The fundamental concept in a capital budgeting decision analysis is the time value of money. Would you rather receive a $100 gift check from a relative today, or would you rather receive a letter prom-ising the $100 in a year? Most of us would rather have the cash now. There are two possible reasons for this attitude. First, if we receive the money today, we can spend it on that new sweater now in-stead of waiting a year. Second, as an alternative strategy, we can invest the $100 received today at 10 percent interest. Then, at the end of one year, we will have $110. Thus, there is a time value as-sociated with money. A $100 cash flow today is not the same as a $100 cash flow in 1 year, 2 years, or 10 years.
Compound Interest Suppose you invest $100 today (time 0) at 10 percent interest for one year. How much will you have after one year? The answer is $110, as the following analysis shows:
0 0 1
$ 0
0 1
$ (.10)($100) $110
Time
Time 0 Year 1 Time 1
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
The $110 at time 1 (end of one year) is composed of two parts, as shown below.
Principal, time 0 amount ... $100 Interest earned during year 1 (.10 3 $100) ... 10 Amount at time 1 ... $110
Thus, the $110 at time 1 consists of the $100 at time 0, called the principal , plus the $10 of interest earned during the year.
acceptance-or-rejection decision, 2
accounting-rate-of-return method, 14
accumulation factor,* 34 after-tax cash flow, 19 annuity, * 35
capital-budgeting decision, 2
capital-rationing decisions, 2 capital cost allowance
(CCA), 21 CCA tax shield, 21 compound interest, * 33 discounted-cash-flow
analysis, 3 discount rate, * 34
future value * 33
hurdle rate ( or minimum desired rate of return), 3 internal rate of return ( or
time-adjusted rate of return), 3
investment opportunity rate, 7
net present value, 3
payback period, 12
postaudit ( or reappraisal), 11 present value, * 34
principal, * 32
profitability index ( or excess present value index), 18 undepreciated capital cost
(UCC), 22 working capital, 24 *Terms appear in Appendix A to this chapter.
Key Terms
For each term’s definition refer to the indicated page, or turn to the glossary at the end of the text.
APPENDIX A TO CHAPTER 15
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Chapter 15 Capital Expenditure Decisions 33
Now suppose you leave your $110 invested during the second year. How much will you have at the end of two years? As the following analysis shows, the answer is $121:
$100 $100 (.10)($100) $110 $110 (.10)($110) $121
Time
Time 0 Year 1 Time 1 Year 2 Time 2
⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭ ⎫⎪⎪⎪⎪⎪⎪⎪⎪⎪⎬⎪⎪⎪⎪⎪⎪⎪⎪⎪⎭
We can break down the $121 at time 2 into two parts as follows:
Amount at time 1 ... $110 Interest earned during year 2 (.10 3 $110) ... 11 Amount at time 2 ... $121
Notice that you earned more interest in year 2 ($11) than you earned in year 1 ($10). Why? Dur-ing year 2, you earned 10 percent interest on the original principal of $100 and you earned 10 per-cent interest on the year 1 interest of $10. Interest earned on prior periods’ interest is called compound interest . Exhibit 15–14 shows how your invested funds grow over the five-year period of the investment. As the Exhibit shows, the future value of your initial $100 investment is $161.05 after five years.
As the number of years in an investment increases, it becomes more cumbersome to compute the future value of the investment using the method in Exhibit 15–14.
Fortunately, the simple formula shown below may be used to compute the future value of any investment.
Fn P(1r)n
(1) where P denotes principal
r denotes interest rate per year n denotes number of years
Exhibit 15–14
Compound Interest and Future Value
Time 0
Year 1
Time 1
Year 2
Time 2
Year 3
Time 3
Year 4
Time 4
Year 5
Time 5
Principal, time 0 amount
Interest earned during year 1 (.10 × $100) Amount at time 1
Interest earned during year 2 (.10 × $110) Amount at time 2
Interest earned during year 3 (.10 × $121) Amount at time 3
Interest earned during year 4 (.10 × $133.10) Amount at time 4
Interest earned during year 5 (.10 × $146.41) Amount at time 5
$100.00
10.00
$110.00
11.00
$121.00
12.10
$133.10
13.31
$146.41
14.64
$161.05
Time
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34 Chapter 15 Capital Expenditure Decisions
Using formula (1) to compute the future value after five years of your $100 investment, we have the fol-lowing computation:
Use formula (1) and the values in Table I to compute the future value after 10 years of an $800 investment that earns interest at the rate of 12 percent per year. 3
Present Value In the discussion above, we computed the future value of an investment when the original principal is known. Now consider a slightly different problem.
Suppose you know how much money you want to accumulate at the end of a five-year investment.
Your problem is to determine how much your initial investment needs to be in order to accumulate the desired amount in five years. To solve this problem, we start with formula (1):
Fn P(1r)n
Let’s try out formula (2) on your investment problem, which we analyzed in Exhibit 15–14. Sup-pose you did not know the value of the initial investment required if you want to accumulate $161.05 at the end of five years in an investment that earns 10 percent per year. We can determine the present value of the investment as follows:
Thus, as we knew already, you must invest $100 now in order to accumulate $161.05 after five years in an investment earning 10 percent per year. The present value of $100 and the future value of
$161.05 at time 5 are economically equivalent , given that the annual interest rate is 10 percent. If you are planning to invest the $100 received now, then you should be indifferent between receiving the present value of $100 now or receiving the future value of $161.05 at the end of five years.
When we used formula (2) to compute the present value of the $161.05 cash flow at time 5, we used a process called discounting . The interest rate used when we discount a future cash flow to compute its present value is called the discount rate . The value 1/(1 + r ) n , which appears in formula (2), is called the discount factor . Discount factors, for various combinations of r and n , are tabulated in Table III of Appendix B.
Suppose you want to accumulate $18,000 to buy a car in four years, and you can earn interest at the rate of 8 percent per year on an investment you make now. How much do you need to invest now?
Use formula (2) and the discount factors in Table III of Appendix B to compute the present value of the required $18,000 amount needed at the end of four years. 4
Present Value of a Cash-Flow Series The present-value problem we just solved involved only a sin-gle future cash flow. Now consider a slightly different problem. Suppose you just won $5,000 in the state lottery. You want to spend some of the cash now, but you have decided to save enough to rent a beach condominium during spring break of each of the next three years. You would like to deposit enough in a bank account now so that you can withdraw $1,000 from the account at the end of each of the next three years. The money in the bank account will earn 8 percent per year. The question, then, is how much do you need to deposit? Another way of asking the same question is, what is the present hiL68241_ch15_001-049.indd Page 34 8/20/09 8:56:07 PM user-s176
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Chapter 15 Capital Expenditure Decisions 35
value of a series of three $1,000 cash flows at the end of each of the next three years, given that the discount rate is 8 percent?
Present Value
? $1,000 $1,000 $1,000
Time
0 1 2 3
One way to figure out the answer to the question is to compute the present value of each of the three
$1,000 cash flows and add the three present-value amounts. We can use formula (2) for these calcula-tions, as shown in panel A of Exhibit 15–15. Notice that the present value of each of the $1,000 cash flows is different, because the timing of the cash flows is different. The earlier the cash flow will occur, the higher is its present value.
Examine panel A of Exhibit 15–15 carefully. We obtained the $2,577 total present value by adding three present-value amounts. Each of these amounts is the result of multiplying $1,000 by a discount factor. Notice that we can obtain the same final result by adding the three discount factors first, and then multiplying by $1,000. This approach is taken in panel B of Exhibit 15–15. The sum of the three discount factors is called an annuity discount factor , because a series of equivalent cash flows is called an annuity . Annuity discount factors for various combinations of r and n are tabulated in Table IV of Appendix B.
Now let’s verify that $2,577 is the right amount to finance your three spring-break vacations. Ex-hibit 15–16 shows how your bank account will change over the three-year period as you earn interest and then withdraw $1,000 each year.
Future Value of a Cash-Flow Series To complete our discussion of present-value and future-value concepts, let’s consider the series of $1,000 condo rental payments from the condo owner’s perspec-tive. Suppose the owner invests each $1,000 rental payment in a bank account that pays 8 percent interest per year. How much will the condo owner accumulate at the end of the three-year period? An equivalent question is, What is the future value of the three-year series of $1,000 cash flows, given an annual interest rate of 8 percent? Exhibit 15–17 answers the question in two ways. In panel A of the exhibit, three separate future-value calculations are made using formula (1). Notice that the
$1,000 cash flow at time 1 is multiplied by (1.08) 2 , since it has two years to earn interest. The $1,000 cash flow at time 2 has only one year to earn interest, and the time 3 cash flow has no time to earn interest.
Exhibit 15–15
Present Value of a Series of Cash Flows
A. Present Value of Cash-Flow Series Using Three Independent Present-Value Calculations Present-value formula (formula (2)):
Present value of time 1 cash flow: $ 925.90
Present value of time 2 cash flow: $ 857.30
Present value of time 3 cash flow: $ 793.80
Total: Present value of series of three cash flows $2,577.00
Sum of Three Discount Factors Is the Annuity Discount Factor
B. Present Value of Cash-Flow Series Using the Annuity Discount Factor
Present value of series of three cash flows 5 $1,000(2.5770) 5 $2,577.00 P F n 1
(1r )n
⎛
⎝⎜
⎞
⎠⎟
$ ,1 000 1 $ , (. )
1 0009259
(1.08)1
⎛
⎝⎜
⎞
⎠⎟
$ ,1 000 1 $ , (. )
1 000 8573
⎛
⎝⎜ ⎞
⎠⎟
(1.08)2
$ ,1 000 1 $ , (. )
1 000 7938
⎛
⎝⎜ ⎞
⎠⎟
(1.08)3
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36 Chapter 15 Capital Expenditure Decisions
In panel B of the exhibit, the three-year annuity accumulation factor is used. This factor is the sum of the three accumulation factors used in panel A of the exhibit. The annuity accumulation factors for various combinations of r and n are tabulated in Table II of Appendix B.
Using the Tables Correctly When using the tables in Appendix B to solve future-value and present-value problems, be sure to select the correct table. Table I is used to find the future present-value of a single cash flow, and Table III is used to find the present value of a single cash flow. Table II is used in finding the future value of a series of identical cash flows; Table IV is used in finding the present value of a series of identical cash flows. Be careful not to confuse future value with present value or to confuse a single cash flow with a series of identical cash flows.
Exhibit 15–16
Verification of Present-Value Calculation for Cash-Flow Series
Time 0 Deposit $2,577 ... $2,577.00
Earn interest of $2,577(.08) ... 206.16*
Year 1
Accumulation at time 1 ... $2,783.16 Time 1 Withdrawal to cover time 1 beach trip ... 1,000.00 Amount remaining to earn interest in year 2 ... $1,783.16
Year 2 Earn interest of $1,783.16(.08) ... 142.65*
Accumulation at time 2 ... $1,925.81 Time 2 Withdrawal to cover time 2 beach trip ... 1,000.00 Amount remaining to earn interest in year 3 ... $ 925.81*
Year 3 Earn interest of $925.81(.08) ... 74.06
Accumulation at time 3 ... $ 999.87†
Time 3 Withdrawal to cover time 3 beach trip ... $ 999.87†
Time
*Rounded
†This amount does not equal $1,000 exactly because of rounding error in the computation of each year’s interest.
{ {
{
Exhibit 15–17
Future Value of a Series of Cash Flows
A. Future Value of Cash-Flow Series Using Three Independent Future-Value Calculations Future-value formula (formula (1)): Fn 5 P (1 1 r )n
Future value of time 1 cash flow: $1,000(1 1 .08)2 5 $1,000(1.1664) 5 $1,166.40
Future value of time 2 cash flow: $1,000(1 1 .08)1 5 $1,000(1.0800) 5 $1,080.00
Future value of time 3 cash flow: $1,000 5 $1,000(1.0000) 5 $1,000.00
Total: future value of series of three cash flows $3,246.40 Sum of Three Accumulation Factors Is the Annuity Accumulation Factor
B. Future Value of Cash-Flow Series Using the Annuity Accumulation Factor Future value of series of three cash flows 5 $1,000(3.2464) 5 $3,246.40 hiL68241_ch15_001-049.indd Page 36 8/20/09 8:56:26 PM user-s176
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Chapter 15 Capital Expenditure Decisions 37