Question

1. A stock just paid a dividend of $0.75. This quarterly dividend is expected to grow at a rate of 4% for the next 10 years, after which it will grow at a rate of -2% in perpetuity. What is the price of the stock if the required return is 12% (all rates are APR with quarterly compounding)?

2. A firm has a P/E ratio of 18.5, a payout ratio of 50%, and a required return of 12% per annum. What is an estimate of this firm's perpetual earnings growth rate?

3. An analyst expects earnings per share to grow next year by 5% from its current level of $5.60 per share. What is the expected stock price next year given that the P/E ratio for the stock remains constant at it's current level of 15.0?

4. A firm has long maintained a payout ratio of 30% of earnings available to shareholders. If an analyst expects a firm to also maintain their current return on equity (ROE) of 22% per year, what earnings growth rate does this imply?

Answer 1

1). D0 (last quarterly dividend paid) = 0.75

Quarterly growth rate for the next 10 years (g1) = annual growth rate/4 = 4%/4 = 1%

D40 (dividend after 10 years or 40 quarters) = D0*(1+g1)^40 = 0.75*(1+1%)^40 = 1.12

Quarterly perpetual growth rate (g2) = -2%/4 = -0.50%

D41 = D40*(1+g2) = 1.12*(1-0.5%) = 1.11

Quarterly required return (k) = annual return/4 = 12%/4 = 3%

Perpetual value (V1) = D41/(k-g2) = 1.11/(3%+0.50%) = 31.74

Present Value (PV) of perpetual value = V1/(1+k)^40 = 31.74/(1+3%)^40 = 9.73

PV of dividends in the first 10 years (using PV of growing annuity formula) = D1/(k-g1)*[1 - ((1+g1)/(1+k))^40]

= (0.75*(1+1%)/(3%-1%))*[1 - ((1+1%)/(1+3%))^40] = 20.59

Stock price = total PV of all future dividends = 20.59 + 9.73 = 30.32 (Note: Due to rounding off, the answer can also come out to be 30.31. Please check for both answers.)

2). P/E ratio = payout ratio*(1+g)/(k-g) where g = growth rate and k = required return

18.5 = (50%*(1+g))/(12%-g)

Solving for g, we get

g = 9.05% (earnings growth rate)

3). Next year's earnings (E1) = current year's earnings*(1+growth rate) = 5.60*(1+5%) = 5.88

Expected stock price next year = P/E ratio*E1 = 15*5.88 = 88.20 per share

4). Growth rate = (1-payout ratio)*ROE

= (1-30%)*22% = 15.40%