[MUSIC] Until now, we will assume that interest is calculated only once a year. In other words, interest is compounded annually. What if the bank compounds it more frequently, say once a month? How does it affect our present and future value calculations? In this video, we will distinguish between an annual percentage rate, APR, and effective annual rate, EAR. You will also see how to calculate an EAR given an APR. Compounding is the idea of earning interest on previously earned interest. Even though banks quote interest rates in annual terms, interest is usually not computed annually. Consider a simple example. You deposit $100 in the bank, which pays you an interest of 10% a year. After the first year, you would have earned $10 and have a balance of $110. In the second year, you would earn $11 in interest, of which $10 is the interest for your initial deposit of $100, and $1 is the interest on the $10 interest you earned in the first year. The additional $1 in interest that you earn in the second year, when compared to the first year, is because of compounding. That is, earning interest on interest. Let's look at the example a little differently. The bank pays an interest of 10% a year, but computes interest every 6 months. This means that the bank will pay you an interest of 5% every 6 months on your deposit of $100. So, if you deposit $100 today, your balance will increase to $105 after 6 months. The $105 will earn interest at 5% for the next 6 months. Interest earned will be $105 times 0.05, which equals 5.25. After one year, you will have $105 plus 5.25, which equals $110.25. This is 25 cents more than in the previous example, where interest was compounded only once a year. The difference is because of compounding interest more frequent. The more frequently it is done, the higher is the future value of your investment. Conversely, the more frequently the compounding is done, the lower is the present value of your investment. We can modify all the present and future value formulae as we saw earlier by replacing r by r/m. Where m is the number of times interest is compounded each year. And n is redefined as the total number of payments made in an annuity, or the total number of periods over which the cash flows is discounted. One interesting thing to note in our example is that when the bank paid interest annually your $100 became $110. A return of $110 minus $100 divided by 100, which equals 10%. But when it paid interest every 6 months your $100 became $110.25 after a year, which is a return off $110.25 minus $100 divided by 100, which equals 10.25%. This 10.25% is the effective annual rate, EAR, of your investment. The following formula relates EAR to the interest rate, r. EAR equals one plus r over m, the whole raised to the power of m minus one. With annual compounding EAR = r. But for all other compounding frequencies, EAR will be greater than r. r is usually referred to as the annual percentage rate, APR. Let's look at an example. Given an APR of 10% per year and a monthly compounding frequency, what is your EAR? We are given that r is equal to 0.10 and m is equal to 12. Then EAR equals 1 plus 0.10 divided by 12, the whole raised to the power of 12 minus one which equals 10.47%. What this says is that if you invest $100 today in a bank account that pays 10% interest a year and compounds interest every month, your balance will be $110.47 at the end of the year. Let's see how changing the compounding frequency affects the periodic payments of an annuity. Going back to our earlier example on the loan you took to buy the condo, let's say that you have to make monthly payments to the bank and interest is compounded monthly. Remember, PVA sub-0 equals 500,000, m is equal to 12 months, n now is equal to 30 years times 12 months a year, which is 360 payments, r, or the periodic interest rate, equals 0.10 over 12. We now have 500,000 equals PMT divided by 0.10 over 12, times 1 minus 1 over 1 plus .01 divided by 12, the whole raised to the power of 360. Solving for PMT, we get $4,387.86. You will pay $4,387.86 every month for 30 years to repay your loan to the bank. Let's list all the formulas now. Modified to reflect a compounding frequency, n. PV sub-0 is equal to FV divided by 1 plus r over m, the whole thing raised to the power of n. FV sub-n equals PV sub-0 times 1 plus r over m, whole thing raised to the power of n. FVA sub-n equals PMT times 1 plus r over m, whole raised of n minus 1, the whole thing divided by r over m. PVA sub-0 equals PMT divided by r over m times 1- 1 over 1 plus r over m, whole raised to the power of n. In the next video, we will see how to calculate the present and future values of an annuity due, and how they relate to those of ordinary annuities. [MUSIC]
