Question

Machine Initial cost today $50 million $57 million $47 million $75 million Annual costs each year $2.5 million $3.5 million $2.0 million $2.0 million Lifespan 5 years 8 years 4 years 10 years The cost of capital for the new machine is 10%. a. Use the same common end-year approach to rank the machines in order from best to worst. (Using Excel will help with this.) b. Use the equivalent annual annuity approach to rank the machines in order from best to worst. 3. You want to buy Super Tech Inc. stock. They pay quarterly dividends, with the next dividend of $0.40 per share being paid later today. You believe that, during the next 6 years, their quarterly dividends will grow by 20% APR, compounded annually. But after 6 years, their quarterly dividends will grow more slowly... only at 4% APR, compounded annually. If the discount rate on this stock is 12% APR, compounded annually, then what would be a fair price for this stock right now?

Answer 1

(2)

(a)

NPV of costs

NPV_A = - 50 - 2.5( 1.1^-1 + 1.1^-2 + 1.1^-3+ 1.1^-4+1.1^-5 ) = -50 - 2.5{(1 - 1.1^-5) / 0.1} = - 59.477  

NPV_B = -57 - 3.5( 1.1^-1 + 1.1^-2 + 1.1^-3+ ...........+ 1.1^-8) = -57 - 3.5{(1 - 1.1^-8) / 0.1} = -75.672

NPV_C = -47 - 2( 1.1^-1 + 1.1^-2 + 1.1^-3+ 1.1^-4 ) = -53.340

NPV_D = -75 - 2( 1.1^-1 + 1.1^-2 + 1.1^-3 +.................+1.1^-10) = -87.289

The ranking from best to worst is in the decreasing order of NPV i.e. C, A, B, D

(b)

The equivalent annual annuity(EAA) values can be obtained as follows

EAA_A{(1 - 1.1^-5) / 0.1} = -59.477

or EAA_A = -15.690

EAA_B{(1 - 1.1^-8) / 0.1} = -75.672

or EAA_B = -14.184

EAA_C{(1 - 1.1^-4) / 0.1} = -53.340

or EAA_C = -16.827

EAA_D{(1 - 1.1^-10) / 0.1} = -87.289

or EAA_D = -14.206

The ranking from best to worst is in the decreasing order of EAA i.e. B, D, A, C

(3)

Dividend growth rate for first six years is e = 1.2^0.25 - 1 = 4.6635 % compounded quarterly.

Dividend growth rate after first six years is f = 1.04^0.25 - 1 = 0.985341 % compounded quarterly.

Discount rate is i = 1.12^0.25 -1 = 2.87373 % compounded quarterly.

Fair price of the stock

= Present value of the dividend stream

= 0.4{ 1 + (1+e)v + (1+e)²v²+ (1+e)³ v³ +........+ (1+e)²⁴ v²⁴} + 0.4(1+e)²⁴(1+f)v²⁵{1 + (1+f)v + (1+f)²v² +..........}

where v = (1+i)^-1

= 0.4{ (1 - (1+e)²⁵v²⁵) / (1 - (1+e)v)} + 0.4(1+e)²⁴(1+f)v²⁵{1/(1 - (1+f)v)}

=0.4{ (1- 1.046635²⁵x1.0287373^-25) / (1 - 1.046635x1.0287373^-1)}

+ {0.4x1.046635²⁴x1.00985341x1.0287373^-25/(1 - 1.00985341x1.0287373^-1)}

= 12.3949 + 32.3598

= $44.75