The rule of 6: a way to estimate interest earned when making monthly deposits

Last week I explained how to calculate interest using the APY from a bank or savings account. The short summary was this: the APY tells you the percent interest you can earn if you make a lump sum deposit for one year. The basic method for calculating interest is:

Interest earned = lump sum deposit * APY

For example, if you deposit $1,000 in a savings account with a 1% APY, then you can expect to earn $10 (1 percent of $1,000) in interest in one year’s time.

This is useful knowledge if you deposit a large amount of money and then don’t touch it. But often people do not have a large chunk of money to deposit at once.

Many people save by depositing smaller amounts on a regular basis, say by making fixed monthly deposits out of their paychecks. So a more practical question is this: if you make fixed monthly deposits, how much interest can you expect to earn?

For example, if you deposit $100 per month at a 5% APY, how much interest would you earn in one year? (assume the interest is credited monthly)

This is not a simple calculation, but it is not that hard either with a spreadsheet.

Below I will explain a decent approximation that I call “the rule of 6″ to come up with a quick answer. I will also explain the way you can calculate the interest exactly.

The rule of 6

I don’t think the spreadsheet method described below is very difficult, but there is a way to get a quick answer using mental math.

The procedure is this: multiply the monthly amount you deposit by 6 and then multiply that by the APY.

In other words:

Interest earned estimate = 6 * monthly deposit * APY

Here is how it works. Suppose you are depositing $100 a month. The rule of thumb is this: multiply the deposit amount by 6 to get $600, and then multiply by the APY of 5%. The approximation says you will earn $30 in interest throughout the year.

This is actually a pretty decent approximation: as calculated in detail below, the actual interest earned is equal to $32.26.

Another example: if you invest $100 at 10 percent, the estimated interest earned in one year is $60 (which is 6 x 100 x 10%) This is very close to the exact answer of $64.06.

Why does this estimate work, and where does the 6 come from? The idea is this. If you deposit $m per month, then you end up depositing $12m by the year end. So your average balance is roughly $6m, and so multiply the average balance by the APY, the term 6 x m x APY, is an estimate of the interest earned in one year. The factor of 6 is merely a result of the fact we have 12 months in a year.

Obviously the approximation is off because it does not account for the intricacies of compounding. But while it’s not perfect, but the rule of 6 is useful because it is easy to calculate.

You’ll appreciate the rule of 6 more once you read the part below on the steps needed to calculate the interest earned exactly.

Calculating the exact interest earned: 3 steps

The calculation below is about using a spreadsheet to determine the exact interest earned.

The way I think about it, there are three general steps to figuring out the interest earned. The steps are: (1) write out the monthly deposits, (2) calculate the monthly interest payments, and (3) keep a running total for the entire year.

Step 1 is to write out the monthly deposits. This is simple: you deposit $100 at the beginning of each month.

Step 2 is to calculate monthly interest payments. Notice that if we have a 5% APY, then that means we have a monthly interest rate of 0.41% (this is essentially 5% divided into the 12 months, but the exact formula involves exponents and is 0.41% = (1+5%)^(1/12)-1. The details for why this is are written in this article).

So every month we will get interest equal to 0.41% of our account balance.

Step 3 is to keep a running total for the entire year. In the first month we have $100 in our account so we get 41 cents. In the second month, we start add $100, so we have $200.41 in our balance to start the month. The interest payment on that amount is 82 cents and that gets added on. We keep doing this to figure out what we will end up with after 12 months.

Below is an image of a spreadsheet that puts all the steps together. You’ll see that we end up with interest payments of $32.26.

Remember the approximation was $30, and that only took seconds to calculate. You can decide on a case by case basis whether calculating the exact answer is worth the hassle.