Solutions to Practice Problems – Chapter 4

 

 

1 a)   P0 = 2.50/.09  = 27.78

b)      P1 = present value at time 1 of D2 onwards.  These future dividends are still a perpetuity of $2.50.  So the price at the end of the year is the same:

         P1 = 2.50/.09  = 27.78

         (For a constant growth stock, stock price grows at the same rate as dividends.  When the dividend growth rate is zero, the stock price does not grow either.)

 

2.      Payout ratio = 1.4/3.5 = 40%

         Growth rate = (1 - .4) * .15 = 9%

         D0 = 1.40

         D1 = 1.40 *1.09 = 1.526

         P0 = D1/(r – g) = 1.526/(.12 - .09) = $50.87

 

3.      Treat D3 onwards as a growing perpetuity     =>    P2 =       D3     = 14.50

                                                                                                  .16 - .08

=>     P0  =    D1    + D2+ P2   =  0.90 +  1.05 + 14.55  = 12.33

                   1.16        1.162         1.16           1.162

 

4.      The growth rate = .5 * .2 = 10%

         The next dividend = D1 = D0 * (1+g) = 3.28 * 1.1 = 3.608

         Required return = D1/P0 + g  = 3.608/65 + .10 = 15.5508%

 

5.      We can treat D5 onwards as a growing perpetuity  

=>     P4 = D5/(.14 - .06) = (1.00*1.06)/.08 = 13.25

         The price today is then:

         P0 = D3/1.143 + (D4 + P4)/1.144  =  0.50/1.143 + (1.00 + 13.25)/1.144   = 8.77

          

 

6.      D1 = 2.50

         D2 = 2.50*1.08 = 2.70

         D3 = 2.50*1.082= 2.916

         D4 = 2.50*1.083= 3.149

         D5 = 2.50*1.083*1.04 = 3.275

         D6 = 2.50*1.083*1.042

         etc.

 

Treat D5 onwards as a growing perpetuity     =>    P4 =  D5/(r - g)  = 3.275/(.12 - .04) = 40.94

=>   P0  =    D1    +    D2    +    D3    +     D4 + P4

                 1.12        1.122      1.123          1.124  

            = 2.5/1.12 + 2.7/1.122 +2.916/1.123 +(3.149+40.94)/1.124 = 34.48

 

 

7 a)   Treat D6 onwards as a growing perpetuity.

         Then P5 =         D6    = 75

                            .14 - .10

         And  P0  =    D1    +    D2    +    D3    +    D4    +    D5 + P5

                           1.14        1.142     1.143      1.144         1.145        

 

                   =   1.6   +    1.4    +   1.3    +     2     +   2.50 + 75  =  44.79

                       1.14       1.142      1.143     1.144         1.145        

 

b)      Your expected return will simply equal the required return, i.e. 14%.  

 

8 a)   Plowback ratio = 0.3

         Growth rate = 0.3 * .15 = 4.5%

         D1 = .7 * 8.50 = 5.95

         P0 = D1/(r – g) = 5.95/(.12 - .045) = 5.95/.075 = 79.33

b)      Existing assets generate a perpetuity, so PV(existing assets) = 8.5/0.12 = 70.83.

         PVGO = P0 - PV(existing assets) = 79.33 - 70.83 = 8.50

c)      With no positive NPV investments, the return they earned on their reinvestment would have to be the same as the required return, namely 12%.

         The growth rate would be .03 * .12 = 3.6%

         [We can verify that at this growth rate, the stock price just equals PV(existing assets):  5.95/(.12 - .036) = 5.95/.084 = 70.83]

 

9 a)      The expected return = required return = D1/P0 + g  = 4/100 + .4 * .1 = 8%

b)         EPS1 = D1/payout ratio = 4/(1 - .4) = 6.6667

c)            PV(existing assets) = EPS1/r = 6.6667/.08 = 83.33

            P0 = D1/(r – g) = 4/(.08 - .04) = 100

            PVGO = 100 – 83.33 = 16.67

d)         The growth rate becomes .6 * .1 = 6%

            P0 = 4/(.08 - .06 = 200

            PVGO = 200 – 83.33 =116.67

            The stock price doubles; percentage increase in PVGO = 100/16.67 = 600%.

            (A relatively small improvement in positive NPV investment opportunities can have a pretty dramatic impact on the stock price!)

 

10 a) g = plowback ratio * ROE = .35*.16 = 5.6%

b) r = div yield + g = .08 + .056 = 13.6%

c) P0 = D1/(r – g) = 2.22/(.136 - .056) = 27.75

(alternatively, div yld = D1/P0  => P0 = D1/ div yld = 2.22/.08 = 27.75)

d) PV(existing assets) = EPS/r

Since the dividend of 2.22 is 65% of the earnings, EPS = 2.22/.65 = 3.415

PV(existing assets) = EPS/r = 3.415/.136 = 25.113

PVGO = P0 - PV(existing assets) = 27.75 – 25.113 = $2.64

(alternatively, first compute the growth rate if retained earnings are invested at zero NPV: .35 * .136 = 4.76%

PV(existing assets) = stock price if retained earnings are invested at zero NPV = 2.22/(.136-.0476) = 25.113)

 

11 a)  Follow table 4.3 from the text:

  Time 1 Time 2 Time 3 Time 4
Equity 40 40 + 12*.9 = 50.8 50.8+15.24*.9=64.52 64.52+12.90*.4=69.68
EPS 40*.3 = 12 50.8*.3 = 15.24 64.52*.2=12.90 69.68*.2=13.94
Div. 12*.1 =1.2 15.24*.1=1.524 12.90*.6=7.742 13.94*.6=8.36
g   27% 408% 8%

If you continue the table for time 5, equity will be 69.68 + 13.94*.4 = 75.25

EPS will be 75.25*.2 = 15.05

The dividend will be 15.05*.6 = 9.03, which is again an 8% growth rate.  After time 4, dividends settle down to constant growth at 8% (which is just the ROE of 20% times the plowback ratio of 40%)  

b) Treat D4 onwards as a growing perpetuity     =>    P3 = D4/(r - g) = 8.36/(.15 - .08) = 119.45   

=>   P0  =    D1    +    D2    +     D3 + P3

                 1.15        1.152          1.153  

            = 1.2/1.15 + 1.524/1.152 + (7.742 + 119.45)/1.153 = 85.82

 

c) The earnings generated by the time 0 investment are as follows:

EPS1 = EPS2 = 40*.3 = 12

EPS3 onwards = 40*.2 = 8 

So EPS3 onwards constitutes a perpetuity; the PV of these earnings at time 2 is 8/.15

PV0 of existing assets (per share) = 12/1.15 + 12/1.152  + (8/.15)*(1/1.152) = 59.84

PVGO =  P0  - PV0 of existing assets = 85.82 – 59.84 = 25.99

 

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