CHAPTER 7 Cash Flows and the Time Value of Money 231 is added the depreciation effect of $16,667. As we’ll see later, introducing a vari- able pattern of periodic cash flows can significantly influence the analytical re- sults. Level periodic flows are easiest to deal with, and are generally found in financial contracts of various kinds, but they are quite rare in the business setting. Uneven cash flows are more common and they make the analysis a little more complex—but such patterns can be handled readily for calculation purposes, as we’ll demonstrate. Economic Life The third element, the time period selected for the analysis, is commonly referred to as the economic life of the investment project. For purposes of investment analysis, the only relevant time period is the economic life, as distinguished from the physical life of equipment, or the technological life of a particular process or service. Even though a building or a piece of equipment might be perfectly usable from a physical standpoint, the economic life of the investment is finished if the market for the product or service has disappeared. Similarly, the economic life of any given technology or service is bound up with the economics of the market- place—the best process is useless if the resulting product or service can no longer be sold. At that point, any resources still usable will have to be repositioned, which requires another investment decision, or they might be disposed of for their recovery value. When redeploying such resources into another project, the net investment for that decision would, of course, be the estimated recovery value after taxes. In our simple example, we have assumed a six-year economic life, the period over which the product manufactured with the equipment will be sold. The depreciation life used for accounting or tax purposes doesn’t normally reflect an investment’s true life span, and in this case we’ve only made it equal to the eco- nomic life for simplicity. As we discussed earlier, such write-offs are based on standard accounting and tax guidelines, and don’t necessarily represent the in- vestment’s expected economic usefulness. Terminal (Residual) Value At the end of the economic life an assessment has to be made whether any resid- ual values remain to be recognized. Normally, if one expects a substantial recov- ery of capital from eventual disposal of assets at the end of the economic life, these estimated amounts have to be made part of the analysis. Such recoveries can be proceeds from the sale of facilities and equipment (beyond the minor scrap value assumed in our example), as well as the release of any working capital as- sociated with the investment. Also, there are situations in which an ongoing value of a business, a facility, or a process is expected beyond this specific analysis hel78340_ch07.qxd 9/27/01 11:19 AM Page 231 232 Financial Analysis: Tools and Techniques period chosen. This condition is especially important in valuation analyses, which we’ll discuss in Chapters 11 and 12. For our simple illustration no terminal value is assumed, but later we’ll demonstrate the handling of this concept. Methods of Analysis We’ve now laid the groundwork for analyzing any normal business investment by describing the four essential components of the analysis. Our purpose was to focus on what must be analyzed. We’ll now turn to the question of how this is done— the methods and criteria of analysis that will help us judge the economics of the decision. How do we relate the four basic components— • Net investment • Operating cash inflow • Economic life • Terminal value —to determine the project’s attractiveness? First we’ll dispose quickly of some simplistic methods of analysis, which are merely rules of thumb that intuitively (but incorrectly) grapple with the trade-off between investment and operating cash flows. They are the payback and the simple rate of return, both of which are still used in practice occasionally despite their demonstrable shortcomings. Our major emphasis will be on the measures employing the time value of money as discussed earlier, which enable the analyst to assess the trade-offs be- tween relevant cash flows in equivalent terms, that is, regardless of the timing of their incidence. Those key measures are net present value, the present value pay- back, the profitability index, and the internal rate of return (yield), and in addition, the annualized net present value. We’ll focus on the meaning of these measures, the relationships between them, and illustrate their use on the basis of simple ex- amples. In Chapter 8, we’ll discuss the broader context of business investment analysis, within which these measures play a role as indicators of value creation, and discuss more complex analytical problems. As part of this broader context, we’ll also deal with risk analysis, ranges of estimates, simulation, probabilistic reasoning, and risk-adjusted return standards. Simple Measures Payback This crude rule of thumb directly relates assumed level annual cash inflows from a project to the net investment required. Using the data from our simplified ex- ample, the calculation is straightforward: hel78340_ch07.qxd 9/27/01 11:19 AM Page 232 CHAPTER 7 Cash Flows and the Time Value of Money 233 Payback ϭ ϭ ϭ 4 years The result is the number of years required for the original outlay to be re- paid, answering the question, How long will it be until I get my money back? It’s a rough test of whether the amount of the investment will be recovered within its economic life span. Here, payback is achieved in only four years versus the esti- mated economic life of six years. Recovering the capital is not enough, of course, because from an economic standpoint, one would hope to earn a return on the funds while they are invested. Visualize a savings account in which $100 is deposited, and from which $25 is withdrawn at the end of each year. After four years, the principal will have been repaid. If the bank statement showed that the account was now depleted, the saver would properly demand to be paid the 4 or 5 percent interest that should have been earned every year on the declining balance in the account. We can illustrate these basics of investment economics in Figure 7–2, where we’ve shown how both principal repayment and earnings on the outstanding bal- ance have to be achieved by the cash flow stream over the economic life. We’re again using the simple $100,000 investment, with a level annual after-tax operat- ing cash flow. If the company typically earned 10 percent after taxes on its in- vestments, part of every year’s cash flow would be considered as normal earnings return, with the remainder used to reduce the outstanding balance. The first row shows the beginning balance of the investment in every year. In the second row, normal earnings of 10 percent are calculated on these balances. In the third row are operating cash flows which, when reduced by the normal earnings, are applied against the beginning balances of the investment to calculate every year’s ending balance. The result is an amortization schedule for our simple investment that extends into the sixth year—requiring about two more years of $100,000 $25,000 Net investment Average annual operating cash flow FIGURE 7–2 Amortization of $100,000 Investment at 10 Percent Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Beginning balance . . . . . $100,000 $85,000 $68,500 $50,350 $30,385 $ 8,424 Normal company earnings @ 10% . . . . . 10,000 8,500 6,850 5,035 3,039 842 Operating cash inflows of project . . . . . . . . . . . 25,000 25,000 25,000 25,000 25,000 25,000 Ending balance to be recovered . . . . . . . . 85,000 68,500 50,350 30,385 8,424 Ϫ15,734 Simple payback (4 ϫ $25,000) . . . . . . . Year 4 hel78340_ch07.qxd 9/27/01 11:19 AM Page 233 234 Financial Analysis: Tools and Techniques annual benefits than the simple payback measure would suggest. If the project ended in Year 4, an opportunity loss of about $30,400 would be incurred, and in Year 5, the loss would be about $8,400. Only in Year 6 will the remaining princi- pal balance have been recovered and an economic gain of about $15,700 achieved. As we’ll see shortly, all modern investment criteria are based on the basic rationale underlying this example, with some refinements in the precise cal- culations used. We can now quickly dispose of the payback measure as an indicator of in- vestment desirability: It’s insensitive to the economic life span and thus not a meaningful criterion of earnings power. It’ll give the same “four years plus some- thing extra” reading on other projects that have similar cash flows but 8- or 10-year economic lives, even though those projects would be clearly superior to our example. It implicitly assumes level annual operating cash flows, and cannot properly evaluate projects with rising or declining cash flow patterns—although these are very common. It cannot accommodate any additional investments made during the period, or recognize capital recoveries at the end of the economic life. The only situation where the measure has some applicability is in compar- ing a series of simple projects with quite similar cash flow patterns, but even then it is more appropriate to apply the economic techniques that are readily available on calculators and spreadsheets. However, it’s possible to make use of a refined concept of payback that is expressed in economic terms, but this measure requires the discounting process to arrive at the so-called present value payback. It’s one of the indicators of invest- ment desirability that build a return requirement into the analysis, and we’ll dis- cuss it in detail later. Simple Rate of Return Again, only passing comments are warranted about this simplistic rule of thumb, which in fact is the inverse of the basic payback formula. It states the desirability of an investment in terms of a percentage return on the original outlay. The method shares all of the shortcomings of the payback, because it again relates only two of the four critical aspects of any project, net investment and operating cash flows, and ignores the economic life and any terminal value: ϭ ϭ ϭ 25% What this result actually indicates is that $25,000 happens to be 25 percent of $100,000, because there’s no reference to economic life and no recognition of the need to amortize the investment. The measure will give the same answer whether the economic life is 1 year, 10 years, or 100 years. The 25 percent return indicated here would be economically valid only if the investment provided $25,000 per year in perpetuity—not a very realistic condition! $25,000 $100,000 Average annual operating cash flow Net investment Return on investment hel78340_ch07.qxd 9/27/01 11:19 AM Page 234 CHAPTER 7 Cash Flows and the Time Value of Money 235 Economic Investment Measures Earlier, we described business investment analysis as the process of weighing the economic trade-off between current dollar outlays and future net cash flow bene- fits that are expected to be obtained over a relevant period of time. This economic valuation concept applies to all types of investments, whether made by individuals or businesses. The time value of money is employed as the underlying methodol- ogy in every case. We’ll use the basic principles of discounting and compounding discussed earlier to explain and demonstrate the major measures of investment analysis. These measures utilize such principles to calculate the quantitative basis for making economic choices among investment propositions. Net Present Value The net present value (NPV) measure has become the most commonly used indi- cator in corporate economic and valuation analysis, and is accepted as the pre- ferred measure in the widest range of analytical processes. It weighs the cash flow trade-off among investment outlays, future benefits, and terminal values in equiv- alent present value terms, and allows the analyst to determine whether the net balance of these values is favorable or unfavorable—in other words, the size of the economic trade-off involved relative to an economic return standard. From the standpoint of creating shareholder value, a positive net present value implies that the proposal, if implemented and performing as expected, will add value because of the favorable trade-off of time-adjusted cash inflows over outflows. In contrast, a negative net present value will destroy value due to an excess of time-adjusted cash outflows over inflows. As a basic rule one can say the higher the positive NPV, the better the value creation potential. To use the tool, a rate of discount representing a normal expected rate of re- turn first must be specified as the standard to be met. As we’ll see, this rate is commonly based on a company’s weighted average cost of capital, which em- bodies the return expectations of both equity and debt providers of the company’s capital structure, as described in Chapter 9. Next, the inflows and outflows over the economic life of the investment proposal are specified and discounted at this return standard. Finally, the present values of all inflows (positive amounts) and outflows (negative amounts) are summed. The difference between these sums rep- resents the net present value. NPV can be positive, zero, or negative, depending on whether there is a net inflow, a matching of cash flows, or a net outflow over the economic life of the project. Used as a standard of comparison, the measure indicates whether an invest- ment, over its economic life, will achieve the expected return standard applied in the calculation, given that the underlying estimates are in fact realized. Inasmuch as present value results depend on both timing of the cash flows and the level of the required rate of return standard, a positive net present value indicates that the cash flows expected to be generated by the investment over its economic life will: hel78340_ch07.qxd 9/27/01 11:19 AM Page 235 236 Financial Analysis: Tools and Techniques • Recover the original outlay (as well as any future capital outlays or recoveries considered in the analysis). • Earn the specified return standard on the outstanding balance. • Provide a “cushion” of economic value over and above meeting the minimum standard. Conversely, a negative net present value indicates that the project is not achieving the return standard and thus will cause an economic loss if imple- mented. A zero NPV is value neutral. Obviously, the result will be affected by the level of benefits assumed, the specific timing pattern of the various cash flows, and the relative magnitudes of the amounts involved. Another word should be said at this point about the rate of discount. From an economic standpoint, it should be the rate of return an investor normally enjoys from investments of similar nature and risk, as we explained in our discussion of the time value of money. In effect, this standard represents an opportunity rate of return. In a corporate setting, the choice of a discount rate is complicated both by the variety of investment possibilities and by the types of financing provided by both owners and lenders. The corporate return standard normally used to dis- count business investment cash flows should reflect the minimum return require- ment that will provide the normally expected level of return on the company’s investments, under normal risk conditions. The most commonly employed standard is based on the overall corporate cost of capital, which takes into account shareholder expectations, business risk, and leverage. As we’ve mentioned before, shareholder value can be created only by making investments whose returns exceed the cost of capital. Therefore, the actual standard established by a company will often be set above the cost of cap- ital, reflecting a specific management objective to achieve returns higher than the cost of capital. Sometimes a corporate return standard is separated into a set of multiple discount rates for different lines of business within a company, in order to recognize specific risk differentials. We’ll deal with these concepts in greater depth in Chapters 8 and 9. For purposes of this discussion, we’ll assume that man- agement has chosen an appropriate return standard with which to discount invest- ment cash flows, and we’ll focus on how present value measures are used to assess potential investments on an economic basis. As a first step, it’s generally helpful to lay out the pertinent information pe- riod by period to give us a proper time perspective. Ahorizontal time scale match- ing spreadsheet patterns should be used, on which the periods are marked off, as Figure 7–3 shows. Positive and negative cash flows are then inserted as arrows at the appropriate positions in time, scaled to the size of the dollar amounts. Note that the time scale begins at point 0, the present decision point, and extends out as far as the project’s economic life requires. Any events that occurred prior to the decision point (shown as negative periods) are not relevant to the analysis, unless the decision specifically causes a recovery of past expenditures, such as the sale of old assets. hel78340_ch07.qxd 9/27/01 11:19 AM Page 236 TEAMFLY Team-Fly ® CHAPTER 7 Cash Flows and the Time Value of Money 237 To illustrate the process, let’s return to the simple investment example used earlier in the chapter. We’ll show the numerical information as a table in Fig- ure 7–4. Note the similarity in approach to the simple amortization process we used in Figure 7–2 (p. 233). Figure 7–4 demonstrates that the pattern in our sam- ple net investment of $100,000, with six annual benefit inflows of $25,000 from Year 1 through Year 6, results in a net present value of almost $16,000. This as- sumes that our company considers the relatively low rate of 8 percent after taxes a normal earnings standard. The total initial outflow will have been recovered over the six-year period, while 8 percent after taxes will have been earned all along on the declining investment balance outstanding during the project life. The positive net present value shows that a value creation of about $15,600 in equiva- lent present value dollars can be expected if the cash flow estimates are correct and if the project does live out its full economic life. FIGURE 7–3 Generalized Time Scale for Investment Analysis Cash inflows Decision point (present) –2 –1 Time 0 12345678 Cash outflows FIGURE 7–4 Net Present Value Analysis at 8 Percent* Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Totals Investment outlay (outflow) . . . . . . . . $Ϫ100,000 000000$Ϫ100,000 Benefits (inflows) . . 0 $25,000 $25,000 $25,000 $25,000 $25,000 $25,000 150,000 Present value factors @ 8%** . . 1.000 0.926 0.857 0.794 0.735 0.681 0.630 Present values of cash flows . . . . . . Ϫ100,000 23,150 21,425 19,850 18,375 17,025 15,750 15,575 Cumulative present values . . . . . . . . . Ϫ100,000 Ϫ76,850 Ϫ55,425 Ϫ35,575 Ϫ17,200 Ϫ175 15,575 Net present value @ 8% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $ 15,575 *This exhibit is available in an interactive format (TFA Template)—see “Analytical Support” on p. 250. **From Table 7–I (p. 252), which assumes benefits occur at year-end. Because the inflows are level, we could instead use an annuity factor of 4.623 from Table 7–II (p. 253) for an identical result. hel78340_ch07.qxd 9/27/01 11:19 AM Page 237 238 Financial Analysis: Tools and Techniques In the simple payback concept we discussed earlier, we referred to the re- covery of the original investment plus “something extra.” The critical difference between simple payback and net present value, however, is the fact that net pres- ent value has a built-in return requirement in addition to full recovery of the in- vestment. Thus, the value “cushion” implicit in a positive net present value is truly a calculated economic value gain that goes beyond satisfying the required return standard. In fact, we can see from the cumulative present value line that if the project performs as expected, the cash flows are sufficient to recover the principal and earn 8 percent by the end of period five, where the cumulative present value is very close to zero. If a higher earnings standard had been required, say 12 percent, the results would be those shown in Figure 7–5. The net present value remains positive, but the size of the value creation has dramatically decreased to only $2,800. We would expect such a decrease, because at the higher discount rate, the present value of the future cash inflows must decline, with all other circumstances un- changed. Note that this time the present value payback requires almost the full six years. At an assumed earnings standard of 14 percent, the net present value shrinks even further. In fact, it is transformed into a negative result ($25,000 ϫ 3.889 Ϫ $100,000 ϭ Ϫ$2,775). These results reflect the great sensitivity of net present value to the choice of earnings standards, especially at higher rates. The cumulative present value row in the two sets of calculations illustrates the importance of the length of the economic life of the investment. We can ob- serve that the time required for the cumulative present value to turn positive (and thus achieve a present value payback) was lengthened as the earnings standard FIGURE 7–5 Net Present Value Analysis at 12 Percent* Year 0 Year 1 Year 2 Year 3 Year 4 Year 5 Year 6 Totals Investment outlay (outflow) . . . $Ϫ100,000 0 0 0 0 0 0 $Ϫ100,000 Benefits (inflows) . . . 0 $Ϫ25,000 $Ϫ25,000 $Ϫ25,000 $Ϫ25,000 $ 25,000 $25,000 150,000 Present value factors @ 12%** . . . 1.000 0.893 0.797 0.712 0.636 0.567 0.507 Present values of cash flows . . . . . . Ϫ100,000 22,325 19,925 17,800 15,900 14,175 12,675 2,800 Cumulative present values . . . $Ϫ100,000 $Ϫ77,675 $Ϫ57,750 $Ϫ39,950 $Ϫ24,050 $Ϫ9,875 $ 2,800 Net present value @ 12% . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . $Ϫ 1 2,800 *This exhibit is available in an interactive format (TFA Template)—see “Analytical Support” on p. 250. **As in Figure 7–4, we could use 4.112 times $25,000 from Table 7–II, p. 253. hel78340_ch07.qxd 9/27/01 11:19 AM Page 238 CHAPTER 7 Cash Flows and the Time Value of Money 239 was raised. At 8 percent, the economic life had to last about five years for the switch to occur (the net present value after the benefits of Year 5 is almost zero), while at 12 percent, most of the sixth year of economic life was necessary for a positive turnaround (approximately $10,000 of negative present value remaining at the end of Year 5 has to be recovered from the benefits of Year 6). At 14 percent more time than the economic life was required to achieve a positive net present value, making the project uneconomic. In our example we assumed a level operating cash inflow of $25,000. Un- even cash flow patterns have a notable impact on the results, although the method of calculation remains the same. The net present value approach can, of course, accommodate any combination of cash flow patterns without difficulty. The reader is invited to test this, using a cash inflow pattern that rises from, say, $15,000 to $40,000, and one that falls from $40,000 to $15,000, each totaling $150,000 over six years. When using a spreadsheet, the reader can select the npv function, specify the discount rate, enter the cash flows from Year 1 through 6, and subtract the net investment from the result to arrive at the net present value. Net present value is a direct measure of value creation as well as a screen- ing device that indicates whether a stipulated minimum return standard, such as the cost of capital, can be met over an investment proposal’s economic life. We stated before that when net present value is positive, there is potential for a return in excess of the standard and therefore, economic value creation. When net pres- ent value is negative, the minimum return standard and capital recovery cannot be achieved with the projected cash flows. When net present value is close to or exactly zero, the return standard has just been met. In this case the investment will be value neutral. All of these conditions, of course, hold only on the assumption that the cash flow estimates and the projected life will in fact be achieved. The graphic representation of net present value in Figure 7–6 demonstrates the three outcomes. While net present value is the most frequently used tool in evaluating investment alternatives, it doesn’t answer all our questions about the economic FIGURE 7–6 A Representation of Net Present Value Cumulative positive and negative present values of different cash flow levels + = – Initial outlay Time Terminal value Cash flows Present values of cash flows hel78340_ch07.qxd 9/27/01 11:19 AM Page 239 240 Financial Analysis: Tools and Techniques attractiveness of capital outlays. For example, when comparing different projects, how does one evaluate the respective size of the value creation calculated with a given return standard, particularly if the investment amount differs signifi- cantly? Also, to what extent is achieving the expected economic life a factor in such comparisons? Furthermore, how does one quantify the potential errors and uncertainties inherent in the cash flow estimates, and how does the measure assist in investment choices if such deviations are significant? Finally, one can ask what specific re- turn the project will yield if all estimates are in fact realized? Further measures and analytical methods are necessary to answer these questions, and we’ll show later how a combination of techniques helps to narrow the choices to be made. Present Value Payback We’ve already referred to this measure during our discussion of net present value. As we saw, the concept establishes the minimum life necessary for an investment to operate as expected, and still meet the return standard of the present value analysis, a break-even condition in value creation. In other words, present value payback is achieved at the specific point in time when the cumulative positive present value of cash benefits equals the cumulative negative present value of all the cash outlays—in fact, a zero net present value condition. It’s the point in the project’s economic life when the original investment has been fully amortized and a return equal to the built-in return standard has been achieved on the declining balance—the point at which the project becomes economically attractive and can begin to create value. Figure 7–7 below provides a visual representation of this concept similar to the one for net present value. Recall that in Figures 7–4 and 7–5, we included a row for the cumulative net present value of the project. It served as a visual check for determining the point at which net present value turned positive. The present value payback for our example, using a discount rate of 8 percent was about 5 years, while a 12 percent standard required almost 6 years, just about the full economic life of FIGURE 7–7 A Representation of Present Value Payback Cumulative present values to point of payback = Initial outlay Time Margin for risk and value creation Terminal value Cash flows Present values of cash flow s Achieved payback hel78340_ch07.qxd 9/27/01 11:19 AM Page 240 [...]... 5.954 6. 047 6. 128 6. 198 6. 259 6. 312 6. 359 6. 399 6. 434 6. 464 6. 491 6. 514 6. 534 6. 551 6. 566 6. 617 6. 642 6. 654 6. 661 6. 665 0. 862 1 .60 5 2.2 46 2.798 3.274 3 .68 5 4.039 4.344 4 .60 7 4.833 5.029 5.197 5.342 5. 468 5.575 5 .66 9 5.749 5.818 5.877 5.929 5.973 6. 011 6. 044 6. 073 6. 097 6. 118 6. 1 36 6.152 6. 166 6. 177 6. 215 6. 234 6. 242 6. 2 46 6.249 0.847 1. 566 2.174 2 .69 0 3.127 3.498 3.812 4.078 4.303 4.494 4 .65 6 4.793 4.910... 7. 469 7. 562 7 .64 5 7.718 7.784 7.843 7.8 96 7.943 7.984 8.022 8.055 8.1 76 8.244 8.282 8.304 6. 324 0.877 1 .64 7 2.322 2.914 3.433 3.889 4.288 4 .63 9 4.9 46 5.2 16 5.453 5 .66 0 5.842 6. 002 6. 142 6. 265 6. 373 6. 467 6. 550 6. 623 6. 687 6. 743 6. 792 6. 835 6. 873 6. 9 06 6.935 6. 961 6. 983 7.003 7.070 7.105 7.123 7.133 7.140 0.870 1 .62 6 2.283 2.855 3.352 3.784 4. 160 4.487 4.772 5.019 5.234 5.421 5.583 5.724 5.847 5.954 6. 047... 10 .67 5 10.810 10.935 11.051 11.158 11.258 11 .65 4 11.925 12.108 12.234 12.3 76 0.909 1.7 36 2.487 3.170 3.791 4.355 4. 868 5.335 5.759 6. 145 6. 495 6. 814 7.103 7. 367 7 .60 6 7.824 8.022 8.201 8. 365 8.514 8 .64 9 8.772 8.883 8.985 9.077 9. 161 9.237 9.307 9.370 9.427 9 .66 4 9.779 9. 863 9.915 9. 967 0.893 1 .69 0 2.402 3.037 3 .60 5 4.112 4. 564 4. 968 5.328 5 .65 0 5.937 6. 194 6. 424 6. 628 6. 811 6. 974 7.120 7.250 7. 366 7. 469 ... 29.490 31.424 34. 761 0. 962 1.8 86 2.775 3 .63 0 4.452 5.242 6. 002 6. 733 7.435 8.111 8. 760 9.385 9.9 86 10. 563 11.118 11 .65 2 12.1 16 12 .65 9 13.134 13.590 14.029 14.451 14.857 15.247 15 .62 2 15.983 16. 330 16. 663 16. 984 17.292 18 .66 5 19.793 20.720 21.482 22 .62 3 0.952 1.859 2.722 3.545 4.329 5.075 5.7 86 6. 463 7.108 7.722 8.307 8. 863 9.393 9.898 10.379 10.838 11.274 11 .69 0 12.0 86 12. 463 12.821 13. 163 13.489 13.799... 14 .64 3 14.898 15.141 15.372 16. 374 17.159 17.774 18.2 56 18.929 0.943 1.833 2 .67 3 3. 465 4.212 4.917 5.582 6. 210 6. 802 7. 360 7.887 8.384 8.853 9.295 9.712 10.1 06 10.477 10.828 11.158 11.470 11. 764 12.042 12.303 12.550 12.793 13.003 13.211 13.4 06 13.591 13. 765 14.498 15.0 46 15.5 46 15. 762 16. 161 0.9 26 1.783 2.577 3.312 3.993 4 .62 3 5.2 06 5.747 6. 247 6. 710 7.139 7.5 36 7.904 8.244 8.559 8.851 9.122 9.372 9 .60 4... 0.158 0.1 46 0.135 0.125 0.1 16 0.107 0.099 0. 066 0.0 46 0.031 0.021 0.010 0.909 0.8 26 0.751 0 .68 3 0 .62 1 0. 564 0.513 0. 467 0.424 0.308 0.350 0.319 0.290 0. 263 0.239 0.218 0.198 0.180 0. 164 0.149 0.135 0.123 0.112 0.102 0.092 0.084 0.0 76 0. 069 0. 063 0.057 0.0 36 0.022 0.014 0.009 0.002 0.893 0.797 0.712 0 .63 6 0. 567 0.507 0.452 0.404 0. 361 0.322 0.287 0.257 0.229 0.205 0.183 0. 163 0.1 46 0.130 0.1 16 0.104 0.093... 26 27 28 29 30 35 40 45 50 60 0.990 0.980 0.971 0. 961 0.951 0.942 0.933 0.923 0.914 0.905 0.8 96 0.887 0.879 0.870 0. 861 0.853 0.844 0.8 36 0.828 0.820 0.811 0.803 0.795 0.788 0.780 0.772 0. 764 0.757 0.749 0.742 0.7 06 0 .67 2 0 .63 9 0 .60 8 0.550 0.980 0. 961 0.942 0.924 0.9 06 0.888 0.871 0.853 0.837 0.820 0.804 0.788 0.773 0.758 0.743 0.728 0.714 0.700 0 .68 6 0 .67 3 0 .66 0 0 .64 7 0 .63 4 0 .62 2 0 .61 0 0.598 0.5 86. .. 2.937 3.0 76 3.184 3. 269 3.335 3.387 3.427 3.459 3.483 3.503 3.518 3.529 3.539 3.5 46 3.551 3.5 56 3.559 3. 562 3. 564 3. 566 3. 567 3. 568 3. 569 3. 569 3.571 3.571 3.571 3.571 3.571 0. 769 1. 361 1.8 16 2. 166 2.4 36 2 .64 3 2.802 2.925 3.019 3.092 3.147 3.190 3.223 3.249 3. 268 3.283 3.295 3.304 3.311 3.3 16 3.320 3.323 3.325 3.327 3.329 3.330 3.331 3.331 3.332 3.332 3.333 3.333 3.333 3.333 3.333 0.741 1.289 1 .69 6 1.997... Page 253 2% 0.990 1.970 2.941 3.902 4.853 5.795 6. 728 7 .65 2 8. 566 9.471 10. 368 11.255 12.134 13.004 13. 865 14.718 15. 562 16. 398 17.2 26 18.0 46 18.857 19 .66 0 20.4 56 21.243 22.023 22.795 23. 560 24.3 16 25. 066 25.808 29.408 32.835 36. 094 39.1 96 44.955 11:19 AM 1% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 35 40 45 50 60 9/27/01 Number of Periods hel78340_ch07.qxd Present... 4.203 4. 265 4.315 4.357 4.391 4.419 4.442 4. 460 4.4 76 4.488 4.499 4.507 4.514 4.520 4.524 4.528 4.531 4.534 4.541 4.544 4.545 4.545 4.545 0.8 06 1.457 1.981 2.404 2.745 3.020 3.242 3.421 3. 566 3 .68 2 3.7 76 3.851 3.912 3. 962 4.001 4.033 4.059 4.080 4.097 4.110 4.121 4.130 4.137 4.143 4.147 4.151 4.154 4.157 4.159 4. 160 4. 164 4. 166 4. 166 4. 167 4. 167 0.800 1.440 1.952 2. 362 2 .68 9 2.951 3. 161 3.329 3. 463 3.571 . 11:19 AM Page 241 242 Financial Analysis: Tools and Techniques estimates. It sharpens the analyst’s understanding of the relationship of economic life and acceptable performance, and is a much improved. 235 2 36 Financial Analysis: Tools and Techniques • Recover the original outlay (as well as any future capital outlays or recoveries considered in the analysis). • Earn the specified return standard. 11:19 AM Page 231 232 Financial Analysis: Tools and Techniques period chosen. This condition is especially important in valuation analyses, which we’ll discuss in Chapters 11 and 12. For our simple