Compound interest explained, with the rule of 72

By Zulu Calculator · Updated 19 October 2026 · Money

Compound interest means your interest earns interest. Each time interest is added, the next lot is worked out on the bigger balance, so the money grows faster the longer you leave it. For example, 10,000 at an example rate of 5% a year, compounded monthly, grows to 16,470.09 after 10 years. Simple interest at the same rate would give only 5,000 of interest, not 6,470.09.

This guide covers the formula, how often interest is added, APY, regular savings, the rule of 72 and inflation. To try your own numbers, use the compound interest calculator. The rates below are examples for the maths only, not a forecast of what any account or investment will pay.

What is the difference between simple and compound interest?

Simple interest is worked out on the starting amount only. You get the same amount every year, and the total grows in a straight line. Compound interest adds each lot of interest to the balance, so next time it earns interest too.

Take 10,000 at 8% a year for 5 years. Simple interest is 10,000 × 0.08 × 5 = 4,000. Compounded yearly, it is 10,000 × 1.08⁵ − 10,000 = 4,693.28. That is 693.28 more, just from interest earning interest. The simple interest calculator shows this comparison for any rate.

The gap is small at first and grows over time. Here is 10,000 at an example rate of 6% a year, compounded monthly:

After 30 years, compounding has added more than three times the starting amount on top of simple interest. This is why time matters so much: the longer the money is left, the bigger the gap.

What is the compound interest formula?

balance = starting amount × (1 + yearly rate ÷ times a year)^{(times a year × years)}
interest = balance − starting amount
continuous compounding: balance = starting amount × e^{(yearly rate × years)}

"Times a year" is how often interest is added: 1 for yearly, 2 for half-yearly, 4 for quarterly, 12 for monthly, 52 for weekly and 365 for daily. Write the rate as a decimal, so 5% is 0.05. "e" is the number 2.71828…, found on most calculators as e^x.

Does compounding monthly or daily make a big difference?

More often is better, but the gain shrinks quickly. Here is 10,000 at an example rate of 6% a year for 10 years:

Going from yearly to monthly adds 285.49 over 10 years. Going from monthly to daily adds only 26.32 more. A slightly higher rate, or a lower fee, usually matters more than how often interest is added.

What are APY, AER and the effective annual rate?

They are the same thing: how much your money really grows in a year once compounding is counted. In the UK it is called the AER (annual equivalent rate). In finance it is the EAR or effective annual rate. The APR, also called the nominal or stated rate, is the yearly rate before compounding.

APY = (1 + APR ÷ times a year)^{times a year} − 1
continuous: APY = e^APR − 1

A 5% APR compounded monthly: 5% ÷ 12 = 0.416667% a month, and 1.00416667¹² = 1.051162, so the APY is 5.1162%. Daily, it is 5.1267%, and continuously 5.1271%. Turned round, to earn a 5% APY with monthly compounding the APR must be 4.8889%. The APY calculator converts either way.

When you compare savings accounts, compare the APY, not the APR, because the APY already allows for how often interest is added.

Worked examples

Example 1: a lump sum of 10,000 for 10 years

10,000 at an example rate of 5% a year, compounded monthly, left for 10 years with nothing added.

• Rate each month: 5% ÷ 12 = 0.416667%
• Number of months: 12 × 10 = 120
• Growth factor: 1.00416667¹²⁰ = 1.647009
• Balance: 10,000 × 1.647009 = 16,470.09, so the interest is 6,470.09

Compounded yearly instead, it would be 10,000 × 1.05¹⁰ = 10,000 × 1.628895 = 16,288.95.

Example 2: adding 100 every month

The same 10,000 at 5% compounded monthly, plus 100 added at the end of every month for 10 years. Each addition grows from the month it goes in, so early additions grow for nearly 10 years and the last one not at all.

• You put in: 10,000 + (100 × 120) = 22,000
• Final balance: 31,998.32
• Interest: 31,998.32 − 22,000 = 9,998.32
• Of that, 6,470.09 comes from the starting amount (as in Example 1) and 3,528.23 from the monthly additions

Timing matters a little. Paying the 100 at the start of each month instead gives each one an extra month of growth, and the balance becomes 32,063.02. If you raise the monthly amount by 5% each year, as a pay rise might allow, you put in 25,093.47 and end with 35,613.76.

Example 3: the same savings after tax and inflation

Go back to Example 1 (10,000 at 5%, compounded monthly, 10 years), but say an example 30% tax is taken from the interest and prices rise by an example 3% a year.

• Rate after tax: 5% × (1 − 0.30) = 3.5% a year, compounded monthly (an effective 3.5567% a year)
• Balance after tax: 14,183.45, so tax costs about 16,470 − 14,183 = 2,287
• Inflation over 10 years: 1.03¹⁰ = 1.343916
• In today's money: 14,183.45 ÷ 1.343916 = 10,553.82

So the balance looks like 14,183.45, but it buys only about what 10,553.82 buys today. Tax rules differ by country and by account, and some savings are tax-free, so use the rate that applies to you.

How does the rule of 72 work?

The rule of 72 is a quick way to see how long money takes to double: divide 72 by the yearly rate. At 8% a year, 72 ÷ 8 = 9 years. The exact answer with yearly compounding is ln 2 ÷ ln 1.08 = 0.693147 ÷ 0.076961 = 9.0065 years, so the rule is off by only a couple of days.

quick rule: years to double ≈ 72 ÷ yearly rate (in %)
exact, compounded yearly: years = ln 2 ÷ ln(1 + rate ÷ 100)

With yearly compounding the rule is exact at about 7.85%. At low rates, or when interest is added daily or continuously, the rule of 70 or 69.3 is closer: with continuous compounding the doubling time is exactly ln 2 ÷ (rate ÷ 100), which is about 69.3 ÷ rate. For high rates, an adjusted rule, 72 + (rate − 8) ÷ 3, does better. To see how long money takes to triple, use 114 instead of 72; to quadruple, use 144. The rule of 72 calculator shows the rule and the exact answer side by side.

How does inflation affect compound interest?

Inflation does not change your balance, but it changes what the balance can buy. To see a future balance in today's money, divide it by (1 + inflation) for each year:

today's money = future balance ÷ (1 + inflation rate)^years

16,470 in 10 years, with 6% inflation, buys what about 9,197 buys today. The rule of 72 also works for prices: at 6% inflation, prices double in about 72 ÷ 6 = 12 years (exactly 11.90), so cash left under the mattress buys half as much in that time. Your interest rate after tax needs to beat inflation for your money to grow in real terms.

Common mistakes

• Comparing an APR with an APY. A 4.9% APR compounded daily (5.0217% APY) beats a 5% APR compounded yearly (5% APY). Compare like with like.
• Treating an example rate as a promise. Calculators assume a fixed rate. Real rates change, and investment returns go up and down.
• Forgetting tax, fees and inflation. They all lower what you really end up with. A small monthly fee can wipe out a better rate on a small balance.
• Expecting daily compounding to work miracles. At 6%, daily instead of monthly adds only 26.32 on 10,000 over 10 years.
• Using the rule of 72 at very low or high rates. At 1% it says 72 years when the truth is 69.66; at 20% it says 3.6 when the truth is 3.80.
• Mixing up start and end of the month. Additions paid at the start of each period get one extra period of growth.

What if my bank's figure is different?

Small differences are normal. Banks may round each month's interest, count days on a 365-day or 360-day year, or credit interest on different dates. Some use the "365/360" method, which divides the yearly rate by 360 but adds interest on all 365 days, so it pays a little more: 5.1998% APY at a 5% APR. If the numbers do not match:

1. Check the APY or AER on the account terms, not just the headline rate.
2. Ask how often interest is added and whether it uses a 365-day or 360-day year.
3. Check for fees and tax taken from the interest, which our calculators only include if you enter them.
4. Check whether the rate is fixed. A variable rate will not match a fixed-rate projection.

Quick summary

• Compound interest pays interest on interest; simple interest does not.
• Balance = starting amount × (1 + rate ÷ times a year)^{times a year × years}.
• More frequent compounding helps, but monthly to daily adds very little.
• Compare savings by APY (AER), not APR.
• Time matters most: the gap over simple interest grows every year.
• Rule of 72: years to double ≈ 72 ÷ rate.
• Tax and inflation lower what your money is really worth.

Ready to try your own numbers? Use the compound interest calculator, the APY calculator to compare accounts, the rule of 72 calculator for doubling time, or the simple interest calculator for loans and deposits that do not compound.

Formulas and figures as used by our compound interest, APY, rule of 72 and simple interest calculators (checked 19 October 2026). General information only, not financial or tax advice.

This guide explains the rules in plain English for planning. It is not legal, financial or flight-operations advice; check your contract, the official source linked above, your employer or your operator.