Compound Interest Formula: How Money Grows Over Time

A common form is A = P(1 + r/n)^(nt). Here A is the amount after compounding, P is the starting principal, r is the annual interest rate written as a decimal, n is the number of compounding periods per year, and t is the number of years. The exponent shows that growth is applied repeatedly.

Suppose you deposit 1,000 at 5 percent annual interest compounded once per year. After one year, the balance is 1,050. In the second year, the 5 percent is applied to 1,050 rather than only to the original 1,000, giving 1,102.50. The extra 2.50 compared with simple interest comes from earning interest on earlier interest.

This repeated growth is the heart of the formula. Compounding does not add the same amount every period. It multiplies the current balance by a growth factor, so the absolute increase tends to get larger as the balance grows.

If interest is compounded more than once per year, the annual rate is divided among more periods. Monthly compounding uses 12 periods per year, quarterly compounding uses 4, and daily compounding uses many more. More frequent compounding can produce a slightly larger final amount when the stated annual rate is otherwise the same.

For example, with monthly compounding, the periodic rate is the annual decimal rate divided by 12, and the number of periods is 12 times the number of years. Keeping those two changes together prevents a common mistake. Do not divide the rate by 12 while forgetting to multiply the number of periods by 12.

When using the formula, also check what kind of interest rate you have been given. Financial products may quote rates using different conventions, and fees or taxes can change real outcomes. In classroom problems, the wording usually tells you exactly which annual rate and compounding frequency to use.

Compounding becomes more noticeable as time passes:

• •Principal is the starting amount.
• •The interest rate must usually be converted from a percent to a decimal.
• •Compounding frequency tells you how often growth is applied.
• •Time determines how many total growth periods occur.
• •Longer periods give interest more chances to earn interest.

The same mathematics can work against a borrower when unpaid interest is added to debt. A balance that compounds at a high rate can grow quickly if payments do not cover the interest and reduce principal. That is why comparing borrowing costs requires more than looking at one small monthly percentage.

The formula is also a useful lesson about exponential growth. Equal percentage changes create unequal numerical increases because each new change begins from a different base. This pattern appears in many areas of economics, population modelling, and science.

A handy shortcut called the rule of 72 estimates how long compound growth takes to double money. Divide 72 by the annual percentage rate, and the result is roughly the number of years needed. At 6 percent, doubling takes about 12 years; at 9 percent, about 8. You can check this with the formula: 1.06 raised to the power 12 is just over 2. The rule works best for rates between a few percent and around ten percent and becomes less accurate far outside that range, but it is excellent for quick mental comparisons.

Compounding frequency is also why savings accounts publish an effective annual rate, called AER in the UK and APY in the United States. It shows how much the balance actually grows in a year once compounding is included. A nominal 12 percent compounded monthly means 1 percent per month, and 1.01 raised to the power 12 is about 1.127, so the effective rate is about 12.7 percent. If compounding became continuous, the growth factor would approach e raised to the rate times the time, where e is about 2.718, the natural limit of ever more frequent compounding.

The formula can also be turned around to answer a planning question. Suppose you want 5,000 in five years, and an account pays 4 percent compounded once a year. How much must you deposit now? Rearranging A = P(1 + r)^t gives P = A divided by (1 + r)^t. Here 1.04 raised to the power 5 is about 1.2167, and 5,000 divided by 1.2167 is roughly 4,110. Depositing that amount today should grow to the target, ignoring fees and taxes.

That starting figure is called the present value of the future 5,000, and the process of working back from a future amount is called discounting. It is one of the most widely used ideas in finance. Businesses use it to compare investments that pay out at different times, and it explains why money promised in the future is worth less than the same amount in your hand today, since money held now could be earning interest. The higher the interest rate or the longer the wait, the smaller the present value becomes.

Real saving rarely means one deposit left alone. More often you add a fixed amount at regular intervals, and each payment then compounds for a different length of time. If you pay in an amount PMT at the end of each year for n years at rate r, the total is PMT multiplied by ((1 + r)^n - 1) divided by r. Saving 1,000 a year for 10 years at 5 percent gives 1,000 times (1.6289 - 1) divided by 0.05, which is about 12,578. You paid in 10,000, so roughly 2,578 is interest, most of it earned by the earliest deposits.

Timing then becomes dramatic. Imagine one saver pays in 1,000 a year for 10 years and then stops, leaving the 12,578 to grow at 5 percent for another 10 years. It becomes about 12,578 times 1.6289, or roughly 20,490. A second saver waits 10 years and then pays in 1,000 a year for the following 10 years, finishing with 12,578. Both deposited exactly 10,000, yet the early saver ends up with nearly 8,000 more. The difference comes entirely from the extra decade during which the first saver's balance kept earning interest on interest.

Benjamin Franklin set up a real test of long-term compounding. When he died in 1790 he left 1,000 pounds each to Boston and Philadelphia, to be lent at 5 percent interest to young tradesmen starting out. His will set out the expected arithmetic: after 100 years each fund should have grown to about 131,000 pounds, part of which would be spent on public works, with the rest compounding for another century. Reality fell well short of the formula, largely because many borrowers failed to repay and the funds were not always well managed. Even so, by the time they were wound up around 1990 they were worth several million dollars.

The constant e was itself discovered through a compound interest puzzle. In 1683 the Swiss mathematician Jacob Bernoulli asked what happens to 1 unit at 100 percent annual interest as compounding becomes more frequent. Compounded once a year it becomes 2. Twice a year, at 50 percent each time, it becomes 1.5 squared, or 2.25. Monthly gives (1 + 1/12) to the power 12, about 2.613, and daily gives about 2.7146. The results keep rising but more and more slowly, closing in on a limit of about 2.71828. Bernoulli showed that the limit lies between 2 and 3, and the number was later named e.

The compound interest formula describes repeated percentage growth on a changing balance. Identify the principal, rate, compounding frequency, and time, then keep the units consistent. The biggest idea is not the symbols. It is that each period builds on the result of the one before, so time can make a large difference.