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1、Example: P=$500 A=$140 n=10 NPV=? i() 0 10 20 25 30 40 NPV 900 360 87 0 -67 -162 -500,ROR的意义： 从收益的观点看，ROR就是项目所能达到的最高收益水平。,ROR的意义： 从收益的观点看，ROR就是项目所能达到的最高收益水平。,NPV是折现率ic的函数，ic连续，则NPV可导。 一阶导数： NPV函数曲线单调递减。 二阶导数： NPV函数曲线凸向原点。,当ic=0， 当ic趋向于无穷大，NPV=NCF0,3.4 Rate of Return Analysis The rate of return analy
2、sis 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 investment brochures will use rate of return on your investment as a criterion to show how good a given investment
3、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 understand, the technique has some major drawbacks. In this section, in addition to explaining how to calcu
4、late 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 the unpaid balance of a loan such that the payment schedule makes the unpaid balance equal to zero when
5、 the final payment is made.” Consider a simple example to illustrate this definition.,Assume that you take 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 t
6、he principal amount of $l,000. Therefore, at the end of four years the unpaid balance is zero. The rate of 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
7、the unrecovered investment such that the payment schedule makes the unrecovered investment equal to zero at the 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
8、 year, you withdraw $600 in interest and at the end of five years, you withdraw $10,000. The investment in the 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. 3.10
9、,Mathematically, 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
10、 project is given 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
11、项式) in iR which 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 s
12、earching for 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,
13、000 2,500 1,800 1,300 900 Solution Using 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 v
14、alues. By linear interpolation(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 respecti
15、vely. In our example,Therefore, We can calculate the NPV at 29.3%. NPV=-66.5 Although close to zero, we can try one more interpolation between 15% and 29.3%. NPV at 28.3% = -10.2 We would assume this value to be close enough to zero. You may note that higher is the difference between the i+ and i-,
16、bigger will be the deviation(背离) 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 per
17、iod of six years. 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=1.70 Therefore, 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 i
18、s very difficult 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 le
19、ss than the initial 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
20、bought a house for $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? Solution Let us assume the ROR to be .017/month. where 120 is the number of months in which the rent is collected. Therefore, the
21、ROR is l.7%/month. or 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 o
22、f $25,000 in an oil property 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. Solution In this example we have a geometric series. Given: A = $6,000, n = 20 years,
23、 g = -0.08 Using the equation 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. A
24、s shown in Fig.3.ll, by connecting the points, we can calculate the ROR corresponding to a point on the curve where NPV is equal to zero.,Figure 3.11: ROR Determination,3.4.l Economic Criteria As stated before, the ROR technique is probably the most used technique in economic analysis. It is easy to
25、 understand. Since every one understands the interest rate, rate of return is equated to return on investment in terms of an interest rate that would be earned. Intuitively(直观地讲), when comparing two investments, one fetching a higher ROR is always more attractive. In a corporate structure, to evalua
26、te the feasibility of a project, we need to compare the ROR to the minimum rate of return (MROR). If the RORMROR, the project is selected; if the RORMROR, the project will be rejected.,Example 3.23 The following two alternatives are considered for a project, (a) (b) Initial Investment $50,000 $500,0
27、00 Annual Benefit $25,000 $125,000 Life, Years 5 5 Salvage Value $50,000 $500,000 If the MROR is 20%, which alternative should be selected?,Solution The first step is to estimate the ROR of the individual alternatives and compare the ROR with the MROR. If the ROR is less than the MROR, the alternati
28、ve(s) should be rejected. In this example, since the salvage value is equal to the initial investment, using Eq.3.9. Since the RORa 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 i
29、nitial investment for both alternatives is not the same. One of the drawbacks of the ROR analysis is its inability(无能) to account for the investment amount. To properly(完全) account for the investment, we need to conduct incremental analysis. That is, to find out by investing additional (incremental)
30、 $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),This number indicates that the ROR on incremen
31、tal 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, we will earn only MROR on that additional amount. Therefore, it is more attractive
32、 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 alternative b, ince (NPV)b (NPV)a, alternative b should be chosen. This is consistent w
33、ith 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 should select an alternative requiring a larger investment. If RORMROR, assume that
34、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 the incremental investment by subtracting the smaller investment from the larger
35、 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 alternative from further analysis. c. Take the remaining alternative and compare
36、 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 the alternative requiring the smaller investment. Remove the rejected alternative from further analysis. d. Repeat ste
37、p (c) till only one alternative remains.,Examp1e 3.24 The following three alternatives are considered for a project. If MROR is 15%, select the appropriate alternative.,Solution Since ROR for all the alternatives is greater than the MROR, all are feasible. In step (b) take the two alternatives requi
38、ring the smallest investment. In this example, we will consider alternatives (a) and (b) for incremental analysis,Therefore, Since RORb-a MROR, select (b) over (a). Eliminate alternative (a) from further analysis. In the next step (step c), compare (b) with the remaining alternative (c). For increme
39、ntal analysis,Therefore, Since ROR.c-b MROR, select (c) over (b). After eliminating (b), we are left with only alternative (c). This will be our choice.,To summarize, the economic criterion applied for the rate of return analysis for a single project: if the RORMROR, the project is selected; if the
40、RORMROR, the project is rejected. For a project having multiple alternatives, an incremental analysis needs to be conducted so long as there is a difference in the cash flow profiles of two projects. Only after applying the incremental analysis, the solution will be consistent with the NPV analysis.
41、,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 alternative requires an initial investment which is followed by future benefits. For this type of alternative, the cash flow
42、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 result in a $300 annual benefit for the next six years with a $500 salvage value at the end of six years, the cash profile c
43、an 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 calculation requires solving a polynomial of i. We calculate the value of i for which the NPV is zero. For economic analysi
44、s, 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, we can only obtain one or zero positive solutions.,In some instances, however, the sign changes more than once in a cash
45、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 never greater than the number of sign changes. That is, if we have two sign changes, we may obtain a maximum of two rate
46、s 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 -100% and .,Solution To calculate the maximum number of possible real solutions between -100% and , we can calculate the
47、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. Similarly, for cash flow C, there are four sign changes, and for cash flow D, there are five sign changes. As stated be
48、fore, 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 be either 4, 3, 2, l. or zero.,The number of possible real solutions can be narrowed down even further by applying cumulat
49、ive 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 only 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.