Solutions to Problem Set #2

1 a) g = plowback ratio * ROE = .65*.13 = 8.45%

b) r = div yield + g = .0255 + .0845 = 11%

c) P₀ = D₁/(r – g) = 0.68/(.11 - .0845) = 26.6667

(alternatively, div yld = D₁/P₀  => P₀ = D₁/ div yld = 0.68/.0255 = 26.6667)

d) PV(existing assets) = EPS/r

Since the dividend of 0.68 is 35% of the earnings, EPS = 0.68/.35 = 1.9429

PV(existing assets) = EPS/r = 1.9429/.11 = 17.6623

PVGO = P₀ - PV(existing assets) = 26.6667 – 17.6623 = $9.0043

(alternatively, first compute the growth rate if retained earnings are invested at zero NPV: .35 * .11 = 7.15%

PV(existing assets) = stock price if retained earnings are invested at zero NPV = 0.68/(.11-.0715) = 17.6623)

2 a)  Follow table 4.3 from the text:

If you continue the table for time 5, equity will be 86.58745 + 15.58574*.6= 95.9389

EPS will be 95.9389*.18 = 17.269

The dividend will be 17.269*.4 = 6.9076, which is again a 10.8% growth rate.  After time 4, dividends settle down to constant growth at 10.8% (which is just the ROE of 18% times the plowback ratio of 60%)

b) Treat D₄ onwards as a growing perpetuity     =>    P₃ = D₄/(r - g) = 6.2343/(.16 - .108) = 119.8903   

=>   P₀  =    D₁    +    D₂    +     D₃ + P₃

                 1.16        1.16²          1.16³ 

            = 2.64/1.16 + 3.1469/1.16² + (5.6266 + 119.8903)/1.16³ = 85.0279

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

EPS₁ = EPS₂ = 55*.24 = 13.20

EPS₃ onwards = 55*.18 = 9.90 

So EPS₃ onwards constitutes a perpetuity; the PV of these earnings at time 2 is 9.9/.16 = 61.875

PV₀ of existing assets (per share) = 13.20/1.15 + (13.20 + 61.875)/1.16²  = 67.1723

PVGO =  P₀  - PV₀ of existing assets = 85.0279 – 67.1723 = 17.8556

3.  These are borrowing cashflows => you can solve for IRR either using a financial calculator or using the formula for a quadratic equation.

In either case, the equation for IRR is:        8,800 – 4,400/(1+IRR) – 5,510/(1+IRR)² = 0

Solving by the quadratic formula, 88x² – 44x – 55 = 0    (where x = 1+IRR)  

      =>  8x² – 4x – 5 = 0

     =>  x = 4 +/- [4² - 4*8*(-5)]^0.5  =  1.0792, -0.5792         

                               16

     => IRR = 7.92% (dropping the negative value of –157.92%)

Since we have borrowing cashflows, the project is good if IRR < required return.  Since here the IRR > required return, we reject this project.

4.         First compute IRR for each project.  Any project with IRR < required return can be dropped right away.  Only projects have IRR > required return need to be compared by looking at the IRR of incremental cash flows.

Project A : IRR  is given by:     173.6(1+IRR)² - 117.95(1+IRR) - 88.55 = 0

     Using a financial calculator, IRR = 13.06% > 11.375%

     => A has a positive NPV; it stays in the running.

Project B : IRR  is given by:     60(1+IRR)² - 30.5(1+IRR) - 39.7 = 0

     Using a financial calculator, IRR = 10.638% < 11.3752%

     => B has a negative NPV; it should be dropped.

Project C : IRR  is given by:      82.5(1+IRR)² - 52.8(1+IRR) - 51.15 = 0

     Using a financial calculator, IRR = 16.994% > 11.375%

     => C has a positive NPV; it stays in the running.  

Check first if the NPV curves for A and C intersect.  

Y-axis intercept for A = sum of the cashflows = 32.9

Y-axis intercept for C = sum of the cashflows = 21.45

The project with the larger intercept has the smaller IRR, so the curves do intersect.  So now we need to compare A and C.  The incremental cash flows for A - C are conventional cashflows:            -91,100, 65,150 and 37,400

IRR is given by:     91.1(1+IRR)² - 65.15(1+IRR) - 37.4 = 0

Using a financial calculator, IRR = 9.133% < 11.375%

Since the IRR for A – C is less than the required return, the NPV of A – C is negative.  A isn’t better than C, and so we should accept C.  

If the projects have different required returns, it will not be possible to make the decision using IRR alone.  When we compute the IRR of the incremental cashflows, we do not know what number to compare the IRR to.  In other words, the required return of the incremental cashflows  is not known.

5.         Both projects have conventional cashflows. 

For A, the equation for IRR is:             -145,600 + 49,280/(1+IRR) + 98,560/(1+IRR)² + 60,000/(1+IRR)³= 0

Using a financial calculator, IRR_A = 19.42%, which is > required return.

For B, the equation for IRR is:             -65,600 -140,320/(1+IRR) + 210,560/(1+IRR)² + 60,000/(1+IRR)³= 0

Using a financial calculator, IRR_B = 19.33%, which is > required return.

Check first if the NPV curves for A and C intersect.  

Y-axis intercept for A = sum of the cashflows = 62.24

Y-axis intercept for B = sum of the cashflows = 64.64

The project with the larger intercept has the smaller IRR, so the curves do intersect.  So now we need to compare A and B,  by looking at the incremental cashflows:

             A - B            -80,000    189,600        -112,000       

These are non-conventional cashflows; solve using quadratic formula.

            -80,000 + 189,600/(1+IRR) – 112,000/(1+IRR)² = 0

=>    80x² – 189.6x + 112 = 0    (where x = 1+IRR)

=>  1+IRR = 189.6 +/- (189.6² - 4*80*112)^0.5 = 1.12, 1.25      => IRR = 12% and 25%

                                       160

The only way to decide whether A-B is a good project or not, is to figure out what A-B’s NPV curve looks like.  It passes through the x-axis at two points: 12% and 25%.  The sum of the cashflows is negative.  So the NPV starts out negative, turns positive at 12% and becomes negative again beyond 25%.  At 15% the NPV is positive => A-B is a good project, so we should pick A over B. 

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