Compound Interest Calculator

Compound interest is interest paid on the principal and on the interest already earned. That is the whole definition the Securities and Exchange Commission gives in its glossary, and it fits on one line. Everything difficult about it comes later, when a real plan meets the small choices that a one-line definition leaves out.

The SEC’s own classroom example shows the mechanism in its plainest form. Put 100 in an account that pays 5 percent a year. After one year there is 105. After the second year there is 110.25, because the second year’s 5 percent is earned on 105, not on 100. The extra 25 cents is interest on interest. Left alone for 10 years the same 100 passes 162, and after 25 years it is close to 340.

A very old formula

People have been working these sums for a long time. A Babylonian clay tablet from roughly 2000 to 1700 BC may be the first record of a compound interest problem. The Florentine merchant Francesco Balducci Pegolotti printed a table of compound interest in his merchants’ handbook around 1340, covering rates from 1 to 8 percent for up to 20 years.

In 1494 Luca Pacioli gave the rule of 72: divide 72 by the rate in percent to estimate how many years money takes to double. In 1613 Richard Witt published a whole book on the subject, with 124 worked examples. And in 1683 Jacob Bernoulli found the constant e while studying a question about compound interest.

The modern statement is short. A principal P at a nominal annual rate r, compounded n times a year for t years, grows to P (1 + r/n) raised to the power n t. As n grows without limit the growth becomes P e to the power r t, which is continuous compounding. Regular deposits add a second term. Deposits of D at the end of each of N periods, each earning a periodic rate i, are worth D ((1 + i) to the N, minus 1) divided by i at the end. That is the future value of what actuaries call an annuity-immediate, or ordinary annuity.

Here is a case to try in the calculator above. Start with 10,000 at 5 percent compounded monthly, add 200 at the end of every month, and run it for 10 years. The starting sum grows to 16,470.09. The 120 deposits, 24,000 in all, grow to 31,056.46. The formula gives 47,526.55, of which 34,000 is money put in and the rest is interest.

The ledger on this page, which rounds each month’s interest to the cent, ends at 47,526.49. Six cents of rounding over 120 months is the price of counting in real coins.

Timing and frequency

The first detail people skip is when the deposit lands. The same formula with each deposit made at the start of its period, an annuity-due, is the ordinary result multiplied by one more period of growth, 1 + i. Rent and insurance premiums are usually paid in advance, so they follow this pattern; loan repayments are usually paid in arrears, so they follow the other.

In the example, moving each 200 from the last day of the month to the first adds 129.48 over ten years. That is not a rounding error. It is exactly one month of interest on everything deposited, compounded to the end, and it follows from nothing more than the order of two lines in a ledger.

The second detail is that deposits and compounding do not have to share a calendar. An account can compound daily while its owner saves monthly, and Canadian mortgages are usually compounded twice a year while payments are monthly. Applying the annual rate divided by twelve to a monthly deposit is then simply wrong, because that is not the rate the account pays per month.

The honest conversion keeps the yearly growth the same. A rate compounded n times a year grows money by (1 + r/n) to the power n in a year, so the growth over one deposit period of a twelfth of a year is (1 + r/n) to the power n/12, and the effective monthly rate is that factor minus one. This calculator works every period’s interest at exactly that rate and shows the factor with your numbers, so the step that usually happens out of sight is printed on the page.

Frequency itself matters less than most people expect. The same plan compounded once a year ends at 47,161.53; compounded continuously, at 47,561.02. The gap between yearly and monthly compounding is about 365. The gap between monthly and the theoretical limit is about 35. Most of the benefit of frequent compounding is used up long before daily.

The rate on the label and the rate you get

Because compounding frequency changes the outcome, a nominal rate alone does not say what an account pays. The effective annual rate does. It is the growth of one unit over a year, minus the unit: (1 + r/n) to the power n, minus one, or e to the power r, minus one, when compounding is continuous. Markets give it many names, among them annual equivalent rate and effective annual rate. In US deposit disclosures it is the annual percentage yield.

At 5 percent compounded monthly the effective rate is 5.1162 percent. A nominal 6 percent compounded monthly gives about 6.17 percent, because each month applies 0.5 percent and twelve such months multiply to roughly 1.0617.

The federal rule for deposit accounts, Regulation DD, defines the yield from money rather than from a rate. Its appendix gives the formula as 100 times ((1 + interest divided by principal) raised to 365 divided by days in term, minus 1). Its first example is an account that pays 61.68 on 1,000 over a 365-day year, which works out to 6.17 percent. Another of its examples pays 5.25 percent compounded daily on 1,000; the regulation says that earns 53.90 in a year, an annual percentage yield of 5.39 percent. The model behind this page reproduces both figures to the cent.

That same appendix shows a subtlety worth knowing. For a two-year certificate paying 6 percent compounded daily in the first year and 6.5 percent in the second, it states total interest of 133.13. Multiplying the two years’ exact growth and rounding once gives 133.14; crediting the first year’s interest as 61.83 and then compounding gives 133.13. A ledger that rounds to the cent along the way and a formula that rounds once at the end can differ by a cent or so, which is why this calculator shows both and states the most that rounding can add up to.

Real money and real limits

A balance in ten years is counted in the money of ten years from now. The Federal Reserve aims for inflation of 2 percent a year over the longer run, measured by the annual change in the price index for personal consumption expenditures. That is an aim, not a promise about any particular decade. To express a future amount in today’s money, divide it by the growth of the price level over the same years.

With 2.5 percent inflation for 10 years that divisor is 1.2801, so the example’s 47,526.49 is worth about 37,127.62 in starting-year money. The Fisher relation says the same thing in rates: one plus the nominal rate equals one plus the real rate times one plus inflation, and the difference between the nominal and inflation rates is a close approximation when all three are small.

Time does more than rate at the start. In the example, one more percentage point of interest adds about 3,443 over ten years, while five fewer years of the same saving removes about 21,092. Early years carry both more deposits and more rounds of compounding.

The rule of 72 shows why. At 9 percent money doubles in about 8 years, the SEC notes, so a plan with 40 years to run sees five doublings while one with 24 years sees three. The rule is an estimate: for continuous compounding a constant near 69.3 is accurate, because the natural logarithm of 2 is about 0.693.

None of this is a forecast. The formula assumes one rate for the whole term, deposits that always arrive on time, no fees and no tax. Real rates move, deposits slip and costs apply, which is why the SEC’s own calculator offers a range of rates above and below the one entered. The output is arithmetic about a stated plan.

That is the case for showing the working. A result with its formula filled in, every period’s interest listed, and the effect of each choice set out beside it can be checked line by line. A single large number on a screen cannot, and it is precisely the details it hides, the timing, the frequency, the rate definition and the price level, that decide what the money is really worth.