monthly payments. The same formula covers car loans, personal loans and mortgages; only the numbers change.
Worked example: 250,000 at 6 percent a year over 30 years. The monthly rate is 0.06/12 = 0.005 and n is 360. (1.005)360 is 6.0226, so M = 250,000 × 0.005 × 6.0226 ÷ 5.0226 = 1,498.90. Over the term you pay 539,604, of which 289,604 is interest — more than the amount borrowed.
Why the early years feel like they achieve nothing
In the first month of that mortgage, interest is 250,000 × 0.005 = 1,250. The payment is 1,498.90, so only 248.90 goes to the principal. After one month you owe 249,751. After a year you have paid nearly 18,000 and still owe about 246,900.
This is not a trick; it is the arithmetic of charging interest on a large balance. The crossover — the point where more of the payment goes to principal than to interest — happens around the halfway point of the term for a level-payment loan. From there the balance falls faster each month, which is why the last few years of a mortgage retire a surprising amount of debt.
What an extra payment is worth
An extra payment goes straight to principal, so it removes every future interest charge on that amount. On the same 250,000 mortgage, paying 100 extra every month clears the loan in about 305 months instead of 360 — four and a half years early — and saves roughly 51,000 in interest.
Two caveats. First, check whether the lender charges an early-repayment penalty; many fixed-rate mortgages allow only a limited overpayment each year. Second, keep an emergency fund before overpaying: money paid into a mortgage is hard to get back, while money in savings is available.
APR, fees and the comparison that actually works
The interest rate is only part of the cost. Arrangement fees, valuation fees, compulsory insurance and broker commissions all add to what you pay, and APR folds most of them into a single annual percentage. For two loans with the same term, the lower APR is almost always the cheaper loan.
Comparing loans of different lengths with APR is less reliable, because APR assumes you hold the loan to term and make every payment on schedule. A five-year loan with a low APR can cost less in total than a ten-year loan with the same APR simply because you pay interest for half as long — even though the monthly payment is higher.
Fixed, variable, interest-only
| Type | How payments behave | What to watch |
|---|---|---|
| Fixed rate | Set for a period, so the payment is predictable | Usually slightly more expensive at the start. |
| Variable rate | Follows a benchmark, so payments rise and fall | Budgeting risk on a large loan. |
| Interest-only | Covers interest but not principal, so the balance stays put | Useful as a bridge; dangerous as a long-term plan. |
| Negative amortisation | Set below the interest due, so the balance grows | Legal in some markets; read the terms very carefully. |
The same maths, working for you
Compound interest is not a property of debt; it is a property of growth. A savings account, a bond or a pension compounds in exactly the same way, and the rule of 72 gives a quick estimate of doubling time: divide 72 by the annual percentage return. At 6 percent, money doubles in about 12 years. At 3 percent, about 24. At 9 percent, about 8.
Remember to think in real terms. If savings earn 4 percent and inflation is 3 percent, purchasing power grows by roughly 1 percent a year, not 4. The nominal number on the statement is not the number that matters when you spend it.
Frequently asked questions
Sources
- Consumer Financial Protection Bureau — loan estimate and APR guidance for US consumer credit.
- Financial Conduct Authority, Consumer Credit sourcebook (CONC) — APR calculation rules for UK credit.
- Standard amortisation mathematics: level-payment annuity formula, as set out in any financial mathematics textbook.
These tools are educational estimates, not financial advice. Lender rules, fees, taxes and penalties vary, and a regulated adviser can see details a calculator cannot.