Interest is the price of using someone else's money
Whenever money is borrowed or lent, the person using the money pays a fee for the privilege, and the person supplying the money receives that fee. This fee is called interest. If you deposit money in a savings account, the bank is effectively borrowing your money and pays you interest. If you take out a loan to buy a warehouse, you are borrowing the lender's money and you pay them interest. The amount you start with — the original sum being lent, borrowed, or invested — is called the principal. Interest is always expressed as a percentage of the principal per period of time, most commonly per year (an annual interest rate). The question this topic answers is: exactly *how* is that interest calculated over multiple periods? There are two fundamentally different answers, and the difference between them compounds (literally) into one of the most important ideas in all of finance.
Simple interest: the same dollar amount, every single period
Simple interest is interest calculated only on the original principal, every period, for the entire life of the loan or investment. It never takes into account any interest that has already accumulated. This means that under simple interest, the dollar amount of interest earned (or owed) is identical in every period.
Worked Example 1 — Simple interest, step by step. Suppose you invest $10,000 in an account that pays 6% simple interest per year, and you leave it there for 3 years. Every single year, the interest is calculated on the same original $10,000:
- Year 1 interest: $10,000 x 0.06 = $600 - Year 2 interest: $10,000 x 0.06 = $600 (still based on the original $10,000, not on $10,600) - Year 3 interest: $10,000 x 0.06 = $600
Total interest after 3 years = $600 + $600 + $600 = $1,800. The account's final balance = $10,000 (original principal) + $1,800 (total interest) = $11,800. Notice the balance grows by exactly the same $600 every year — a straight line, not a curve.
Simple Interest
I = P x r x t
- I
- — Total interest earned or owed
- P
- — Principal — the original amount of money
- r
- — Annual interest rate, expressed as a decimal (6% = 0.06)
- t
- — Time, expressed in years (or fraction of a year)
Multiply the principal by the rate by the number of years. There is no compounding — the formula never looks at how much interest has already piled up. This is why simple interest is common for very short-term loans, certain bonds, and situations where lenders intentionally want a simple, easy-to-verify calculation.
Worked example: A 6-month (0.5 year) loan of $5,000 at 4% simple interest: I = $5,000 x 0.04 x 0.5 = $100 of total interest, so the borrower repays $5,000 + $100 = $5,100.
Compound interest: interest that earns interest
Compound interest is interest calculated on the principal *plus* all interest that has already accumulated. In other words, once interest is earned in one period, it gets added to the balance, and the next period's interest is calculated on that larger balance. This is often summarized as 'earning interest on your interest.'
Worked Example 2 — Compound interest, step by step, using the exact same numbers as Example 1. Take that same $10,000 invested for 3 years, but now at 6% *compound* interest, compounded once per year:
- Year 1: Interest = $10,000 x 0.06 = $600. New balance = $10,000 + $600 = $10,600. - Year 2: Interest = $10,600 x 0.06 = $636.00 (notice: calculated on $10,600, not the original $10,000). New balance = $10,600 + $636.00 = $11,236.00. - Year 3: Interest = $11,236.00 x 0.06 = $674.16. New balance = $11,236.00 + $674.16 = $11,910.16.
Compare the two results after 3 years, same principal, same rate: simple interest produced $11,800.00, while compound interest produced $11,910.16 — a difference of $110.16. That difference seems small over 3 years, but it grows dramatically over longer periods, because every additional year adds interest-on-interest on top of an already-larger base.
Compound Interest (Future Value)
FV = P x (1 + r)^n
- FV
- — Future value — the balance after compounding
- P
- — Principal — the original amount of money
- r
- — Interest rate per compounding period, as a decimal
- n
- — Number of compounding periods
Add 1 to the rate, raise that sum to the power of the number of periods, and multiply by the principal. Raising to a power is the mathematical way of repeating the 'multiply, then add back in' process from Example 2 automatically, for as many periods as you specify.
Worked example: Using Example 2's numbers directly in the formula: FV = $10,000 x (1 + 0.06)^3 = $10,000 x (1.06)^3 = $10,000 x 1.191016 = $11,910.16 — exactly matching the year-by-year calculation above.
Compounding frequency changes the outcome, even at the same stated rate
Compound interest doesn't have to compound just once a year — it can compound semiannually, quarterly, monthly, daily, or even continuously. Each time it compounds, that period's interest gets folded into the balance and starts earning its own interest sooner.
Worked Example 3 — Monthly compounding. Suppose $1,000 is invested at a stated annual rate of 12%, but compounded *monthly* rather than annually. To compound monthly, first convert the annual rate to a monthly rate by dividing by 12: 12% / 12 = 1% per month. Then apply the compound interest formula with n = 12 monthly periods: FV = $1,000 x (1 + 0.01)^12 = $1,000 x (1.01)^12 = $1,000 x 1.126825 = $1,126.83. Total interest earned = $1,126.83 − $1,000.00 = $126.83 — more than the $120.00 you would get from simple interest at 12% for one year ($1,000 x 0.12 x 1 = $120), because monthly compounding lets interest start earning interest within the same year. (This connects directly to the next topic in this block, Nominal vs. Effective Interest Rates, which is entirely about this exact phenomenon.)
Simple vs. Compound Interest at a Glance
| Feature | Simple Interest | Compound Interest |
|---|---|---|
| Calculated on | Original principal only | Principal + all accumulated interest |
| Growth pattern | Straight line (linear) | Curves upward (exponential) |
| Dollar interest per period | Same every period | Increases every period |
| Formula | I = P x r x t | FV = P x (1 + r)^n |
| Typical real-world use | Short-term loans, some bonds, per-diem interest calculations | Savings accounts, mortgages, investment growth, most CRE loan calculations |
| Who benefits from it | Favors the borrower on longer loans (less owed than compounding) | Favors the lender/saver on longer horizons (more earned or owed) |
Common beginner mistake: assuming a stated rate is the whole story
The single most common beginner error is treating a quoted interest rate — '6% interest' — as if it fully describes the deal. It does not. You must also know whether the interest is simple or compound, and if compound, how often it compounds (annually, monthly, daily). Two loans quoted at the identical 6% rate can have meaningfully different costs depending on these details, as Example 3 demonstrated. Always ask: simple or compound, and compounded how often?
Module Check
You invest $8,000 at a simple interest rate of 5% per year. How much total interest will you have earned after 4 years?