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1、技术经济学英文版演示文稿C32Stillwatersrundeep.流静水深流静水深,人静心深人静心深Wherethereislife,thereishope。有生命必有希望。有生命必有希望 ROR的意义：的意义：从收益的观点看，从收益的观点看，ROR就是项目所能达到的最高收益水平。就是项目所能达到的最高收益水平。ROR的意义：的意义：从收益的观点看，从收益的观点看，ROR就是项目所能达到的最高收益水平。就是项目所能达到的最高收益水平。NPV是折现率是折现率ic的函数，的函数，ic连续，则连续，则NPV可导。可导。一阶导数：一阶导数：NPV函数曲线单调递减。函数曲线单调递减。二阶导数：二阶导数
2、：NPV函数曲线凸向原点。函数曲线凸向原点。当当ic=0，当当ic趋向于无穷大，趋向于无穷大，NPV=NCF03.4 Rate of Return AnalysisThe rate of return analysis is probably the most popular criterion in economic analysis. Its popularity stems from the ease with which a common person can understand the meaning of rate of return. Most of the investmen
3、t brochures will use rate of return on your investment as a criterion to show how good a given investment opportunity is. It is much easier to understand that a project will provide 20% return on your investment than the project will result in a NPV of $5,000. Unfortunately, although simple to under
4、stand, the technique has some major drawbacks. In this section, in addition to explaining how to calculate the rate of return (ROR), we will discuss the advantages and disadvantages of this technique .Rate of return has two definitions. One definition can be stated as “the interest rate earned on th
5、e unpaid balance of a loan such that the payment schedule makes the unpaid balance equal to zero when the final payment is made.” Consider a simple example to illustrate this definition. 1001,000$1,000100 1001 2n-1 nFigure 3.9: A Loan of $1,000 with a Uniform Payment of InterestAssume that you take
6、a loan of $1,000 from a bank at an interest rate of 10% for a period of four years. Every year, including last year, you pay an interest of $100 to the bank. At the end of four years, you pay the principal amount of $l,000. Therefore, at the end of four years the unpaid balance is zero. The rate of
7、return for the bank is (l,00/1000=)10%. Schematically, the cash flow is shown in Fig. 3.9.This definition can be turned around to state that the rate of return is the interest rate earned on the unrecovered investment such that the payment schedule makes the unrecovered investment equal to zero at t
8、he end of the life of the investment. Using a similar example as before, let us assume that you have invested $10,000 in the bank at an interest rate of 6% for five years. At the end of each year, you withdraw $600 in interest and at the end of five years, you withdraw $10,000. The investment in the
9、 bank at the end of five years is, therefore, zero. You can consider that the rate of return on the investment is (600/10,000=) 6%. Schematically, the cash flow profile is shown in Fig. 3.10Figure 3.10: Investment of $10,000 with a Uniform Payment of Interest1 2n-1 n60010,000$10,000600 600Mathematic
10、ally, the rate of return (ROR) is defined as the rate at which net present worth (NPV) for a given investment is equal to zero. In equation form, the rate at which, (3.4)is the rate of return.In other words, the rate at which NPV = 0 3.5)If we assume that the cash flow for a particular project is gi
11、ven by Aj where Aj represents the cash flow in year j, we can write the equation for NPV as, (3.6)If we define the rate iR corresponding to the rate at which NPV is zero, we can write the equation for iR as, (3.7)Observing Eq.3.7, we notice that the equation represents a polynomial(多多项项式式) in iR whi
12、ch may result in n possible solutions for iR which will satisfy Eq.3.7. In economic analysis, we are only interested in real solutions. Although negative rate of return is a real value, we may not be interested in an investment of negative rate of return. As a practical matter, we are searching for
13、positive, real solutions of this equation. In most instances, we will obtain only one positive, real solution which represents the rate of return. This is shown in the following examples.Example 3.17 Calculate the rate of return for the following cash flow. Year 0 1 2 3 4 Cash Flow -4,000 2,500 1,80
14、0 1,300 900SolutionUsing the cash flows, we can write the equation for NPV as,Since this is a polynomial equation in i , we will have to solve it by trial and error, Since the value of NPV changes a sign between i = 15% and i = 35%, the rate of return should fall in between the two values. By linear
15、 interpolation(线形内插法线形内插法), we can write an approximate equation for the rate of return (ROR) as, (3.8)where i+ and i- respectively represent the vial values which resulted in positive and negative NPV values, and NPV+ and NPV _ represent the positive and the negative NPV values respectively. In our
16、 example,Therefore,We can calculate the NPV at 29.3%. NPV=-66.5Although close to zero, we can try one more interpolation between 15% and 29.3%. NPV at 28.3% = -10.2We would assume this value to be close enough to zero. You may note that higher is the difference between the i+ and i-, bigger will be
17、the deviation(背背离离) between the true ROR and the interpolated value. Therefore, the interpolation may have to be carried out more than once to obtain a correct value of the ROR.Example 3.18 By investing $10,000 in a project, you are promised that you will earn $2,700 per year for a period of six yea
18、rs. What is the ROR for this investment?Solution For i = 10%, For i =20%, Using Eq.3.8, For 16.3%, NPV = -129 At i = 15.8% NPV=1.70Therefore, the rate of return is 15.8%.From the above examples, one can see that the ROR calculation has to be done by trial and error. Many times, it is very difficult
19、to assume the initial value of interest rate. One way to overcome this problem is to use a ratio of periodic payment to initial investment. We can show that if the initial investment is equal to the salvage value, the ROR can be calculated as,Using Eq.3.9, if the salvage value is less than the initi
20、al investment, On the other hand, if the salvage value is greater than the initial investment, Eq.3.9 through Eq.3. 11 are applicable only if the investment is made at the beginning of the project and the periodic payments are equal to each other.Example 3.19 As an investment, you bought a house for
21、 $50,000. If you can rent the house for $800 per month, and can sell the house for $70,000 at the end of ten years, what is the ROR on your investment?SolutionLet us assume the ROR to be .017/month. where 120 is the number of months in which the rent is collected. Therefore, the ROR is l.7%/month. o
22、r 20.4%/year.As can be seen from the above example, by using the correct initial guess. we did not have to use too many trial and errors. A similar equation can be developed for geometric series as explained in the example below.Example 3.20 A proposal calls for an investment of $25,000 in an oil pr
23、operty which will result in an initial income of $6,000 per year declining at a rate of 8% per year over the next twenty years. What is the rate of return? Assume the salvage value to be zero.SolutionIn this example we have a geometric series.Given: A = $6,000, n = 20 years, g = -0.08Using the equat
24、ion for geometric series,After one additional trial and error, the ROR = 15.7%.The ROR can also be calculated using a graphical procedure. For a typical investment scenario, we call assume different interest rates and calculate the NPV as a function of the interest rate. As shown in Fig.3.ll, by con
25、necting the points, we can calculate the ROR corresponding to a point on the curve where NPV is equal to zero. Figure 3.11: ROR Determination3.4.l Economic CriteriaAs stated before, the ROR technique is probably the most used technique in economic analysis. It is easy to understand. Since every one
26、understands the interest rate, rate of return is equated to return on investment in terms of an interest rate that would be earned. Intuitively(直直 观观 地地 讲讲 ), when comparing two investments, one fetching a higher ROR is always more attractive.In a corporate structure, to evaluate the feasibility of
27、a project, we need to compare the ROR to the minimum rate of return (MROR). If the RORMROR, the project is selected; if the ROR MROR and RORb MROR, both alternatives satisfy the feasibility criterion.Intuitively, since RORa RORb, one may be inclined to select (a) over (b), but notice that the initia
28、l investment for both alternatives is not the same. One of the drawbacks of the ROR analysis is its inability(无无能能) to account for the investment amount. To properly(完完全全) account for the investment, we need to conduct incremental analysis. That is, to find out by investing additional (incremental)
29、$450,000 in alternative (b), what incremental benefit are received? Subtracting values related to alternative (a) from alternative (b), we obtain,For incremental investment, we can calculate the ROR by Eq. 3.9 (since the investment = the salvage value),a - bInitial Investment$450,000Annual Benefit$1
30、0,000Life, Years5Salvage Value$450,000This number indicates that the ROR on incremental investment is 22.2% which is greater than the MROR. In other words, by investing an additional $450,000, we will earn a ROR of 22.2%. On the other hand, if we do not invest an additional $450,000 in alternative b
31、, we will earn only MROR on that additional amount. Therefore, it is more attractive to invest the additional $450,000 in alternative b. That is, to select alternative b over a.This analysis can be easily confirmed by calculating the NPV for both the alternatives at MROR.For alternative a,For altern
32、ative b,ince (NPV)b (NPV)a, alternative b should be chosen. This is consistent with the answer we obtained from the incremental analysis.To generalize, if two alternatives requiring different amounts of investment need to be compared, we should carry out an incremental analysis. If RORMROR, we shoul
33、d select an alternative requiring a larger investment. If RORMROR, assume that the alternative is feasible and retain it for further incremental analysis. If the RORMROR, remove the alternative from further analysis.b. Take two alternatives requiring the smallest investments. Calculate the ROR on th
34、e incremental investment by subtracting the smaller investment from the larger investment. We denote the ROR on incremental analysis as ROR. If RORMROR, select the alternative requiring the larger investment; if RORMROR, select the alternative requiring the smaller investment. Remove the rejected al
35、ternative from further analysis.c. Take the remaining alternative and compare it with the alternative requiring the next largest investment. Calculate the incremental ROR. If RORMROR, select the alternative requiring the larger investment; if RORMROR, select (b) over (a). Eliminate alternative (a) f
36、rom further analysis.In the next step (step c), compare (b) with the remaining alternative (c). For incremental analysis,(b)- (a)Initial Investment$2,000Annual Benefit$700Life, Years10Salvage Value$2,000Therefore, Since ROR.c-b MROR, select (c) over (b). After eliminating (b), we are left with only
37、alternative (c). This will be our choice.(c)- (b)Initial Investment$3,000Annual Benefit$800Life, Years10Salvage Value$3,000To summarize, the economic criterion applied for the rate of return analysis for a single project: if the RORMROR, the project is selected; if the ROR0ic=0=ic0MROR, we consider
38、the alternative to be feasible. If 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, is reproduced below.Year0123456A030201814106B-2060406420B-A
39、-203020-12-10-8-6 Solving for GROR, GROR=21% YearB-ASTEPS12340-20-20-20-20-2013030*1.220000220202020203-1231.231.2*1.20004-10-1027.4427.44*1.2005-8-8-824.9324.93*1.206-6-6-6-623.91 F=23.91+20(1+0.2)4=65.3856Using the equation, F=P(1+GROR)n 65.3856=20(1+GROR)nSolving, GROR = 21.8%.Although in this ex
40、ample, we could cover the subsequent negative cash flows with our positive cash flows, in some cases in between positive cash flows 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 ex
41、pense and should be converted to the present value in the GROR analysis. Example 3.30 An investment in a producing property results in the following cash profile due to price fluctuations. If the MROR is 15%, calculate the GROR.Year01234567Cash Flow(in thousands)-3015-2015101086 SolutionWe have nega
42、tive 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. Investing $15,000 at a rate of 15% will result in, 15, 000(1+.15) = 17,250Adding the positive cash flow to -$20,000, th
43、e net cash flow in year 2, -20,000 + 17,250 = -2,750This negative cash flow cannot be covered from any previously generated revenues. Therefore, it is considered out-of-pocket expenses. The new cash flow profile is presented below.Year01234567Cash Flow(in thousands)-300-2.7515101086 F= 15(1+.15)4+10
44、(l+.15)3+10(l+.15)2+8(l+.15) + 6 =69.9Therefore, 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 additional knowledge of reinvestment rate. However, if such information is known
45、, 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 investment ratio (PIR) is the ratio of the NPV at MROR to the present va
46、lue 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, only if the subsequent investment is not covered by prior benefits, that
47、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 with out of pocket expenses. The out of pocket expense being an addit
48、ional 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 producing property for a price of $l million. It is expected to generat
49、e $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 b o u g h t ? U s e P I R a n a l y s i s . Example 3.32 The following cash flow profile is expect
50、ed for an investment. If the MROR is 10%, check the feasibility of the project using PIR criterion.Period0123456Cash flow-10080-100200100503080(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 3.33 The follow
51、ing two alternatives are considered for a project. Based on the PIR analysis, select the best alternative. Assume the MROR to be 10%.(a)(b)Initial Investment$10,000$20,000Annual Benefit$3,000$7,000Life, Years55Salvage Value$8,000$3,000 Although PIRa PIRb , we need not select (a) unless we carry out
52、one additional calculation. Since projects (a) and (b) require different amounts of investment, we will have to calculate the PIR for incremental investment, Since PIR0, an alternative with a higher investment should be selected; i.e., we should select (b) over (a). If PIR0, an alternative requiring
53、 a smaller investment would have been selected. a. Calculate the PIR for all the alternatives. If the PIR is positive, retain the alternative for future consideration; otherwise reject it from additional consideration. b. Select the two alternatives requiring the smallest investments. Calculate the
54、PIR. If PIR0, select the alternative requiring a bigger investment. Reject the remaining alternative from further consideration. c. Select the alternative requiring the next biggest investment and compare it with the retained alternative from the previous step. Calculate PIR. If PIR0, select the alt
55、ernative requiring a bigger investment. d. Repeat step (c) until only one alternative remains. The following example illustrates the application. Example 3.34 The following three alternatives are considered for buying a workstation. Based on PIR analysis, select the appropriate alternative. Assume t
56、he MROR to be 10%.ABCInitial Cost$15,000$25,000$50,000Net Annual Benefit$6,000$9,000$16,000Life, Years777Salvage Value$4,000$4,000$8,000 A method similar to the PIR method is often used in the literature to evaluate mutually exclusive alternatives. The method uses a ratio defined as,PIR=B/C-1 Exampl
57、e 3.35 Information for two mutually exclusive alternatives is given as follows.ABInitial Cost$12,000$18,000Annual Benefit6,0007,500Life, Years77Salvage Value$1,000$1,300If the MROR is 10%, using B/C method, choose the correct alternative. Confirm the answer with PIR method. SolutionAlternative AAlternative BWe can confirm this analysis by PIR method.Alternative ANPV=PVbenefits PVcosts = 29,724 12,000=17,724 Alternative BNPV= 37,180 18,000=19,180
