Final answer:
To find the fair price of the stock today, we discount the expected future dividends and stock price using the required rate of return of 13%. After calculating the present values of each cash flow, we find that the fair price of the stock today is $38.93.
Step-by-step explanation:
To calculate the fair price of the stock today, also known as its present value, we need to discount the expected future cash flows (dividends and stock price after the third dividend) back to their present value using the stock's required rate of return. The future dividends are $2.54, $2.49, and $2.49 for the next three years, and the expected stock price immediately after the third dividend is $47.25. Using a discount rate of 13%, we can calculate the present value of each of these cash flows separately and then sum them up to get the total present value, which will be the fair price of the stock today.
The formula for the present value (PV) of each cash flow is:
PV = CF / (1 + r)^t
where CF is the cash flow, r is the rate of return, and t is the time in years. We apply this formula for each of the three dividends and the stock price:
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- PV of first dividend = $2.54 / (1 + 0.13)^1
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- PV of second dividend = $2.49 / (1 + 0.13)^2
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- PV of third dividend = $2.49 / (1 + 0.13)^3
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- PV of the stock price = $47.25 / (1 + 0.13)^3
Summing up these present values will give us the fair price of the stock today. Calculating and adding up these present values:
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- PV of first dividend = $2.54 / 1.13 = $2.25
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- PV of second dividend = $2.49 / (1.13^2) = $1.95
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- PV of third dividend = $2.49 / (1.13^3) = $1.73
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- PV of the stock price = $47.25 / (1.13^3) = $33.00
The fair price of the stock today = $2.25 + $1.95 + $1.73 + $33.00 = $38.93 (rounded to the nearest penny).