Question

Problem 7-05 Nonconstant Growth Valuation A company currently pays a dividend of $2 per share (D0 = $2). It is estimated that the company's dividend will grow at a rate of 20% per year for the next 2 years, and then at a constant rate of 8% thereafter. The company's stock has a beta of 1.85, the risk-free rate is 6.5%, and the market risk premium is 3%. What is your estimate of the stock's current price? Do not round intermediate calculations. Round your answer to the nearest cent.

Answer 1

Solution :

Calculation of required rate of return of the stock:

As per Capital Asset Pricing Model, required return of a stock is calculated using the following formula:

R_S = R_F + [ β * ( R_M - R_F ) ]

Where

R_S = Required return of stock ; R_F = Risk free rate of return   ; β = Beta of the stock ;

( R_M – R_F ) = Market risk Premium

As per the information given in the question we have

R_F = 6.5 % = 0.065 ;

( R_M - R_F ) = Market Risk Premium = 3 % = 0.03 ; β = 1.85

R_S = To find ;  

Applying the above values in the formula we have

= 0.065 + [ 1.85 * 0.03 ]

= 0.065 + 0.0555

= 0.1205

= 12.05 %

Thus the required rate of return = 12.05 %

Calculation of Current price of a stock :

The current price of a stock = present value of dividends + present value of stock at year n where the firm experiences a non constant growth rate

Thus the current price of the stock = [ D₁ * ( 1 / ( 1 + r) ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ P₂ * ( 1 / ( 1 + r)² ) ]

Calculation of Dividend per share Years 1 to 2 :

As per the information given in the question we have

D₀ = $ 2 ; g₁ = 20 %   ; g₂ = 20 %   ;

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

Thus we have D₁ = $ 2.40 ;    D₂ = $ 2.88   ;

Calculation of price of share at year 2:

Price of the share at year 2 where the firm expects a constant growth rate of 8 %

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

P₂ = [ D₂ * ( 1 + g ) ] / ( K_e – g )

We know that

D₂= $ 2.88 ; g = 8 % = 0.08   ; K_e = 12.05 % = 0.1205 ;

P₂ = [ $ 2.88 * ( 1 + 0.08 ) ] / ( 0.1205 – 0.08 )

= ( $ 2.88 * 1.08 ) / ( 0.1205 – 0.08 )

= ( $ 2.88 * 1.08 ) / 0.0405

= $ 3.1104 / 0.0405

= $ 76.80

Thus the price of the share at year 2 = $ 76.80

Calculation of price of stock today :

Thus the current price of the stock = [ D₁ * ( 1 / ( 1 + r) ) ] + [ D₂ * ( 1 / ( 1 + r)² ) ] + [ P₂ * ( 1 / ( 1 + r)² ) ]

Applying the available information in the formula we have the price of the stock as follows :

= [ $ 2.40 * ( 1 / 1.1205 )¹ ] + [ $ 2.88 * ( 1 / 1.1205 )² ] + [ 76.80 * ( 1 / 1.1205 )² ]

= [ $ 2.40 * 0.892459 ] + [ $ 2.88 * 0.796483 ] + [ $ 76.80 * 0.796483 ]

= $ 2.141901 + $ 2.293870 + $ 61.169862

= $ 65.605632

= $ 65.61 ( when rounded off to the nearest cent )

Thus the current price of the stock = $ 65.61