Simple vs. Compound Interest
When discussing interest and interest calculating there are two primary forms of interest we are discussing: simple and compound. Simple interest is, as one would guess, the simpler of the two types of interest. Interest accrues based entirely on the principal, the original amount of money loaned or invested. So, if you borrowed $100 at 10% interest, you would owe $10 in interest on top of the original amount loaned.
Compound interest involves the adding of interest accrued to the principal. If, for example, you invested $10,000 with a 6% return per year and at the end of each year added the interest back into the investment, it would be compounded. After the first year you would have $600 in interest, added to the $10,000 gives you $10,600, from which total you will earn 6% the next year.
In investments the more often compound interest is calculated the higher the advantage to you. If, for example, it is compounded monthly, your total will increase every month and thus increase your returns on the next month. This of course works the opposite way with loans, if interest is accrued every month and you do not pay it off, you will end up owing even more money.
Calculating Simple Interest
When calculating simple interest we have a basic formula that we use:
I = Prt
where I is the Interest, P is the Principal, r is the rate and t is the time for which the money has been generating interest. Most often the interest is calculated per year.
For example, say that you take out a $1,500 personal loan to be paid out over 1 year on a monthly basis with 15% interest. $1,500 would be the principal, the rate would be 0.15 and the time would be 1 year. To calculate just how much interest you are paying, simply multiply 1,500 by 0.15 and 1 (Prt). This gives you a total of $225 that you are paying in interest on the loan. Over the year, then, you will be paying $1,755 back to the lender for the original $1,500 loan. Separated out over 12 months, you will be paying $146.25 a month.
Dealing with interest you will sometimes want to know what the present value of an investment is. If we invest $5,000 in an investment accruing 4.5% interest and want to know how much we’ve made after 3 and a half years we use the present value formula for simple interest. This formula looks like this:
Looking at our example, the principal is $5,000. The rate is 0.045 and the time is 3.5. Plugging these numbers into our formula we get $5,000 times 1.1575 (1 plus 0.045 times 3.5). When we multiply it out we get a figure of $5,787.50, the current value of our investment. We have made a total of $787.50 on our investment.
Calculating Compound Interest
Interest is not always so simple, however. And we don’t always want it to be so simple. Looking at the last example from above, we earned a total of $787.50 in 3 and a half years. Imagine, though, that we had compounded the interest, even annually (where the interest is added at the end of every year).
At the end of the first year, we would have earned $225 in interest, giving us a total of $5,225. The next year we would then earn 4.5% interest on $5,225 for the next year, giving us a return of $235.13, added to our total giving us $5,460.13 at the end of the second year. The end of the 3rd year we would have earned another $245.71, added to our total giving us $5,705.84. In the next half year we would earn another $128.38, for a total of $5,834.22, $46.72 more than we earned with simple interest, giving us more bang for our buck.
In order to calculate compound interest, though, especially over long periods of times, it could be very tedious to go through so many individual calculations. Rather than go through all of that, we use the compound interest formula:
Here once again P is the principal and i is the interest rate. n represents the number of times the interest is compounded. When interest is compounded annually n is simply the number of years the money is invested. If interest is compounded semi-annually, quarterly or monthly we will have to go a little further to find n.
For example, if we invest $4,000 at 7% interest compounded quarterly for two years, we would calculate n like this. First we take the number of years (2), and multiply it by the number of times per year interest is compounded (4), to get 8.
We might think that our equation would then be S=4000(1+0.07)^8, but there is still one more step to take. When we are compounding the interest quarterly, we are not adding the entire 7%, but only a quarter of that. To find out the interest being earned each quarter we must first divide the interest by 4. 0.07 divided by 4 = 0.0175. Our equation is then S=4000(1+0.0175)^8. At the end of two years then we have $4595.53.
Using this formula along with the simple interest formula can greatly help us in determining the value of investments over the long run, estimate payments on loans and help us look for the best deals wherever we go. Understanding the basics of interest calculating can greatly help us in almost all matters of our financial lives.