The Foundation
4.2 Future Value of a Present Amount
Someone investing a sum of money today at a given interest rate for a given period of time would expect to have a larger sum of money at that future date. We can determine that future amount using the future value of a present amount formula. We define future value as the value of a present amount at a future date after applying compound interest over a specified period. The amount invested today is the present value. Equation 4.1 shows how to compute the future value of a present amount:
FV = PV (1 + r)n (4.1)
where FV is the future value of an amount n periods from now; PV is the present value of an amount – it represents the amount invested today; r is the rate of return or interest rate per period; and n is the compounding term or interval, which is the number of periods between the PV and the FV.
Thus, future value calculations answer the question: How much will an investment be worth at a specific time in the future, compounded at a specific rate of interest? Future value is a compounded value because of compounding of interest. Another common name for future value is terminal value.
Example 4.1 illustrates how to compute the future value of a present amount using Equation 4.1.
Example 4.1 Future Value of a Single Amount
An investor has the opportunity to earn an 8 percent annual interest rate for the next 15 years. Assuming an investment of $5,000, how much will the investor have 15 years from today?
Solution: This problem requires computing a future value, given a present value of
$5,000, an annual interest rate of 8 percent, and a compounding term of 15 years.
Substituting into Equation 4.1 we get:
FV = PV (1 + r)n
FV = $5,000(1.08)15= $15,860.85
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Effects of Interest Rates and Compounding Terms on Future Values
The level of the interest rate (r) and the length of the compounding term (n) affect future values. Specifically, future values increase as the compound interest rate increases and/or the compounding term becomes longer. As illustrated in Example 4.2 below, substantial increases in future values can result from small changes in interest rates, especially when the compounding term is long.
Example 4.2 Effect of Interest Rates on Future Values
Sue Wang is considering two long-term investment options for an inheritance of
$50,000 she has recently received. Under Option A, she expects an annual return of 10 percent for the next 30 years; under Option B, she expects an annual return of 12 percent over the same period. What are the expected future values of these two plans 30 years from today when she retires?
Option A: FV = $50,000(1.10)30 = $872,470.11 Option B: FV = $50,000(1.12)30= $1,497,996.11
Solution: This example shows that a relatively small difference in interest rates results in a substantial difference in the future values 30 years later. The future value under Option B is about 72 percent higher than Option A.2 This substantial difference in future value amounts explains why young investors, who have long investment horizons for their retirement savings, are often encouraged to seek the higher expected returns from long-term growth equity investments.
Time Diagrams
Using a time diagram is helpful to illustrate the timing of cash flows, especially for situations involving cash flows at different points in time that are not equal. For Example 4.1, our time diagram would look like this:
Year
Amount
0 15
$5,000 $15,860.85
2 Of course, the choice of Option B would need to take into account any perceived differences in risk between the two investments.
Compounding of Interest
Interest earned and paid in future years depends not only on the initial amount, but also on any interest earned that has not been withdrawn from the investment account. Thus, with compounding, interest is earned on interest, and the future value amount will increase geometrically over time, not linearly.
To illustrate, assume an investment of $100 today at 10 percent per year. One year from today, the future value grows to $110, computed as [$100 + (0.10)($100)]. Assuming no interest is withdrawn, the investor will earn interest on $110 during the second year and receive a year-end interest payment of $11.00. The future value at the end of the second year will be $121.00, computed as [$110 + (0.10)($110)]. Assuming that no interest is withdrawn, the investor will earn $12.10 during the third year and will have a future value of $133.10 at the end of three years, computed as [$121 + (0.10)($121)]. Table 4.1 summarizes these interest and future value amounts.
By compounding rates annually, the interest is earned throughout the year, but it is paid at the end of the year. As just discussed, the first interest payment of $10 on the $100 initial investment would be paid at the end of the first year. If the investor withdrew the principal before the year-end interest payment date, the investor would not receive any partial interest payment. The second interest payment of $11 would be paid at the end of the second year, and the third interest payment of $12.10 would be paid at the end of the third year. Thus, no partial intra-year interest payments are made under annual compounding. Later in this chapter, we examine situations where interest is compounded more frequently than once per year.
Writing the future value equation (Equation 4.1) with subscripts on the FV and PV terms is sometimes helpful. For a present value invested at time t, the equation for a future value n periods from time t may be written:
FVn = PVt(1 + r)n
For example, suppose an investor expects to receive a sum of money 3 years from today and plans to immediately invest the funds in a savings account at a fixed interest rate for 5 years.
We can write the future value equation to show the timing of the cash flows:
FV8 = PV3(1 + r)5
where PV3 is the amount to be invested 3 years from now and FV8 is the future value 5 years later (8 years from today).
Table 4.1 Future value of $100 at an interest rate of 10 percent
Year Future value of $100 investment at 10% Interest earned
1 $110.00 $10.00
2 $121.00 $11.00
3 $133.10 $12.10
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Future Value of Multiple Amounts
When multiple present values need to be compounded to obtain a future value, these compounded future value amounts can be summed together. For example, if an investment provides a cash flow of $100 in year 0, $250 in year 1, and $475 in year 2, the future value in year 3 could be computed by summing the future value of each. Assuming an annual interest rate of 10 percent, we get:
FV3 = PV0(1 + r)3 + PV1(1 + r)2 + PV2(1 + r)1
FV3 = $100(1.10)3 + $250(1.10)2 + $475(1.10)1 = $958.10
Thus, future value of present amount calculations are additive. Of course, these calculations get tedious when we have many present value amounts. If the present value amounts are equal and periodic, they constitute what is called an annuity and the future value calculation is simplified. We address future value of annuity calculations in Section 4.4.
Using the BA II PLUS® Financial Calculator
Skillful use of financial calculators can greatly simplify and quicken our time value of money calculations. Throughout this chapter, we illustrate how to solve time value of money cal- culations on the Texas Instruments (TI) BA II PLUS® financial calculator. However, before attempting any problems, we need to check and perhaps change an important setting on your calculator. This calculator comes with a preset factory setting of 12 P/YR to accommodate the many users who compute monthly loan payments. We need to change this setting to 1 P/YR by sequentially pressing [2nd] [P/Y] “1” [ENTER] [2nd][QUIT]. These keystrokes will also automatically change the calculator setting to one compounding period per year. We recommend keeping the setting on 1 P/Y because it is easy to forget to change the setting back to 1 P/Y for future calculations.
Now we can proceed to an example. Across the third row (from the top) of this calcu- lator, there are five TVM keys that correspond to the basic time value of money concepts:
N, I/Y, PV, PMT, and FV keys.
• N = number of compounding periods.
• I/Y= interest rate per compounding period.
• PV= present value.
• FV= future value.
• PMT = annuity payments or constant periodic amount.
Refer to Example 4.1 and follow these steps:
1 2nd CLR TVM 2 15 N
3 8 I/YR 4 5,000 +/− PV
5 CPT FV (CPT refers to the compute key in the top row.)
Steps 2 through 5 look like the following on the BA II PLUS® calculator:
The answer is 15,860.85. Note the negative sign on PV. Specifically, the $5,000 represents the funds needed to create the account (an outflow) and the $15,860.85 represents the value of the account at the end of the period (an inflow). Although this step is unnecessary, it makes the FV come out as a positive number. If PV is entered as a positive number, the negative sign that appears on the FV should be ignored.
Practical Financial Tip 4.1
Using the BA II PLUS® calculator requires understanding the sign convention. This cal- culator uses the convention that cash outflows (expenditures) have negative signs (−) and cash inflows (receipts) have positive signs (+). Failing to follow this convention may yield nonsensical answers or error messages. Thus, PV and FV must have opposite signs.
Practical Financial Tip 4.2
Always start time value of money (TVM) computations on the BA II PLUS® calculator with [2nd] [Clear TVM]. The keys N, I/Y, PV, PMT, and PV are TVM memory registers.
Many problems use only four of the five keys (three known variables and one unknown variable). This calculator has continuous memory that retains numbers in storage registers even when the calculator is turned off. Forgetting to clear the calculator’s TVM memory after each use could result in the calculator’s internal program solving the current problem using information stored from a previous problem. This may result in an incorrect answer. Another way to handle this situation is to enter 0 in any unused TVM memory registers such as entering 0 PMT in a single sum TVM problem.
Concept Check 4.2
1 What is meant by the term compounding of interest?
2 How do the length of the compounding term and the interest rate affect future values?
15 N
8 I/Y
5,000 +/−
PV PMT
CPT FV
(press this key as the last step)
80 THEFOUNDATION