Solutions to Problem Set # 3

1 a)  Applying the SML equation, 0.16354 = . R_f  + 1.48 [.1225 -  R_f ]     =>    R_f (1 - 1.48)  =   .16354 - 1.48 * .1225

=>      R_f = (-.01776)/(-0.48) = 3.7%

b)  Risk premium for the market portfolio = E(R_m) -  R_f = .1225 - .037 = 8.55%

c)  Risk premium for EI's stock= E(R_i) -  R_f = .16354 - .037 = 12.654%

d) The extra return on the market is .046 - .1225 = -7.65%.  The extra return on the stock will be 1.48 * (-.0765) = -11.322%.  This means the return we now expect from the stock  = .16354 - .11322 = 5.032%

The abnormal return = .075 - .05032) = 2.468%

2 a) We have to apply the CAPM equation twice, once to RC's stock and once to the stock portfolio:

            0.00561 = R_f - 0.26* RP_m  

            0.10626 = R_f + 0.84* RP_m  

Subtracting the first equation from the second gives:

           .10065 = 1.10 * RP_m =>  RP_m = .10065/1.1 = 9.15%

Substituting this back in the first equation:

           0.00561 = R_f - 0.26* .0915   =>   R_f = .00561 + 0.26*.0915 = 2.94%

For the second stock, we then have:  E(R_i) = .0294 + 1.2*.0915 = 13.92%

b) (I though this was covered in the notes.  I now find that it isn't there.  I had mentioned it quickly in class, but not enough to stress it.  So I'll just drop this part of the question.)  What we're assuming here is market efficiency.  The CAPM, in itself, gives required return.  We can treat that as the expected return only if we assume that markets are efficient.

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] 

    = .0265 + 0.72*(0.1326 - 0.0451) = 8.95%

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 = .0488 - (.0642-.0451) = 2.97%

avg. historical long-term R_f = avg. historical short-term R_f = .0451

=> R_i = .0297 + 0.72*(0.1326 - 0.0451) = 9.27%

5 a) The total value of both divisions is $842 million

The asset beta of the firm = (569/842)*1.24 + (273/842)*0.59 = 1.0293

The equity beta will then be:    

        b_e  = b_a + (D/E)[ b_a - b_d] = 1.0293 + (.36/.64)*(1.0293 - 0.23) =  1.4788

b) Original debt = .36*842 = 303.12; original equity = 842 - 303.12 = 538.88

New debt = 303.12 - 57 = 246.12; new equity = 538.88 + 57 = 595.88

New debt equity ratio = 246.12/595.88 = 0.4130

When the firm reduces its debt-equity ratio, the debt will become less risky than before.  Any reasonable assumption about how much the debt beta drops is acceptable.  Let's assume the debt beta becomes 0.2.  The new equity beta will then be:

        b_e  = 1.0293 + 0.413*(1.0293 - 0.2) = 1.3718

5) First row:     when there's no debt, b_e = b_a = 0.962

Last column:    since we're holding the assets fixed, the asset beta does not change.  b_a = 0.962 in each row.

Second row:    0.25*b_d + 0.75*1.2093 = 0.962 => b_d = (0.962 - 0.75*1.2093)/.25 = 0.2201

Third row:        b_e = 0.962 + (.5/.5)*(0.962 - .41) = 1.514

Last row:          (D/V)* 0.68 + (1 - D/V)* 1.8080 = 0.962  => D/V = (0.962 - 1.808)/(0.68 - 1.808) = .75

So the completed table will be:

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