How interest is calculated

A balance is a number, and a rate is an instruction

A balance is a number attached to a moment in time. It is how much money is sitting in one place right now. It changes when money goes in, when money comes out, and when interest is added to it.

An interest rate on its own is not money. It is an instruction for a calculation, and it has two parts: a percentage, and the period that percentage applies over. Ten per cent per year means the balance multiplied by 0.10 is the interest for one year. On a balance of $1,000, that calculation gives $100.

Two facts sit outside the rate itself, and they do most of the work in everything below. The first is which number the percentage gets applied to. The second is how often the calculation is run. Documents differ in whether they make either one plain.

One more thing, about direction. Interest can be added to a balance someone holds, or to a balance someone owes. The arithmetic that follows runs the same way in both cases. What differs is the direction the money travels.

Simple interest keeps going back to the original amount

Simple interest means every calculation is applied to the amount at the start, and never to the interest that has been added since.

With $1,000 and a rate of 10% per year, the first year's interest is $100. The second year's interest is also $100, because the calculation is still being applied to the original $1,000. So is the third year's, and the tenth's. After ten years, $1,000 of interest has been added, and the balance is $2,000. After twenty years, $2,000 of interest has been added, and the balance is $3,000.

Written out, simple interest is one multiplication: starting amount, times rate, times number of periods. The amount added each period never changes, so the balance climbs in a straight line.

Compound interest changes the number the calculation is applied to

Once interest has been added to a balance, the balance is a different number than it was. Compounding is what happens when the next calculation is applied to that new number rather than the old one.

Same $1,000, same 10% per year. The first year's interest is $100, and it is added, so the balance is $1,100. The second year's calculation is applied to $1,100, not to $1,000, so the interest is $110 and the balance becomes $1,210. The third year's calculation is applied to $1,210, which gives $121 and a balance of $1,331.

The extra $10 in the second year, and the extra $21 in the third, are interest calculated on interest that was added earlier. That is the whole of what the word compounding refers to. Nothing else changed: same starting amount, same rate.

There is a shorthand for it. Adding 10% to a balance is the same as multiplying that balance by 1.10. Doing that for ten years is multiplying by 1.10 ten times over, which is written as 1.10 to the power of ten. The general form is the starting amount, multiplied by (1 plus the rate), raised to the number of periods.

A worked example, which is arithmetic and not a projection

Everything in this section is invented arithmetic. The $1,000 is a round number, chosen to keep each step readable on the page, and 10% was chosen because it keeps the multiplication visible at every step. Neither figure describes any account, product or rate anywhere, and none of it is a statement about what will happen in the future. It is the same kind of calculation as working out the area of a rectangle.

At 10% per year on $1,000, over ten years: simple interest gives a balance of $2,000, and annual compounding gives $2,593.74. The distance between those two figures is $593.74.

The same arithmetic over twenty years: simple interest gives $3,000, and annual compounding gives $6,727.50. The distance is now $3,727.50.

Those cents are exact for the calculation described, and only for that calculation. A real balance also moves when money is put in or taken out, when fees are charged, when tax is applied to interest, and when the rate itself changes. Documents differ in whether the rate they set is held for a term or able to move. The arithmetic above holds one rate still for twenty years, which is a convenience for showing the mechanism rather than a description of anything.

Where the amount sits, and where the periods sit

The shorthand again: starting amount, times (1 plus the rate), raised to the number of periods. The starting amount is a multiplier sitting out the front. The number of periods is an exponent. They are two different positions in the same expression.

That difference shows up in the arithmetic. Doubling the starting amount from $1,000 to $2,000, with ten years and 10% held fixed, doubles the answer as well: $5,187.48 instead of $2,593.74. The input is multiplied by two, and so is the output.

With the amount left at $1,000 and the number of years doubled instead, from ten to twenty, the figure moves from $2,593.74 to $6,727.50. That is a multiplication by about 2.59, the same factor the first ten years produced. In this arithmetic, the second ten-year block multiplies the balance by that 2.59 rather than adding the dollar amount the first block added.

That is a property of exponents: the number of periods sits in the exponent, and the starting amount sits out the front as a multiplier. It holds in this arithmetic whether the balance is one somebody holds or one somebody owes, and it holds regardless of who they are.

How often the calculation runs changes the answer

So far the calculation has been run once a year. It can be run more often than that. The lines below divide the yearly percentage by the number of periods in the year, which is one way of setting the rate for each period.

At 10% per year on $1,000, with the length of time held at one year: calculated once at the end of the year, the balance is $1,100. Calculated twice, at 5% each time, it is $1,102.50. Calculated four times, at 2.5% each time, it is $1,103.81. Calculated twelve times, at one twelfth of 10% each time, it is $1,104.71.

The reason those differ is the reason from earlier in a smaller frame. When interest is added part-way through the year, the calculations for the rest of the year are applied to the larger balance.

Two terms get used for this. The nominal rate is the headline percentage per year, which is 10% on every line above. The effective rate expresses what a full year of the calculation comes to, as a single percentage: the twelve-times-a-year line above works out to 10.47% across the year. Four items describe how interest is set out in a document: the rate, the period it applies over, how often interest is calculated, and how often it is added to the balance. Documents differ in how many of them they state, and the last two are not necessarily the same as each other. One arrangement calculates interest daily and adds it monthly.

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