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1、ROR（IRR) 的优缺点： 易理解；与基准点无关； 在项目寿命期内任意时刻，使项目收益换算值之和等于费用换算值之和的利率称为ROR (IRR)。 所以，ROR的计算，可以用NPV(i)=0, NFV(i)=0, NAV(i)=0 进行计算。,3.4.2 Multiple Rates of Return In addition to the requirement of incremental analysis, the ROR analysis method also has another drawback. This method works well when a given alte
2、rnative requires an initial investment which is followed by future benefits. For this type of alternative, the cash flow profile can be shown as negative cash flow in the first year followed by positive cash flow in the future years. For example, if we consider an investment of $1,000 which will res
3、ult in a $300 annual benefit for the next six years with a $500 salvage value at the end of six years, the cash profile can be written as.,In this profile, there is only one sign change in cash profile between Years 0 and 1. Such profile is amenable to conventional ROR analysis. Note that the ROR ca
4、lculation requires solving a polynomial of i. We calculate the value of i for which the NPV is zero. For economic analysis, we are only interested in obtaining positive, real values of i for which the NPV is equal to zero. When there is only one sign change in the cash flow profile, as shown above,
5、we can only obtain one or zero positive solutions.,In some instances, however, the sign changes more than once in a cash flow profile. Under these circumstances, we may obtain more than one real ROR. The rule of signs for polynomial solution states that the number of real solutions between -l and is
6、 never greater than the number of sign changes. That is, if we have two sign changes, we may obtain a maximum of two rates of return values between -100% and . The following example illustrates the calculation of the number of feasible solutions.,Example 3.25 For the following four cash ROR between
7、-100% and .,Solution To calculate the maximum number of possible real solutions between -100% and , we can calculate the number of sign changes. For cash flow A, there is only one sign change between period 0 and l. For B, there are three sign changes; between periods 0 and l, periods l and 2, and 3
8、. Similarly, for cash flow C, there are four sign changes, and for cash flow D, there are five sign changes. As stated before, the number of sign changes will indicate the maximum number of possible real solutions. That is, for cash flow profile C, the number of real solutions between 100% and can b
9、e either 4, 3, 2, l. or zero.,The number of possible real solutions can be narrowed down even further by applying cumulative cash flow sign test. If we assume Aj to be a cash flow in period j, then we can define the cumulative cash flow Cj as, If Cj starts with a negative number and changes sign onl
10、y once, we will obtain only one positive solution. This cumulative cash flow method may allow us to narrow down the number of possible solutions for the ROR.,Example 3.26 Reconsider the cash flows provided in Example 3.23. Applying the cumulative cash flow sign test, investigate the possibility of n
11、arrowing the number of positive ROR solutions. Solution We can calculate the cumulative cash flows for each of the profiles as follows:,As a sample calculation, for period 3 for Project A, we can calculate. C3 = -100+20+20+30 =-30 For period 6, C6 = -100+20+20 +30+20+30+30=50 Looking at the cash flo
12、w profiles, for cash flow profiles A and D, there in only one sign change in the cumulative cash flow profile. That is, we will obtain only one unique positive value of the ROR. For cash flow profiles B and C, the results of cumulative cash flow profiles are inconclusive. We cannot reduce the possib
13、le number of solutions by using the cumulative cash flow profile for these two profiles.,Example 3.27 An in-fill drilling project is being considered for an existing oil field to accelerate oil recovery. The following two scenarios, based on two alternatives (no in-fill drilling versus in-fill drill
14、ing) are predicted. Which alternative would you select? The numbers are in millions. Assume that MROR is 20%.,Solution The first step in ROR analysis is to compare individual RORs for each alternative with the MROR. For alternative A, there is no sign change in the cash flow profile. Therefore, the
15、ROR for alternative A is . For alternative B, ROR can be shown to be greater than 20% (the ROR for alternative B is 260%). Therefore, both alternatives satisfy the requirement that the ROR be greater than the MROR. The next step is to conduct the incremental analysis. The cash flow profile for incre
16、mental values can be written as,The cash flow profile shows more than one sign change. The cumulative cash flow profile also shows more than one sign change. This indicates the possibility of more than one positive ROR solution. NPV for any interest rate can be calculated as,Fig. 3. 13 shows a plot
17、of NPV as a function of i.,Figure 3.13: Plot of NPV vs. i for Example 3.27,As stated before, the ROR is the rate at which the NPV is equal to zero. Based on Fig. 3.13, two RORs are possible; 11% and 72%. If we assume ROR to be 11%, then alternative A (an alternative requiring a smaller investment) s
18、hould be selected ( ). If we assume ROR to be 72%, then alternative B (an alternative requiring larger investment) should be selected ( ). Obviously, our answer changes depending upon the selected value of ROR.,One easy way to confirm this analysis is to calculate the NPV at the MROR (=20%) for incr
19、emental cash flow. Since NPV is positive, alternative B should be selected. This is the same predicted in the previous paragraph. It is obvious that, for such problems (where more than one sign change occurs in cash flow analysis), the ROR analysis is difficult to adopt. A better alternative would b
20、e to use the present worth analysis.,必须是连续的正现金流量,必须是连续的负现金流量,ir再投资利率 if 融资成本,3.5 Growth Rate of Return Analysis A related technique to the ROR analysis is the growth rate of return. Unlike ROR calculations, which are independent of what we do with the future benefits, growth rate of return depends o
21、n the reinvestment of the future benefits. It assumes that the future benefits are reinvested at certain interest rates and calculates the future value of all the benefits at the end of the useful life of the project.,Let us illustrate this schematically. In Fig. 3.14, we have a cash flow profile fo
22、r calculation of the rate of return. We invested $l,000 in the bank at an interest rate of 10% per year, and withdrew interest of $100 per year for 10 years finally withdrawing the initial investment of $l,000. In this simple scenario, the rate of return on the investment is 10%. The ROR value is in
23、dependent of what we did with the $100 we received at the end of each year. We could have gambled it away or could have reinvested it in buying stocks. The ROR would still be 10%. This is why the ROR is sometimes called internal rate of return. It only depends on internally generated revenues, not o
24、n external rates.,Contrast this cash flow profile with a case where the $100 revenue per year is reinvested in treasury bills at a 6% interest rate. The schematic diagram is shown in Fig. 3.15. We will assume that as soon as the annual payment is received, it is reinvested at a 6% interest rate. At
25、the end of 10 years, all the accumulated sum is withdrawn including the initial investment of $1000, 6% rate is called an external rate of return.,1000,In this case, we can calculate the future value of reinvested amount by knowing the relationship between the future value and the periodic payment.
26、For this example,In addition to $l,318, we will also receive the original investment back resulting in a total future value of $2,318. If we know that the $l,000 investment has resulted in a cumulative the asset of $2,318, we can calculate the rate at which our investment has grown by knowing the re
27、lationship between the future and the present values.,Therefore, i = 8.8%,Example 3.28 Assume the same values as given in the Example 3.17. Assume further that the annual benefit earned is reinvested at a rate of 10%. Calculate the GROR. Solution Given: Initial investment = $1,000 and annual benefit
28、 = $2,700 for 6 years. We can calculate the future value of the annual benefits by assuming that the benefits are invested at 10%. Using Eq.2.5,Solving for GROR, GROR = 13%,Growth rate of return is a useful calculation if the rate at which future benefits are invested is known. One possibility is th
29、at the reinvestment rate can be assumed to be equal to the MROR. Another possibility is to chose a rate which reflects the most conservative investment; i.e., treasury bills. Since any corporation is assumed to have perpetual existence, the GROR is an accurate indicator of the rate at which the trea
30、sury of a corporation will grow.,One major advantage of the GROR calculation is that it eliminates the trial and error procedure required for the conventional ROR analysis. The procedure also eliminates the possibility of multiple RORs for a given cash flow. If there is more than one sign change in
31、the cash flow profile, all positive cash flows are transferred at the end of the useful life by assuming that the positive cash flows are invested at the rate of reinvestment. This step will result in elimination of multiple sign changes and will result in only one sign change.,In applying the growt
32、h rate of return (GROR) as a criterion, we need to compare the calculated GROR with the MROR. If GRORMROR, we consider the alternative to be feasible. If GRORMROR, we will reject the alternative. It should be understood that the GROR technique does not eliminate the need for incremental analysis. Li
33、ke the ROR method, if we are presented with more than one alternative, we need to consider the incremental analysis.,Once we calculate GROR, we can compare it with the investment rate (it may be MROR); the rate at which the future benefits are invested. If GROR is greater than the investment rate, w
34、e select the alternative requiring a bigger investment. If the value of GROR is less than the investment rate, we select the alternative requiring a smaller investment. The following example illustrates the application.,Example 3.29 Assume the same data as given in Example 3.27. If we assume that al
35、l the revenues are invested at the same rate as the MROR, which option is preferable?,Example 3.27 An in-fill drilling project is being considered for an existing oil field to accelerate oil recovery. The following two scenarios, based on two alternatives (no in-fill drilling versus in-fill drilling
36、) are predicted. Which alternative would you select? The numbers are in millions. Assume that MROR is 20%.,Solution In this example, we are considering two alternatives. One alternative requires no additional drilling; the other alternative requires in-fill drilling. The cash flow profiles are provi
37、ded for both the alternatives.,For both these cash flow profiles, we can easily show that the GRORMROR. Therefore, both alternatives are feasible alternatives. This brings us to the next step. The cash flow profile for the incremental analysis, along with the cash flow profile for each alternative,
38、is reproduced below.,Solving for,GROR,GROR=21%,F=23.91+20(1+0.2)4=65.3856 Using the equation, F=P(1+GROR)n 65.3856=20(1+GROR)n Solving, GROR = 21.8%. Although in this example, we could cover the subsequent negative cash flows with our positive cash flows, in some cases in between positive cash flows
39、 may not be able to cover all the subsequent negative cash flows. If that happens, then the net negative cash flows in a given year should be treated as out of pocket expense and should be converted to the present value in the GROR analysis.,Example 3.30 An investment in a producing property results
40、 in the following cash profile due to price fluctuations. If the MROR is 15%, calculate the GROR.,Solution We have negative cash flows in year 0 and year 2. Before we consider the actual out of pocket expenses, we need to cover the negative cash flow in year 2 with the positive cash flow in year l.
41、Investing $15,000 at a rate of 15% will result in, 15, 000(1+.15) = 17,250 Adding the positive cash flow to -$20,000, the net cash flow in year 2, -20,000 + 17,250 = -2,750 This negative cash flow cannot be covered from any previously generated revenues. Therefore, it is considered out-of-pocket exp
42、enses. The new cash flow profile is presented below.,F= 15(1+.15)4+10(l+.15)3+10(l+.15)2+8(l+.15)1 + 6 =69.9 Therefore, to calculate the GROR, F = P(l +GROR)n 69.9 = 32.l(l + GROR)7 GROR = 11.8%,To summarize the GROR method, it is a modification of the rate of return method. It requires an additiona
43、l knowledge of reinvestment rate. However, if such information is known, the method eliminates the need of trial and error procedure as required for the ROR method. Further, by using the GROR method, we eliminate the possibility of multiple rates of return.,3.6 Profit to Investment Ratio Profit to i
44、nvestment ratio (PIR) is the ratio of the NPV at MROR to the present value of out of pocket investment. We can write it as,This number is an indication of the efficiency of the investment. In other words, PIR is the amount of money earned per dollar invested. As in the case of GROR calculations, onl
45、y if the subsequent investment is not covered by prior benefits, that investment is included in the present value of investments. Since the future benefits can only be received if we initiate the project, it is critical that we try to cover the subsequent costs by prior benefits before we cover them
46、 with out of pocket expenses. The out of pocket expense being an additional expense should be reflected in the denominator of Eq.3. 14. For a project to be feasible, the PIR has to be greater than zero. The following examples illustrate the application.,Example 3.31 An oil company intends to buy a p
47、roducing property for a price of $l million. It is expected to generate $280,000 net income in the first year declining at 10% per year. The property will be held for at least 10 years with an expected salvage value of $200,000. If the MROR is 15%, should the property be bought? Use PIR analysis.,Ex
48、ample 3.32 The following cash flow profile is expected for an investment. If the MROR is 10%, check the feasibility of the project using PIR criterion.,80(l+0.l) = 88 -100+88 = -12,Total out of pocket expenses are calculated as,Since this value is greater than 0, the project is feasible.,Example 3.3
49、3 The following two alternatives are considered for a project. Based on the PIR analysis, select the best alternative. Assume the MROR to be 10%.,Although PIRa PIRb , we need not select (a) unless we carry out one additional calculation. Since projects (a) and (b) require different amounts of investment, we will have to calculate the PIR for incremental