# 留学生辅导 AF2617, AF3625, and AF3901. Let us move on to the first topic in Unit 7. – cscodehelp代写

Unit 7: Evaluation and Selection of Alternative Projects
Topics Page
TOPICS …………………………………………………………………..1 YOUR OBJECTIVES ………………………………………………………………………… 2 SELF EXERCISES…………………………………………………………………………….2

7.1: THE MINIMUM ATTRACTIVE RATE OF RETURN (MARR)2
7.2: THE METHODS OF PRESENT WORTH, FUTURE WORTH, AND ANNUAL WORTH ……………………………………………………………….. 5
7.2.1: THE PRESENT WORTH METHOD ……………………………………………… 5 7.2.2: THE FUTURE WORTH METHOD ……………………………………………….8 7.2.3: THE ANNUAL WORTH METHOD …………………………………………….12 7.2.4: THE CAPITALIZED WORTH METHOD ……………………………………… 14
7.3: THE INTERNAL RATE OF RETURN METHOD…………………15
7.3.1 DEFINING AND DERIVING IRR ……………………………………………….16 7.3.2 USING THE IRR METHOD ………………………………………………………19
7.4: BASIC CONSIDERATIONS IN THE COMPARISON AND SELECTION OF ALTERNATIVES …………………………………………….. 21
7.4.1: TYPES OF MUTUALLY EXCLUSIVE ALTERNATIVES (MEAS) ………21 7.4.2: THE STUDY (ANALYSIS) PERIOD …………………………………………… 22 7.4.3: REPEATABILITY OF THE PROJECTS …………………………………………27
MUST READ BEFORE MOVING ON…………………………………………33 RESPONSES TO SELF EXERCISES……………………………………………34
November 2016

In this unit you will learn how to apply the techniques you have acquired from Unit 6 to answer the questions “To do or not to do?” and “Which of the alternatives to choose?” by conducting cost-benefit analyses.
In Unit 6, the interest rate used in the derivation of economic equivalence of a given cash flow is always given. This is also the case in the calculation of the cost of capital for the cost of production in Topic 3. In this unit I will briefly explain what determines the level of interest rate.
While the Principles of Economics does not consider where the money comes from to pay for the cost of capital, Engineering Economics considers the alternative sources of finance (i.e. money). A firm pays different interest rates to different types of suppliers of such financial resources. How the source of finance affects the interest rate will be briefly examined in the first topic in this unit. Then we will look at various methods that evaluate and compare the worthiness of individual investment projects.
There are four main topics in Unit 7.
Topic 7.1: The Minimum Attractive Rate of Return (MARR)
This topic briefly studies the major determinants of the interest rate used in the evaluation of investment projects, which is known as the Minimum Attractive Rate of Return (MARR).
Topic 7.2: The Methods of Present Worth, Future Worth, and Annual Worth
In this topic we look at the three basic methods to evaluate a given project or alternative, namely, the present worth method, the future worth method, and the annual worth method. They correspond to the present equivalent, future equivalent, and annual equivalent introduced in Unit 6.
Topic 7.3: The Internal Rate of Return Method
In this topic we will introduce a slightly different method that derives the break-even interest rate of a given project. The basic idea is still based on the establishment of economic equivalence of a given cash flow.
Topic 7.4: Basic Considerations in the Comparison and Selection of Alternatives
Using the various methods introduced in Topics 7.2 and 7.3, this topic will discuss the major considerations in the comparison and selection of mutually exclusive alternatives that can perform the same tasks for a firm.
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When you have worked through this unit you should be able to complete the following objectives.
1. Identify the major determinants of the appropriate interest rate for evaluating a given investment project.
2. Apply the present worth (PW) method to evaluate individual investment projects and compare alternative projects.
3. Apply the future worth (FW) method to evaluate individual investment projects and compare alternative projects.
4. Apply the annual worth (AW) method to evaluate individual investment projects and compare alternative projects.
5. Understand the special feature of the internal rate of return (IRR) method that evaluates individual investment projects.
6. Identify the major considerations that determine the appropriate method to compare and select alternatives that are mutually exclusive.
Self Exercises
There are six (6) self exercises in this unit. Note that all of them are former examination questions in AF2617, AF3625, and AF3901. Let us move on to the first topic in Unit 7.
7.1: The Minimum Attractive Rate of Return (MARR)
This topic briefly studies the major determinants of the interest rate used in the evaluation of investment projects, which is known as the Minimum Attractive Rate of Return (MARR).
Figure 7.1
Determination of MARR
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MARR is usually determined by the top management based on the demand and supply of financial resources to meet the needs of a firm to make capital investments. Figure 7.1 can help you understand what is involved in its determination.
Does Figure 7.1 remind you of the demand-supply diagram? It does indeed involve the same concepts of marginal benefit and marginal cost underlying the demand-supply analyses introduced in Unit 1. Remember why the demand curve is downward sloping for an individual consumer? It is because of the diminishing marginal benefit a consumer can derive from each additional unit of a good. As explained in Unit 1, a consumer has to decide whether to buy a good and how many units to buy. Similarly, a firm has to decide whether to invest and how many projects to invest. A rational decision maker for the firm will first invest in the project that brings the highest marginal benefit, which is measured by the rate of profit. Figure 7.1 illustrates a hypothetical case in which the firm has nine projects (A to I) available for consideration. They are ordered in descending order (i.e. from the highest to the lowest) of their profitability. Note that the amount of investment required for each project is different. For example, project A requires \$10 million whereas Project B requires \$15 million. The rate of profit is usually measured by the Internal Rate of Return (IRR), which will be discussed later in Topic 7.3. Subject to the available financial resources, the most profitable alternatives will be selected.
The availability of financial resources is indicated by the upward-sloping curve in Figure 7.1 that indicates the cost of capital supplied. It is upward sloping because as more and more financial resources are needed, the firm will have to go to the sources that demand higher and higher interest rate. For example, if the firm only considers projects A and B, it will need only \$25 million. Perhaps there will be enough retained earnings (profits retained by the firm after distributing dividends to its shareholders) to finance these two projects and the cost of capital will only be the implicit cost of forgone interest from lending to the bank (i.e. deposits in the bank). If the firm needs to raise more money to invest in more projects, then it may have to borrow from the bank paying higher interest rates (borrowing rate are higher than lending rates). The money borrowed from banks or from issuing bonds is called debt capital. To raise even larger amount of investment funds, the firm will have to issue new shares to the shareholders (on the stock market if the firm is publicly listed). This type of funds is called equity capital. Because the suppliers of equity capital have to bear greater risk (they will be paid dividends only if the firm makes profits) than the suppliers of debt capital, suppliers of equity capital demand a higher return to their funds. Thus the cost of equity capital will be higher than the cost of debt capital. This is why the supply curve of financial capital is upward sloping, just like the supply curve of goods and services that you encountered in Topic 1.
If a firm faces a capital supply curve as shown in Figure 7.1, then the MARR is similar to the concept of equilibrium price that equates the demand and supply of investment funds. As shown in Figure 7.1, the MARR is 18%. Only Projects A to F will be selected because, as you have learned in Unit 1, “a rational decision maker will only take those actions for
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which marginal benefit exceeds or equals marginal cost.” Projects G, H, and I will be rejected because their rates of profit are below the MARR.
In Unit 1 you have learned that the demand and supply curves will shift if their determinants change over time. As a result, the equilibrium price will also change. Similarly, MARR will change by one of the following factors:
1. The supply of financial resources, i.e. how much is available and how expensive? The larger (and cheaper) the supply, the lower will be the MARR. The smaller (and more costly) the supply, the higher will be the MARR. Sometimes a firm may be subject to capital rationing, in which case there is only a fixed amount of funds available even if the firm is willing to pay higher interest for additional funds. For example, assume the firm in Figure 7.1 only has a supply of \$40 million. (The supply curve of capital will be a vertical line at \$40 million.) This amount is only sufficient to finance Projects A, B, C, and D. As a result, the firm has to sacrifice the opportunity of earning 20% of profit from Project E. The appropriate MARR in this case will be 20%, which reflects the opportunity cost of the next best alternative investment opportunity.
2. The demand for investment funds, i.e. is the firm considering more and fewer investment projects? The greater the demand, i.e. the more the investment opportunities and the higher their expected profitability, the higher will be the MARR.
3. This is a determinant not considered before. It is the level of risk associated with the investment opportunities and the administrative costs estimated by the firm. The higher the risk and administrative cost, the higher will be the corresponding cost of capital and the higher will be the MARR.
4. This is another determinant not considered before. It is the nature of the organization. For non-profit making organizations such as government departments that make decisions regarding investments in infrastructure, their MARR is usually lower than those adopted by private enterprises that aim at making profits.
Before we proceed to the next topic to evaluate the worthiness of an investment project, we need to introduce briefly the concept of Weighted Average Cost of Capital (WACC). You have been introduced to the various sources of financial capital or investment funds in the discussions of MARR. Equity capital, which also includes retained earnings, is from internal source and debt capital is from external source. The costs of these two sources of capital are quite different although both of them are market- determined. Depending on their respective weights in the total supply of financial capital for a particular firm, we can calculate the WACC of the firm. In general, the WACC only establishes the lower bound of MARR. A firm usually sets higher MARR to compensate for the risk involved in the projects under consideration.

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The derivation of WACC is part of the topic on Capital Budgeting, which will not be covered in the self study notes. Please refer to the relevant chapter of the textbook for better understanding.
7.2: The Methods of Present Worth, Future Worth, and Annual Worth
In this topic we look at the three basic methods to evaluate a given project or alternative, namely, the present worth method, the future worth method, and the annual worth method. They correspond to the present equivalent, future equivalent, and annual equivalent introduced in Unit 6.
7.2.1: The Present Worth Method
As already mentioned in Unit 6, to conduct cost benefit analysis (CBA) for an investment project, we must first convert the cash inflows (receipts) and cash outflows (expenses) at different points in time to their corresponding values at the same point in time based on the time value of money. For the present worth (PW) method, all these cash flows will be converted to their economic equivalence at period 0. The PW (sometimes also known as Net Present Worth, NPW) of a project is actually the present value of the economic surpluses generated by the project over its lifetime. It is equal to the PW of all the benefits minus the PW of all the costs of the project, which is equivalent to the sum of discounted cash inflows minus the sum of discounted cash outflows. Note that the benefits generated by the project may not be the same as the cash inflows while the costs incurred may not be the same as the cash outflows. For example, a manufacturer spends \$1 million to install a new production line and intends to replace it in 10 years. It is estimated that the salvage value of the used machines will be \$100,000 at the end of the 10th year. The \$100,000 of cash inflow at the end of the project life is not the benefit but a value to be subtracted from the cost. The distinction between benefit and cost may be important for the understanding of economic concepts. But for decision making in engineering economics, it may be more practical and easier to simply refer to the cash flows.
After correctly identifying the amount and the timing of each cash flow, we can derive the present worth of the project under consideration. If the value is non-negative, then the project is economically justified and worthwhile. If its PW is negative, then it should be rejected. Again, this is the same criterion of CBA introduced in Unit 1.
Consider the following simplified example.
Example 1: To Do or Not to Do
A PolyU graduate from the Faculty of Engineering is considering starting a consultancy firm after working for the government for 25 years. His plan is to operate the business in for 5 years and then migrate to Canada for early retirement. The initial investment (the first cost) will be \$5 million, to be financed by his saving. He expects to recover 40% of this
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amount at the end of the 5th year when he wraps up the business. The annual after-tax profit, which is net of all the relevant explicit costs as well as his forgone salary and forgone rent, will be \$1 million every year for 5 years. Assume all the cash flows occur at the end of each year. Using the present worth method at MARR = 10%, do you think this PolyU graduate should start the consultancy firm?
Drawing the following cash flow diagram will help you do the analysis more easily.
The only amount in the above diagram that needs to be clarified is the cash inflow at the end of year 5, which is the sum of \$1 million of profit and \$2 million of the initial investment recovered. The amount of recovered capital may include the salvage value of any equipment, appliances or furniture. Usually it also includes the amount of working capital that the owner of a business has to put aside throughout the project life to pay for the wages, rent, or other expenses of operation before receiving any revenues. Once the project is terminated, that amount of money will be recovered and become available for other uses.
One important point is worth noting here. That is, the treatment of the cost of capital in this unit is different from what you have learned in Unit 3. In Units 3 and 4, you are told to include the explicit interest cost and the implicit interest cost in the total cost and subtract them from revenue to derive economic profit. In this unit, however, only the cash outflows such as wages, rent, material costs, energy costs, etc., and the implicit cost unrelated to the cost of capital, such as forgone salary and forgone rent of the owner, will be subtracted from the revenue or cash inflows to derive the net cash flows. Both the implicit interest cost and explicit interest cost need not be subtracted because they will be reflected by the MARR. And the calculation of economic equivalence will take care of all the interest costs and depreciation, the two components of the cost of capital as described in Topic 3.1.3. You will have a better understanding of the proper treatment of the cost of capital after learning the concept of capital recovery (CR) in Topic 7.2.3.

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The derivation of the PW of the consultancy business in Example 1 is straightforward. One way to do it is to use the uniform series present worth factor ( P / A, i %, N ) and the single payment present worth factor
(P/F,i%,N)as follows.
PW(10%) = \$[5 + 1 ( P / A,10%,5) + 2 ( P / F ,10%,5) ] million
= \$[ 5 + 1  3.7908 + 2  0.6209] million
As the value of PW is positive, this investment project is economically justified and worthwhile.
Apart from applying this straightforward criterion of cost-benefit analysis to evaluate the worthiness of the project, do you know exactly what the finding means? Think about this question first. I will explain the answer after we have studied the future worth method in the next topic.
The first self exercise gives you an opportunity to apply the PW method to evaluate a real-life investment opportunity. First of all, let me give you some background knowledge.
As mentioned in Unit 6, one way for a firm to acquire financial resources for its investment projects is to issue bond. Bond is an IOU (debt contract) that the buyer (lender) agrees to lend a specific amount to the issuer (borrower) for a specific period of time. In return, the buyer/bondholder will receive periodic interest payments and get back the principle when the issuer redeems the bond at its maturity. Not all buyers of newly issued bonds hold them until maturity. Instead, they may sell the bonds on the bond market before maturity. How much such “second-hand” bonds can be sold for depends on the market interest rate at the time of the transaction. Now try the following exercise.
A corporate bond was issued two years ago at a par value (or face value) of \$10,000 for a period of 5 years. The bond rate or coupon rate is 3% for every 6 months (i.e. the bondholder will be paid \$300 of interest every 6 months until the maturity of the bond). Now this bond is available for sale on the market. If you are interested in this investment opportunity, what will be the maximum amount you are willing to pay for the bond if the market interest rate (i.e. the best alternative rate of return you can get) is 4% per year? What if the market interest rate is 8%?
A response to this self exercise is on page 34.
Self Exercise 1
Market Value of Bond
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