Learning Outcomes. After watching this video you will be able to distinguish between Annual percentage rate, that is APR, and Effective annual rate, that is EAR. And then calculate EAR given an APR and vice versa. >> Compounding frequency and effective annual rate. Until now, we have assumed 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? Compounding is the idea of interest earning 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've earned $10 and have a balance of $110. In the second year, you would earn $11 in interest, of which $10 is interest for an initial deposit of $100 and $1 is the interest on the $10 interest you earn 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 six months. This means that the bank will pay you an interest of 5% every six months on your deposit of $100. So if you deposit a $100 today, your balance will increase to $105 after six months. The $105 will earn interest at 5% over the next six months. Interest earned, it will be $105 x 0.05 which equals $5.25. After one year, you will have $105 + $5.25 which equals $110.25. This is $0.25 more than in the previous example where interest was compounded only once a year. The difference is because of compounding interest more frequently. The more frequently interest is compounded, the higher is the future value of your investment. Conversely, the more frequently compounding is done, the lower is the present value of your investment. We can modify all the present and future value formulas we saw earlier by replacing r with r/m, where m is the number of times interest is compounded each year. And now n is redefined as the total number of payments made in an annuity or the total number of periods over which the cashflow is discounted. One interesting thing to note in our example is that when the bank paid interest annually, your $100 became $110, which is a return of $110 minus $100 divided by $100, which is a 10% return. But when it paid interest every six months, your $100 became $110.25 after a year, which is a return of $100.25 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) = (1 + r/m), the whole thing raised to the power of m- 1. 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. To conclude, let's list all the formulas modified to reflect a compounding frequency of m. PV0 = FVn / (1 + r/m) raised to the power of n. FVn = PV0(1 + r/m) raised to the power of n. The future value of an ordinary annuity, FVAn = PMT [(1+r/m) raised to the power of n-1]. The whole thing divided by r/m. The present value of an ordinary annuity, PVA sub zero, is equal to PMT divided by r/m [1-1/1+r/m the whole thing raised to the power of n]. In the next video, we will see how to calculate the present and future value of annuity dues. 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 = 0.1 and m = 12. Then EAR = (1 + 0.1 /12) the whole thing raised to the power of 12- 1 = 10.47%. What this says is that if you invest $100 today in a bank 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 loan and interest is compounded monthly. Remember, PVA sub 0 is equal to 500,000, m is equal to 12 months, n is now 30 years, 12 months a year, so 360 payments. r/m is 0.1/12. We now have 500,000 = PMT/0.1/12 [1- 1/(1 + 0.1/12) the whole thing raised to the power of 360]. Solving for PMT, we get $4,387.86. That is you will pay $4,387.86 every month for 30 years to repay your loan to the bank.