As explained in earlier chapters, the present value of a future cash flow is the amount that a knowledgeable investor would pay today for the right to receive that future amount. Arriving at a present value figure depends on (1) the amount of the future cash flow, (2) the length of time that the investor must wait to receive the cash flow, and (3) the rate of return required by the investor. Discounting is the process by which the present value of cash flows (referred to as the discounted cash flows ) is determined.
The use of present value tables to discount future cash flows is demonstrated in Appendix B (at the end of this text). Those who are not familiar with the concept of present value or with present value tables should read the appendix before continuing with this chapter.
For your convenience, the two present value tables presented in the appendix are repeated in this chapter. Exhibit 26–3 shows the present value of a single lump-sum payment of $1
Evaluate capital investment proposals using (c) discounted cash flows.
L e a r n i n g O b j e c t i v e t
e
LO3
Present Value of $1 Due in n Periods*
Number of Discount Rate Periods
(n) 1% 11⁄2% 5% 6% 8% 10% 12% 15% 20%
1 .990 .985 .952 .943 .926 .909 .893 .870 .833
2 .980 .971 .907 .890 .857 .826 .797 .756 .694
3 .971 .956 .864 .840 .794 .751 .712 .658 .579
4 .961 .942 .823 .792 .735 .683 .636 .572 .482
5 .951 .928 .784 .747 .681 .621 .567 .497 .402
6 .942 .915 .746 .705 .630 .564 .507 .432 .335
7 .933 .901 .711 .665 .583 .513 .452 .376 .279
8 .923 .888 .677 .627 .540 .467 .404 .327 .233
9 .914 .875 .645 .592 .500 .424 .361 .284 .194
10 .905 .862 .614 .558 .463 .386 .322 .247 .162
20 .820 .742 .377 .312 .215 .149 .104 .061 .026
24 .788 .700 .310 .247 .158 .102 .066 .035 .013
36 .699 .585 .173 .123 .063 .032 .017 .007 .001
*The present value of $1 is computed by the formula p 1/(1 i )n, where p is the present value of $1, i is the discount rate, and n is the number of periods until the future cash flow will occur. Amounts in this table have been rounded to three decimal places and are shown for a limited number of periods and discount rates. Many calculators are programmed to use this formula and can compute present values when the future amount is entered along with values for i and n.
Exhibit 26–3
PRESENT VALUE OF $1 PAYABLE IN n PERIODS
Present Value of $1 to Be Received Periodically for n Periods Number of
Periods Discount Rate
(n) 1% 11⁄2% 5% 6% 8% 10% 12% 15% 20%
1 0.990 0.985 0.952 0.943 0.926 0.909 0.893 0.870 0.833
2 1.970 1.956 1.859 1.833 1.783 1.736 1.690 1.626 1.528
3 2.941 2.912 2.723 2.673 2.577 2.487 2.402 2.283 2.106
4 3.902 3.854 3.546 3.465 3.312 3.170 3.037 2.855 2.589
5 4.853 4.783 4.329 4.212 3.993 3.791 3.605 3.352 2.991
6 5.795 5.697 5.076 4.917 4.623 4.355 4.111 3.784 3.326
7 6.728 6.598 5.786 5.582 5.206 4.868 4.564 4.160 3.605
8 7.652 7.486 6.463 6.210 5.747 5.335 4.968 4.487 3.837
9 8.566 8.361 7.108 6.802 6.247 5.759 5.328 4.772 4.031
10 9.471 9.222 7.722 7.360 6.710 6.145 5.650 5.019 4.192
20 18.046 17.169 12.462 11.470 9.818 8.514 7.469 6.259 4.870
24 21.243 20.030 13.799 12.550 10.529 8.985 7.784 6.434 4.937
36 30.108 27.661 16.547 14.621 11.717 9.677 8.192 6.623 4.993
Exhibit 26–4
PRESENT VALUE OF A $1 ANNUITY RECEIVABLE EACH PERIOD FOR n PERIODS
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to be received in n periods (years) in the future. Exhibit 26–4 shows the present value of a $1 annuity —that is, $1 to be received each year for n consecutive years. For illustrative purposes, both tables have been kept short. They include only selected discount rates and only extend for a limited number of periods. However, they contain the appropriate rates and periods for all of the problem material in this chapter.
The discount rate may be viewed as an investor’s required rate of return. The present value of an investment’s future cash flows is the maximum amount that an investor should be willing to pay for the investment and still expect to earn the required rate of return. Therefore, an investment is considered desirable when its cost is less than the present value of its future cash flows. In such cases, the expected rate of return exceeds the rate of return required by the investor. Conversely, when the cost of an investment exceeds the present value of its future cash flows, its expected return is less than that required by the investor.
The higher the discount rate being used, the lower the resulting present value figure will be. It follows that, the higher the required rate of return for a particular investment, the less an inves- tor will be willing to pay for the investment. The appropriate discount rate (or required rate of return) for determining the present value of a specific investment depends on the nature of the investment, the alternative investment opportunities available, and the investor’s cost of capital.
The required rate of return is adjusted in many companies for a variety of strategic reasons.
For example, management may allow a lower required rate of return when there is a strategic necessity to penetrate a new market or to acquire new technology. Also, for certain capital expenditures, such as new technology, estimating the cash flows and the timing of those cash flows can be extremely difficult. Managers know that establishing a high required rate of return will place projects with higher cash flows occurring in the more distant future at a dis- advantage. Using a high discount rate for projects where high net cash flows are not received until several years in the future will result in low net present values.
Let us now apply the concept of discounting cash flows to our example. We shall assume that the Stars require a 15 percent annual rate of return on all capital investments. As shown in Exhibit 26–5 , the 10 vending machines are expected to generate annual net cash inflows of
$24,000 for five years. Exhibit 26–4 shows that the present value of $1 to be received annu- ally for five years, discounted at 15 percent, is 3.352. Therefore, the present value of $24,000 received annually for five years is $24,000 3.352, or $80,448. Notice in Exhibit 26–5 that, even though the total annual cash inflows are $120,000, their present value is only $80,448.
In addition to these annual cash flows, Wilson expects that VendiCorp will repurchase the machines from the Stars at the end of five years for $5,000 (their salvage value). Referring to Exhibit 26–3 , we see that the present value of $1 to be received in five years, discounted at 15 percent, is .497. Thus, the present value of $5,000 to be received at the end of five years
TIME
DISCOUNTING
DISCOUNTING Year 1 5
$24,000 Initial cash
outflow 5
$75,000
Present value of annual cash inflows 5 $80,448
Present value of Year 5 inflow 5 $2,485
Year 5 salvage inflow 5 $5,000 Sum of annual cash inflows 5
$120,000
Year 2 5
$24,000
Year 3 5
$24,000
Year 4 5
$24,000
Year 5 5
$24,000
Exhibit 26–5 PRESENT VALUE OF CASH FLOWS FOR VENDICORP
Rev. Confirming Pages
Capital Investment Decisions 1121
is $5,000 .497, or $2,485. Using the information in Exhibit 26–5 , we may now analyze the proposal to invest in the 10 vending machines in the following manner:
Present value of expected annual cash flows ($24,000 3.352) . . . $80,448 Present value of proceeds from disposal ($5,000 .497) . . . 2,485 Total present value of investment’s future cash flows . . . $82,933 Cost of investment (payable in advance) . . . 75,000 Net present value of proposed investment . . . $ 7,933
Investment’s net present value
This analysis indicates that the present value of the vending machines’ future cash flows, discounted at a rate of 15 percent, amounts to $82,933. This is the maximum amount that the Stars could invest in these machines and still expect to earn the required annual return of 15 percent. As the actual cost of the investment is only $75,000, the machines have the poten- tial to earn a rate of return in excess of 15 percent.
The net present value of VendiCorp’s proposal is the difference between the total present value of the net cash flows and the cost of the investment. If the net present value is equal to zero, the rate of return is equal to the discount rate. A positive net present value means that the investment is expected to provide a rate of return greater than the discount rate, whereas a negative net present value means that the investment is likely to yield a return less than the discount rate. In financial terms, proposals with a positive net present value are considered acceptable and those with a negative net present value are viewed as unacceptable. These relationships are summarized in Exhibit 26–6 .
Discuss the relationship between net present value and an investor’s required rate of return.
L e a r n i n g O b j e c t i v ee
LO4
On the basis of our cash flow analysis, purchase of the vending machines appears to be an acceptable proposal. However, there are numerous nonfinancial issues that might be consid- ered before making a decision based purely on the numbers.
For instance, all of the revenue and expense estimates used in determining these financial measures were supplied by VendiCorp. It is entirely possible that these estimates may be overly optimistic. Furthermore, Wilson knows nothing about VendiCorp’s business reputa- tion. What assurances does he have that VendiCorp will honor its agreement to stock the machines with fresh merchandise before each game, maintain the machines when they break down, and repurchase the machines for $5,000 at the end of five years? Has Wilson obtained bids from other suppliers of vending machines? Or has he considered an arrangement with an outside catering service to provide concessions at the Stars’ home ball games? Finally, perhaps there are unrelated investment opportunities to consider, such as investing in a new pitching machine, team uniforms, or new stadium seats.
Net Present Value (NPV) Interpretation Action
NPV Zero Return exceeds the discount rate. Accept
NPVZero Return is equal to the discount rate. Accept
NPV Zero Return is less than the discount rate. Reject
Exhibit 26–6
SUMMARY OF
RELATIONSHIPS AMONG NPV, THE DISCOUNT RATE, AND PROJECT ACCEPTABILITY
You are attending your first meeting with the management team for the Maine LobStars.
Your job is to discuss planned capital budgeting projects to get management’s approval.
Management, including the owner, Steve Wilson, is accustomed to looking at payback period and return on average assets. However, you have also prepared net present value information for management’s review. Steve Wilson complains that the net present value information is redundant and unnecessary. How will you respond?
(See our comments on the Online Learning Center Web site.)
Y O U R T U R NY O U R T U R N You as a Chief Financial Officer
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