Rearden Metals expects to have earnings this coming year of $2.50 per share.Rearden plans to retain all of its earnings for the next year.For the subsequent three years,the firm will retain 50% of its earnings.It will then retain 25% of its earnings from that point onward.Each year,retained earnings will be invested in new projects with an expected return of 20% per year.Any earnings that are not retained will be paid out as dividends.Assume Rearden's shares outstanding remains constant and all earnings growth comes from the investment of retained earnings.If Rearden's equity cost of capital is 10%,then Rearden's stock price is closest to:

Question

Quizplus · Accepted Answer

To calculate the stock price, we need to discount all future cash flows to present value.
First, we calculate the expected earnings for the next four years:

- Year 1: $2.50 per share
- Year 2: 50% of earnings retained = 0.5 × $2.50 = $1.25 per share, expected earnings = $1.25 × 1.20 = $1.50 per share
- Year 3: 50% of earnings retained= $1.50 × 0.5 = $0.75 per share, expected earnings = $0.75 × 1.20 = $0.90 per share
- Year 4: 50% of earnings retained= $0.90 × 0.5 = $0.45 per share, expected earnings = $0.45 × 1.20 = $0.54 per share

Next, we calculate the present value of all four years of expected earnings:
PV = ($2.50 / (1+0.10)^1) + ($1.50 / (1+0.10)^2) + ($0.90 / (1+0.10)^3) + ($0.54 / (1+0.10)^4) = $3.87 per share

Then, we add the present value of any future dividends. Since the company retains all earnings for the first year, the first dividend will be paid at the end of year 2. After that, the dividend payout ratio is 100% - retention ratio:

- Year 2: 50% of earnings paid as dividend = $1.25 × 0.5 = $0.625 per share
- Year 3: 75% of earnings paid as dividend = $0.75 × 0.25 = $0.1875 per share
- Year 4: 75% of earnings paid as dividend = $0.45 × 0.25 = $0.1125 per share

PV of dividends = ($0.625 / (1+0.10)^2) + ($0.1875 / (1+0.10)^3) + ($0.1125 / (1+0.10)^4) = $0.84 per share

Finally, we add the present value of the terminal value, which is the present value of all future retained earnings beyond year 4:
Terminal value = ($0.54 / (0.10 - 0.20)) / (1+0.10)^4 = $3.02 per share
PV of terminal value = $3.02 / (1+0.10)^4 = $2.11 per share

The total stock price is the sum of the present values of all expected future cash flows:
Stock price = PV of expected earnings + PV of dividends + PV of terminal value = $3.87 + $0.84 + $2.11 = $6.82 per share

So the closest stock price to $6.82 is choice B ($44.60).

Rearden Metals Expects to Have Earnings This Coming Year of $2.50