Compound Interest, Worked Out by Hand

Compound interest gets described as a wonder often enough that the phrase has lost its force. The arithmetic has not. Here it is, worked through slowly.

Simple growth versus compound growth

Simple interest pays a fixed amount on the original principal each period. $1,000 at 7% simple interest earns $70 every year, forever. After 30 years you have $1,000 plus $2,100, so $3,100.

Compound interest pays on the principal and on the interest already earned. Year one earns $70. Year two earns 7% of $1,070, which is $74.90. Year three earns 7% of $1,144.90.

After 30 years, that same $1,000 has become about $7,612.

Same rate, same starting amount, same period. The difference is $4,512, and all of it comes from interest earning interest.

The formula

The standard expression is:

A = P(1 + r/n)^(nt)

Where P is the principal, r is the annual rate as a decimal, n is how many times a year interest compounds, t is the number of years, and A is what you end up with.

For $1,000 at 7% compounded annually for 30 years: 1000 times 1.07 to the power of 30, which gives $7,612.

Compounding frequency matters less than people expect

Running $1,000 at 7% for 30 years at different frequencies:

Annually: about $7,612
Monthly: about $8,116
Daily: about $8,166

Moving from annual to monthly compounding adds around $500. Moving from monthly to daily adds about $50. There are diminishing returns, and the gap between daily and continuous compounding is negligible.

The rate and the time horizon do far more work than the frequency. If you are choosing between accounts, a higher rate compounded annually beats a lower rate compounded daily almost every time.

The Rule of 72

A useful shortcut: divide 72 by the annual rate to approximate how many years it takes for money to double.

At 4%: 72 divided by 4 = 18 years
At 7%: about 10 years
At 10%: about 7 years

It is an approximation, most accurate between roughly 5% and 12%, but it is close enough for mental arithmetic and it makes the effect of rate differences immediate. At 7%, money doubles three times in 30 years - eight-fold. At 4%, it doubles less than twice.

Why starting early beats contributing more

This is the part worth internalising.

Two people both retire at 65, both earn 7%:

A invests $200 a month from age 25 to 35, then stops. Ten years of contributions, $24,000 total.
B invests $200 a month from age 35 to 65. Thirty years of contributions, $72,000 total.

B contributes three times as much. A ends up with more.

A's smaller pot has 40 years to compound, thirty of them with no further contributions at all. B's larger contributions never get the same runway. The variable doing the work is time, and it is the one variable you cannot buy back later.

The same mechanism in reverse

Compounding is indifferent to which direction it runs. Carried credit card balances compound daily against you at rates that dwarf typical investment returns - which is why clearing a 24% APR balance is mathematically the strongest "return" most people have access to. We cover the mechanics in how credit card interest is calculated.

Try it with your own numbers

Abstract percentages do not persuade anyone. Your own figures might. Put your actual starting amount, monthly contribution and time horizon into our investment calculator and look at the gap between what you contribute and what it becomes.

The gap is the whole argument.

Sources

Current as of August 2026. The 7% return is an illustrative assumption for the arithmetic, not a forecast. Investments can lose value.