1.2 Compounding and Discounting a Single Sum

Compounding involves finding the future value of a cash flow (or set of cash flows) using a given discount or interest rate. Whether we are moving that cash flow forward in time 1 year or 100 years, the process is the same. We will start our discussion of compounding, and of time value of money calculations in general, by calculating the future value of a single sum.

Suppose you deposited $100 in a bank account today and that you earned 6% on the money for one year. How much would you have at the end of the year? Most likely, you could have done this problem even before beginning this topic, and you already know that you would have $106. We can state this mathematical relationship as follows:

$\text{Future Value}=\text{Present Value}\times (1+i)$

where i = discount rate.

What would happen if you left your $100 in the bank for two years? Would you earn $6 in interest each year for a total future value of $112? Actually, you would have more than $112. Why? Because of the effect of compounding. At the end of the first year, you would have earned $6 in interest. Hence, during the second year, you would have $106 earning interest instead of just the original $100. The compounded future value would be $112.36. This additional $0.36 is interest on interest. While it might not look like much here, you will see that the impact of compounding can be very significant over a longer time period.

We can state the more complete mathematical relationship for future value calculations as follows:

$\text{Future Value}=\text{Present Value}\times (1+i{)}^n$

where n = number of compounding periods.

Calculators: The Beginning Finance Student’s Best Friend

While the mathematical equations show us how to think about time value calculations, the process of using the formulas can become very cumbersome. For instance, there are 360 time periods in the typical 30-year mortgage (more on this later). This means that we will have to solve problems with a sum raised to the 360th power. For most of us, this surpasses our mental-math capabilities. Hence, you will need some calculation assistance for most TVM calculations. For most college finance classes, this comes in the form of a financial calculator. (Note: We also include Excel functions in this topic.)

Using the calculator greatly simplifies the time value of money calculation process. Look at the future value equation above. How many variables do you see? There are four variables: FV, PV, i, and n. Since there is only one equation, there can be only one unknown. While this equation is only one of several equations we will review, you will soon see that most TVM calculations are single equations and contain four variables. Given the use of a calculator or computer, solving for any of the four variables in the equation with a financial calculator becomes what we will call a “3 find 4” game. Almost all time value of money problems we do in this text will be “3 find 4” games.

A “3 find 4” game means that we enter values for three of the four variables (say, PV, i, and n) into our calculator, then solve for the fourth variable (FV). With any financial calculator, this involves three easy steps: (1) set up your calculator (see Figure 1.2), (2) enter the three known variables in any order, and (3) solve for the unknown variable. In the problem we just solved, the three known variables were PV = $100, i = 0.06, and n = 1, and we solved for the fourth variable, FV. Alternatively, if we had known the PV, FV, and n, we could have solved for the interest rate that links those three. The same is true of solving for PV and n. Knowing values for any three variables enables us to solve for the fourth.

There are at least two cautions to remember when doing TVM calculations on a financial calculator. First, ensure that your calculator is set to the correct number of payments per year for the problem you are solving. In almost all of the examples in this text we will assume that your calculator is set to just one payment per year. Figure 1.2 shows you how to set the number of payments per year for the HP 10B (or HP 10bII+) and TI BA II Plus calculators. Second, develop the habit of setting up your calculator before every TVM problem. Setting up your calculator involves clearing its memory registers (again, see the bottom panel of Figure 1.2). Neglecting to clear the calculator’s memory registers will result in using the information from the previous problem to solve the current problem with potentially erroneous results. Thus, step one for every time you use a calculator to solve a TVM problem is setting up your calculator by clearing its memory and making sure it is in the correct payments mode.

If you use a TI-83, TI-83 Plus, or TI-84 Plus calculator, you’re in luck! The examples in this topic are worked out by keystroke. Additionally, directions to set up your calculator are provided in the sidebar to the right.

Note that some calculators include the buttons P/Y and I/Y, while others use P/YR and I/YR. P/Y and P/YR both mean payments per year, and I/Y and and I/YR both mean the annual interest rate. We will use P/Y and I/Y in our examples in this text unless we are discussing a specific calculator that uses P/YR or I/YR.

Example: Future Value Calculation

Let’s try a quick example with your financial calculator. (Before beginning, consult Figure 1.2 to make sure that your calculator is set to one payment per year and that you are in the END mode.) Suppose you invested $1,000 in the stock of XYZ Corp. After 30 years, you hear a TV commentator saying, “The average appreciation of XYZ stock over the last 30 years has been 14.5%.” What is the value of your $1,000 investment after the 30 years? Complete the following steps to find out.

1. Set up your calculator.

2. Enter the three known values in any order.

   N = 30

   I = 14.5 (enter this in percent form, not decimal form)

   PV = –1000

3. Solve for the unknown variable (future value).

   HP 10B: push FV

   TI BA II Plus: push CPT FV

   FV = $58,098.46

4. All the inputs must be stated in the same terms: annually, monthly, and so on. In this example, all of our inputs are stated on an annual basis—number of years, an annual interest rate, and annual cash flows. Later on, we will discuss periods other than annual.

You may have noticed that we entered the present value as a negative number. The sign indicates the direction of the cash flows. To keep the signs straight, think of the cash flows from this investment as money going in and out of your wallet. So if, as the example states, you were going to spend $1,000 to purchase the stock today, that would be money flowing out of your wallet; from your wallet’s perspective that’s a negative cash flow. At the end of the 30 years you could sell the stock and put the $58,098.46 back into your wallet. That would be a positive cash flow in year 30. While we can put in a positive $1,000 in this example and still obtain the right answer (–$58,098.46), that won’t always work. Specifically, we will encounter examples where we will enter more than one dollar-denominated variable (i.e., entering both PV and FV). In these cases, we must be very careful with the signs associated with the cash flows. In fact, if you ever work a problem and get “No Solution” on your calculator display, then the most likely error is that you failed to keep track of the signs on the cash flows.

Discounting

Now that we have mastered the compounding of single sums, let’s move on to discounting single sums. Compounding involves moving money away from us in time, that is, finding a time-adjusted equivalent sometime further in the future. Discounting is just the opposite. When we discount, we move money closer to us in time. Keep in mind that the present is at time zero on the timeline. To solve for the present value of a future single sum, we use the same formula we used to find the future value, Future Value = Present Value × (1 + i)ⁿ. This time, however, we will algebraically manipulate this equation to solve for PV:

Once again, since we have only one equation and four variables, we have to know values for three of the variables to solve for the unknown variable.

Example: The Present Value of a Single Sum

If you will receive $100 one year from now, what is the present value (PV) of that $100 today if the opportunity cost is 6%? What if the $100 is to be received five years from now?

Once again, we can solve this problem mathematically either using the equation above or with a financial calculator. We know values for three of our four variables: FV = $100, n = 1, and i = 6%. Plugging these three known values into the equation results in:

$\text{PV}=\$100÷(1.06{)}^1=\$94.34$

With a calculator we must first clear the memory and ensure that it is in one payment per year mode. Inputting those same three known values into our calculator yields the following:

Now let’s assume that we will not receive our $100 until five years from today. How does that change affect its present value today? Recalculating the present value with n = 5 gives

As shown in the example, at a 6% discount rate the $100 in five years is worth $74.73 today. In other words, if we were to deposit $74.73 in the bank today at a 6% interest rate, in five years we would have exactly $100. Another way to interpret this value is to realize that if 6% really is our personal rate for transferring/deferring consumption through time, we would be indifferent between receiving $74.73 today or $100 in five years. They would both be equivalent to us because of the time value of money.

In review, we have learned how to compound or discount single sums. Given that we have one equation and four variables (FV, PV, i and n), we must know values for three of these variables in order to solve for the fourth. We can do this either mathematically or in three simple steps on a financial calculator. You should plan for some extra practice time on this topic—on test day it is easy to tell students who practice TVM problems from those who don’t!