Solutions  to MIDTERM 1 (Yellow)

1 a) He enters college 16 years from today, i.e. at time 16.  The savings run from time 1 through 16.  The first amount of $25,600 will be paid at the beginning of his first year, which is time 16.  So the 4 payments run time 16 through 19.  The graduation gift of $6,000 is paid at the end of his 4th year, which is time 20.   So cashflows are as follows:

                                           25.6  25.6  25.6  25.6    6

              S        S        S       S                

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

    0        1        2        15      16     17     18     19     20 

Solve for S by setting the PV of the expenses at time 16 = the FV of the savings at time 16.

PV₁₆(Expenses) = 25,600 + 25,600/1.05 + … + 25,600/1.05³ + 6,000/1.05⁴ = 100,261.3644

=> FV₁₆(Savings) = S + S*1.05 + S*1.08² + … + S*1.05¹⁵ = 100,261.3644

Solving for S will give $4,237.6159

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 16, the real cashflows will just be 25,600.  (For example, for the second payment, the nominal cashflow is 25,600*1.015.  When we deflate it back to time 16 at the inflation rate of 1.5%, we just end up with 25,600.) 

The real discount rate = 1.05/1.015 – 1 = 3.4483%

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

        PV₁₆ = 25,600 + 25,600/1.034483 + … + 25,600/1.034483³ = 97,392.8296

2 a) Since payments are made every month, we need to discount using a 1-month effective rate.

Stated rate = 8.8% compounded quarterly =>   Actual rate = 8.8/4 = 2.2% over 3 months

1-month effective rate = (1.022)^1/3 - 1 = 0.728%

PV₀ = 550/1.00728 + 550/1.00728² + . . . + 550/1.00728³⁶ = 17,362.6898  

b) We need to set FV₉(Annuity) = PV₉(Perpetuity)

=>        300 [1 – 1/(1+R)⁹]*(1+R)⁹   = 500  

              R                                         R   

=>        [1 – 1/(1+R)⁹]*(1+R)⁹   = 5/3

=>        (1+R)⁹ = 8/3   =>  R = 11.5141%

3 a)   Plowback ratio = 0.7 

         Growth rate = 0.7 * .12 = 0.084%

         D₁ = .3 * 3.65 = 1.095

         P₀ = D₁/(r – g) = 1.095/(.09 - .084) = 182.50  

b)      For a constant growth stock, price just grows at the growth rate, g   =>  P₁ = P₀*1.084 = 182.5 * 1.084 = 197.83

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

         D₂ = 1.095*1.084 = 1.187

         P₁ = D₂/(r – g) = 1.187/(.09 - .084) = 187.83  

c)      Existing assets generate a perpetuity of 3.65, so PV(existing assets) = 3.65/0.09 = 40.56.

         PVGO = P₀ - PV(existing assets) = 182.50 - 40.56 = 141.94

c)      With no positive NPV investments, the return they earn on their reinvestment will just equal the required return, namely 9%.

         The growth rate would be .7 * .09 = 6.3%

4 a)      IRR for project A is given by:

             -123.5 + 113.89/(1+ IRR_A) + 51.972/(1+ IRR_A)² =  0  

Using a financial calculator, IRR_A = 25.6978%

IRR for project B is given by:

             -78 + 42/(1+ IRR_B) + 80/(1+ IRR_B)² =  0  

Using a financial calculator, IRR_B = 31.7146%

IRR for project C is given by:

             -140 + 101.234/(1+IRR_C) + 104.321/(1+ IRR_C)²  =  0  

Using a financial calculator, IRR_C = 29.7428%

Since only project B, has an IRR greater than the required return of 30%, it's the only one with a positive NPV.  

 => 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                                  94,000

-       Mfg. cost                                 53,500

-       Incremental overheads               3,725

-       Tax Depreciation                       5,500       

=      Pre-tax income                        32,300

-       Tax (@ 35%)                          11,305 

=      N.I.                                        20,995

+      Tax Depreciation                      5,500

+      Recovery of W.C.                  19,000  

+      After-tax salvage value             33,025

=      Net cashflows                         78,520

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

PV of costs = 5,000/1.065 +  7000/1.065² +  9,000/1.065³ +  11,000/1.065⁴  = 28,867.6465

EAC_O is given by              EAC_O/1.065 + … + EAC_O/1.1065⁴  = 28,867.6465

Solving for the payment that makes the PV of a 4-year annuity equal to 28,867.6465, we get EAC₀ = 7,842.7396

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

NPV of costs for new machine = 4,000/1.065 +  … + 4,000/1.065⁶ + 18,000 = 37,364.0542

EAC_N is given by              EAC_N/1.065 + … + EAC_N/1.065⁶  = 37,364.0542

Solving for the payment that makes the PV of a 6-year annuity equal to 37,364.0542, we get EAC_N = 7,718.2296

The cashflows for the different replacement alternatives are then:

            Time                    C₁        C₂       C₃         C₄        C₅       C₆

               0                   7718      7718    7718     7718     7718    7718  . . .

               1                   5000      7718    7718     7718     7718    7718  . . .

               2                   5000      7000    7718     7718     7718    7718  . . .

               3                   5000      7000    9000     7718     7718    7718  . . .

               4                   5000      7000    9000   11000     7718    7718  . . .

Clearly, it is best to replace at time 2 (at the end of the second year)   

b) Assumptions:

7 a) 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.

b) Nominal cashflows are inflation-adjusted cashflows; real cashflows are not.  F The reverse is true.

c) When a company makes positive NPV investments, its Market-to-Book ratio increases.  T Every positive NPV investment increase the Market-to-book ratio.

d) For a constant growth stock, the dividend yield tomorrow is the same rate as the dividend yield today.   T The dividend yield today is D₁/P₀.  Since numerator and denominator both grow at the same rate g, this ratio doesn't change over time.  (Alternatively, dividend yield for a constant growth stock equals r - g, which is the same each year.)

e) If a company with a required return of 11% reinvests a part of its earnings each year and earns an NPV of 11% on the reinvested earnings, the stock will exhibit growth but the growth will not be meaningful. F Since they are making positive NPV investments, the growth is meaningful (PVGO is positive).

f)  If the incremental cashflows for A-B have 2 IRRs, 6% and 16%, this means that the NPV curve for A and the NPV curve for B intersect the x-axis at both 6% and 16%. F The NPV curve for A-B intersect the x-axis at 6% and 16%, or the NPV curves for A and B intersect each other at 6% and 16%.

g) If two mutually exclusive projects have different required returns, you cannot choose between them by looking at the IRR of the incremental cashflows.  T We don't have a required return to compare the IRR of the incremental cashflows to.

h) 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 The NPV is the same whether you discount real cashflows at the real rate or nominal cashflows at the nominal rate.

i) 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.

j) 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.