## Interest Rate Compounding & Effective Rates

In this video, we will go over what interest rate compounding exactly means, how to understand it and we will learn what effective interest rates are. Let’s start. So what does compounding semi-annually, monthly, and quarterly and so on mean? Let’s look at a real life example. Let’s say you are looking for a \$10,000 loan for one year and you see 2 rates being offered by the bank, one is j2 = 12%, and the other one is j4 = 12%. Now which one is better? Are they the same? Now what you really care about is how much money will you owe at the end of the year to the bank? Let’s see how this works. Let’s look at the interest rate j2 first. Here is our year. You borrow \$10,000 in the beginning. The rate is compounded semi-annually so the periodic rate is half of j2, half of 12% and is equal 6%. So after half a year there is 6% added to your debt. And then at the end of the year there is another 6% added but this time it will be 6% of a larger amount. In the first half a year, the bank adds 6% of 10,000, or 600 to the loan. Now you owe 10,600. At the end of the year, the bank adds 6% AGAIN, BUT THIS time it is 6% of 10,600, which is \$636. You can see how it’s more than what was added at first. This is exactly what’s called compounding. Your debt increases as the interest builds and your interest becomes higher each compounding period. So at the end of it all you owe 11,236 to the bank. You can see that it is a bit more than simply adding 12% to 10,000 because of the compounding. Now what if our rate was j4 = 12%. This means that every compounding period, or every quarter, we would add 3% to the loan, because 12 divided by 4 is 3. 3% is our periodic quarterly rate. Again, each time your interest would be calculated as 3% of a higher and higher amount, so your interest would go up every time. After the first quarter you’d owe 10,300, after the second quarter it would be 10,609, and eventually at the end of the year you’d owe \$11,255.09 You can do this on your calculator if you like. The easiest way to add 3% to a number is to multiply it by 1.03. Now let’s compare the 2 situations we just considered. When the rate was j2 = 12%, we ended up with \$11,236. When the rate was j4 = 12% (same percentage, but different compounding), we ended up with \$11,255.09 Now you see that the more frequent the compounding, the higher the rate is. You can guess that if we had j12 = 12% and we did the same exercise we would end up with an even higher amount at the end of the year. This is where effective rate comes in. We don’t really care about how the compounding happens, all we want to know is how much we owe at the end of the year. This is what effective rate tells us: by what % does your loan increase over one year. Now what if our rate was j4 = 12%. This means that every compounding period, or every quarter, we would add 3% to the loan, because 12 divided by 4 is 3. 3% is our periodic quarterly rate. Again, each time your interest would be calculated as 3% of a higher and higher amount, so your interest would go up every time. After the first quarter you’d owe 10,300, after the second quarter it would be 10,609, and eventually at the end of the year you’d owe \$11,255.09 You can do this on your calculator if you like. The easiest way to add 3% to a number is to multiply it by 1.03. Now let’s compare the 2 situations we just considered. When the rate was j2 = 12%, we ended up with \$11,236. When the rate was j4 = 12% (same percentage, but different compounding), we ended up with \$11,255.09 Now you see that the more frequent the compounding, the higher the rate is. You can guess that if we had j12 = 12% and we did the same exercise we would end up with an even higher amount at the end of the year. This is where effective rate comes in. We don’t really care about how the compounding happens, all we want to know is how much we owe at the end of the year. This is what effective rate tells us: by what % does your loan increase over one year. Let’s say we have an effective rate (remember effective rate is j1, so it’s the rate with annual compounding). If our effective rate j1 is 12%, it simply means that at the end of the year, 12% is added to the loan, so we owe \$11,200 because 12% of 10,000 is 1,200. You can see that in the other 2 examples we looked at the amount at the end was higher because the compounding was more frequent than annually. There are so many ways we can have an interest rate: monthly, quarterly, daily compounding. How do we compare them? To compare interest rates with different compounding, we have to convert them to a common denominator, and the easiest one to use is j1. So when you have different rates you need to compare, you will need to convert them all to j1 to see which one is higher. Converting to j1 allows you to compare apples to apples. Now in theory you could convert to j2 or j4 but j1 is just easier. In the next video, I’ll show you how to convert any interest rate easily using your financial calculator.