Question

Consider a firm that has just paid a dividend of $2. An analyst expects dividends to grow at a rate of 8% per year for the next five years. After that dividends are expected to grow at a normal rate of 5% per year. Assume that the appropriate discount rate is 7%.

i. Calculate the dividends for years 1, 2, and 3.

ii. What is the price of the stock in year 5?

iii. Calculate the present value today of dividends for years 1 to 5. iv. What is the price of the stock today (P0)?

Answer 1

Q1) Dividend in year 1(D1)= Dividend × (1+growth rate)

= 2 (1+0.08)

= 2 (1.08)

= $2.16

Dividend in year 2 (D2) = Dividend of year 1 (1+growth rate)

= 2.16 (1+0.08)

= 2.16 (1.08)

= $2.33

Dividend in year 3 (D3) = Dividend of year 2 ( 1+growth rate)

= 2.33 (1+0.08)

= 2.33 (1.08)

= $2.52

ii) Dividend in year 4= 2.52 (1.08) = 2.72

Dividend in year 5 = 2.72 (1.08) = 2.94

Dividend in year 6 = 2.94 (1.05) = 3.087

price of stock in year 5= D1(1+r)^4 + D2(1+r)^3 + D3(1+r)^2 + D4(1+r) + D5/(1+r)^1 + (D6/r-g)/(1+r)^1

= 2.16(1+0.07)^4 + 2.33(1+0.07)^3 + 2.52(1+0.07)^2 + 2.72(1+0.07) + 2.94/(1+0.07) + (3.087/0.07 - 0.05) / (1+0.07)

= 2.16(1.07)^4 + 2.33(1.07)^3 + 2.52(1.07)^2 + 2.72 (1.07) + 2.94/(1.07) + (3.087/0.02)/(1.07)

= 2.16(1.3108) + 2.33(1.2250) + 2.52(1.1449) + 2.72(1.07) + 2.94/(1.07) + 154.35/(1.07)

= 2.8253 + 2.8543 + 3.65 + 2.91+ 2.7477 + 144.25

= $159.24

iii) Dividend of year 4 = dividend of year 3 × (1+growth rate)

= 2.52 (1+0.08)

= 2.52 (1.08)

= $2.72

Dividend of year 5= Dividend of year 4 (1+growth rate)

= 2.72 (1+0.08)

= 2.72 (1.08)

= $2.94

Present value of dividend from year 1 to 5

= D1/(1+r) + D2/(1+r)^2 + D3/(1+r)^3 + D4/(1+r)^4 + D5/(1+r)^5

= 2.16/(1+0.07) + 2.33/(1+0.07)^2 + 2.52/(1+0.07)^3 + 2.72(1+0.07)^4 + 2.94(1+0.07)^5

= 2.16/1.07 + 2.33/(1.07)^2 + 2.52/(1.07)^3 + 2.72/(1.07)^4 + 2.94/(1.07)^5

= 2.02 + 2.33/1.1449 + 2.52/1.2250 + 2.72/1.3108 + 2.94/1.4026

= 2.02 + 2.035 + 2.057 + 2.0751 + 2.0961

= $10.28

iv) price of the stock today= present value of dividend + (D5 (1+g) /(r-g))/(1+r)^5

= 10.28 + (2.94(1+0.05)/(0.07 - 0.05)/(1+0.07)^5

= 10.28 + (3.087/0.02)/(1.07)^5

= 10.28 + 154.35/1.4026

= 10.28 + 110.05

= $120.33