Whether you know it or not, compound interest calculations affect us every day of our lives. Mortgage payments, credit card debt, savings accounts, retirement funds, etc. They are all affected by compound interest.
Let's have a look at how some of these calculations are made.
The first thing we need to discuss is how to take interest rates that we see quoted everyday on things like mortgages, savings accounts and credit cards, and find out what the actual interest rates are.
For example, a mortgage in the United States is compounded monthly. But the advertised rate is what is known as a nominal interest rate. In other words, the interest rate that is charged before the effect of compounding. When we take into account the effects of compounding on a posted mortgage rate, only then do we arrive at what is known as the EAR or Effective Annual Rate.
Here is a general calculation for converting nominal rates into effective rates.
EAR = (1 + I / N) ^ N - 1
I = nominal interest rate
N = frequency of compounding
Therefore, going back to our discussion of mortgages, a mortgage with a 6% quoted rate, has an EAR of 6.168%. Here is the compound interest calculation for this example.
EAR = (1 +.06 / 12) ^ 12 - 1
EAR = .06167781 or 6.168%
Now that you know how to do this calculation, the rest of the compound interest calculations flow from it.
For our next example, let's learn how to calculate the future sum of an investment. What we need is a formula that allows us to account for different frequencies of compounding (for example daily, weekly, monthly, semi-annually, annually etc.) as well as investments of different lengths. Here is the formula.
S = P * (1 + I / N) ^ (N * T)
P = the initial investment
I = the nominal annual interest rate
N = number of times the interest rate is compounded each year
T = the length of the investment in years
S = the sum of the original principal and the interest earned over time "T"
Therefore if we invested $10,000 at 3% over 4.5 years, compounded quarterly, we would end up with...
S = $10,000 * (1 + .03 / 4) ^ (4 * 4.5)
S = $11,439.60
And it doesn't matter if we change any of the variables, this formula will still work.
The last of the compound interest calculations we'll look at is determining a required nominal interest rate given that we have a sum of money, an investment goal for that money, a time frame, and the compounding frequency. Here is an example of what I mean.
Let's say you have $10,000 to invest over the next 10 years, and you want to end up with $16,000 in principal and interest at the end of those ten years. What nominal rate, compounded monthly, would your investment have to earn in order to achieve this return? The formula is arrived at by rearranging the above formula to solve for I, and it looks like this...
I = [(S / P) ^ (1 / (N * T)) - 1] * N
I = [($16,000 / $10,000) ^ (.00833333) - 1] * 12
I = .04709253 or 4.71%
You would need to invest at a nominal rate of 4.71% compounded monthly in order to achieve your goal.
As you can see from the examples above, compound interest can work for you or against you, depending on whether you are a borrower or a lender.
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