Simple vs Compound Interest

Simple interest and compound interest answer the same classroom question — what happens to money over time at a stated rate? — and then diverge on one rule. Simple interest always multiplies the original principal. Compound interest multiplies the latest balance. If interest is paid out and spent, the two stories collapse together, because nothing is left to compound. If interest stays put, compounding pulls ahead, slowly and then suddenly.

The two expressions

Simple balance after t years at annual rate r (decimal), no contributions:

A_simple = P × (1 + r × t)

Compound balance, compounded n times a year:

A_compound = P × (1 + r/n)^(n × t)

€5,000 at 6% for 3 years.

• Simple: 5000 × (1 + 0.06 × 3) = 5000 × 1.18 = €5,900. Interest = €900, a straight €300 a year.
• Compound annual: 5000 × 1.06^3 = 5000 × 1.191016 = €5,955.08. Extra versus simple: about €55.
• Compound monthly: 5000 × (1 + 0.06/12)^36 ≈ €5,983. Extra versus simple: about €83.

Three years is not a fable. The gap is real and, at 6%, still a few percent of the interest rather than a doubling of it. Stretch the same €5,000 at 6% annual compounding to 20 years: compound balance ≈ €16,036. Simple balance = 5000 × (1 + 0.06 × 20) = €11,000. Now the gap is about €5,000, the size of the original principal. Time, not a cleverer formula, did that. The exponent did it by being allowed to run.

A year-by-year look

€1,000 at 10%, chosen because 10% makes the drift obvious. Real savings rates are often lower; the shape is the lesson.

Year one matches. It has to: there is no prior interest to include. The famous “interest on interest” is a year-two event. Anyone selling compounding as a first-year miracle is selling something else.

Where simple interest is the real contract

Some short-term notes, certain payroll advances, and a few statutory calculations still say interest equals principal × rate × time, with time as a fraction of a year. In that contract, compounding would misquote the amount due. Read the words “simple interest” before you reach for the compound tool.

Day-count quirks sit on top. “Time” might be actual days over 365, or 30/360. A simple-interest calculator that assumes a whole number of years will miss a 40-day loan. For forty days at 8% simple on €2,000: interest ≈ 2000 × 0.08 × (40/365) ≈ €17.53. Compounding that daily for forty days is a few cents different. On short clocks, fighting about compounding is less important than getting the day count right.

Where people use the wrong one

Quoting a multi-year savings goal with simple interest understates the balance if the interest will actually be left on deposit, and the understatement grows with the rate and the years. It is a conservative error, and it is still an error if you are comparing two banks.

Quoting a credit-card balance with simple interest on the original purchase understates the cost if the balance revolves and interest is charged on interest. The compound (or daily-accrual) figure is the scary one because it is closer to the agreement. Pay the statement and the question goes away; the math is not a reason to carry the balance.

Using compound math on an amortizing loan’s total interest double-counts. A loan payment already includes interest on the remaining balance, and the balance is designed to fall. Total interest on a 5-year loan is not “compound the principal for 5 years.” It is the sum of the interest column in the schedule. See how loan EMI works.

Contributions change the ranking, not the definition

If you add money every month, both models can accept deposits. Simple interest on each deposit for the time that deposit has existed is a legitimate “no compounding” world. Compound interest lets each deposit’s interest start earning too. At low rates the practical gap may be smaller than the gap caused by skipping a contribution. The compound interest calculator is the compounding world. If a product explicitly pays simple interest, do not “improve” it in the tool by turning frequency up. Model the contract you can sign.

When the distinction should change a decision

It should change a decision when the horizon is long, the rate is not tiny, and interest will remain inside the account — retirement contributions, a child’s savings plan, a reserve you refuse to skim. It should not dominate a decision between two one-year deposit accounts whose nominal rates differ by a full percent; the rate gap will beat the compounding-frequency gap. Compare effective annual rates when the frequencies differ, using effective annual rate, and compare balances at the horizon you actually have.

A savings goal said both ways

You want €8,000 in 8 years and you will not add monthly deposits. At 4% compounded annually, the principal that grows into €8,000 is 8000 / (1.04^8). 1.04^8 ≈ 1.3686, so you need about €5,845 today. Simple interest at 4% for 8 years multiplies by 1.32, so the same target would seem to need 8000 / 1.32 ≈ €6,061 if someone used the straight-line formula by mistake. The €200 gap is the cost of ignoring interest on interest in the plan, or the surprise if the product really is simple and you planned as if it compounded.

Flip it into a loan-shaped warning without using the loan formula. Leaving €1,000 unpaid at 18% compounded monthly for two years grows by (1 + 0.18/12)^24 − 1 ≈ 43%, to about €1,429. Simple interest at 18% for two years would add 36%, to €1,360. On a nasty rate and a multi-year neglect, compounding is not a textbook flourish. It is why revolving balances hurt. Paying the balance off returns you to a world where the distinction does not get to run.

The rule to keep is almost dull: first year, they match; every later year, compound interest includes last year’s interest and simple interest declines to. If your contract says which world you are in, believe the contract over the more exciting curve. Use the compound interest calculator only for the compounding contract, and label a simple-interest agreement as simple even if the curve looks less impressive.