Math Problem Statement
Let’s look at a company that is currently not paying dividends. You predict that in five years the company will pay a dividend for the first time. The dividend will be $0.50 per share. You expect the dividend to grow at an 8% rate per year indefinitely at that time. The required return on companies like this one is 15%. What is the price of the stock today?
Multiple Choice
$1.75
$3.45
$4.08
$5.39
$7.14
Solution
We can solve this problem using the Dividend Discount Model (DDM) with the Gordon Growth Model to estimate the price of the stock today, given future dividends.
Key Variables:
- Dividend in Year 5,
- Dividend growth rate,
- Required return,
- Time until the first dividend is paid: 5 years
The price of the stock at the end of year 4, just before the dividend in year 5 is paid, can be computed using the Gordon Growth Model:
Substitute the known values:
This is the stock price at the end of year 4. However, we are interested in the price of the stock today, which is 5 years earlier. Therefore, we need to discount this price back to the present value (today's value).
To discount it back, we use the formula for present value of a future sum:
Substitute the values:
Now, calculate :
Now, find :
Since the closest option is $3.45, that is the correct answer.
Final Answer:
$3.45
Would you like further details on this calculation or have any questions?
Here are 5 related questions you could explore:
- How does changing the required return affect the stock price today?
- What if the dividend growth rate were higher or lower than 8%?
- How does the present value formula apply to different time periods?
- What happens to the price today if dividends begin after 10 years instead of 5?
- How would the stock price change if a different dividend amount were expected?
Tip: Always discount future cash flows to their present value when dealing with long-term investments like stocks.
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Math Problem Analysis
Mathematical Concepts
Dividend Discount Model
Gordon Growth Model
Present Value of Future Cash Flows
Formulas
Price at year 4: P_4 = D_5 / (r - g)
Present value formula: P_0 = P_4 / (1 + r)^t
Theorems
Gordon Growth Model
Present Value Theorem
Suitable Grade Level
Undergraduate Finance / Advanced High School Economics
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