# Mike's Toolbox

## Interest

The fuel that keeps an economy running is interest. Banks entice you to deposit your money with them by offering to pay you a small fee based on the amount of money you give to them. They then turn around and lend your money to someone else at a higher interest rate. This is a winning situation for everyone involved: the person borrowing from the bank is able to purchase a home that they could not afford if they had to pay the full price all at once; your money grows in value, which it wouldn't do if you simply stuffed it in your mattress; and the bank makes a profit on the difference between the amount of interest it charges the borrower and the amount of interest it pays you.

When I opened my first bank account, you could expect to earn 6 percent on your savings. An interest rate of 6 percent means that the bank will pay you \$6 for every \$100 you deposit into your account. The total amount of interest earned every year therefore depends on how much money you have in the account; the more money in the account, the more interest you get.

To calculate how much money you can expect to earn in your account every year, you multiply your account balance times the interest rate. Your account balance is the amount of money you have in your savings account; this is also sometimes called the principal. For example, if you have \$500 in your savings account and the interest rate is 6 percent, the bank will pay you \$30 at the end of the year:

Interest = \$500 × 0.06 = \$30

Another way to think of it is your account balance at the end of the year will be 106 percent of the starting value. This is because you will still have 100 percent of your money at the end of the year, but the bank will add to that an additional 6 percent:

New Balance = \$500 × 1.06 = \$530

A calculator with a % button can compute the amount of interest and new balance for you easily. Press the buttons from left to right to calculate your new balance. Notice that after you press the % button, the amount of interest is displayed, and the final balance is shown after pressing the = button:

## Compounding

If you keep your money in the bank for a second year, it continues to grow at a rate of 6 percent. But your account balance is larger than your initial deposit of \$500, since you earned \$30 in interest. That \$30 earns 6 percent interest in the second year just as your original \$500 continues to do. This effect is called compounding, where previously earned interest earns more interest in subsequent years. Each year your account grows faster than the previous year, so to maximize your earning potential over your lifetime, you should put as much money to work as you can as early and often as possible.

To determine your account balance after earning interest for two years, you need to determine the starting balance for the second year and scale it by 106 percent as we did for the first year:

 1st Year Starting Balance = \$500 1st Year Ending Balance = \$500 × 1.06 2nd Year Starting Balance = \$500 × 1.06 2nd Year Ending Balance = \$500 × 1.06 × 1.06

The starting balance in the second year is just the ending balance of the first year. Can you guess what your balance will be after 3 years? It's easy – just multiply again by 1.06:

 3rd Year Starting Balance = \$500 × 1.06 × 1.06 3rd Year Ending Balance = \$500 × 1.06 × 1.06 × 1.06

A pattern seems to be emerging; to calculate your account balance after a number of years, simply multiply the initial balance by 1.06 once for each year that has passed. Since we've been working with an interest rate of 6 percent, the scaling factor is 1.06, but in general it will be 1 plus the interest rate.

While it was helpful to us to find the pattern, writing 1.06 × 1.06 × 1.06 is tedious and takes up a lot of space. This sort of pattern shows up in mathematics often enough that a notation has been created for it: 1.063. The number 3 is called the exponent and simply means multiply 1.06 together three times. Three is also called a power; you would say that 1.063 is "1.06 to the third power."

Using the notation for exponents, we can write an equation that calculates your account balance after it has earned interest for a number of years:

B = P × (1 + r)t

In the equation, B is the final balance we're trying to determine. P represents your initial deposit (the principal), which in our running example is \$500. The interest rate is r which we've set at 6 percent or 0.06. And the exponent t is the time the principal has been earning interest in years, perhaps 3.

Plugging in all the numbers gives us

B = \$500 × (1 + 0.06)3

Performing this computation is difficult without the aid of a calculator, so let's see how to use one. The important button you will need is the one for computing exponents and is usually labeled yx. First you enter the number that is to be multiplied together, such as 1.06. Then you press the yx button followed by the exponent, such as 3. Finally press the = button to calculate the answer. Try it out:

The value displayed is the amount of money a single dollar will become after earning 6 percent interest for 3 years. It is a little more than \$1.19. To determine the final balance, multiply by the initial deposit:

## Monthly Compounding

In the real world, people don't deposit money into their account or withdraw money out of their account at yearly intervals. The bank needs to be able to handle the cases where you make a deposit into your account at some point during the year, or take some out to make a purchase. The first thing that the bank does is shorten the time period between interest payments from one year to one month. Then to handle transactions during the month, the bank tracks your average balance for the month and pays interest on the average.

Let's continue using the previous example where you deposit \$500 into a savings account earning 6 percent interest. The rate of 6 percent is the annual percentage rate, often abbreviated APR, but the bank is now paying you interest every month. Since there are 12 months in a year, the bank simply pays you 1/12th of the APR every month. In our example, you would earn 1/2 percent interest per month. We still use the same formula for compound interest as before, but the rate is divided by 12, and the exponent is now specified as the number of months:

 B = P × (1 + r )t 12

After one month, your new balance would be:

 B = \$500 × (1 + 0.06 )1 12 = \$500 × 1.005 = \$502.50

The \$2.50 earned in the first month is simply 1/12th of the \$30 we received when the interest was compounding yearly. But just as we saw with yearly compounding, the amount of interest we earn in the second and subsequent months increases because our balance is increasing. This effect means that because of monthly compounding, we will actually earn more than 6 percent interest during the year. How much more? Just solve the equation with an exponent of 12 months:

 B = P × (1 + 0.06 )12 12 = P × 1.00512 = P × 1.061678

Our effective interest rate is therefore 6.1678 percent, which is better than the advertised APR of 6 percent. This higher percentage is called the annual percentage yield, abbreviated APY. Depending on the situation, a bank may advertise the APY instead of the APR to make their interest rate appear more competitive to those who don't understand the difference.

The final piece of the puzzle is determining the average monthly balance of your account. It is relatively straightforward; if you deposit your \$500 in the middle of the month, your average balance is half of your deposit, which is \$250. So in the first month, you will receive 1/2 percent interest on \$250. The following month, your \$500 has been in the account the entire time, so the amount of interest paid is 1/2 percent of the \$500 plus the small amount of interest you earned in the first month.

Now that we have an equation that calculates compound interest, we might want to use it to answer interesting questions such as, "how long will it take to double my money?" The equation can help us answer that question, but first we need to talk a bit about salary.