What it means
Put $1,000 in an account paying 5% a year. After one year it holds $1,050.
Under simple interest, the second year's interest is again $50, on the original $1,000, and after ten years the balance is $1,500. Under compound interest, the second year's interest is 5% of $1,050, or $52.50, and after ten years the balance is $1,628.89.
After thirty years: simple interest $2,500, compound $4,321.94. After fifty years: $3,500 against $11,467.40.
The difference is interest on interest, and it grows without limit. The mechanism is multiplication.
Each period the balance is multiplied by one plus the rate; after n periods it has been multiplied by that factor n times. The growth is exponential, and the human intuition that growth is linear is what makes compound interest surprising: doubling takes a fixed time at a fixed rate (about 72 divided by the percentage rate: 14.4 years at 5%, 7.2 years at 10%), and each doubling adds as much as everything before it.
Frequency matters at the margin. A nominal 6% rate compounded annually gives 6%; compounded monthly, each month adds 0.5% and the year adds 6.17%; compounded daily, 6.18%; compounded continuously, 6.18%.
The effective annual rate is the true annual growth after compounding, and it is the figure to compare across products with different compounding frequencies. Lenders quote nominal rates; borrowers pay effective rates.
The applications run throughout finance. Savings and investments grow by compounding of returns, and the case for starting early rests on it: $5,000 a year invested from age 25 to 35 and then left alone can exceed $5,000 a year invested from 35 to 65, at the same rate, because the early money compounds longer.
Debts compound against the borrower: credit card balances at 20% double in under four years if unpaid, and interest on arrears, if capitalised, does the same. Loan repayment schedules are built on compound interest: each payment covers the period's interest on the outstanding balance and reduces the principal, so early payments are mostly interest and later ones mostly principal.
Bond prices, discounted cash flow valuations, pension calculations, lease accounting and growth rates all use the same arithmetic in one direction or the other: compounding forward to a future value, or discounting back to a present value. For financial decisions, the practical lessons are: compare rates on an effective annual basis; recognise that time is the most powerful variable and start early, whether saving or repaying; understand that a percentage point of return compounds into a large difference over a working life; and treat any debt that compounds at a high rate as urgent.
In practice
Real-world examples.
Example
A pension fund's 20-year return of 6.5% a year compounds a $1 million contribution to $3.52 million.
Example
A company's revolving credit charges interest monthly on the outstanding balance including unpaid interest, so a facility drawn at 8% nominal costs 8.3% effective.
Example
A country's debt at 5% interest with no repayments doubles in about 14 years, which is why persistent deficits compound into crises.
Think of it
“Compound interest is interest on interest-your earnings grow faster because they keep earning.
Formula
Calculation
Future Value = Principal x (1 + r) to the power n, where r is the rate per period and n the number of periods
With compounding m times a year at nominal annual rate i: Future Value = Principal x (1 + i/m) to the power (m x years)
Effective Annual Rate = (1 + i/m) to the power m minus 1
Continuous compounding: Future Value = Principal x e to the power (i x years)
Present Value = Future value / (1 + r) to the power n
Rule of 72: Years to double is approximately 72 / Rate in percent
Worked example 1, savings. $10,000 deposited at 6% a year.
- Simple interest after 20 years: $10,000 + 20 x $600 = $22,000
- Annual compounding: $10,000 x 1.06 to the power 20 = $10,000 x 3.207 = $32,071
- Monthly compounding: $10,000 x (1 + 0.06/12) to the power 240 = $10,000 x 1.005 to the power 240 = $10,000 x 3.310 = $33,102
- Effective annual rate with monthly compounding = 1.005 to the power 12 minus 1 = 6.17%
- Interest on interest over 20 years (annual compounding) = $32,071 minus $22,000 = $10,071, nearly half the total interest earned
Worked example 2, starting early. Two savers each earn 7% a year. Saver A puts in $6,000 a year from age 25 to 35 (10 years, $60,000 total) and then nothing. Saver B puts in $6,000 a year from age 35 to 65 (30 years, $180,000 total).
- A at 35: $6,000 x [(1.07 to the power 10 minus 1) / 0.07] = $6,000 x 13.816 = $82,900; left to compound 30 years to 65: $82,900 x 1.07 to the power 30 = $82,900 x 7.612 = $631,000
- B at 65: $6,000 x [(1.07 to the power 30 minus 1) / 0.07] = $6,000 x 94.461 = $566,800
- A invested a third as much and finished with more, because ten extra years of compounding on the early contributions outweighed twenty years of additional contributions
Worked example 3, debt. A credit card balance of $8,000 at 22% a year, compounded monthly (1.833% a month), with no payments.
- After one year: $8,000 x 1.01833 to the power 12 = $8,000 x 1.2436 = $9,949
- After three years: $8,000 x 1.01833 to the power 36 = $8,000 x 1.923 = $15,385
- Doubling time by the rule of 72: 72 / 22 = 3.3 years; exact: ln(2) / ln(1.01833) = 38.2 months
- With a minimum payment of 2% of the balance each month, the balance falls very slowly: interest of about $147 in the first month against a $160 payment reduces principal by only $13; clearing the debt at that rate takes decades and costs several times the original balance
Worked example 4, comparing rates. Two loans: Bank A quotes 7.9% compounded monthly; Bank B quotes 8.0% compounded annually.
- A effective rate = (1 + 0.079/12) to the power 12 minus 1 = 1.006583 to the power 12 minus 1 = 8.19%
- B effective rate = 8.00%
- Bank B is cheaper despite the higher nominal rate. On a $200,000 loan the difference is about $380 a yearCase study
Seen in the real world.
A small company borrowed $400,000 from a private lender at 1.5% a month when its bank declined, intending to repay within six months from a contract receipt. The contract was delayed, then disputed, and the company paid nothing for twenty months while it waited. The owner understood the loan as "18% a year", which on $400,000 was $72,000 a year, and expected to owe about $520,000.
The loan agreement compounded monthly: $400,000 x 1.015 to the power 20 = $400,000 x 1.347 = $538,800, and the lender's statement showed $538,800 plus default fees. More seriously, the owner had not noticed that the agreement compounded the default interest at 3% a month after a missed payment date, so that from month twelve the rate had doubled: the true balance was about $600,000.
The company settled by selling a property. The owner's later summary was that he had understood the interest rate and had not understood the compounding, and that the difference had cost him $80,000 and a building.
Watch out
Common mistakes.
- Comparing loans or investments on nominal rates without converting to effective annual rates, which differ by compounding frequency.
- Assuming growth is linear. Compounding is exponential; the late years of any long period contribute far more than the early ones, and the early money matters most.
- Ignoring the compounding of arrears and default interest on debts, which can double a balance in a few years.
Questions
People also ask.
What is the difference between simple and compound interest?
Simple interest is charged on the principal only; compound interest is charged on the principal plus accumulated interest. Over any period longer than one, compound interest produces more.
What is the rule of 72?
A quick estimate: the number of years for a sum to double at a compound rate is about 72 divided by the percentage rate. At 6%, about 12 years; at 9%, about 8.
Does compounding frequency matter much?
A little: 6% compounded monthly gives 6.17% a year against 6% annually. It matters more at high rates and over long periods, and it matters most when comparing products quoted on different bases.
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