Well, I am back! I was out of the country for a while and catching up on work for the other part, but I am hopefully back to writing three times a week or more. I figured a good place to start anew would be an interest of mine: interest – compound interest to be precise.

Today we will explore the difference and connection between an annual percentage yield a bank advertises and the interest rate that adds to your account every month. These two terms apply to your checking accounts, savings accounts, certificates of deposit, almost all investment products that banks offer.

When you sign up for a savings account, you are allowing the bank to hold your money for a period of time in exchange for an extra percentage of money over that period of time. This percentage is your interest rate. The banks usually give you an interest payment every month (they call it a period – you will see why later.) They calculate this by taking the interest rate and dividing it by 12 (because there are 12 months – or periods – in a year) and multiplying it by the amount of money in the account or the principle. This gives you your interest payment for the month. This happens quite the same next month, but the amount of money in your account is higher – because of last month’s interest payment. In addition to the interest on your initial principle, you are now earning interest on the interest you earned last month. This concept is what is known as compounding interest. Your money is now working hard to earn you more money. Over the course of a year, you earn interest on your interest (in addition to the interest on your initial principle) over and over again as the months go by. The annual percentage yield is the percentage your principle has increased over the course of one year compared to the initial principle.We are going to figure out what the annual percentage yield would be for a 10% Interest Rate.We need to organize our variables first:

Interest Rate (I) = 10.00% per year

Period (P) = 1 month

Number of Periods (N)= 12 (12 months in a year)

Principle ($) = Does not matter right now…we will just use $ as a placeholder.

Annual Percentage Yield (APY) = ???

This math may look daunting if you scan though it, but if you take it step by step, it will make sense.

OK. Let’s start by figuring out the first month. It will look something like this:

We start with the initial principle: $ At the end of the month the bank will give you interest (I) on this money, but they won’t give you 10%, because it is 10% per year. Instead they will divide 10% by 12 (because there is 12 months or periods (P) in a year.) They will then take that monthly interest percentage and multiply it by your principle. So our equation will look like this:

$ + $*(10%/12) = New1$ (New Principle after 1 month/period) Or with only variables from above, it will look like this:

$ + $(I/P) = New1$ We can factor out the principle ($)

$ * (1 + (I/P)) = New1$

Next month/period it will look like this because our $ after this month now equals $*(1 + (I/P)):

New1$ * (1 + (I/P)) = New2$ (New Principle after 2 months). But since we know what New1$ equals, let’s substitute it in and simplify:

[$ * (1 + (I/P))] * (1 + (I/P)) = New2$ Simplify.

$ * (1 + (I/P))^2 = New2$. (^2 means squared or to the power of 2)

The next month will look almost the same:

New2$ * (1 + (I/P)) = New3$ (New Principle after 3 months). Since we know what New2$ equals, let’s substitute it in and simplify:

[$ * (1 + (I/P))^2] * (1 + (I/P)) = New3$ Simplify.

$ * (1 + (I/P))^3 = New3$. (^3 means cubed or to the power of 3)

We can see a pattern:The (1 + (I/P)) part is raised to the power of the number of periods that you wish to calculate: $ * (1 + (I/P))^N This is our formula! The best thing about this formula is that it is a simple formula for complex mathematics. The initial principle stays the same from beginning to end and only the percentage changes over N periods. (1 + (I/P))^N will give you the yield for any number of periods you desire. If we want to see the rate for 12 months or periods we need to raise it to the power of 12. We started out trying to find the annual percentage yield for 10%. Lets give it a shot:

Since 10% = .10,$ * (1 + (.10/12))^12 We need a calculator for this. .10/12 = .008333. So:

$ * (1 + .008333)^12 or:

$ * (1.008333)^12

Using a calculator we see that 1.008333^12 = 1.10471.

So we would have $ * 1.10471, or an increase of $ * .10471 over our original principle.

This means that we have now 110.471% of the money we had initially, thereby making our Annual Percentage Yield 10.471%!

In other words, an interest rate of 10% earns you 10.471% over the course of one year employing the compound interest.

Thanks for stopping by again after my short hiatus. Next time, we will test this formula to ensure it’s accuracy, and we will use this equation to calculate how a given principle will grow over a certain number of periods with a certain interest rate. If anything was unclear about today’s long post, then hopefully the next post will clear it up. CD

Possibly related posts: (automatically generated)