The Foundation
4.3 Present Value of a Future Amount
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How should we interpret this present value amount? The $14,945,163.46 present value amount (today) is financially equivalent to $20 million five years from today at an annual interest rate of 6 percent. Assuming no preference for spending the money today or 5 years from now, an individual would be indifferent between receiving $14,945,163.46 today or $20 million five years from today at an annual interest rate of 6 percent.
Suppose the borrower and lender decide to settle this future liability today. If they both agree that 6 percent is the appropriate annual interest rate over the next 5 years, a
$14,945,163.46 payment would be the appropriate (present value) payment today.
The process of computing the present value of a future amount is called discounting.
The interest rate used to compute the present value is called the discount rate, but may also be referred to as the opportunity cost, required rate of return, and the cost of capital.
Regardless of the term used, it represents the compound rate of return that an investor can earn on an investment. In Example 4.3, we would say that $14,945,163.46 is the discounted value of a $20 million payment 5 years from now at a discount rate of 6 percent. Time value of money analysis is often referred to as discounted cash flow analysis.
Present values decrease as the discount rate increases. Thus, present values are inversely related to interest rates. The logic is simple. Suppose a person needs to have $50,000 five years from today, and wants to invest money today to meet that goal. The higher the interest rate, the lower the (present value) amount needed to be invested today. This inverse relationship is an important concept that we will return to several times in this the book, especially when we discuss valuing securities issued by firms and capital investment projects that firms adopt.
Using the BA II PLUS® Financial Calculator
The steps used for computing the future value of a present amount on the BA II PLUS® cal- culator are similar to those used for computing the present value of a future amount in Section 4.2. We illustrate these steps using the previous Example 4.3 where we solved for the present value of a $200,000 future value amount 5 years from now at a discount rate of 6 percent.
1 2nd CLR TVM 5
2 5 N 4
3 6 I/YR 6 Steps 2–4 can be done in any order.
4 20,000,000 +/− FV 4
5 CPT PV 7
Steps 2 through 5 will look like the following on a BA II PLUS® calculator:
5 N
6 I/Y
CPT
PV PMT
20,000,000 +/−
FV
(press this key last)
82 THEFOUNDATION
The answer should appear as $14,945,163.46. Recall from Example 4.1 that when using the BA II PLUS® calculator, entering a negative (positive) future value amount results in a positive (negative) present value amount due to the sign convention.
Present Value of Multiple Future Cash Flows
The present value of multiple future value amounts can be computed by simply summing the present value of each individual future value amount. The following example illustrates the important property that present values are additive.
Example 4.4 Present Value of Uneven Amounts
The Crusty Crumbs Company expects to receive the following payments from one of its delinquent customers: $500 one year from today, $800 two years from today, and
$950 three years from today. At an interest rate of 8 percent, what is the present value of these future value payment amounts?
Year Amount
Solution: We obtained the present value of this stream of uneven amounts as follows:
PV FV
r FV
r FV
0 1 1 2 2 3 r3
1 1
1 1
1
= 1
+
⎡
⎣⎢ ⎤
⎦⎥ + +
⎡
⎣⎢ ⎤
⎦⎥ + +
⎡
⎣⎢ ⎤
⎦⎥ ( )
( )
( )
PV $
( . ) $
( . ) $
( . ) $ , .
= ⎡
⎣⎢ ⎤
⎦⎥ + ⎡
⎣⎢ ⎤
⎦⎥ + ⎡
⎣⎢ ⎤
⎦⎥ =
500 1
1 08 800 1
1 08 950 1
1 08 1902 97
1 2 3
While computing the present value of multiple future value amounts is straightforward, the computations become tedious and time consuming as the number of cash flows increases.
When the multiple future value amounts are equal and periodic, they constitute what is called an annuity. Present value of annuity calculations are discussed in Section 4.5. When the amounts are uneven (not equal or periodic), follow the steps in the example above.
Using the BA II PLUS® Financial Calculator
We illustrate two approaches for solving this example. The more tedious approach is to compute the present value of each future amount and then sum the results, as shown below:
PV1: FV = 500 +/−; I/Y = 8; N = 1; CPT → PV = PV1 = 462.96 PV2: FV = 800 +/−; I/Y = 8; N = 2; CPT → PV = PV2 = 685.87 PV3: FV = 950 +/−; I/Y = 8; N = 3; CPT → PV = PV3= 754.14 PV of uneven amounts = ∑PV = 1,902.97
0 1 2 3
PV $500 $800 $950
We can also solve for the present value on the BA II PLUS® calculator by entering the actual cash flows into the calculator and pressing the NPV key.3 These steps are detailed in Table 4.2 below.
3 NPV refers to net present value, a concept we address in Chapter 8. A net present value subtracts the cash flow in year 0 from the present value of the future cash flows. Since we have no cash flow in year 0 in this example, the NPV is simply equal to the present value of the future cash flows.
Table 4.2 Calculating the PV of multiple future amounts using the BA II PLUS® calculator
Key strokes Explanation Display
[2nd] → [Format] → [2] → [Enter] Display 2 decimals DEC = 2.00 (Need to do this only once)
[CF] → [2nd] → [CLR WORK] Clear Memory Registers CF0= 0.00
0 → [+/−] → [ENTER] Initial Cash Outlay CF0= 0.00
[↓] → 500 → [ENTER] Period 1 Cash Flow C01 = 500.00
[↓] Frequency of Cash Flow 1 F01 = 1.00
[↓] → 800 → [ENTER] Period 2 Cash Flow C02 = 800.00
[↓] Frequency of Cash Flow 2 F02 = 1.00
[↓] → 950 → [ENTER] Period 3 Cash Flow C03 = 950.00
[↓] Frequency of Cash Flow 3 F03 = 1.00
[NPV] → 8 → [ENTER] 8% Discount Rate I = 8.00
[↓] → [CPT] Calculate NPV NPV =1,902.97
Concept Check 4.3
1 What steps are needed to solve for the present value of a future amount?
2 What is meant by the terms discounting and discount rate?
3 How do the length of the compounding term and the interest rate affect present values?