Question

12/(Two-stage growth) It is common practice in security analysis to modify the basic div- idend growth model by allowing more than one stage of growth, with the growth factors being different in the different stages. As an example consider company Z, which currently distributes dividends of $10M annually. The dividends are expected to grow at the rate of 10% for the next 5 years and at a rate of 5% thereafter (a) Using a dividend discount approach with an interest rate of 15%, what is the value of the company? (b) Find a general formula for the value of a company satisfying a two-stage growth model. Assume a growth rate of G for k years, followed by a growth rate of g thereafter, and an initial dividend of D1

Answer 1

a) Let us assume dividend initially at 1^st year is D₁, Dividend at 2nd year is D₂, Dividend at k^th year is D_k  so on.

Likewise Present value initially is P_0, Present value after 1 year is P_1, Present value after K^th   year is P_k so on.

Given Data, Annual Dividend(D₀) = $10 M , Growth rate of G (10%) for 5 years, followed by growth rate of g(5%) thereafter. Interest rate (r) = 15%.

value of the company at the end of the k i.e. 5th year = D₆ /(r-g)

= D₅*(1+g)/(r-g)

= D₀(1+G)⁵(1+g)/(r-g)

= $ 10M (1.1)⁵(1.05)/(15%-5%)

= $ 169,103,550

value of the company (P₀) = D₁/(1+r)¹ + D₂/(1+r)² + D₃/(1+r)³ + D₄/(1+r)⁴ + (D₅ +P₅)/ (1+r)⁵

⁼ (D₁/r-G) * [1 - ((1+G)/(1+r))⁵] + P₅/(1+r)⁵

= (11,000,000/.15-.10) * [1 - (1.1/1.15)⁵] + 169,103,550 / 1.15⁵

⁼ 43844217.13 + 84074350.92

= $ 127,918,568.10

Hence value of the company = $ 127,918,568.10

b) Let us assume dividend intially at 1^st year is D₁, Dividend at 2nd year is D₂, Dividend at k^th year is D_k  so on.

Likewise Present value initially is P_0, Present value after 1 year is P_1, Present value after K^th   year is P_k so on.

Growth rate of G for k years, followed by growth rate of g thereafter, interest rate = r%.

value of the company at the end of the kth year = D_k+1/(r-g)

= Dk*(1+g)/(r-g)

= D₀(1+G)^k(1+g)/(r-g)

value of the company (P₀) = D₁/(1+r)¹ + D₂/(1+r)² + .......... + (D_k+P_k)/ (1+r)^k

⁼ (D₁/r-G) * [1 - ((1+G)/(1+r))^k] + P_k/(1+r)^k

  

Hence the general formula for the value of the company satisfying a two stage growth model is

(D₁/r-G) * [1 - ((1+G)/(1+r))^k] + P_k/(1+r)^k where D₁ is the initial Dividend , growth rate of G for k years followed by growth rate of g thereafter for interest rate r%.