Solutions  to MIDTERM 2 (Pink)

1 The total output per machine is 12,000 (peak season) + 6,600 (off season) = 18,600 shovels

If you keep the old machines:

Annual operating cost = 2*18,600*8.75= 325,500

PV of operating costs = 325,5000/0.08 = 4,068,750

If you replace both machines:

Initial investment = 2*300,000 = 600,000

Annual operating cost = 2*18,600*7.25 = 269,700

PV of costs = 600,000 + 269,700/0.08 = 3,971,250  

    (Thus, it is cost-effective to replace the old machines, i.e. to pay $300,000 per new machine to obtain output of 18,600 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 13,200.  You’ll use the new machine at full capacity to get 12,000 shovels from it; the old machine will be used only to produce the remaining 1,200.  In fall/winter, you’ll use both machines at full capacity, to produce 12,000 shovels each.  

Output of new machine = 12,000 + 12,000 = 24,000; output of old machine = 13,200

Initial investment = 300,000

Annual operating cost = 24,000*7.25 + 13,200*8.75 = 28 9,500

PV of costs = 300,000 + 28 9,500/0.08 = 3,918,750  

=> It’s best to replace just one machine (it's cost-effective to replace the first old machine, since you pay $300,00 for a new machine and get output of 24,000 from it; it's not cost-effective to replace the second one machine, since you pay the same $300,00 but you only get output of 13,200 from it.) 

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

        .1312 =  R_f + b_i * RP_m = .0272 +  b_i * .0816  

=>   b_i = (.1312 - .0272)/.0816  =  1.2745

b)  Expected return on the market portfolio = RP_m + R_f  = .0816 + .0272 = 10.88%

c)  Risk premium for the stock = E(R_i) - R_f = .1312 - .0272 = 10.40%

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

The extra return on the stock will be 1.2745 * 0.0272 = 3.4667%.  

The return we now expect from the stock  = .1312 + .034667 = 16.5867%  

The abnormal return = .1528 - .165867 = -1.3067%

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

R_i = short-term R_f today + beta * [avg. historical stock return - avg. historical short-term R_f] 

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

    = .0275 + 0.49*(0.1264 - 0.0397) = 6.9983%

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).

R_i = long-term R_f today + beta * [avg. historical stock return - avg. historical long-term R_f]

long-term R_f today = 20-year t-bond yld today - avg. risk premium for 20-year t-bond = .0510 - (.0623-.0397) = 2.84%

avg. historical long-term R_f = avg. historical short-term R_f = .0397

=> R_i = .0284 + 0.49*(0.1264 - 0.0397) = 7.0883%  

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

               b_a = (D/V) b_d + (E/V) b_e  

Since D/E = 0.46, this means that D = 0.46*E   =>  D/V = D/(D+E) = .46/1.46 = 0.3151

=>    0.3151*b_d + 0.6849*1.02 = 0.73   =>    b_d = (0.73 - 0.6849*1.02)/.3151 = .0996

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

       b_e  = b_a + (D/E)[ b_a - b_d] = 0.73 + 0.62*(0.73 - 0.14) =  1.0958

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

    Firm P: b_a = (D/V) b_d + (E/V) b_e = (155/567)*0.19 + (412/567)*1.46 = 1.1128

    Firm Q: b_a = (183/599)*0.21 + (416/599)*1.59 = 1.1684

   Firm R: b_a = (219/950)*0.16 + (731/950)*1.49 = 1.1834

The project beta is then the average asset beta = (1.1128 + 1.1684 + 1.1834)/3 = 1.1549

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 

    b_e  = b_a + (D/E)[ b_a - b_d] = 1.1549 + 0.6667*(1.1549 - 0.26) =  1.7515

6 a) We have three separate decisions here:

a) whether to make the initial R&D investment of $2,400,000 at time 0

b) whether to make the additional R&D investment of $1,280,000 at time 1 if the first round R&D effort doesn't succeed

c) whether to make the $14,400,000 investment to produce the yo-yos once R&D efforts succeed.

                                                              LAUNCH    4,040,677.97 

                                        SUCCESS  /  AT TIME 1

                                    /    prob 0.6     \ DON'T     0                                          LAUNCH     4,040,677.9

                                  /    4,040,677.97                                      SUCCESS /  AT TIME 2

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

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

      /       -65,120.68      \                    /       464,970.74        \  FAILURE    0

      /                                    \   FAILURE   /                                          prob 0.65

    /                                          prob 0.4      \

464,970.74  \  ABANDON   0

 DON'T INVEST  0

First question: is it worth making the $14,400,000 investment at time t if the R&D efforts succeed:

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

Next: is it worth investing investing $800,000 at time 1 in additional R&D efforts:

=> the NPV at time 1 of making the additional investment = (0.35*4,040,677.97)/1.118 - 800,000 = 464,970.74

=> 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 $2,400,000 at time 0 to launch the first round of R&D efforts?

=> the NPV at time 0 of launching the first round of R&D efforts = (0.6*4,040,677.97 + 0.4*464,970.74)/1.118 - 2,400,000 = -65,120.68

=> 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 4,040,677.97                                      

                                  /                                                 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 464,970.74

If you buy the information at time 1:

=>  NPV at time 1 of making second round investment = 4,040,677.97/1.118 - 800,000)  = 2,814,202.12 

      (you invest 800,000 at time 1 to get a time 2 NPV of 4,040,677.97)

=>  NPV at time 1 of making buying the information = .35*2,814,202.12 - 160,000 = 824,970.74

      (you spend 160,000 at time 1; with a probability of .35 you get a time 1 NPV of 2,814,202.12)

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

=> the NPV at time 0 of investing in the first round of R&D = (0.6*4,040,677.97 + 0.4*824,970.74)/1.118 - 2,400,000 = $63,680.75

=> 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 a stock and a portfolio have the same beta, the portfolio has less risk since it is more diversified, so it will have a lower required return.  F  The same beta means the same required return.

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 randomly positive or negative just like firm-specific news; it’s not clear why one type of news is diversifiable and one is not. F  It's true that all news/information is randomly positive or negative, but it's perfectly clear why one is diversifiable and the other is not.  On a given day, firm-specific news sends half the firms up and half the firms down, and these movements cancel out.  On a given day, economy-wide news affects most firms the same direction, so these movements don't cancel out.

d)   No rational investor will ever hold a risky asset if it actually has a negative expected return.  F  If the beta is negative enough, the expected return will be negative. Investors hold such stocks anyway because the stocks greatly reduce the risk of their portfolios (just liek an insurance policy, for example).

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 interest rates are expected to stay the same, the yield on long-term treasuries today will equal the yield on short-term treasuries today.  F  The expectations factor will be zero, but the risk premium for interest rate risk will still make the yield on long-term treasuries higher than the yield on short-term treasuries.

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)   As a firm increases its debt ratio by making a pure capital structure change, b_a - b_d will keep shrinking until it eventually becomes negative.  F  Debt is always less risky than equity and the asset beta, being the weighted average of the debt and equity betas, always lies between the two.  This means that the debt beta is always less than the asset beta, so b_a - b_d can never become negative.

j)    If the PV today of the time 1 gold price is $420, and the discount rate is 5%, then the PV today of the time 2 gold price is $400. F PV today of the time 2 gold price is also $420.

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)    In practice, markets are expected to be semi-strong form efficient but not strong form efficient. T  Only public information is available to the market, so the best we can expect is that all public information is reflected in the price, i.e. that the market is semi-strong form efficient.  Markets cannot be expected to be strong form efficient. 

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