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.

What time actually does

The often-repeated point that starting early matters is usually asserted rather than shown. Here it is with numbers. Assume 7% annual growth and a target retirement at 65.

The first two rows are the argument. Ten years of delay costs more than $400,000, while the amount actually contributed differs by only $36,000. The gap is not the missing contributions; it is the growth those early contributions would have generated for four decades.

The fourth row is the more useful comparison. Someone starting at 35 must contribute double every month for thirty years and still finishes slightly behind the person who started at 25 at half the rate. Time is doing work that money struggles to replicate.

Where the growth comes from

Splitting the first row into contributions and growth shows the shape clearly.

For the first decade, this looks like a savings account: the balance is mostly money you put in, and progress feels slow. That is the period in which people conclude it is not working and stop.

By the final decade, growth is producing far more each year than the contributions do. The account gains more in some single years near the end than was contributed in the entire first decade. The mechanism did not change; it simply needed a base to work on.

The practical implication is about persistence rather than optimisation. The early years feel unrewarding by design, and they are the years that matter most.

Frequency, quantified

Compounding more often produces more, but the effect is much smaller than the marketing around it suggests. $10,000 at 7% for one year:

Moving from annual to daily compounding gains about $25 on $10,000. There is a hard ceiling: continuous compounding is the limit, and daily is already essentially at it.

The useful conclusion is that compounding frequency is a rounding detail. A quarter of a percentage point of rate, or one year of time, each matter more than any change in frequency.

The same arithmetic, working against you

Nothing in the mechanism cares about direction. A credit card balance at 22.99% compounds daily on the amount owed, including interest already added.

$5,000 left untouched at 22.99%, compounding:

The balance triples in five years with nothing added. This is why paying off a 22.99% balance is a better guaranteed return than almost any investment: you are stopping compound growth that is running against you at a rate no portfolio reliably matches.

Clearing high-rate debt before investing is not a conservative choice. It is the same arithmetic, applied where it pays most.

A caution about the assumption

Every figure above assumes a smooth 7% every year. Real markets do not deliver that. They deliver sequences that average out over long periods while being sharply negative in some years and sharply positive in others.

That distinction matters in two ways:

• Over long horizons, the averaging works and projections like these are a reasonable guide to the shape of the outcome, though not to the exact number
• Over short horizons, or when you are drawing money out, the order of returns matters as well as the average, and a bad sequence early in retirement does real damage

So use these projections to understand the mechanism and to compare decisions — starting now versus later, contributing more versus less — rather than as a forecast of what you will have.

Run your own figures through our investment calculator, and see how investment fees compound for the same mechanism applied to costs, which is where a great deal of the growth above can quietly go.

The formula, and what each part does

The standard expression is future value equals principal times one plus the rate, raised to the number of periods.

• Principal scales the result linearly. Double it and you double the outcome
• Rate sits inside the base being raised to a power, so small changes have outsized effects
• Time is the exponent, which is why it dominates everything else

That last point is the whole reason the age comparison earlier is so lopsided. Doubling your contribution doubles the result; doubling your time frame squares part of it.

The Rule of 72

For quick mental arithmetic, dividing 72 by the annual return gives roughly the number of years to double.

It is an approximation, most accurate between about 5% and 10%, but it is accurate enough to reason with. It also works in reverse on debt: a balance at 24% doubles in about three years if left alone.

Sources

• SEC Investor.gov: Compound interest calculator
• SEC Investor.gov: Introduction to investing

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