Compound Interest Calculator

Compound interest is one formula and one idea. The idea is that interest, once paid, stops being a payout and becomes principal, so the next payment is calculated on a larger sum than the last one. Everything else on this page, the compound interest formula itself, the compounding frequency, the effective annual rate, the Rule of 72 and the gap against simple interest, is that single sentence written in different notation.

Where the formula comes from

Start with one year and no algebra. Put $1,000 somewhere that pays 10% a year. At the end of the year you have $1,000 plus $100, which is $1,100. Factor it and that is $1,000 multiplied by 1.10. Do it again and you have $1,000 multiplied by 1.10 twice. After five years you have $1,000 multiplied by 1.10 five times, which is $1,610.51 rather than the $1,500 that five separate $100 payments would have given you.

The growth factor 1.10 is just 1 plus the rate, so the general statement is A equals P multiplied by (1 + r) raised to the power t. That is the whole derivation. The only complication left is that most accounts do not pay once a year.

If interest is credited n times a year, each credit uses a slice of the annual rate, r divided by n, and there are n multiplied by t of those credits in the whole term. Substituting gives the formula every calculator quotes:

A equals P multiplied by (1 + r/n) raised to the power nt. Two mistakes account for most wrong answers. The first is leaving r as a percentage, so that 5 goes in where 0.05 belongs. The second is dividing the rate by n but forgetting to multiply the years by n, which quietly converts a ten year sum into a ten month one.

Compounding frequency, all the way to the ceiling

Because every extra credit starts earning immediately, a higher n always pays more. The question nobody answers with a number is how much more. Take $10,000 at a 5% nominal rate for ten years and change nothing but the schedule. Compounded annually it reaches $16,288.95. Compounded monthly, $16,470.09. Compounded daily, $16,486.65.

The entire distance from annual to daily is $197.70 on a gain of $6,288.95, which is a little over 3% of the interest earned. Moving from annual to monthly captures $181.14 of that, or 91% of everything frequency has to offer. The remaining rungs are rounding errors with marketing attached.

Push n higher and the pattern becomes obvious. Hourly compounding, at 8,760 periods a year, gives a growth factor of 1.051270946. Compounding every minute gives 1.051271094. The sequence is not running away, it is converging on a number, and that number is e raised to the power 0.05, which is 1.051271096. This is continuous compounding, written A equals P multiplied by e raised to the power rt, and it is the ceiling rather than an option. At this rate over this term it is worth $16,487.21, which is 56 cents more than daily compounding and $198.26 more than annual.

The constant e is not decoration here. Jacob Bernoulli found it by asking exactly this question, what happens to (1 + r/n) raised to the power n as n grows without bound, and the answer is e raised to the power r. So the whole frequency ladder has a mathematical roof, and every real account sits close to it. A bank that advertises continuous compounding is offering you less than a dollar more than daily on a ten thousand dollar balance.

The effective annual rate, which is the only number that compares two offers

A quoted rate is a nominal rate. It does not say what a year actually pays, because that depends on the schedule. The effective annual rate, EAR, collapses both into a single figure: EAR equals (1 + r/n) raised to the power n, minus 1.

At a 5% nominal rate the effective rate runs 5.0000% annually, 5.0625% semi-annually, 5.0945% quarterly, 5.1162% monthly, 5.1267% daily and 5.1271% continuously. Those spreads look trivial, and at 5% they are. At 18% compounded monthly the effective rate is 19.56%, and at a nominal 100% compounded daily it is 171.46%. Frequency matters in proportion to the rate, which is why it is a footnote on a savings account and a serious number on a credit card.

This is the figure on the advert. In the United States the annual percentage yield is defined by Regulation DD Appendix A as 100 multiplied by the quantity (1 + interest divided by principal) raised to the power 365 over the days in the term, minus 1. It is annualised on a 365 day year and it assumes nothing is withdrawn. That is why a certificate paying a 4.65% nominal rate compounded monthly is advertised as 4.75% APY. If you are comparing two accounts, compare the effective rates and ignore the schedule entirely, because the effective rate already contains it.

One trap survives even that. A rate quoted on an ACT/360 basis divides by 360 rather than 365, so a 5% quote held for a full 365 days accrues 5.0694% as simple interest and 5.1998% if it compounds daily, against 5.0000% and 5.1267% on ACT/365. The ratio between the two bases is 365 divided by 360, or 1.0139. Short term US dollar instruments are usually quoted on ACT/360 and sterling on ACT/365, which means the same printed number is not the same rate.

Compound against simple interest

Simple interest never lets interest become principal. Its formula is A equals P multiplied by (1 + rt), and the balance climbs in a straight line. Compound interest curves. On the same $10,000 at 5%, here is what the divergence costs:

Read the last column rather than the second. The two methods are identical after one period and differ by $25 after two, which is why compound interest feels like an exaggeration in the first few years. By year 30 the difference is larger than the original deposit, and by year 40 it is four times the entire simple interest return. Time, not rate and certainly not frequency, is the lever that moves this table.

The Rule of 72, and the three places it breaks

Doubling time has an exact answer: t equals the natural log of 2, divided by n multiplied by the natural log of (1 + r/n). Since ln 2 is 0.693147 and ln(1 + r) is roughly r for small r, that collapses to about 69.3 divided by the rate in per cent. The shortcut in circulation uses 72 instead, because 72 divides cleanly by 2, 3, 4, 6, 8, 9 and 12 and can be done in your head. It has been in print since Luca Pacioli set it down in 1494.

It is an approximation, and it is honest only in the middle of its range:

So the rule breaks in three ways. Below about 4% it runs slow, and the Rule of 70 is the better shortcut there because at 2% it is out by 0.01%. Above about 20% it runs fast and the error grows without limit, reaching 28% at a 100% return. And it silently assumes annual compounding: at 4% a balance doubles in 17.67 years compounded annually but 17.36 years compounded monthly, a shift of about four months that the rule cannot see. If you need the rule outside its comfortable band, the usual adjustment is to move the 72 by one for every three percentage points away from 8%, so 76 at 20%. If you need the real figure, use the logarithm.

Working backwards

The same formula answers the two questions people usually arrive with. To find the deposit that reaches a target, divide rather than multiply: P equals A divided by (1 + r/n) raised to the power nt. Reaching $2,000 in five years at 10% takes $1,241.84 today. To find the rate a known result implies, take the root: r equals n multiplied by the quantity (A over P) raised to the power 1 over nt, minus 1. A painting bought for $2,000 and sold for $3,000 six years later returned 6.9913% a year, which is a far less impressive sentence than "it went up by half". And to find the term, use the doubling formula with your own multiple in place of 2.