Sample MIDTERM 1 : Solutions

1 a)      Since the nominal cashflows grow at the inflation rate, the real cashflows are just constant.  The real cashflow corresponding to C₁ = 328/1.03 = 318.45.  The remaining real cashflows are the same.

b) The real discount rate for the first 2 years = 1.08/1.025 – 1 = 5.3659%

The real discount rate for the next 2 years = 1.08/1.04 – 1 = 3.8462%

So we have to use 5.3659% per year between time 0 and time 2, and 3.8462% between time 2 and time 4.

We can first discount the last two cashflows to time 2, using a discount rate of 3.8462%.  Then this PV₂ along with the first two cashflows can be discounted back to time 0, using a discount rate of 5.3659%.

PV₂(C₃, C₄) = 8000/1.038462 + 8000/1.038462² = 15,122.09

PV₀ of all the cashflows = 8000/1.053659 + (8000 + 15,122.09)/1.053659² = 28,419.62

2 a) Actual rate = 10.10/600 – 1 = 1.6833% over 2 months

Effective annual rate involves converting a 2-month rate to a 12 month rate:

(1 + .016833)^12/2 –1 = 10.5347%

b) Actual rate = 2% over 3 months

Since payments are monthly, we need to discount using a 1-month effective rate.

Converting the three month rate to a one month rate, we have:

(1 + .02)^1/3 – 1 = 0.6623%

We have an annuity with 36 payments of $600 each, to be discounted at a rate of 0.6623% per period.

=> PV₀ = 600/1.006623 + 600/1.006623² + . . . + 600/1.006623³⁶ = 19,161.96

3 a)      Payout ratio = .42/1.20 = 35%

Plowback ratio = 1 - Payout ratio = 65%

Growth rate = .65 * .14 = 9.1%

D₁ = D₀*(1+g) = .42 * 1.091 = .4582

P₀ = D₁/(r – g) = .4582/(.10 - .091) = 50.91

Existing assets will generate earnings of 1.20*1.091 = 1.3092 at time 1.  

If the firm makes no more reinvestment, earnings will be constant at 1.3092 forever.  

In other words, the existing assets will generate a perpetuity of 1.3092.  So PV of existing assets = 1.3092/.1 = $13.09

=> PVGO = 50.91 – 13.09 = 37.82

b) You would first compute the stock price using the actual dividends.  You would then re-compute dividends assuming that reinvested earnings earn only the required return (i.e. all future investments are zero NPV investments), and compute the resulting stock price.  The difference between these two stock prices is PVGO.

4 a)      IRR for project A is given by:

             -130 + 44/(1+IRR_A) + 88/(1+ IRR_A)² + 50/(1+ IRR_A)³ =  0  

Using a financial calculator, IRR_A = 18.43%

IRR for project B is given by:

             -80 - 74.5/(1+IRR_B) + 158/(1+ IRR_C)² + 50/(1+ IRR_C)³ =  0  

Using a financial calculator, IRR_B = 18.36%

Since both IRRs are greater than the required return of 10%, both projects have positive NPV.  We need to look at the incremental cashflows:

                        Project              C₀                C₁               C₂  

                        B - A              50,000       -118,500       70,000      

IRR_B-A is given by:     5x² - 11.85x + 7 = 0

     =>  x = 11.85 ± (11.85² - 4*5*7)^0.5 = 1.12, 1.25      =>   IRR = 12% and 25%

                                2*5

Since there are two IRRs, we will have to consider what the NPV curve for B-A looks like.  Since the sum of the cashflows is positive, the NPV curve has a positive y-axis intercept.  NPV therefore starts out positive, turns negative at 12% and turns positive again at 25%.

At a required return of 10%, B-A will have a positive NPV, so we will choose project B.

b) A would be chosen at any discount rate higher than 12% and less than 18.43%.   12% comes from the fact that at discount rates greater than 12%, B-A has a negative NPV, so A has a higher NPV than B.  18.43% comes from the fact that at discount rates greater than 18.43%, A has a negative NPV.  The fact that B has a more negative NPV doesn’t help; we still don’t want to take A. 

5.   Since the land will be used for the project, we have to charge the project the opportunity cost of the land, which will be it’s market value today. 

So at time zero, the net cashflow = –150,000 (capital investment) – 25,000 (opportunity cost of the land, treated as part of the capital investment) - 6,000 (initial working capital) = -181,000. 

In computing net cashflows for the last year, we ignore interest and allocated overheads but include incremental overheads. 

Working capital requirements each year are:

First year             6,000

Second year        6,720

Third year            7,526.40

Fourth year          8,429.57

At the end of the project, we will recover the working capital used during the last year, i.e. $8,429.57.

At time 4, the fixed assets will be fully depreciated, and will be sold for $22,000.  The entire sale price is taxable, so tax = .35*22,000 = 7,700.  The after-tax salvage value = 22,000 – 7,700 = 14,300

The net cashflows are for the last year are:                                               

        Revenue                                 520,000

-       Variable cost                           428,000

-       Fixed cost                                 23,500

-       Incremental overheads               3,600

-       Depreciation                             11,115             (.0741*150000)

=      Pre-tax income                          53,785

-       Tax (@ 35%)                             18,824.75 

=      Modified N.I.                              34,960.25

+      Depreciation                              11,115

+      Recovery of W.C.                        8,429.57         

+      After-tax salvage value              14,300

=      Net cashflows                            68,804.82

6 a)         First compute Equivalent Annual Cashflow for the new machine:

NPV = 18,000/1.09 +  … 18,000/1.09⁴ - 50,000 = 58,315.96 - 50,000 = 8,315.96

EAC is given by              EAC/1.09 + … + EAC/1.09⁴  = 8,315.96

Solving for the payment that makes the PV of a 4-year annuity equal to 8,315.96, we get EAC = 2,566.57

Since the existing machine will last for 3 more years, it must be replaced by time 3.

The cashflows for the different replacement alternatives are then:

                           C₀                  C₁                      C₂                    C₃                     C₄

Replace at 0       0              2,566.57              2,566.57          2,566.57           2,566.57 . . .

Replace at 1       0              5,000                   2,566.57          2,566.57           2,566.57 . . .

Replace at 2       0              5,000                   2,500               2,566.57           2,566.57 . . .

Replace at 3       0              5,000                   2,500               1,250                2,566.57 . . .

Cashflows beyond time 3 are 2,566.57 each year in each case, and can be ignored.

It is clear that:

-- replacing at time 2 is better than replacing at time 3 (higher cashflow at time 3; all other cashflows equal.)

-- replacing at time 1 is better than replacing at time 2 (higher cashflow at time 2; all other cashflows equal.)

-- replacing at time 1 is better than replacing at time 0 (higher cashflow at time 1; all other cashflows equal.)

=> the machine should be replaced at time 1 (i.e. the end of the first year or the beginning of the second year).

b) Assumptions:

7.         The total output per machine is 15,000 (peak season) + 6,750 (off season) = 21,750 shovels

If you keep the old machines:

Annual operating cost = 2*21,750*5 = 217,500

PV of operating costs = 217,500/0.06 = 3,625,000

If you replace both machines:

Initial investment = 2*150,000 = 300,000

Annual operating cost = 2*21,750*4.2 = 182,700

PV of costs = 300,000 + 182,7000/0.06 = 3,345,000

Replace just one machine:

The new machine is cheaper per unit ($4.20 instead of $5).  In the spring/summer you’ll use only the new machine to get the 13,500 shovels you need.  In fall/winter, you’ll use both machines, to produce 15,000 shovels each.

Output of new machine = 13,500 + 15,000 = 28,500; output of old machine = 15,000

Initial investment = 150,000

Annual operating cost = 28,500*4.20 + 15,000*5 = 194,700

PV of costs = 150,000 + 194,700/0.06 = 3,395,000  

=> It’s best to replace both machines; that’s what results in the lowest PV of costs. 

8 a) In the context of a corporation the term “agency problem” refers to the fact that managers will act in the interests of stockholders only instead of balancing the interests of all the stakeholders in the firm (including bondholders, employees, customers, suppliers, etc).  F It refers to the fact that although managers are supposed to act in the interests of stockholders, sometimes there will be a conflict of interests and they will act in their own interests.

b)   If you have a capital market in which you can lend but not borrow, you can compute a FV of a time 0 cashflow, but you cannot compute a PV today for a future cashflow.  F You cannot compute either PV of FV.

c)   Computing stock price as the present value of just the future dividends ignores the fact that stockholders receive part of their return through capital gains.  F Capital gains are implicitly captured.  Each stockholder receives a specific number of dividends, and the capital gain they have when they sell is captured by the remaining dividends.

d)   Holding other things constant, a company with valuable growth opportunities (i.e. positive PVGO) will have a high P/E ratio.  T Holding other things constant, having valuable growth opportunities increases the stock price, and therefore increases the P/E ratio.

e)   If a company whose required return is10% has future investment opportunities each year with an NPV equal to 8% of the investment, their stock would be considered a growth stock.  T Since the NPV is positive, PVGO is positive, so it is a growth stock.

f)    When you are considering just a single project, and the project has non-conventional cash flows, the investment decision can be made using IRR alone, if we reverse the IRR rule to make it: "Accept when IRR is less than required return."  F That works only with borrowing cashflows, not with all non-conventional cash flows.

g)   If short term and long-term interest rates are not equal, investment decisions cannot be made using IRR alone.  T IRR can be used only when there is a flat term structure (yield curve is horizontal, or short term and long-term interest rates are equal).

h)   If working capital is a fixed proportion of revenue, and revenues and costs increase at the inflation rate, nominal cashflows will grow at the inflation rate and real cashflows will be constant.  F Depreciation will not increase at the inflation rate, so nominal cashflows as a whole will not grow at the inflation rate.

i)    When there is capital rationing, and following the Profitability Index rule does not fully utilize the budget, we look for the best combination that fully utilizes the budget.  F The objective is not to fully utilize the budget, but to generate the highest total NPV.  We look for the best combination regardless of whether it fully utilizes the budget.

j)    A project’s cashflows can increase simply because of inflation; only when capital budgeting is done with real cashflows do we get a reliable assessment of a project’s worth.  F You get the same NPV whether you work with nominal cashflows or real cashflows; both methods give you an equally reliable assessment.

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