Solutions  to MIDTERM 2 (Yellow)

 

1 The total output per machine is 8,000 (peak season) + 4,400 (off season) = 12,400 shovels

If you keep the old machines:

Annual operating cost = 2*12,400*8.75= 217,000

PV of operating costs = 217,000/0.08 = 2,712,500

If you replace both machines:

Initial investment = 2*200,000 = 400,000

Annual operating cost = 2*12,400*7.25 = 179,800

PV of costs = 400,000 + 179,800/0.08 = 2,647,500  

    (Thus, it is cost-effective to replace the old machines, i.e. to pay $200,000 per new machine to obtain output of 12,400 per machine.)

Replace just one machine:

The new machine is cheaper per unit ($7.25 instead of $8.75).  In the spring/summer, you need total output of 8,800.  You’ll use the new machine at full capacity to get 8,000 shovels from it; the old machine will be used only to produce the remaining 800.  In fall/winter, you’ll use both machines at full capacity, to produce 8,000 shovels each.  

Output of new machine = 8,000 + 8,000 = 16,000; output of old machine = 8,800

Initial investment = 200,000

Annual operating cost = 16,000*7.25 + 8,800*8.75 = 193,000

PV of costs = 200,000 + 193,000/0.08 = 2,612,500  

=> It’s best to replace just one machine (it's cost-effective to replace the first old machine, since you pay $200,00 for a new machine and get output of 16,000 from it; it's not cost-effective to replace the second one machine, since you pay the same $200,00 but you only get output of 8,800 from it.) 

 

2 a) Applying the SML equation to PPI's stock:

        .1312 =  Rf + bi * RPm = .0227 +  bi * .0861  

=>   bi = (.1312 - .0227)/.0861  =   1.2602

b)  Expected return on the market portfolio = RPm + Rf  = .0861 + .0227 = 10.88%

c)  Risk premium for the stock = E(Ri) - Rf = .1312 - .0227 = 10.85%

d) The extra return on the market is .136 - .1088 = 2.72%.  

The extra return on the stock will be 1.2602 * 0.0272 = 3.4276%.  

The return we now expect from the stock  = .1312 + .034276 = 16.5476%  

The abnormal return = .1528 - .165476 = -1.2676%

   

3 a) Regardless of what other information is given, we use the yield on 3-month t-bills for the short-term riskfree rate.

Ri = short-term Rf today + beta * [avg. historical stock return - avg. historical short-term Rf] 

    = 3-month t-bill yld today + beta * [avg. historical stock return - avg. historical return on 3-month t-bills] 

    = .0257 + 0.73*(0.1236 - 0.0379) = 8.8261%

b)  There isn't an equally strong convention governing which t-bond should be used for the long-term riskfree rate, but we stick with using 20-year t-bonds (regardless of what other information is given).

Ri = long-term Rf today + beta * [avg. historical stock return - avg. historical long-term Rf]

long-term Rf today = 20-year t-bond yld today - avg. risk premium for 20-year t-bond = .0509 - (.0573-.0379) = 3.15%

avg. historical long-term Rf = avg. historical short-term Rf = .0379

=> Ri = .0315 + 0.73*(.1236 - .0379) = 9.4061%  

 

4 a) Solve the following equation to get the debt beta:

               ba = (D/V) bd + (E/V) be  

Since D/E = 0.34, this means that D = 0.34*E   =>  D/V = D/(D+E) = .34/1.34 = 0.2537

=>    0.2537*bd + 0.7463*0.79 = 0.62   =>    bd = (0.62 - 0.7463*0.79)/.2537 = .12

b) Lever up the asset beta using the new debt-to-equity ratio: 

       be  = ba + (D/E)[ ba - bd] = 0.62 + 0.48*(0.62 - 0.16) =  0.8408

5 a) First we take each firm's equity beta, and unlever to get the asset beta:

    Firm P: ba = (D/V) bd + (E/V) be = (219/931)*0.11 + (712/931)*1.36 = 1.0660

    Firm Q: ba = (183/599)*0.19 + (416/599)*1.64 = 1.1970

   Firm R: ba = (155/526)*0.17 + (371/526)*1.49 = 1.1010

The project beta is then the average asset beta = (1.066 + 1.197 + 1.101)/3 = 1.1213

 

b) Using the asset beta estimated in part a, we lever it up to get the estimated equity beta for firm S:

If D/V = 0.4, then E/V = 0.6    =>   D/E = 0.4/0.6 = 0.6667 

    be  = ba + (D/E)[ ba - bd] = 1.1213 + 0.6667 *(1.1213 - 0.26) =  1.6956

 

6 a) We have three separate decisions here:

a) whether to make the initial R&D investment of $1,500,000 at time 0

b) whether to make the additional R&D investment of $500,000 at time 1 if the first round R&D effort doesn't succeed

c) whether to make the $9,000,000 investment to produce the yo-yos once R&D efforts succeed.

                                                                           

                                                              LAUNCH    2,525,423.73

                                        SUCCESS  /  AT TIME 1

                                    /    prob 0.6     \ DON'T     0                                          LAUNCH     2,525,423.73

                                  /    2,525,423.73                                      SUCCESS /  AT TIME 2

               INVEST IN    /                                INVEST MORE IN /    prob 0.35  \ DON'T           0

        /       R&D AT 0    \                            /   R&D AT TIME 1   \

      /       -40,700.43      \                    /       290,606.71        \  FAILURE    0

      /                                    \   FAILURE   /                                          prob 0.65

    /                                          prob 0.4      \

290,606.71  \  ABANDON   0

 DON'T INVEST  0

 

First question: is it worth making the $9,000,000 investment at time t if the R&D efforts succeed:

Expected cashflows each year, starting at t+1 = 0.8*1,500,000 + 0.2*800,000 = 1,360,000

PVt of future cashflows = 1,360,000/0.118 = 11,525,423.73

NPVt = 11,525,423.73 - 9,000,000 = 2,525,423.73

=> if the R&D efforts succeed, it is worth investing to  produce the yo-yos.

 

Next: is it worth investing $500,000 at time 1 in additional R&D efforts:

With a probability of 0.35, you will start production at time 2, and generate a time 2 NPV of 2,525,423.73

With a probability of 0.65, you abandon the project and end up with zero

=> the NPV at time 1 of making the additional investment = (0.35*2,525,423.73)/1.118 - 500,000 = 290,606.71

=> if the first round of R&D efforts do not succeed, you will go ahead and invest in additional R&D efforts

 

Next: is it worth investing $1,500,000 at time 0 to launch the first round of R&D efforts?

With a probability of 0.6, you will succeed and launch production at time 1, to generate a time 1 NPV of $2,525,423.73

With a probability of 0.4, you won't succeed at time 1.  As computed above, this means you end up with a time 1 NPV of 290,606.71

=> the NPV at time 0 of launching the first round of R&D efforts = (0.6*2,525,423.73 + 0.4*290,606.71)/1.118 - 1,500,000 = -40,700.43

=> launching the R&D efforts is a negative NPV project, and should not be undertaken.

 

b) You have the opportunity to buy information at time 1 if the first round of R&D efforts doesn't succeed.  

 

Earlier, without the information, the time 1 decision was to go ahead and make the second round R&D investment.

If it's worth making this investment even when you don't know whether it will lead to success or not, it is certainly worth making if you buy the information and find that it will succeed.

It is equally clear that if you buy the information and find that the second round R&D investment will not lead to success, then there is no point making the second round investment.  So you just abandon the R&D efforts (and the project).

The decision tree then looks like:

                                                            

                                        SUCCESS  same outcome as before

                                    /    prob 0.6   launch and get time 1 NPV of 2,525,423.73                                      

                                  /                                                 POSITIVE       =>   invest in 2nd round at time 1

              INVEST IN     /                              GET INFO /  prob 0.35               and in production at time 2

             R&D TODAY  \                          /  AT TIME 1  \                         

                                  \                  /                     \ NEGATIVE  => ABANDON

                                           \   FAILURE /                             prob 0.65

                                                prob 0.4 \

                \  DON'T GET  => same outcome as before;

                      INFO               invest in 2nd round and get

                                              time 1 NPV of 290,606.71

If you buy the information at time 1:

you pay $100,000 for the information at time 1

with probability of 0.35, the information is positive; you invest $500,000 in the second round of R&D at time 1 and pick up a time 2 NPV of $2,525,423.73 when you make the time 2 investment for producing yo-yos.

with a probability of 0.65, the information is negative; you abandon the project and get nothing

=>  NPV at time 1 of making second round investment = 2,525,423.73/1.118 - 500,000)  = 1,758,876.32 

      (you invest 500,000 at time 1 to get a time 2 NPV of 2,525,423.73)

=>  NPV at time 1 of making buying the information = .35*1,758,876.32 - 100,000 = 515,606.71

      (you spend 100,000 at time 1; with a probability of .35 you get a time 1 NPV of 1,758,876.32)

                                                                        

So when we consider whether to invest in the first round of trials:

you invest $1,500,000 at time 0

with a probability of 0.6 you end up with a time 1 NPV (from launching production) of 2,525,423.73

with a probability of 0.4 you end up with a time 1 NPV (from buying the information) of 515,606.71

=> the NPV at time 0 of investing in the first round of R&D = (0.6*2,525,423.73 + 0.4*515,606.71)/1.118 - 1,500,000 = $39,800.47

 

=> Clearly, they should buy the information.  Buying the information turns what used to be a negative NPV project into a positive NPV project.

 

7 a)  If stock A has double the systematic risk of stock B, it should have double the risk premium. T  Risk premium is proportional to systematic risk.

b)   Ex-post expected return reflects ex ante E(R) plus the impact of economy-wide news plus the impact of firm-specific news. F  It only reflects ex ante E(R) plus the impact of economy-wide news.

c)   Economy-wide news is undiversifiable even though it is randomly positive or negative just like firm-specific news.  T  All news/information is randomly positive or negative; economy-wide news is undiversifiable because a given piece of news affects most firms the same way.

d)   A stock with an E(R) less than the riskfree rate tends to go up when there is bad news about the economy. T  A negative beta stock will have an E(R) less than the riskfree rate, and it moves up when the market as a whole goes down.

e)   Positive NPV projects plot to the left of the SML and negative NPV projects plot to its right. T  "Above the SML" and "to the left of the SML" mean the same thing.

f)    If short-term rates are expected to increase, today’s long-term Rf will be greater than today’s short-term Rf. T  The long term riskfree rate equals the short term riskfree rate plus an expectations factor.

g)   In a given industry, for some firms the risk of equity is greater than business risk, and for some firms the risk of equity is less than business risk.  F  The risk of equity consists of business risk plus financial risk; financial risk is either positive (if a firm issues debt) or zero (if there's no debt).  So the risk of equity can never be less than business risk.

h)   If a firm holds its assets constant and increases its debt ratio, the debt beta, equity beta and asset beta will all increase. F  The debt beta and equity beta will go up but the asset beta will stay the same.

j)    If the PV today of the time 1 gold price is $530, and the discount rate is 6%, then the PV today of the time 2 gold price is $500.  F  PV today of the time 2 gold price is also $530.

k)   In a semi-strong form efficient market, future returns cannot be predicted using inside information. F  Inside information is not reflected in the price, so future returns can be predicted using inside information. 

l)     If the market does a perfect job of reflecting available information in the price, then the market will be strong form efficient. F  Only public information is available to the market, so if the market does a perfect job of reflecting available information in the price, the market will be semi-strong form efficient.

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