Compound Interest Basics

Compound interest is a repeated multiplication. You earn a rate on whatever is there, the interest stays (or is charged, if you are the borrower), and the next round uses the new total. Simple interest refuses that second step and always uses the original principal. The gap between them is small over a short time at a low rate, and it is the plot of every long savings chart.

The compound interest calculator projects a balance from a principal, a nominal annual rate, a compounding frequency, a time span, and an optional contribution each period. This page explains what that projection is assuming, so a smooth curve does not get mistaken for a promise.

The formula, named in pieces

If P is principal, r is the nominal annual rate as a decimal, n is the number of compounding periods per year, and t is years, then with no extra contributions:

A = P × (1 + r/n)^(n×t)

The piece (1 + r/n) is the growth factor for one period. You multiply by it once per period. The exponent n×t counts the periods.

€10,000 at 5% for 10 years, compounded monthly. r/n = 0.05/12. The exponent is 120. The factor is about 1.647. The balance is about €16,470. Interest earned is about €6,470.

The same money, simple interest at 5%. Interest each year is 10,000 × 0.05 = €500, and it does not join the principal. Over 10 years that is €5,000. Balance = €15,000. Compounding paid about €1,470 more in this example, roughly a 9% larger pile of interest, not a different universe. The story gets louder if you stretch the years. At 30 years the compound balance at 5% monthly is about €44,600, while simple interest would leave €25,000. Same rate label, very different patience.

Simple versus compound stays with that contrast, including the cases (short treasury-style quotes, some loan fees) where simple is the contract you actually signed.

Frequency is a smaller lever than the rate

Five percent, €10,000, 10 years:

The jump from annual to monthly is about €180 on €10,000 over a decade. The jump from monthly to daily is pocket change by comparison. Chasing a daily-compounding advertisement when the nominal rate is lower is a bad trade: 4.8% compounded daily loses to 5% compounded annually over long horizons. Compare effective rates, or just compare the balances the formula produces at your real horizon. Effective annual rate is the single-number version of this table.

A nominal rate is the one in the contract before frequency is applied. An effective rate is what a year of compounding actually does to a balance with no contributions. They are equal only when n = 1.

Contributions, the part that dominates

The no-contribution formula is a clean classroom object. Households rarely match it. Suppose the same €10,000, 5% monthly, 10 years, plus €100 contributed at the end of each month.

There are 120 contributions. They do not all grow for 10 years. The first one compounds for 119 months, the last one for none. The future value of that series is:

PMT × (((1 + r/n)^(n×t) − 1) / (r/n))

when the payment lands at the end of the period and the rate per period is not zero. Plugging in gives a contribution pile of about €15,530 of future value, of which €12,000 was money you added and about €3,530 is growth on the contributions. Add the principal’s €16,470 and the account is near €32,000.

More than half the ending balance, in this sketch, came from the contributions and their growth, not from the opening cheque. That is normal when the monthly amount is serious relative to the starting principal. It is also why “I don’t have a lump sum, so compound interest isn’t for me” misses the formula’s second term. The calculator’s contribution field is not a decoration.

Timing assumptions hide here. End-of-period versus beginning-of-period contributions shift the result by roughly one period of growth. A calculator should say which one it uses. If it does not, treat the output as a plan, not as a bank statement. PercentBox’s tool follows the standard end-of-period series shown above for the contribution term, on top of a principal that compounds for the full span.

A rate that is not a savings rate

The same multiplication describes money you owe, if interest is compounding and you are not paying it down. A balance of €2,000 at 20% a year, compounded monthly, left alone for 3 years, becomes 2000 × (1 + 0.20/12)^36 ≈ €3,620. Nobody should leave a 20% balance untouched; the point is that “20%” without a payment is a compound path, not a one-time fee of €400.

Loans that you do pay monthly are a different formula: the payment is set so the balance hits zero, and interest is charged on the remaining balance each month. That is amortization, explained in how loan EMI works, not a savings projection with a minus sign. Using the compound-interest tool to “see a loan” by typing a negative contribution will not reproduce an amortization schedule unless the contribution happens to equal the exact payment and the compounding convention matches. Use the loan tool for loans.

What the smooth curve assumes

• The nominal rate does not change. Real savings rates move. A projection at today’s rate is a scenario.
• Every interest credit actually stays in the account. If you skim the interest, you have converted the plan back toward simple interest on the principal, plus whatever you did with the skimmed money.
• Contributions actually happen. A skipped year is not a rounding error; remove those payments from the series.
• Taxes and fees are absent unless you lower the rate or the contribution to mimic them. A 5% return taxed so that you keep 4% should be modeled at 4% if you want the spendable balance, or modeled gross if the account is tax-sheltered and you will think about tax later. Do not mix those stories in one unlabeled number.
• Inflation is absent. €16,470 in ten years buys less than €16,470 today. Inflation and percent change is how to name that second percent. A compound projection that ignores it is still useful as a nominal balance. It is a poor answer to “will I feel richer?”

A worked comparison you can redo

You can save €200 a month for 15 years. You assume 4% nominal, monthly compounding, starting from zero. The series formula gives a future value around €49,200. You contributed 200 × 180 = €36,000. Growth is the rest, about €13,200.

At 7% the same deposits land near €63,000. The extra 3 percentage points are not “3% more money.” They are a different exponent on every contribution. The gap, roughly €14,000 in this sketch, is why rate assumptions deserve a range: run 3%, 5%, and 7%, and treat the middle as a plan only if you can say why that rate is plausible after fees.

None of those runs is a forecast of markets. A bank account can honestly offer a rate. A stock portfolio cannot honestly offer 7% as a contract. If you type 7% you are choosing a scenario, and the calculator is doing arithmetic on it. Label it that way when you show someone the picture.

When to use the compound interest calculator

Use it to compare frequencies when a bank advertises “daily” as if it were a different product from “monthly” at the same nominal rate. Use it to see how much of an ending balance is contributions versus growth, which keeps the glory attached to the habit rather than to a magic rate. Use it to test “what if I start five years later?” by shortening t, which is usually a harsher result than shaving a percent off the rate.

Do not use it for a fixed monthly loan payment — that is the loan calculator. Do not use it as an investment performance report. And do not read a single run as the future. Two rates and a note about tax will teach you more than one precise-looking cent.