Question

Common stock value—Variable growth Lawrence​ Industries' most recent annual dividend was ​$1.27 per share ​(D0equals = $1.27​), and the​ firm's required return is 10​%. Find the market value of​ Lawrence's shares when dividends are expected to grow at 15​% annually for 3​ years, followed by a 5​% constant annual growth rate in years 4 to infinity.

Answer 1

Solution:

The Market value of a share of stock = Present value of dividends earned for Years 1 to ‘n’ + Present value of stock at year ‘n’ where the firm experiences a constant growth rate

Thus the market of a share of stock = [ D₁ * ( 1 / ( 1 + r)¹ ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ D₃ * ( 1 / ( 1 + r)³) ] + [ D₄ * ( 1 / ( 1 + r)⁴ ) ] + [ P₄* ( 1 / ( 1 + r)⁴ ) ]

Calculation of Dividend per share Years 1 to 4 :

As per the information given in the question we have

D₀ = $ 1.27 ; g₁ = 15 %   ; g₂ = 15 %   ; g₃ = 15 %   ; g₄ = 5 %   ;

Thus the Dividend per year can be calculated as follows :

D₁ = D₀ * ( 1 + g₁ ) = $ 1.27 * ( 1 + 0.15 ) = $ 1.27 * 1.15 = $ 1.4605

D₂ = D₁ * ( 1 + g₂ ) = $ 1.4605 * ( 1 + 0.15 ) = $ 1.4605 * 1.15 = $ 1.679575

D₃ = D₂ * ( 1 + g₃ ) = $ 1.679575 * ( 1 + 0.15 ) = $ 1.679575 * 1.15 = $ 1.931511

D₄ = D₃ * ( 1 + g₄ ) = $ 1.931511 * ( 1 + 0.05 ) = $ 1.931511 * 1.05 = $ 2.028087

Thus we have D₁ = $ 1.4605   ; D₂ = $ 1.679575; D₃ = $ 1.931511 ; D₄ = $ 2.028087 ;

Calculation of market price of share at year 4:

Price of the share at year 4 where the firm expects a constant growth rate of 5 %

The formula for calculating the price of the share at year 4

P₄ = [ D₄ * ( 1 + g ) ] / ( K_e – g )

We know that

D₄ = $ 2.028087 ; g = 5 % = 0.05   ; K_e = 10 % = 0.10 ;

P₄ = [ $ 2.028087 * ( 1 + 0.05 ) ] / ( 0.10 – 0.05 )

= ( $ 2.028087 * 1.05 ) / ( 0.10 – 0.05 )

= ( $ 2.028087 * 1.05 ) / 0.05

= $ 2.129491 / 0.05

= $ 42.589827

Thus the market price of the share at year 4 = $ 42.589827

Calculation of Market value of a share of stock :

Thus the market value of a share of stock = [ D₁ * ( 1 / ( 1 + r)¹ ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ D₃ * ( 1 / ( 1 + r)³) ] + [ D₄ * ( 1 / ( 1 + r)⁴ ) ] + [ P₄* ( 1 / ( 1 + r)⁴ ) ]

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

= [ $ 1.4605 * ( 1 / 1.10 )¹ ] + [ $ 1.679575 * ( 1 / 1.10 )² ] + [ $ 1.931511 * ( 1 / 1.10 )³ ] + [ $ 2.028087 * ( 1 / 1.10 )⁴ ] + [ $ 42.589827 * ( 1 / 1.10 )⁴ ]

= [ $ 1.4605 * 0.909091 ] + [ $ 1.679575 * 0.826446 ] + [ $ 1.931511 * 0.751315 ] + [ $ 2.028087 * 0.683013 ] + [ $ 42.589827 * 0.683013 ]

= $ 1.327727 + $ 1.388078 + $ 1.451173 + $ 1.385210 + $ 29.089406

= $ 34.641594

= $ 34.64 ( when rounded off to two decimal places )

Thus the market value of​ Lawrence's shares = $ 34.64