Solutions  to MIDTERM 1 (Pink)

1 a) He enters college 18 years from today, i.e. at time 18.  The savings run from time 1 through 18.  The first amount of $28,500 will be paid at the beginning of his first year, which is time 18.  So the 4 payments run time 18 through 21.  The graduation gift of $7,500 is paid at the end of his 4th year, which is time 22.   So cashflows are as follows:

                                           28.5  28.5  28.5  28.5   7.5

              S        S        S       S                

    |         |         |  .....   |        |        |        |        |       |

    0        1        2        17      18     19     20      21     22 

Solve for S by setting the PV of the expenses at time 18 = the FV of the savings at time 18.

PV₁₈(Expenses) = 28,500 + 28,500/1.06 + … + 28,500/1.06³ + 7,500/1.05⁴ = 100,621.5430

=> FV₁₆(Savings) = S + S*1.05 + S*1.08² + … + S*1.05¹⁸ = 100,621.5430

Solving for S will give $3,579.3304

b)  Since the nominal cashflows grow at just the inflation rate, the real cashflows are constant.  Since we are computing the real cashflows at time 18, the real cashflows will just be 28,500.  (For example, for the second payment, the nominal cashflow is 28,500*1.025.  When we deflate it back to time 18 at the inflation rate of 2.5%, we just end up with 28,500.) 

The real discount rate = 1.06/1.025 – 1 = 3.4146%

So the PV at time 16 of just the 4 years' expenses is:

        PV₁₆ = 28,500 + 28,500/1.034146 + … + 28,500/1.034146³ = 108,477.04

2 a) Since payments are made every quarter, we need to discount using a 3-month effective rate.

Stated rate = 8.8% compounded semi-annually =>   Actual rate = 8.8/2 = 4.4% over 6 months

3-month effective rate = (1.044)^1/2 - 1 = 2.1763%

PV₀ = 1400/1.021763 + 1400/1.021763² + . . . + 1400/1.021763¹² = 14,646.4190  

b) We need to set FV₆(Annuity) = PV₆(Perpetuity)

=>        500 [1 – 1/(1+R)⁶]*(1+R)⁶   = 300  

              R                                         R   

=>        [1 – 1/(1+R)⁶]*(1+R)⁶   = 3/5

=>        (1+R)⁶ = 8/5   =>  R = 8.1484%

3 a)   Plowback ratio = 0.75

         Growth rate = 0.75 * .15 = 0.1125%

         D₁ = .25 * 4.20 = 1.05

         P₀ = D₁/(r – g) = 1.05/(.12 - .1125) = 140  

b)      For a constant growth stock, price just grows at the growth rate, g   =>  P₁ = P₀*1.125 = 140 * 1.1125 = 155.75

         (You can always compute this the long way round too:

         D₂ = 1.05*1.1125 = 1.168

         P₁ = D₂/(r – g) = 1.187/(.12 - .1125) = 155.75  

c)      Existing assets generate a perpetuity of 4.20, so PV(existing assets) = 4.20/0.12 = 35

         PVGO = P₀ - PV(existing assets) = 140 - 35 = 105

c)      With no positive NPV investments, the return they earn on their reinvestment will just equal the required return, namely 12%.

         The growth rate would be .75 * .12 = 9%

4 a)      IRR for project A is given by:

             -126.4 + 112.604/(1+ IRR_A) + 72.522/(1+ IRR_A)² =  0  

Using a financial calculator, IRR_A = 32.4151%

IRR for project B is given by:

             -90 + 54/(1+ IRR_B) + 96/(1+ IRR_B)² =  0  

Using a financial calculator, IRR_B = 37.5484%

IRR for project C is given by:

             -120 + 85.234/(1+IRR_C) + 88.321/(1+ IRR_C)²  =  0  

Using a financial calculator, IRR_C = 28.3653%

Both A and B have an IRR greater than the required return of 30%.  First check if the NPV curves intersect.  

Y-axis intercept for A = sum of the cashflows = 58,726

Y-axis intercept for B = sum of the cashflows = 60,000

Since B has a higher IRR and a higher Y-axis intercept, the two curves don't intersect,  So B has the higher NPV, and we should choose B.  

5.   We ignore interest and allocated overheads but include incremental overheads.  The working capital recovered at the end of the project = the 19,000 employed over the last year.

When the assets are sold for $36,000 at the end of the project, we need to include the after tax version of the salvage value:

Book Value at time 4= 60,000 - accumulated depreciation = 60,000 - (11,000 + 8,700 + 7,300 + 5,500) = 27,500

Capital Gain = Sale price - Book Value = 36,000 - 27,500 = 8,500

Tax = .35 * 8,500 = 2,975

After-tax salvage value = 36,000 - 2,975 = 33,025

The net cashflows are for the last year are:                                               

        Revenue                                  96,000

-       Mfg. cost                                 54,500

-       Incremental overheads               3,725

-       Tax Depreciation                       5,500       

=      Pre-tax income                        32,275

-       Tax (@ 35%)                          11,296.25 

=      N.I.                                        20,978.75

+      Tax Depreciation                      5,500

+      Recovery of W.C.                  19,000  

+      After-tax salvage value             33,025

=      Net cashflows                         78,503.75

6 a)        EAC for old machine (assume the costs are paid at the end of each year):

PV of costs = 6,000/1.065 + 8000/1.065² + 10,000/1.065³ + 12,000/1.065⁴  = 30,293.445

EAC_O is given by              EAC_O/1.065 + … + EAC_O/1.1065⁴  = 30,293.445

Solving for the payment that makes the PV of a 4-year annuity equal to 30,293.445, we get EAC₀ = 8,842.7396

EAC for new machine (assume annual costs are paid at the end of each year):

NPV of costs for new machine = 5,000/1.065 +  … + 5,000/1.065⁶ + 25,000 = 49,205.0678

EAC_N is given by              EAC_N/1.065 + … + EAC_N/1.065⁶  = 49,205.0678

Solving for the payment that makes the PV of a 6-year annuity equal to 49,205.0678, we get EAC_N = 10,164.208

The cashflows for the different replacement alternatives are then:

            Time                  C₁         C₂          C₃           C₄           C₅      

               0                10,164    10,164    10,164     10,164    10,164  . . .

               1                   6000    10,164    10,164     10,164    10,164  . . .

               2                   6000       8000    10,164     10,164    10,164  . . .

               3                   6000       8000    10,000     10,164    10,164      . . .

               4                   6000       8000    10,000     12,000    10,164  . . .

Clearly, it is best to replace at time 3 (at the end of the third year)   

b) Assumptions:

7 a)   Real cashflows represent the actual number of dollars you receive at any given point in time. F Nominal cashflows represent the actual number of dollars; real cashflows are inflation-adjusted.

b)   If a company reinvests part of its earnings and earns an ROE greater than the required return, then PVGO is greater than the PV of existing assets. F PVGO is only positive; need not be greater than PV of existing assets.

c)   If you use existing assets for a project, you should charge the project the opportunity cost of using those assets; the opportunity cost will be the original purchase price or current market value, whichever is greater. F It is the current market value; that is the potential amount you forego today when you use the existign assets for a project.

d)   If long term rates are different from short term rates, you cannot evaluate investment projects using IRR.  T We won't have a required return number to compare the IRR to.

e)   If there is no capital market, the interest rate is zero; so if you have $100 tomorrow, its PV today will just be $100. F The concept of PV does not exist if there is no capital market.

f)  For a constant growth stock, stock price grows at the same rate as dividends or earnings.  T All three grow at the growth rate, g.

g)   If a company plans to be in business indefinitely, the Equivalent Annual Cost (EAC) of machine A represents the after-tax cost of operating machine A indefinitely. T We assume that machine A is replaced by itself indefinitely.  So the EAC is the after-tax cost each year from using A indefinitely.

h)  If the incremental cashflows for A-B have 2 IRRs, 8% and 20%, this means that the NPV curves for both A and B intersect the x-axis at both 8% and 20%. F The NPV curve for A-B intersect the x-axis at 8% and 20%, or the NPV curves for A and B intersect each other at 8% and 20%.

i)   In computing a project's cashflows, either we can ignore depreciation, or we can first subtract it away and then add it back. F Have to subtract it away before taxes, and then add it back after taxes.  Ignoring it is not the same thing. 

j)    Market-to-Book ratio increases if you make positive NPV investments, but it can increase even without making positive NPV investments. T It also increases just with the passage of time (even if you make no new positive NPV investment).  BV doesn't change, but the MV in the numerator tends to keep increasing (due to a positive expected return).