技术经济学英文版演示文稿C32.ppt
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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
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