Piedmont Enterprises currently pays a dividend (D0) of $1 per share. This dividend
is expected to grow at a 20 percent per year rate for the next 2 years, after which it is
expected to grow at 6 percent per year for the foreseeable future. If you require a
15 percent rate of return on an investment of this type, what price do you expect
the stock to sell for at the beginning of year 5?
To calculate the price of the stock at the beginning of year 5, we need to apply a discounted cash flow (DCF) analysis.
Step 1: Calculate the future dividends for years 1, 2, 3, and 4
Given that the dividend (D0) is $1 per share, and it is expected to grow at a 20% rate for the next 2 years, the dividends for years 1 and 2 can be calculated as follows:
Year 1 dividend (D1) = D0 * (1 + growth rate) = $1 * (1 + 0.20) = $1.20 per share
Year 2 dividend (D2) = D1 * (1 + growth rate) = $1.20 * (1 + 0.20) = $1.44 per share
From year 3 onwards, the dividend is expected to grow at a 6% rate per year. So, the dividends for years 3 and 4 can be calculated as follows:
Year 3 dividend (D3) = D2 * (1 + growth rate) = $1.44 * (1 + 0.06) = $1.53 per share
Year 4 dividend (D4) = D3 * (1 + growth rate) = $1.53 * (1 + 0.06) = $1.62 per share
Step 2: Calculate the present value of the dividends for years 1, 2, 3, and 4
To determine the present value of the dividends, we discount them back to the present using the required rate of return of 15%. The present value (PV) calculation is as follows:
PV = Dividend / (1 + rate of return) ^ number of years
Present value of the Year 1 dividend (PV1) = $1.20 / (1 + 0.15) ^ 1 = $1.04 per share
Present value of the Year 2 dividend (PV2) = $1.44 / (1 + 0.15) ^ 2 = $1.16 per share
Present value of the Year 3 dividend (PV3) = $1.53 / (1 + 0.15) ^ 3 = $1.12 per share
Present value of the Year 4 dividend (PV4) = $1.62 / (1 + 0.15) ^ 4 = $1.10 per share
Step 3: Calculate the price of the stock at the beginning of year 5
The price at the beginning of year 5 is determined by the present value of all future dividends beyond year 4. Since these dividends will grow at a 6% rate indefinitely, we can use a constant growth model called the Gordon Growth Model.
The formula for the Gordon Growth Model is:
P5 = D5 / (rate of return - growth rate)
Where P5 is the price in year 5, D5 is the dividend in year 5, the rate of return is 15%, and the growth rate is 6%.
Since the dividend in year 5 (D5) will be equal to the dividend in year 4 (D4) times the growth rate (6%), we can calculate it as follows:
D5 = D4 * growth rate = $1.62 * 0.06 = $0.0972 per share
Now, we can calculate the price in year 5 (P5) using the Gordon Growth Model:
P5 = $0.0972 / (0.15 - 0.06) = $0.0972 / 0.09 = $1.08 per share
Therefore, we can expect the stock to sell for $1.08 per share at the beginning of year 5.