Question

Kai Zen Motors currently pays a dividend of $2 per share, and this dividend is expected to grow at a 20 percent annual rate for three years, and then at a 11 percent rate for the next two years, after which it is expected to grow at a 6 percent rate forever. What value would you place on the stock if a 16 percent rate of return were required? [5

Answer 1

Solution :

The value of a stock today is calculated as follows :

The value of a stock today = Present value of dividends + Present value of share at year n

Thus the value of a stock today with respect to the details given in the question is calculated using the formula

= [ D₁ * ( 1 / ( 1 + r)¹ ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ D₃ * ( 1 / ( 1 + r)³) ] + [ D₄ * ( 1 / ( 1 + r)⁴ ) ] + [ D₅ * ( 1 / ( 1 + r)⁵ ) ] + [ P₅ * ( 1 / ( 1 + r)⁵ ) ]

Calculation of Dividend per share Years 1 to 5 :

As per the information given in the question we have

D₀ = $ 2.00 ; g₁ = 20 %   ; g₂ = 20 %   ; g₃ = 20 % ; g₄ = 11 %   ; g₅ = 11 % ;

Thus the Dividend per year can be calculated as follows :

D₁ = D₀ * ( 1 + g₁ ) = $ 2 * ( 1 + 0.20 ) = $ 2 * 1.20 = $ 2.40

D₂ = D₁ * ( 1 + g₂ ) = $ 2.40 * ( 1 + 0.20 ) = $ 2.40 * 1.20= $ 2.88

D₃ = D₂ * ( 1 + g₃ ) = $ 2.88 * ( 1 + 0.20 ) = $ 2.88 * 1.20 = $ 3.4560

D₄ = D₃ * ( 1 + g₄ ) = $ 3.4560 * ( 1 + 0.11 ) = $ 3.4560 * 1.11 = $ 3.836160

D₅ = D₄ * ( 1 + g₅ ) = $ 3.836160 * ( 1 + 0.11 ) = $ 3.836160 * 1.11 = $ 4.258138

Thus we have

D₁ = $ 2.40 ; D₂ = $ 2.88   ; D₃ = $ 3.4560   ; D₄ = $ 3.836160   ; D₅ = $ 4.258138 ;

Calculation of price of share at year 5 :

Price of the share at year 5 where the firm expects a constant growth rate of 6 %

The formula for calculating the price of the share at year 3

P₅ = [ D₅ * ( 1 + g ) ] / ( K_e – g )

We know that

D₅ = $ 4.258138 ; g = 6 % = 0.06   ; K_e = 16 % = 0.16 ;

P₅ = [ $ 4.258138 * ( 1 + 0.06 ) ] / ( 0.16 - 0.06 )

= ( $ 4.258138 * 1.06 ) / ( 0.16 - 0.06 )

= ( $ 4.258138 * 1.06 ) / 0.10

= $ 4.513626 / 0.10

= $ 45.136260

Thus the price of the share at year 5 = $ 45.136260

Calculation of value of a stock today :

Thus the current price of the share = [ D₁ * ( 1 / ( 1 + r)¹ ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ D₃ * ( 1 / ( 1 + r)³) ] + [ D₄ * ( 1 / ( 1 + r)⁴ ) ] + [ D₅ * ( 1 / ( 1 + r)⁵ ) ] + [ P₅* ( 1 / ( 1 + r)⁵ ) ]

Applying the available information in the formula we have the value of a stock today as follows :

= [ $ 2.4 * ( 1 / 1.16 )¹ ] + [ $ 2.88 * ( 1 / 1.16 )² ] + [ $ 3.456 * ( 1 / 1.16 )³ ] + [ $ 3.836160 * ( 1 / 1.16 )⁴ ] + [ $ 4.258138 * ( 1 / 1.16 )⁵ ] + [ 45.136260 * ( 1 / 1.16 )⁵ ]

= [ $ 2.4 * 0.862069 ] + [ $ 2.88 * 0.743163 ] + [ $ 3.456 * 0.640658 ] + [ $ 3.836160 * 0.552291 ] + [ $ 4.258138 * 0.476113 ] + [ $ 45.136260 * 0.476113 ]

= $ 2.068966 + $ 2.140309 + $ 2.214114 + $ 2.118677 + 2.027355 + $ 21.489960

= $ 32.059381

= $ 32.06 ( when rounded off to two decimal places )

Thus the value that should be placed on the stock = $ 32.06