Question
If the current dividend \( \left(\mathrm{D}^{0}\right) \) is \( \$ 3.00 \) and the growth rate is \( 6 \% \). How much will the dividend be, at Time 5 ?
Ask by Davies Carroll. in the United States
Mar 10,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The dividend at Time 5 will be approximately \$4.01.
Solution
To find the dividend at Time 5, we can use the formula for compound interest:
\[ A = P \times (1 + r)^n \]
where:
- \( A \) is the amount at Time 5,
- \( P \) is the initial dividend (\( \$3.00 \)),
- \( r \) is the growth rate (\( 6\% \)),
- \( n \) is the number of years (5 years).
Given:
- Initial dividend (\( P \)) = \( \$3.00 \)
- Growth rate (\( r \)) = \( 6\% \) = \( 0.06 \)
- Time (5 years)
Substitute the values into the formula:
\[ A = 3.00 \times (1 + 0.06)^5 \]
Now, we can calculate the dividend at Time 5.
Calculate the value by following steps:
- step0: Calculate:
\(3\left(1+0.06\right)^{5}\)
- step1: Add the numbers:
\(3\times 1.06^{5}\)
- step2: Convert the expressions:
\(3\left(\frac{53}{50}\right)^{5}\)
- step3: Simplify:
\(3\times \frac{53^{5}}{50^{5}}\)
- step4: Multiply:
\(\frac{3\times 53^{5}}{50^{5}}\)
The dividend at Time 5 will be approximately $4.01.
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Simplify this solution Extra Insights
To find the future dividend at Time 5, we can use the formula for calculating future dividends with constant growth: \[ \mathrm{D}^{t} = \mathrm{D}^{0} \times (1 + g)^t \] where \( \mathrm{D}^{0} \) is the current dividend, \( g \) is the growth rate, and \( t \) is the time in years. Plugging in the numbers: \[ \mathrm{D}^{5} = 3.00 \times (1 + 0.06)^5 \] Calculating it out: \[ \mathrm{D}^{5} = 3.00 \times (1.338225) \approx 4.014675 \] So, the dividend at Time 5 will be approximately \( \$ 4.01 \). Feel free to ask if you'd like to dive deeper into dividend growth models or explore other finance-related queries!
