Question

Conlins Limited always pays dividend annually and it has just paid out $4 dividend per share today (t=0). The earnings per share of this firm is expected to be $5 at t=1 (one year from today). Suppose it has some investment opportunities at t=1(one year from today) and t=2 (two years from today). Therefore, at t=1 and t=2, Conlins only distributes 60% of its earnings out as dividends and reinvest the remaining earnings into these investment opportunities. The rate of return of these new investments is 30% and the proper risk-adjusted discount rate for dividend payments is 15% forever. After that (starting from t=3, three years from today), this firm will not have any good projects to reinvest its earnings so that all earnings will be paid out as dividends. That is the annual growth rate for Conlins dividends after the first reinvestment? 2. What is the theoretical stock price of Conlins today? 3. If Conlins does NOT ever do any reinvestments of earnings, what is the stock price of Conlins today? 4. Explain the difference of stock prices that you have computed in Part (2) and Part (3) by calculating the net present values (NPVs) of re-investments of earnings in t=1 and t=2. Show that the sum of NPVs of these investments equals the difference between your answers in Part (2) and Part (3).

Answer 1

Dividend at time1 =D1 = E1* Dividend payout ratio = $5 * 60% = $3

Reinvestment ratio = 1- Dividend payout ratio = 1-0.6 = 0.4

1. Annual Growth rate of Dividends (g)= Return on Investment * Reinvestment ratio = 30%* 0.4 = 12%

2. E2= E1 * (1+g) = 5* (1+0.12) = $5.6

D2= E2*Dividend payout Ratio = 5.6*0.6 = $3.36

E3= E2 * (1+g) = 5.6* (1+0.12) = $6.272

D3= E3 (as all earnings are paid as dividends and there is no reinvestment)

= $6.272

Dividends growth rate after three years = Return on Investment * Reinvestment ratio = 30%* 0 = 0

As there is no growth in dividends , D4=D5=.... = $6.272

So, Stock Price today P0= present value of all Dividends

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

= 3/1.15 + 3.36/1.15^2+ 6.272/1.15^3 + (H3/1.15^3)   

where H3 is the horizon value at the end of 3 years where there is no growth in dividends. By zero growth model

H3= D4/r = 6.272/0.15= 41.8133

So, P0 = 3/1.15 + 3.36/1.15^2+ 6.272/1.15^3 + (41.8133/1.15^3)

= 36.76623 = $36.773.

3. If there are no reinvestments ever

D1= D2= ... = $5 (earnings expected next year)

So, By zero growth model

P0 = D1/r = 5/0.15 = $33.33

4. Reinvestment at time 1 = E1-D1 = $5-$3 =$2

Reinvestment at time 2 = E2-D2 = $5.6-$3.36 =$2.24

Thereafter there are no reinvestments

So, the sum of the Net present value (NPV) of all reinvestments = 2/1.15 + 2.24/1.15^2 = $3.432892

Difference between the Prices calculated in part (2) and part (3) above

= 36.76623 - 33.3333 = $3.432892 = sum of NPV of reinvestments