Final answer:
The price of a share of Bean's stock is closest to $12.41.
Step-by-step explanation:
To calculate the price of a share of Bean's stock, we need to find the present value of its future earnings. For the first three years, all of the earnings will be retained, so we need to find the present value of $2 per share for each of the three years. Using the formula for present value, PV = FV / (1 + r)^n, where PV is the present value, FV is the future value, r is the discount rate, and n is the number of years, we can calculate the present value of the earnings.
- For year 1, PV = 2 / (1 + 0.12) = $1.79
- For year 2, PV = 2 / (1 + 0.12)^2 = $1.60
- For year 3, PV = 2 / (1 + 0.12)^3 = $1.43
After year 3, Bean plans to retain 50% of its earnings for the next two years, and then retain only 25% of its earnings from that point forward. We can calculate the future earnings for these years and then find the present value using the same formula.
- For year 4, earnings = $2 * 0.5 = $1
- For year 5, earnings = $2 * 0.5 = $1
- For year 6 and onward, earnings = $2 * 0.25 = $0.5
Now we can calculate the present value of these future earnings.
- For year 4, PV = 1 / (1 + 0.12)^4 = $0.71
- For year 5, PV = 1 / (1 + 0.12)^5 = $0.63
- For year 6 and onward, PV = 0.5 / (0.12 - 0.2) = $6.25
Finally, we can add up all the present values to find the total present value of the earnings.
Total PV = $1.79 + $1.60 + $1.43 + $0.71 + $0.63 + $6.25 = $12.41
Since Bean plans to retain all its earnings, the stock price is equal to the present value of the earnings. Therefore, the price of a share of Bean's stock is closest to $12.41.