There's a difference in financial terms between an effective interest rate and something known as a nominal interest rate. So I'm going to explain all about that and why you need to differentiate between the two in this screen cast. The example that we're going to solve at the end of this screen cast is what if the compounding period is not the same as the payment period? In all of the examples that we've covered so far, the compounding period has coincided with the payment period. But this doesn't always have to be the case. In this example, we take a $1000 loan with an annual interest rate of 5 percent compounded monthly. Our compounding period is every month. However, we want to make payments quarterly and we wish to pay the loan off in five years. So the payments are every quarter or every three months and the interest is compounded monthly. In this case, we cannot use the simple analysis that we used in previous screen casts where we calculated amortization schedules and payment values. In order to solve this problem, we need to differentiate between something known as the effective interest rate and the nominal interest rate. If $1,000 is invested in at the end of the year, we have earned $50 of interest. For a total of $1,050 in the bank, we can calculate the effective annual interest rate here as the following. We can take $1,050, subtract our initial amount, and divide that by 1,000. So effectively, we have earned five percent interest in that year. The effective annual interest rate would be the interest rate if a loan was compounded once a year. The nominal interest rate, however, is what we have used in some other examples where we have monthly compounding. An annual interest rate combined with a compounding period is known as a nominal interest rate. In other words, the nominal interest rate is the periodic interest rate multiplied by the number of periods per year. Banks' published rates are typically nominal interest rates. For example, five percent annual interest rate compounded monthly. That is the combination of an annual interest rate with a compounding period. This does not mean that $1,000 will earn $50 in a year. The future value of $1,000 invested at five percent compounded monthly in one year. So we're looking at one year. One year has 12 months, the interest rate per month, we have to take the annual percentage rate of 0.05 and divide by 12. When we do that calculation, we get $1,051. So to calculate the effective interest rate of five percent compounded monthly, we can just take the $1,051, subtract the initial amount in that year, and divide by the initial amount of $1,000, and we get an effective interest rate of about 5.11 percent. So we see that the effective interest rate is slightly greater than the annual percentage rate compounded monthly, so the five percent. The effective interest rate can always be determined from the nominal interest rate and the number of compounding periods using this formula. For example, if the interest rate is 4.5 percent and we compound monthly, n is going to be 12. The interest rate is 0.045. So we can put that into this formula here, and we can calculate an effective interest rate a little bit higher than the nominal interest rate of about 4.58 percent. Don't worry if these mathematical formulas scare you. If you don't like math, you can always use Excel's effect and nominal functions. Let's take a look at this in Excel. We have a couple of examples. The first one, we're going to use Excel to determine the effective interest rate if the annual interest rate is 4.5 percent and the interest is compounded quarterly. The second example, $2,000 has been invested in a savings account and has earned $240 at the end of the first year. If interest is compounded monthly, what's the nominal interest rate of this loan? Let's go ahead and solve this in Excel. I've got this starter file, effective verse nominal interest rate, and I've got two sheets. On the first sheet, we have our first two examples. We want to calculate the effective annual interest rate. We have a nominal interest rate, and that's the one that's typically provided by banks. This is annual interest rate but compounded quarterly. I've denoted m as payments per year instead of n, which is the number of years. The effective interest rate, we can always use the effect function in Excel. The nominal rate is our interest rate up here. The number of payments per year is just going be four. It's a pretty simple function. We can go ahead and press enter, and that that means our effective interest rate is about 4.58 percent. The second example, we have $2,000 beginning of the year. We have a future value of $2,240. Ultimately, we want to determine the nominal annual interest rate compounded monthly. So we have 12 payments per year. First though, we have to calculate the effective annual interest rate. I didn't give this to you, but we can always use this formula shown down here at the bottom to calculate the effective. It's just the amount that you have at a future date, minus the initial amount, divided by the initial amount. So to calculate the effective interest rate, I'm going to subtract from our future value, our initial amount, and I'm just going to divide that by the initial amount. That means in one year, our effective interest rate is 12 percent. Now what we want to do is we want to convert that into annual interest rate and nominal annual interest rate if we're making payments monthly. There is a nominal function in Excel where we just have the effective rate. So I'm just going to click up there. Then the number of payments, that's going to be 12 per year. When I press enter, this is the nominal annual interest rate compounded monthly. So let's get back to the original problem that I posed at the beginning of this screen cast. What if the compounding period is not the same as the payment period? We want to determine what's the payment amount each quarter. We're just going to create a quick amortization schedule for this loan over the five years. Instead of making monthly payments, we're going to be making quarterly payments in this amortization schedule. The approach, we have a five percent nominal interest rate for 12 month compounding, where you have to convert this back to an annual effective interest rate. Using the annual effective interest rate, you can then calculate a new nominal interest rate. So we're going to convert this to a nominal interest rate for quarterly compounding, and that's what we're going to use when we solve this problem. I've got this starter over on sheet 2. First thing I'm going to do is I'm going to name these P, i ,and n. I'm going to go up here to the formulas tab, create from selection, go ahead and click okay. The effective interest rate, we have monthly compounding and our rate is 0.05. So we're going to use the effect function. The nominal interest rate is i, and that's over 12 months or 12 payments per year. When I press enter, that means that we get about effective interest rate of 5.1 percent. Now we did that because we need to now calculate what the nominal interest rate, if you compound quarterly is. We're making four payments per year. So I can just use the nominal function in combination with the effective rate that I just calculated, but now we're making four payments a year, so that's quarterly. I can press enter, so that's my new nominal interest rate that I can now use in the payment function. So we have the payment function, our rate is the nominal rate. Now remember you have to divide by the number of payments per year to get a quarterly rate, the number of periods total, we're doing five years. So I want to multiply n times my payments per year, which is four, and there's a total of 20 payments. The present value or the principal is p, and that's a positive value because we've obtained that from the bank. Then the last argument here, we could leave off zero. We want to pay off until the future value is zero. You can either leave that off or include it, and I press enter, which means that we have to make payments every quarter of $56.85 to get our $1,000 loan down to zero in five years. So hopefully, in this screen cast, you learned how we can take a interest rate that is compounded at a different frequency than our payment rate, and you can determine the effective interest rate, convert that to a different nominal interest rate. Here we did compounded quarterly, created a quarterly payment, and then we just created a quick amortization schedule. Thanks for watching.
