1.3 Compounding and Discounting Annuities

An annuity is an equally spaced sequence of equal cash flows. Annuities are particularly common in lending relationships, such as car loans, mortgages, and bonds, as well as in financial contracts, such as insurance contracts and rental agreements. All of these arrangements are based upon equally spaced (e.g., monthly, yearly) payments of the same dollar amount. Figure 1.3 shows a typical annuity. Note that although the first payment in Figure 1.3 is at time 1, annuities may start at any point in time and continue for any length of time. For example, the stream of cash flows shown in Figure 1.4 is also an annuity—though one that starts at time 5 and goes until time 8 (called a deferred annuity).

To calculate the future or present value of an annuity we can simply use the equations found in the previous section to compound or discount each individual payment. With long annuities, however, such as a 30-year mortgage with 360 monthly payments, this would be a rather lengthy and laborious process. To shorten this process, we can instead use a discount factor multiplied by the size of the annuity payment. For future value calculations, this equation is

$\text{FV}=\text{PMT}\times \left\{\frac{(1+i{)}^n-1}{i}\right\}$

where PMT is the annuity payment, n refers to the number of payments, and the other two variables, FV and i, are as defined previously.

Notice that the bracketed portion of the equation is defined solely by n and i. This bracketed term is sometimes called the future value interest factor for an annuity (FVIFA). So, while the equation looks more complicated, note that here we still have only four variables (though i appears twice in the equation, it is the same variable each time) and one equation. In order to solve for any one variable, therefore, we have to know values for the other three. Hence, the future value of an annuity is a “3 find 4” game just like the present and future value calculations for single sums. Consequently, we can use our financial calculators to perform the same three-step process outlined above.

Finding the present value of an annuity involves a similar process. Once again, we have the option of finding the present value of each individual cash flow in the annuity and then summing. Alternatively, we can use the following equation to calculate the present value:

$\text{PV}=\text{PMT}\times \left\{\frac{1-{\left(\frac{1}{1+i}\right)}^n}{i}\right\}$

The bracketed portion in this case is called the present value interest factor for an annuity (PVIFA). Conceptually, the present value of an annuity is today’s time-adjusted equivalent of the stream of future cash flows.

Example: The Future Value of an Annuity

Let’s look at an example. If you invest $1,000 at the end of each of the next three years at 8%, how much would you have after three years?

If starting today, at time 0, we are investing cash at the end of each year our first deposit will be at time 1 (in one year) followed by two more at times 2 and 3. If we make 8% on our invested cash, we want to know how much total cash we will have at time 3. Note that time 3 is also the moment when we make our third and final deposit—thus, conceptually it is as if we are depositing our final $1,000 in the bank and then immediately asking the teller for our account balance. That balance is the future value of this annuity and is the number we want to calculate. As mentioned, we could calculate the future value by individually compounding each of the three cash flows using a single sum equation and then taking the sum of the three individual future values. However, with any annuity of significant length, such a tactic would be impractical and extremely tedious. Instead, we can use the annuity equation as follows:

$\text{FV}=\mathrm{1,000}\times \left\{\frac{(1 + 0.08{)}^3-1}{0.08}\right\}=\$3,246.40$

Using a financial calculator in one payment per year mode, we enter the three known values and solve for future value.

Example: The Present Value of an Annuity

What is the PV of $1,000 at the end of each of the next three years if the opportunity cost is 8%?

We want to find the present value at time 0 of this stream of cash flows. Our equation, with n = 3, i = 8% and PMT = $1,000, yields the following:

$\text{PV}=1000\times \left\{\frac{1-{\left(\frac{1}{1+\mathrm{0.08}}\right)}^3}{0.08}\right\}=\$2,577.10$

Otherwise, we enter the three known values into a financial calculator and solve for PV:

How do we interpret the present value for $2,577.10? This value is the value today of the three future $1,000 cash flows. Assuming that we really have an 8% opportunity cost, we should be indifferent between receiving three annual end-of-year future cash flows of $1,000 and receiving $2,577.10 today. Another way to interpret the present value is to imagine that we deposit $2,577.10 in the bank today. If the cash grows annually at 8%, we will be able to withdraw exactly $1,000 each year for the next three years, leaving nothing in the bank after the third withdrawal. A final application of the present value of an annuity is the case of a loan. Suppose that we borrow $2,577.10 from the bank today at 8% for 3 years. If we are making annual payments, in order to repay the loan by the end of the third year we will have to pay exactly $1,000 per year.

All the annuities that we have seen in this section have been ordinary annuities. An ordinary annuity has a one-period delay between the start of the annuity period and the time of the first payment. For instance, in our previous example the three-year annuity starting at time 0 didn’t make its first payment until time 1. Many consumer annuities are ordinary. When you sign the loan contract for a car loan or mortgage, for example, you will not make a loan payment until the following month. There is a one-month delay. When calculating the present value of an ordinary annuity, make sure to remember that you are calculating the present value one period before the first cash flow. Many students tend to mess up here on finance exams.

Compounding Periods

If you walk into a bank and request information on a car loan, they may tell you something like “three-year loans have an interest rate of 12% APR” (APR stands for annual percentage rate). However, once you apply for the loan you will be required to make monthly payments. This difference between the payment frequency and the period over which the interest rate is stated causes a compounding problem. If we were to calculate the payments on a $10,000 car loan for three years at 12%, we might be tempted to enter data as follows:

This, of course, is correct if we want to calculate annual payments. Since the loan contract calls for monthly payments, the annual payment calculation is not appropriate. The key thing to remember in these compounding problems is that all the variables need to be stated for the same time period. Previously, we calculated annual payments using annually stated interest rates—no problem. Now, however, we need to calculate monthly payments and are given an annual interest rate. To deal with this, we must adjust the interest rate to reflect the monthly nature of the payments. In this case, we are not getting a three-year loan with a 12% annual rate, but rather a 36-month loan (3 years × 12 months per year) at a rate of 1% per month (annual rate of 12% ÷ 12 months per year). So, to calculate our monthly payments, we would enter the following into our financial calculator:

Further, notice that the monthly payment of $332.14 is NOT equal to the annual payment (4,163.49) divided by 12, or $346.96. Why not? Because when you make your first payment one month after the loan is originated, some of the payment is interest and some is principal. Thus, during the second month you have a smaller loan balance on which interest is charged; this means that the amount of interest accrued during the second month is smaller. In essence, making monthly payments rather than annual payments reduces principal faster, which causes your total interest cost to go down, causing the monthly payment to be less than the annual payment divided by 12.

This same logic can be applied to any differences in timing between the payment frequency and the interest rate. For example, many business loans involve quarterly payments (i.e., four times per year). Let’s try an example.

Example: Years to Pay Off

Suppose you are the CFO of a firm that has borrowed $10 million at a rate of 8% (remember, unless stated otherwise, interest rates are always given as annual rates). The quarterly payments are $365,557.48. How many years will it be until the loan is repaid?

Solution: This is a simple present value of an annuity problem where we know the interest rate, the payments, and the present value, so all we have to do is solve for the number of periods and then correctly interpret the calculation. The following keystrokes provide the solution:

Because we entered the data (both interest rate and payments) on a quarterly basis, this solution implies that the loan will be repaid in 40 quarters. However, the original question asks for the number of years until the loan is repaid. Since there are four quarters in a year, we simply divide 40 by 4 to get 10 years.

You may have noticed that we entered the payment as a negative number in the exercise above, while we ignored the cash flow signs in some of our previous examples. In this problem, it is very important to keep track of the signs. If you work the problem again and enter both the present value and the payment as positive numbers, your calculator will say “No Solution.” As discussed above, the reason for this is that the positive sign indicates the entered dollar amount is an inflow. It is not possible to get an inflow today of $10 million and then to get inflows of $365,557.48 each quarter with no outflows, unless you are dealing with a very unusual banker.1 Hence, when you enter multiple dollar amounts in any TVM problem, you need to be careful to decide which are inflows and which are outflows. If there are only two dollar amounts entered, it generally doesn’t matter which is positive and which is negative as long as they don’t have the same sign. As we’ll see in the section on bonds, when we enter three dollar amounts, the tracking of cash flows becomes more critical.

Effective Yield

When you borrow money in the United States, federal law requires a number of disclosures. These include the interest rate, the term of the loan, the total amount of interest to be paid, and the annual percentage yield (APY). In this context, the APY is frequently called the effective yield. Think back to our discussion of compound interest at the beginning of this topic. In essence, we noted that if we invest for multiple periods, we earn interest on interest after the first period. Well, this same principle is at play if we “compound” our interest payments by making more than one payment per year. Incidentally, this same logic is the reason that the monthly car payments calculated in the last section were smaller than the annual payments divided by 12.

To understand effective yields, let’s consider an example where a bank offers a savings program that pays 8% interest compounded quarterly. What does this mean, exactly? Basically, it means that after the first quarter, you will be “paid” the interest that you have earned to that point (in this case, 2%). This, of course, means that the interest received at the end of the first quarter will earn interest for the rest of the year. Likewise, you will receive another interest payment at the end of the second quarter and so forth. The ability to earn interest on interest during the year is what makes the stated rate (in this case, 8% APR) different from the effective rate.

So, if we invest for one year at 8% compounded quarterly, what is the effective rate? There are two ways to approach this question. First, we could use a little algebra to solve for an effective rate equation, resulting in

where:

i = annually stated interest rate

m = number of compounds per year

For our example, this gives an effective rate of

Hence, that 8% compounded quarterly has an effective yield of 8.24%. Stated differently, if you were offered 8% compounded quarterly or 8.24% compounded annually you would be indifferent since both offer the exact same yield.

There is another way to approach this problem. Instead of memorizing a formula, you could use the TVM skills you’ve already learned and a little intuition to get the same answer. Suppose we invest $100 in the account paying 8% compounded quarterly. How much will we have at the end of the year? On your financial calculator, this is equivalent to investing for four periods (N = 4) at 2% per period, or

Realizing that this implies that $100 invested today will grow to $108.24 one year from now means that the effective yield (i.e., the annually compounded rate that will cause $100 to grow to $108.24 in one period) will ultimately be as follows:

Calculating and understanding effective yields is an important skill in many situations in finance.

Calculating and understanding effective yields is an important skill in many situations in finance.