Model answers to all the questions will appear on April 4.

The questions in the first section are worth 4 points each; the questions in the second section are worth 15 points each. Total time for the exam is three hours.

Assume the inflation rate is zero unless the question says it isn't.

Short Answer Section: 4 points each

1. You have a small company that has grown by 15% every year for the last three years. You sell that company for $500,000 in order to start a new company. You borrow $500,000 at 10% interest, and sell $500,000 worth of stock in the new company, to members of the public who were impressed by the performance of your last company. What is the weighted cost of capital to the new company, and what should the pre-tax MARR be?

   The new company has $1.5M; 0.5M at 10% interest, 0.5M from investors who expect 15% growth, and 0.5M from you, where you also expect 15% growth. So the WCC is (1*0.15 + 0.5*0.1)/1.5.

   Your MARR should be at least as great as the WCC.

2. What is the acid-test ratio, and how is it used?

   The acid-test ratio is the ratio of a company's current assets, including cash and accounts receivable, but not including inventory, to the company's current indebtedness. The desirable value for this ratio varies from industry to industry; it gives an indication of the company's abilitiy to pay its debts if credit becomes short (as it might, for example, following a fall in stock prices.)

   `

3. When a proposal is analysed to find the IRR, two solutions are obtained: IRR = 5% and IRR = 45%. If the auxiliary rate of return is 10%, what can you conclude about the ERR?
   1. You can conclude nothing about the ERR
   2. The ERR will be between 5% and 10%.
   3. The ERR will be between 10% and 45%
   4. The ERR will be between 5% and 45%

   (d)

4. Why does ranking proposals in order of their IRR sometimes give misleading results? How can your analysis be modified to allow for this?

   If the proposals being considered are the ONLY ones available, you need to consider which makes the best use of your TOTAl investment funds. It's better to get a medium rate of return on a large sum than an excellent rate of return on a small sum, if the latter choice leaves a lot of funds uninvested. To take this into account, you should do an incremental analysis.

   `

5. Distinguish between the PERT and CPM methods of project analysis.

   PERT is concerned with estimating the time to completion of a project, CPM with making the best cost/time trade-off.`

6. What is a simplex, and what is the simplex method?

   A simplex is a closed, convex polygon in n dimensions. The simplex method is an algorithm that visits the vertices of such a simplex in turn, searching for the optimal solution. `

7. In the simplex method, what is the shadow price of a resource?

   The shadow price of a resource is the premium it would be worth paying for additional supplies of that resource. Its value is conveniently given by the coefficient of the associated slack variable in the objective function of the final tableau of the simplex method.

8. What is the importance of the Central Limit Theorem in the CPM method of project planning?

   The CLT allows us to conclude that the overall completion time of a project, being the sum of the randomly distributed completion times of the activities on the critical path, will itself be a random variable with a normal distribution, its mean the sum of the activity means, its variance the sum of the activity variances. `

9. What is the definition of a critical path?

   The critical path is the sequence of activities required for completion of a project such that delay to any one of the activities will delay completion of the project.

10. The deadline for a project has been calculated using the PERT method. A change in conditions leads to a doubling of the estimated time for all the activities on the critical path; the estimated time for activities not on the critical path is unchanged. What can you conclude about the estimated completion time of the project?
    1. It will be at least twice as long
    2. It will be at most twice as long
    3. It will be exactly twice as long
    4. You can draw no conclusions about the project completion time.

    It will be exactly twice as long.

    Section 2: Long Answer Questions, 15 points each. Answer 4 out of 8.

    1. Your company needs to purchase ten oscilloscopes during the coming year. The oscilloscopes are top-of-the-line models, with a built-in Pentium processor, and cost $80,000 each. You are presenting an argument to Revenue Canada that they should be considered as computers, and depreciated at 30% per annum. Revenue Canada thinks that they're generic electronic equipment, which is depreciated at 20% per annum. Your company pays tax at 50%, and your pre-tax cash flow every year is at least $500,000. Your after-tax MARR is 5%. What is the present value of convincing Revenue Canada that they're computers?

       The present cost of the purchase is 800,000*CCTF. The value of the CCTF factor depends on the depreciation rate, according to the formula provided in the cribsheet. So you need to calculate 800,000(CCTF(20%)-CCTF(30%)).

    2. Your company, which makes gas separation equipment, is considering several alternative strategies. One strategy is to market an existing separation rig. This will generate a revenue of $200,000 at the end of this year, declining by 20% per year as your competitors put better products on the market. A second strategy is to invest in the development of a greatly improved rig. If you do this, you will not have any sales this year (`Year 1'). There is a 50% chance of your development efforts succeeding the following year (`Year 2'); if they don't succeed, there is again a 50% chance that they will succeed the year after that (`Year 3'), and similarly for the following year (`Year 4'). Beyond that it's impossible to predict. As soon as the development efforts succeed, they will yield an income of $300,000 a year, and this income is expected to continue till at least the end of Year 4.

       Your pre-tax MARR is 25%. Which strategy will you choose?

       To solve this, you need to compare the PW of the first strategy with the expected value of the PW of the second strategy, taking a four-year study period. We can speed up the calculation of the PW of the first strategy by lumping together the decline in sales and the discounting of future income: PW(1) = 200,000*1.2*(P/A,0.45,4), where the factor of 1.2 is needed to get the decline in sales and the discounting of future income in step.

       To calculate the expected PW of the second option, construct a probability tree, and find the present worth of each of the leaves of the tree. Then apply the formula

       E(PW)=Sum(probability of outcome i * PW of outcome i)

    3. 1. (8 points) Sketch a diagram showing cash flow within a corporation. The diagram should clearly show which cash flows are subject to tax, and should include dividends to stockholders, income from borrowing, wages, and depreciation allowance, plus whatever other cash flows are necessary to give a complete picture.

       2. (8 points) Your company needs to raise $650,000 to set up a new production facility. It will be four years before the new facility starts generating income, but as soon as it does start generating income, you'll have enough to pay back the cost of setting up. You can raise the money either by taking out a bank loan, selling bonds, or using internal funds.

          The bank charges 10% interest per year. The bonds are sold now for $1000, with a promise to redeem them for $1,500 four years in the future. If you use internal funds for this project, you will have to divert them from other potential investments within the company. Your pre-tax MARR is 15%. How should you raise the money?

       The cash flow diagram is in the notes. The second half of the question is just asking for the cheapest source of capital. Your internal funds cost 15%, the bank loan costs 10%, and the issue of bonds costs i%, where

       1000(1+i)⁴=1,500

       Solving for i, it turns out to be just over 10%. So you should borow from the bank.

    4. A machine has five years of life remaining to it, and its function will be needed for exactly that period. It costs $19,000 per year to operate. Right now its market value is $8,000, and in five years time its salvage value will be $1,000.

       A new machine of advanced design can perform the same function as the current machine for an operating cost of just $12,000 per year. It will cost $24,000 to purchase, and will have a salvage value of $6,000 after 5 years of use.

       The company has a 52% effective tax rate, and its after-tax MARR is 20%. Both machines are in Class 8: declining balance depreciation at 20%. Should the machine be replaced?

       This calls for an after-tax analysis, with a study period of 5 years. We compare the PW of each option.

       Option 1: keep old machine:

       PW = -19,000(1-t)(P/A,i,5)+1,000.CCTF.(P/F,i,5)

       Option 2: Buy new machine

       PW = -12,000(1-t)(P/A,i,5) - (24,000-8,000).CCTF + 6,000.CCTF.(P/F,i,5)

    5. An old wooden bridge over a bay is in danger of collapse. The highway department is considering two alternatives to alleviate the situation and provide for expected increases in future traffic. One plan is a conventional steel bridge, and the other is a tunnel. The department is familiar with bridge construction and maintenance, but has no experience with maintenance costs for tunnels. The following data have been developed for the bridge:

       First cost	$17,000,000
       Painting every 6 years	$1,000,000
       Deck resurfacing every 10 years	$3,000,000
       Structural overhaul after 15 years	$4,000,000
       Annual maintenance	$300,000

       The tunnel is expected to cost $24,000,000 and will require repaving every 10 years at a cost of $2,000,000. If both designs are expected to last 30 years with minimal salvage value, determine the maximum equivalent annual amount for maintenance that could be permitted for the tunnel while holding the total equivalent annual cost equal to that of the bridge. The interest rate is 8%.

       Since the unknown is an equivalent annual cost, we should calculate the equivalent annual cost for each option, using a 30-year study period. We assume that no periodic maintainance is done in the final year. For the bridge, we have:

       EUAC(bridge) = (A/P,0.08,30) * (17 + 1(P/A,i₆,4) + 3(P/A,i₁₀,2) +4(P/F,0.08,15)) + 0.3

       EUAC(tunnel)=(A/P,0.08,30) * ( 24 +2(P/F,0.08,10)+2(P/F,0.08,20)) + x

       This can then be solved for x. Note that i₆ and i₁₀ are the effective interest rates for 6 and 10-year periods respectively.

    6. On completing a doctorate in industrial engineering, a student has been offered two jobs. One, a faculty postion in a university, pays a starting salary of $45,000 with expected annual raises of 8% over the next 5 years. The other, a research position with an aerospace company, has a starting salary of $52,000 with annual raises of 5% over the next 5 years. If inflation is 5% per annum and the market interest rate is 15% per annum, what is the difference in the present worths of the two job offers?

       We take a five-year study period and consider all cashflows to occur at year's end. For the second position, we note that the inflation rate exactly matches the raises, so the present worth is just

       PW(aerospace)=52,000(P/A,0.15,5)

       For the faculty position, the initial payment is 45,000, and the present worth of payments in subsequent years changes by a factor of 1.08/(1.05)(1.15) every year. This can be simplified to a factor of 1/(1+x). So the present worth of the position is

       PW(faculty)=45,000(P/A,x,5)

    7. The following table shows the activities that must be completed to put a new mobile phone on the market. Which activities are on the critical path, and what is the estimated time to project completion?

       Activity	Initial State	Final State	Duration	Description
       1	1	2	2	Overall Design
       2	2	3	2	Design Casings
       3	2	4	4	Write Chip Specifications
       4	3	5	6	Purchase Moulder
       5	4	6	2	Order Chips
       6	4	7	3	Write Battery Specifications
       7	5	8	9	Produce Moulds
       8	6	9	11	Receive Chips
       9	7	10	2	Order Batteries
       10	8	9	13	Receive Moulds
       11	9	11	0	Dummy
       12	10	12	8	Receive Batteries
       13	11	13	7	Install Chip in Casing
       14	12	13	0	Dummy
       15	13	14	10	Install Battery in Casing
       16	14	15	12	Test
       17	15	16	2	Package and Ship

       The first step is to plot a network showing the states and the linking activities. Then mark the activity times on the network. Then take a pass through the nextwork, noting that the earliest we can reach state 1 is at time 0, the earliest we can reach state 2 is at time 2, and so on, till we find the earliest that we can reach the final state, state 17. Now take a backward pass through the network, noting the latest time we can leave each state if the final state is not to be delayed. The difference between the earliest and latest times for each state is the slack for that state, and states with no slack lie on the critical path.

    8. Three Martian colonies, Lowell, Bradbury and Clarke, are planning their crop production for 2098. The colonies have a limited amount of useable land and water, as shown in the following table:

       Colony	Useable Land (Hectares)	Water Allocation (Hectare metres/hectare)
       Lowell	400	600
       Bradbury	600	800
       Clarke	300	375

       Crops adapted for Martian conditions include sour potatoes, bladder-root, and transgenic hemp. There is an upper limit on the total area that can be devoted to each crop by the three colonies together, imposed by the Martian Board of Agriculture. This limit, and the different profitability and water requirement of the crops, are given by the following table:

       Crop	Maximum Quota (Hectares)	Water Consumption (Hectare metres/hectare)	Net Return
       Sour Potatoes	600	3	400
       Bladder-root	500	2	300
       Transgenic Hemp	325	1	100

       The three colonies are developing a joint plan to maximize the profitability of their crops. By selecting suitable decision variables, set up a linear programming problem whose solution will give the most profitable crop mix for the three colonies together. You should put the problem into suitable form for input to a standard simplex solution package, but do not attempt to solve it.

    The hardest part is choosing the right decision variables. There should be nine, representing the acreage of each crop grown at each colony:

    x_l,s, x_l,b,x_l,h, x_b,s,x_b,b, x_b,h,x_c,s, x_c,b,x_c,h

    The objective function is then

    Z = 400*(x_l,s+x_b,s+x_c,s) + 300*(x_l,b+x_b,b+x_c,b) + 100*(x_l,h+x_b,h+x_c,h)

    There are a number of constraints:

    Maximum quotas:

    x_l,s+x_b,s+x_c,s <= 600

    x_l,b+x_b,b+x_c,b <=500

    x_l,h+x_b,h+x_c,h <= 325

    Water consumption

    3*x_l,s+2*x_l,b+x_l,h <= 600

    3*x_b,s+2*x_b,b+x_b,h <= 800

    3*x_c,s+2*x_c,b+x_c,h <= 375

    Useable Land

    x_l,s+x_l,b+x_l,h <= 400

    x_b,s+x_b,b+x_b,h <= 600

    x_c,s+x_c,b+x_c,h <= 300

    These can be converted to standard form by adding slack variables.

    Cribsheet for Final Exam
    

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