Compound Interest and the Rule of 72, Checked Against the Math

Compounding is the reason a modest amount saved at 25 can outgrow a larger amount saved at 45, and also the reason a credit card balance can balloon. The idea is simple, but its effects are easy to underestimate, because growth that doubles on itself is not something our intuition handles well. This guide works through the arithmetic with real numbers so you can see it for yourself.

Simple interest versus compound interest

Simple interest is paid only on the original amount. If you put in 10,000 at 8% simple interest, you earn 800 every year, whether it is year one or year ten. After 10 years: 10,000 + (10 × 800) = 18,000.

Compound interest is paid on the original amount plus all the interest already earned. Year one earns 800, so you start year two with 10,800 and earn 864, and so on. After 10 years at 8% compounded annually you have 10,000 × 1.08¹⁰ = 21,589.25. The extra 3,589 over the simple-interest result is interest earned on interest.

The formula

A = P × (1 + r/m)^m×t

• A is the final amount.
• P is the starting amount.
• r is the annual rate as a decimal (8% is 0.08).
• m is how many times a year interest is added.
• t is the number of years.

For annual compounding, m is 1 and the formula reduces to P × (1 + r)^t.

Does compounding frequency matter?

It does, but less than most people expect. The same 10,000 at 8% for 10 years:

Going from simple interest to annual compounding adds about 3,589. Going from annual to daily adds only about another 664. The jump from “none” to “some” is large; the jump from “some” to “more often” is small. The things that matter far more are the rate and, above all, the time.

One practical note: when comparing savings accounts or loans, use the effective annual rate (often called APY or APR depending on the product and country) rather than the nominal rate, since it already accounts for compounding frequency. The compound interest calculator lets you switch frequency and see the difference for your own numbers.

The Rule of 72

The Rule of 72 is a mental shortcut for estimating how long it takes money to double: divide 72 by the annual percentage rate. At 8%, 72 ÷ 8 = 9 years. Check it: 1.08⁹ = 1.999, so 10,000 grows to about 19,990 after 9 years. Very close.

The exact doubling time is ln(2) ÷ ln(1 + r). Here is how the shortcut compares:

The rule is excellent between roughly 6% and 10%, and drifts a little at the extremes: it slightly overestimates the doubling time at low rates and underestimates it at high ones. It also works backwards. If you want to double your money in 6 years, you need roughly 72 ÷ 6 = 12% a year. It is for quick estimates only; it assumes a steady rate, which real investments do not deliver. The Rule of 72 calculator shows the estimate and the exact figure side by side.

It works just as well on the bad side. Debt growing at 18% a year doubles in about 4 years if nothing is paid.

Why starting early matters more than saving more

Here is a comparison, assuming a steady 7% a year (a simplifying assumption, not a forecast) and contributions made at the end of each year, with both people stopping the clock at the same date, 40 years from the start:

• Early Eva invests 5,000 a year for just the first 10 years, then never adds another cent and leaves the pot to grow. She contributes 50,000 in total.
• Late Luis waits 10 years, then invests 5,000 a year for the following 30 years. He contributes 150,000 in total.

Eva’s 10 years of contributions grow to about 73,918. Left alone for another 30 years at 7%, that becomes about 562,683. Luis’s 30 years of contributions add up to about 505,365. Eva puts in a third of the money and finishes ahead, because her first contributions had four decades to compound.

This does not mean that late starters should give up. Luis would have done much better still by starting late than not starting at all, and Eva would have ended up with far more by continuing. The lesson is only that each year you delay costs you the most growth in the part of the curve where the balance is largest.

Regular contributions (the SIP idea)

A systematic investment plan, or SIP, is just a fixed contribution made on a schedule, usually monthly. Each contribution compounds for a different length of time: the first for the whole period, the last for almost none. With 5,000 a month for 10 years at an assumed 8% a year (compounded monthly, deposits at month end), you pay in 600,000 in total and the projected balance is about 914,730. The projection depends entirely on the assumed rate. Real returns on market-linked investments vary from year to year and can be negative, so treat these figures as arithmetic, not promises. Try your own numbers with the SIP calculator.

Inflation quietly takes a share

A balance that grows is not the same as buying power that grows. Suppose 10,000 earns 6% a year for 20 years: 10,000 × 1.06²⁰ = 32,071.35. If prices rise 3% a year over the same period, what that money buys is closer to 32,071.35 ÷ 1.03²⁰ = 17,757 in today’s terms. The real growth rate is about (1.06 ÷ 1.03) − 1 = 2.9% a year, not 6%. That is why people often compare returns after inflation. The inflation calculator helps you adjust amounts across years.

Compounding works against borrowers too

The same maths that grows savings makes debts heavier when interest is added to the balance and you do not pay it off. Minimum payments on high-interest balances can leave you paying mostly interest for a long time. The related guide on how EMI is calculated shows how a fixed instalment repays interest and principal on a loan.

Honest limitations

• All projections assume a constant rate. Real returns fluctuate.
• Fees, taxes, and inflation all reduce what you actually keep.
• Contribution timing (start versus end of period) changes results slightly.
• A projection is not a recommendation to buy any product.

For a plain-language definition and official calculators, see the sources below.