Solutions to Practice Problems – Chapter 9

1 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] 

    = .018 + .9*[.125 - .037] = 9.72%

b)  There isn't an equally strong convention governing which t-bond should be used for the long-term riskfree rate, but we'll 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 = .046 - (.057-.037) = 2.6%

avg. historical long-term R_f = avg. historical short-term R_f = .037

=> R_i = .026 + .9*[.125 - .037] = 10.52%

c) The second term is always the same, so the two methods yield the same number when the long-term R_f today is equal to the short-term R_f today.  In other words, when the only difference between the 20-year t-bond yield today and the 3-month t-bill yield today is the historical risk premium of 2%.   

In this example though, the difference between today's 20-year t-bond yield and today's 3-month t-bill yield is more than the historical risk premium of 2%. (This basically means that short-term rates are expected to increase over the next 20 years.  The 20-year yield = geometric average of expected 3-month yields over the next 20 years + the historical risk premium of 2%.)

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

            = .02 + 1.25*[.1212 - .0344] = 12.85%

b)  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 = .05 - (.0535-.0344) = 3.09%

avg. historical long-term R_f = avg. historical short-term R_f = .0344

=> R_i = .0309 + 1.25*[.1212 - .0344] = 13.94%

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

            = .023 + 1.1*[.1225 - .039] = 11.485%

b)  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 = .033 - (.056-.039) = 1.6%

avg. historical long-term R_f = avg. historical short-term R_f = .0344

=> R_i = .016 + 1.1*[.1225 - .039] = 10.785%

4 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] 

    = .0165 + 1.25*(0.1222 - 0.0388) = 12.075%

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 = .052 - (.0575-.0388) = 3.33%

avg. historical long-term R_f = avg. historical short-term R_f = .0388

=> R_i = .0333 + 1.25*(.1222 - .0388) = 13.755%

c) The two estimates are equal when the long-term R_f today is equal to the short-term R_f today.  This happens when the difference between the 20-year t-bond yield today and the 3-month t-bill yield today equals the historical risk premium on 20-year t-bonds.  What we are saying here is that the expectations factor is zero. This means that short term rates are expected to remain constant over time.  If short-term rates were expected to increase, the expectations factor would be positive, and the long-term R_f today would be greater than the short-term R_f today.

5)  Viewing the firm as a portfolio of the two divisions, the asset beta is just a weighted average of the betas for the two divisions:   b_a = x_CHICKEN b_CHICKEN + x_EGG b_EGG = (5/8)*1.2 + (3/8) * 0.9 = 1.0875

6) Stock A:   b_a = (D/V) b_d + (E/V) b_e = 0.15 * 0.22 + (1 - 0.15) * 1.53 =  1.3335

Stock B:   b_e  = b_a + (D/E)[ b_a - b_d] = 0.56 + (0.3/0.7)*(0.56 - 0.15) =  0.7357

                (D/E is just D/V divided by E/V)

Stock C:   b_e  = b_a + (D/E)[ b_a - b_d] = 1.22 + (0.25/0.75)*(1.22 - 0.25) =  1.5433

Stock D:   0.90 = 0.2 * b_d + 0.8 * 1.09     =>   b_d = (0.9 - 0.8 * 1.09)/0.2  =  0.14

Stock E:  1.11 = x * 0.32 + (1-x) * 1.76    =>    x = (1.76-1.11)/(1.76 - 0.32) = 0.45

7) There may be a brief moment of panic when you look at the first row of the table and go "What the blank? There are three unknown variables!"  But then after you take a deep breath, here's how it goes:

With a pure capital structure change, asset beta stays constant.  So first we use the second row to compute this firm's asset beta:

         b_a = (D/V) b_d + (E/V) b_e = 0.2 * 0.1 + 0.8 * 0.9 =  0.74

For the first row then, since there is no debt the equity beta equals the asset beta (and there is no debt beta)

For the third row, we solve for b_e :  b_e  = b_a + (D/E)[ b_a - b_d] = 0.74 + (0.4/0.6)*(0.74 - 0.15) =  1.1333

For the fourth row we solve for D/V just like stock E in question 5:

        0.74 = x * 0.2 + (1-x) * 1.4    =>    x = (1.4 - 0.74)/(1.4 - 0.2) = 0.55

For the last row we solve for b_d just like stock D in question 5:

0.74 = 0.7 * b_d + 0.3 * 1.53     =>   b_d = (0.74 - 0.3 * 1.53)/0.7  =  0.401

The completed table would then be:

(A good idea to do a quick check and make sure that debt beta and equity beta both increase as D/V goes up.)

8 a) The asset beta of the firm = 0.7*1.1 + 0.3*0.85 = 1.025

The equity beta will then be:    

        b_e  = b_a + (D/E)[ b_a - b_d] = 1.025 + (28/67)*(1.025 - 0.25) = 1.025 + 0.4179 *(1.025 - 0.25) = 1.3489

b) If they double their debt-equity ratio, the debt will become more risky than before.  A little algebra (or trial and error) will show that the debt-equity ratio doubles when the debt increases to $43.25 million and the equity drops to $51.75 million (the total remaining unchanged at $95 million).  So the debt increases from $28 million to $43.25 million. Any reasonable assumption about how much the debt beta increases is acceptable.  Let's assume the debt beta becomes 0.4.  The new equity beta will then be:

        b_e  = 1.025 + (2*0.4179)*(1.025 - 0.4) = 1.5474

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

    Firm A: b_a = (D/V) b_d + (E/V) b_e = 0.3*0.27 + 0.7*0.999 = 0.7803

    Firm B: b_a = 0.45*0.36 + 0.55*0.869 = 0.64

    Firm C: b_a = 0.2*0.22 + 0.8*0.945 = 0.8

Then we take the average asset beta = (0.7803 + 0.64 + 0.8)/3 = 0.7401

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

If D/V = 0.36, then E/V = 0.64    =>   D/E = 0.36/0.64 = 0.5625

    b_e  = b_a + (D/E)[ b_a - b_d] = 0.7401 + 0.5625*(0.7401 - 0.32) =  0.9764

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

    Firm A: b_a = [D/(D+E)] b_d + [E/(D+E)] b_e = (14/40) 0.2 + (26/40) 1.85 =  1.273

    Firm B: b_a = (24.2/55) 0.3 + (30.8/55) 2.16 =  1.342

    Firm C: b_a = (16.1/46) 0.15 + (29.9/46) 1.89 =  1.281

Then we take the average asset beta = (1.273 + 1.342 + 1.281)/3 = 1.2984

(Firmly resist all temptation to compute a weighted average of the three asset betas.  In finance, it becomes almost a reflex action to compute weighted averages.  But this is not a finance issue.  This is a statistics issue.  We're trying to estimate asset beta for the paper industry; we have three independent estimates; the best overall estimate is the simple average of the three individual estimates.)

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

    b_e  = b_a + (D/E)[ b_a - b_d] = 1.2984 + 0.36*(1.2984 - 0.2) =  1.6938

11)  Once again, just unlever each equity beta, and average the resulting asset betas:

    Firm 1: b_a = 0.35 * 0.32 + 0.65 * 1.874 =  1.330

    Firm 2: b_a = 0.45 * 0.45 + 0.55 * 2.268 =  1.450

    Firm 3: b_a = 0.3 * 0.24 + 0.7 * 1.669 =  1.240

    Firm 4: b_a = 0.28 * 0.27 + 0.72 * 1.812 =  1.380

The average value = 1.350

12 a)   If you look at annual returns for 3-month t-bills and 20-year t-bonds for the last 75 years, half the years the t-bill return will be higher and half the years the t-bond return will be higher. F T-bond returns are systematically higher, since t-bonds are riskier (they have more interest rate risk).  On average, riskier assets generate higher returns.

b)   If you look at annual returns for the last 75 years, the average long-term riskfree rate will just equal the average short-term riskfree rate. T The only systematic difference between t-bill returns and t-bond returns is the risk premium for interest rate risk.  We get the long-term riskfree rate by subtracting away the risk premium form the average t-bond return.  This has to equal the average t-bill return, which is the short-term riskfree rate.

c)   The yield on long-term treasuries today differs from the yield on short-term treasuries for two reasons: a risk premium for interest rate risk and an expectations factor that reflects whether short-term rates are expected to increase or decrease over time. T The first factor is systematic; it doesn't cancel out when we take the average return over many years.  Half the time the expectations factor is positive, and half the time it is negative.  So it cancels out when we take the average return over many years.  But at a given point in time, it is either positive or negative, and affects the long term rate.

d)   If short-term interest rates are expected to increase today, the difference between the yield on short-term Treasuries and the yield on long-term Treasuries today will be more than the average historical difference between these two yields. T The expectations factor will be positive, so the yield on long-term treasuries will equal the yield on short-term treasuries + the risk premium for interest rate risk (the average historical difference between these l-t and s-t yields) + a positive expectations factor.

e)   Positive NPV projects plot above the SML and negative NPV projects plot below it. T Since the SML plots the required return and a positive NPV project has an expected return > the required return.

f)   If you take a given firm and increase its debt ratio while holding assets constant, debt beta goes up (since the debt becomes riskier) so equity beta has to go down to keep asset beta constant.   F Both debt and equity beta go up.  The weighted average stays constant because as the debt ratio increases, we keep shifting weight from the higher number (b_e) to the lower number (b_d).

g)   Even though equity beta represents the systematic risk stockholders will actually bear, we still use asset beta to compute a project's required return.   T Since we assume 100% equity financing in computing cashflows (by ignoring interest expenses) we have to be consistent and make the same assumption in coming up with the discount rate.  So we use the asset beta, which represents the systematic risk stockholders would bear if the project was 100% equity financed.

h)   A firm's debt beta will always be lower than its asset beta.  T Debt holders stand first in line.  Their cashflows are safer than the cashflows to equity holders.  This means b_e > b_d.  Since b_a is the weighted average, it lies in between  =>  b_e > b_a > b_d.  As a result, the asset beta is always greater than the debt beta.  

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