Compound Interest vs Simple Interest: How They Really Differ

What is simple interest?

Simple interest is calculated only on the original principal amount, using the formula I = P x r x t, where P is the principal, r is the annual interest rate expressed as a decimal, and t is the time in years. Because interest is never added back into the amount that earns further interest, a balance under simple interest grows by exactly the same dollar amount in every period. For example, $10,000 invested at a 5% simple annual interest rate earns $500 in year one, $500 in year two, and $500 in every subsequent year, for a total balance of P + (P x r x t).

Simple interest is used in some short-term lending contexts, such as certain auto loans, short-term promissory notes, and some Treasury bill calculations, because it is straightforward to compute and predict. Its defining feature is linear growth: a graph of balance over time under simple interest is a straight line, since each period contributes an identical, unchanging amount of interest regardless of how long the investment has been held.

What is compound interest?

Compound interest is calculated on both the original principal and any interest that has already been added to the balance, using the formula A = P(1 + r/n)^(nt), where A is the final amount, P is the principal, r is the annual interest rate as a decimal, n is the number of times interest compounds per year, and t is the time in years. Because each period's interest becomes part of the base for the next period's calculation, the balance grows at an accelerating rate rather than a constant one.

The compounding frequency, n, materially affects outcomes: n = 1 for annual compounding, n = 12 for monthly compounding, n = 365 for daily compounding, and so on. As n increases toward continuous compounding, the growth factor approaches the mathematical constant e (Euler's number) raised to the power of rt, though the practical difference between monthly and daily compounding is small for most consumer savings and loan products.

Worked example: $10,000 at 5% over 10, 20, and 30 years

The table below compares a $10,000 principal growing at a 5% annual rate under simple interest and under compound interest with monthly compounding (n = 12), calculated using A = P(1 + r/n)^(nt) for the compound column and I = P x r x t for the simple column. The gap between the two methods widens substantially as the time horizon lengthens, illustrating why compounding frequency and time horizon matter more than the headline interest rate alone for long-term outcomes.

The Rule of 72: a quick mental-math shortcut

The Rule of 72 is an approximation used to estimate how many years it takes an amount to double at a given annual compound interest rate, calculated as 72 divided by the interest rate expressed as a whole number. For example, at a 6% annual rate, the Rule of 72 estimates a doubling time of 72 / 6 = 12 years; at 9%, the estimate is 72 / 9 = 8 years; at 12%, the estimate is 72 / 12 = 6 years.

The Rule of 72 is most accurate for annual compound interest rates in roughly the 6% to 10% range, where the approximation error is smallest; at very low or very high rates, the estimate diverges further from the exact doubling time calculated from the compound interest formula. Applied to the 5% example above, monthly compounding produces an effective annual rate of approximately 5.12%, giving an estimated doubling time near 72 / 5.12, or about 14.1 years -- consistent with the balance in the worked table roughly doubling between the 10-year and 20-year marks.

Why compounding frequency matters

For the same stated annual rate, more frequent compounding produces a higher effective annual rate and therefore a larger final balance. A 5% rate compounded monthly (n = 12) produces an effective annual rate of approximately 5.12%, higher than the 5.00% effective rate under annual compounding (n = 1), because interest is added to the balance -- and begins earning its own interest -- twelve times per year instead of once.

This distinction is the reason financial disclosures often separate the nominal (stated) interest rate from the annual percentage yield (APY) or effective annual rate, which accounts for compounding frequency. Two savings products advertising the same nominal rate can produce different actual returns if one compounds daily and the other compounds annually, which is why comparing effective annual rates -- not just nominal rates -- is necessary for an accurate comparison between financial products.

Practical implications for saving and borrowing

Compound interest works in favor of savers and investors: balances left untouched over long horizons benefit disproportionately from the accelerating growth shown in the worked example, which is why starting to save early is generally emphasized more than the size of any single contribution. Retirement accounts, in particular, rely on decades of compounding to turn moderate regular contributions into substantially larger balances by the time they are needed.

The same mathematics works against borrowers: credit card balances, which typically compound daily or monthly at high annual percentage rates, can grow quickly if only minimum payments are made, because unpaid interest is added to the balance and itself begins accruing interest. Understanding that compound interest formulas apply equally to savings growth and debt growth is central to evaluating both investment products and borrowing costs.

Preguntas frecuentes

Referencias

1. U.S. Securities and Exchange Commission, Office of Investor Education and Advocacy. "Compound Interest Calculator." Investor.gov.
2. FINRA Investor Education Foundation. "The Rule of 72." FINRA.org.
3. Ross SA, Westerfield RW, Jaffe J, Jordan BD. Corporate Finance. 12th ed. McGraw-Hill Education, 2019.
4. Fisher I. The Theory of Interest. Macmillan, 1930.