What an interest rate actually is
An interest rate is the price of money over time. When you borrow money, the interest rate is what you pay the lender for the use of their money for a period. When you save or invest money, the interest rate is what you earn for letting someone else use your money for a period. The rate is expressed as a percentage of the principal per unit of time, usually per year. A 5 percent annual interest rate on a $1,000 loan means you pay $50 per year for the use of the $1,000.
Interest rates exist because money now is worth more than money later. This is the time value of money, and it has three causes. First, inflation: a dollar today buys more than a dollar will in ten years, because prices tend to rise over time. Second, opportunity cost: a dollar today can be invested and grow, so a dollar today is worth more than a dollar tomorrow even without inflation. Third, risk: a dollar today is certain, a dollar promised for the future may not arrive, because the borrower may default. The interest rate compensates the lender for all three.
The real interest rate, in economics, is the nominal rate minus inflation. If a savings account pays 5 percent and inflation is 3 percent, the real rate is 2 percent — your money grows 2 percent in purchasing power, not 5 percent. Real rates can be negative, and have been for much of the 2010s in many developed countries, when central banks kept nominal rates near zero while inflation ran at 1 to 2 percent. Negative real rates punish savers and reward borrowers, which is the intended effect when central banks want to stimulate borrowing and spending.
Simple vs compound interest
Simple interest is calculated only on the principal. If you lend $1,000 at 5 percent simple interest for 10 years, you earn $50 per year for 10 years, total $500, and get back $1,500 at the end. The math is linear: the interest each year is constant, because the principal is constant. Simple interest is rare in modern finance; it appears in some short-term loans and bonds, but most saving and borrowing uses compound interest.
Compound interest is calculated on the principal plus all accumulated interest. If you lend $1,000 at 5 percent compounded annually for 10 years, the first year earns $50 (5 percent of $1,000), the second year earns $52.50 (5 percent of $1,050), the third year earns $55.13 (5 percent of $1,102.50), and so on. After 10 years, the total is $1,628.89 — about $129 more than simple interest produced over the same period. The interest earns interest, and the growth accelerates over time.
The compounding frequency matters. The same nominal rate compounded more frequently produces a higher effective rate. 5 percent compounded annually produces an effective annual rate of 5.000 percent. Compounded semi-annually, it produces 5.0625 percent. Compounded monthly, 5.1162 percent. Compounded daily, 5.1267 percent. Compounded continuously, 5.1271 percent. The differences are small at low rates and short periods, but they compound (in the colloquial sense) over long periods and at high rates. A 30-year mortgage at 6 percent compounded monthly has an effective annual rate of 6.168 percent; the difference over 30 years on a $300,000 loan is about $7,000 in additional interest.
APR vs APY, and why the difference matters
APR, short for Annual Percentage Rate, is the yearly cost of borrowing, expressed as a percentage. It includes the nominal interest rate plus certain fees and costs associated with the loan, averaged over the life of the loan and expressed as an annual rate. In the United States, the Truth in Lending Act of 1968 requires lenders to disclose APR for consumer loans, so that borrowers can compare loans on a consistent basis. A mortgage with a 6 percent nominal rate and $3,000 in fees on a $300,000 loan has an APR of about 6.25 percent, because the fees are amortized into the rate.
APY, short for Annual Percentage Yield, is the yearly return on saving or investing, expressed as a percentage, taking into account the effect of compounding. A savings account that pays 5 percent compounded monthly has an APY of 5.116 percent. The APY is higher than the nominal rate because of compounding; the APR is higher than the nominal rate because of fees. The two are not directly comparable: APR is for borrowing (no compounding, just fees), APY is for saving (compounding included, no fees).
The trick to watch for is that lenders quote APR (which is lower because it does not include compounding) and savers quote APY (which is higher because it does include compounding). A credit card with a 19 percent APR has an effective rate, if interest is compounded monthly, of about 20.7 percent. A savings account with a 4.5 percent APY has a nominal rate of about 4.41 percent. The numbers are technically correct but presented in the light most favorable to the institution. When comparing rates, convert them all to the same basis — APY for savings, APR for loans — before comparing.
Continuous compounding and the math behind it
As the compounding frequency increases, the effective rate approaches a limit. This limit is continuous compounding, where interest is calculated and added to the principal at every instant. The formula is A = P × e^(rt), where A is the final amount, P is the principal, r is the nominal annual rate, t is the time in years, and e is the base of the natural logarithm, approximately 2.71828. A $1,000 investment at 5 percent continuously compounded for 10 years grows to $1,000 × e^(0.05 × 10) = $1,000 × e^0.5 = $1,648.72, which is about $20 more than annual compounding produced.
The mathematical curiosity here is that e, the base of natural logarithms, was discovered by Jacob Bernoulli in 1683 while studying exactly this problem: what happens to compound interest as the compounding frequency goes to infinity. Bernoulli was investigating a hypothetical account that paid 100 percent interest per year, and asked what would happen if the interest were compounded annually, monthly, daily, hourly, every minute, every second. He found that the limit was not infinity but a finite number, about 2.718, which is e. The discovery was a foundational moment in the development of calculus.
For practical purposes, continuous compounding is a mathematical convenience, not a financial product. No real account compounds continuously; the highest frequency you will see is daily. But the formula is useful for theoretical work, for option pricing (the Black-Scholes formula uses continuous compounding), and for approximating the effect of high-frequency compounding. The difference between daily and continuous compounding is negligible at most rates and periods, so the continuous formula is a good approximation when daily is not available.
Real vs nominal rates
The nominal interest rate is the number on the page: the 5 percent your savings account pays, the 6 percent your mortgage charges. The real interest rate is the nominal rate minus inflation, and it is what actually matters for purchasing power. If your savings account pays 5 percent and inflation is 3 percent, your real return is 2 percent. Your money grows 5 percent in nominal terms, but prices also rise 3 percent, so your money grows 2 percent in what it can buy.
Real rates have varied dramatically over history. In the high-inflation 1970s, nominal rates were high (mortgage rates peaked above 18 percent in 1981) but real rates were sometimes negative, because inflation was even higher. In the 2010s, nominal rates were near zero, but real rates were also near zero or slightly negative, because inflation was low. In 2023, nominal rates rose sharply (mortgage rates above 7 percent) as central banks fought inflation, and real rates returned to positive territory for the first time in over a decade.
The practical implication is that you cannot evaluate a rate in isolation. A 5 percent savings account looks attractive when inflation is 2 percent (real rate 3 percent) and unattractive when inflation is 7 percent (real rate negative 2 percent). When comparing investments, compare real rates, not nominal rates. The Fisher equation, named after the economist Irving Fisher, gives the precise relationship: (1 + nominal) = (1 + real) × (1 + inflation). For small rates, the approximation real = nominal − inflation is close enough; for large rates, use the full formula.
The Rule of 72 and other mental shortcuts
The Rule of 72 is a quick mental estimate of how long it takes for an investment to double at a given compound interest rate. Divide 72 by the interest rate (as a percentage), and the result is approximately the number of years to double. At 6 percent, money doubles in about 12 years (72 ÷ 6 = 12). At 8 percent, about 9 years. At 10 percent, about 7.2 years. The rule is accurate to within a year for rates between 4 and 15 percent, which covers most practical investments.
The Rule of 72 works because of the mathematics of continuous compounding. The doubling time at rate r is ln(2) ÷ r, which is 0.693 ÷ r. The number 72 is used instead of 69.3 because 72 is divisible by many small integers (2, 3, 4, 6, 8, 9, 12), making mental arithmetic easier. The small error introduced by using 72 instead of 69.3 is negligible at typical rates. For rates above 15 percent, the rule underestimates the doubling time; for rates below 4 percent, it slightly overestimates. For precise work, use the actual formula.
Other useful shortcuts: the Rule of 115 for tripling (115 ÷ rate is approximately the years to triple), the Rule of 70 for halving under inflation (70 ÷ inflation rate is approximately the years for purchasing power to halve). To estimate the effect of small rate differences over long periods, use the approximation that a 1 percentage point difference in annual return produces about a 20 percent difference in final value over 20 years, and about a 50 percent difference over 40 years. This is why seemingly small fees on long-term investments matter: a 1 percent annual fee on a 40-year investment consumes about 30 percent of the final value. The math of compound growth is unforgiving in both directions.