Solutions to Problem Set # 4

1 a) V_U = (ECF of firm)/R_a => ECF of firm = 6,200,000*.108 = 669,600

CF to s/h = ECF of firm - Interest = 669,600 - 275,000 = 394,600

D = interest/R_d = 275,000/.079 = 3,481,012.66

E = V_L - D = V_U - D = 6,200,000 - 3,481,012.66 = 2,718,987.34

R_e = R_a + (R_a – R_d)*(D/E) = .108 + (.108 - .079)*(3,481,012.66/2,718,987.34) = 14.5128%  

b) Alternately, since E = (CF to s/h)/R_e, R_e = 394,600/2,718,987.34 = 14.5128%

c) WACC is not affected by leverage, and always equals R_a.  So WACC is still 10.8%.

2 a) Expected cashflow from the project = .5*750 + .3*1,500 + .2*3,000  = 1,425

NPV of project = 1425/1.06 – 1,100 = 244.34

b) If the project is not taken, cashflows will be as follows:

               State of Economy      Probability      Total CF        CF to b/h       CF to s/h

                        Great                    0.5                16,000           10,500           5,500

                        Okay                     0.3                11,900           10,500           1,400

                        Poor                      0.2                  7,000             7,000                  0

The value of debt is (.8*10,500 + .2*7,000)/1.06  = 9,245.28  

The value of equity is (.5*5,500 + .3*1,400/1.06  = 2,990.57  

If the project is taken, cashflows are as follows:

               State of Economy      Probability      Total CF        CF to b/h       CF to s/h

                        Great                    0.5                16,750           10,500           6,250

                        Okay                     0.3                13,400           10,500           2,900

                        Poor                      0.2                10,000           10,000                  0

The value of debt is (.8*10,500 + .2*10,000)/1.06  = 9,811.32  

The value of equity is (.5*6,250 + .3*2,900/1.06  = 3,768.87  

So the wealth of bondholders increases by 9,811.32 - 9,245.28 = 566.04

Since stockholders contribute the investment of 1,100, the increase in their wealth = 3,768.87 - (2,990.57 + 1,100) = -321.70

Bondholders get the entire NPV of the project, 244.34.  In addition, they get a wealth transfer of 321.70 from the stockholders.  This explains their total increase in wealth of 566.04.

c) Stockholders will not invest in this project, since it reduces their wealth by 321.70.   

3 a) Since the debt-to-value ratio is given (and R_d is not) we need to compute WACC using:

         WACC = R_a * [ 1 – (D/V)* T_c] = .116 * (1 - .3*.35) = 10.382%

Then V_L = (ECF of firm)/WACC = 365,000/.10382 = 3,515,700.25

The straightforward way to compute debt is using the debt-to-value ratio:  D = 3 * 3,515,700.25 = 1,054,710.08

There is only one other way to compute D with the information given.  (Since the interest amount is not given, you cannot go D = Interest/ R_d, after computing R_d).

We have to use V_L = V_U + D* T_c  =>  3,515,700.25 = 365,000/.116 + D*.35 = 3,146,551.72 + D*.35

                                                       => D = (3,515,700.25 - 3,146,551.72)/0.35 = 1,054,710.08

E = V_L - D = 3,515,700.25 - 1,054,710.08 = 2,460,990.18 (or you can go .7*3,515,700.25)

Finally, solve for R_d from the WACC equation:

                .10382 = .3 * (1 - .35) * R_d + .7 * .133 = .195 R_d + .0931

          =>  R_d = (.10382 - .0931)/.195 = 5.4974%

The after-tax cost of debt is then R_d (1 – T_c) = .054974*.65 = 3.5733%

b) When the debt-to-equity ratio changes to .625, debt-to-value = .625/1.625 = 0.3846

WACC = .116 * (1 - .3846*.35) = 10.0385%

Then V_L = 365,000/.100385 = 3,636,015.33

D = 0.3846 * 3,636,015.33 = 1,398,467.43

Or, the other way,  D = (3,636,015.33 - 3,146,551.72)/0.35 = 1,398,467.43

E = V_L - D = 3,636,015.33 - 1,398,467.43 = 2,237,547.89

R_d, however, cannot be computed (since we don't the new value of R_e)

4 a) T^’ = 1 – [(1 - T_pe)(1 - T_c)]/( 1 - T_p) = 1 – (1 - .13)*(1 - .35)/(1 – 0.28) = .214583

V_U = 1,675,000/0.125 = 13,400,000

D = 579,000/0.086 = 6,732,558.14

PVTS = D*T^’ = 6,732,558.14 *.214583 = 1,444,694.77

V_L = 13,400,000 + 1,444,694.77  = 14,844,694.77  

Now we can compute the value of equity: E = V_L– D = 14,844,694.77 - 6,732,558.14 = 8,112,136.63

b) Annual cashflow to s/h = firm's cashflow - interest + annual interest tax shield 

                                         = 1,675,000 - 579,000 + 579,000*.214583 = 1,220,243.75

(Can also compute this the backdoor way: Annual cashflow to s/h = E * R_e.  This would require computing WACC as done below, and then solving for R_e in the other WACC formula.)

c) The original WACC = R_a * [ 1 – (D/V)* T^’] = .125 * [1 – (8.112/14.845)*.214583] = 11.2835%

new WACC = .95 * .112835 = 10.7193%

=> .107193 = .125* [ 1 – (D/V)* .214583] 

=> D/V = (1 - .107193/.125)/.214583 = .6639

The new value of the firm is 1,675,000/0.107193 = 15,625,994.49

The new value of debt is 0.6639 * 15,625,994.49 = 10,373,566.56

5 a) Firm A: 

        WACC = R_a * [ 1 – (D/V)* T^’] = .1234*(1 - .18*.22) = 11.8513%

        .118513 = (D/V) R_d*(1 - T^’) + (E/V) R_e = .18*.057*(1 - .22) + .82*R_e = .008003 + .82 R_e

=> R_e = (.118513 - .008003)/.82 = 13.4769%

Firm B:

        WACC = R_a * [ 1 – (D/V)* T^’]    =>   .0987 = .1111 * [ 1 – (D/V)* .22]  

                                                         =>  D/V = (1 - .0987/.1111)/.22 = .5073

        .0987 = .5073*.R_d*(1 - .22) + .4927*.1293 = .3957 R_d + .063073

=> R_d = (.0987 - .063073)/.3957 = 8.8440%

Firm C:

WACC = (D/V) R_d*(1 - T^’) + (E/V) R_e = .36 * .063 * .78 + .64*.1380 = 10.6010%

R_a = WACC/[ 1 – (D/V)* T^’] = .10601/[1 – .36*.22] = 11.5129%

b) We use the same basic approach as in Ch. 9.  Take the average R_a for the sample of firms, and use the project's D/V ratio to compute the project's WACC:

Average R_a = (.1234 + .1111 + .115129)/3 = 11.6543%

At a D/V ratio of .25, WACC = R_a * [ 1 – (D/V)* T^’] = .116543 * [1 – .25*.22] = 11.0133%

6 b) First compute the project’s APV, and then do WACC.

Base case NPV = 18,000/.106 – 175,000 = -5,188.68

Amount of new debt to be issued = 175,000*.35/(1 - .03) = 63,144.33

Issue costs for debt = .03*63,144.33 = 1,894.33

  (Proceeds of debt issue should be .45 * 175,000 = 61,250.  Verify this:  63,144.33 - 1,894.33 = 61,250)

Amount of new equity to be issued = 175,000*.2/(1 - .05) = 36,842.11

Issue costs for equity = .05*36,842.11 = 1,842.11

  (Proceeds of equity issue should be .2 * 175,000 = 35,000.  Verify this:  36,842.11 - 1,842.11 = 35,000)

Total issue costs = 1,894.33 + 1,842.11  = 3,736.44 

PVTS = D*T^’ = 63,144.33*0.2 = 12,628.87

Adjusted NPV of project = Base case NPV + PVTS - Issue costs = -5,188.68  + 12,628.87 - 3,736.44 = 3,703.75

a) For computing WACC we use the value of the project before issue costs are subtracted away.  In other words, V = NPV + investment + Issue costs = 3,703.75 + 175,000 + 3,736.44 = 182,440.19

WACC = R_a * [ 1 – (D/V)* T^’] = .1 * [1 – (63,144.33/182,440.19)*.2] = 9.8662%

  (Verify that this WACC yields correct NPV:

   NPV = 18,000/.098662 - 175,000 - 3,736.44 = 3,703.75)