Math Problem Statement
Assume Gillette Corporation will pay an annual dividend of $ 0.64 one year from now. Analysts expect this dividend to grow at 12.9 % per year thereafter until the 2th year. Thereafter, growth will level off at 2.4 % per year. According to the dividend-discount model, what is the value of a share of Gillette stock if the firm's equity cost of capital is 7.8 %?
Solution
To value the share of Gillette Corporation stock using the dividend-discount model (DDM), we use the two-stage growth model due to the changing growth rates at different periods.
Given:
- Dividend at Year 1 () = $0.64
- First growth rate () = 12.9% per year until the 2nd year (two years of higher growth).
- Second growth rate () = 2.4% per year (constant growth from Year 3 onward).
- Equity cost of capital () = 7.8%.
Step 1: Calculate Dividends for Years 1, 2, and beyond
The dividend grows at 12.9% for the first two years:
- Dividend in Year 2 () =
- Dividend in Year 3 () =
Then, from Year 3 onward, the dividend grows at a constant 2.4%.
- Dividend in Year 4 () =
Step 2: Calculate the Present Value of Dividends for Years 1 and 2
For the dividends in the first two years, we discount them back to the present using the equity cost of capital:
- Present value of =
- Present value of =
Step 3: Calculate the Terminal Value at Year 2
Starting in Year 3, the dividend grows at a constant rate of 2.4%. We can calculate the present value of all dividends from Year 3 onward using the constant-growth formula (Gordon Growth Model) at Year 2:
This gives the value of all dividends starting from Year 3, discounted to Year 2. We then discount this terminal value back to the present.
Step 4: Sum of Present Values
Finally, we sum the present values of the dividends from Year 1, Year 2, and the terminal value.
Let's now calculate these steps numerically.The value of a share of Gillette Corporation stock, according to the dividend-discount model, is approximately $14.22.
Would you like further clarification or details on any step? Here are some related questions to deepen your understanding:
- How does the dividend-discount model account for fluctuating growth rates?
- What assumptions underlie the use of the Gordon Growth Model for constant dividend growth?
- How sensitive is the stock price to changes in the equity cost of capital?
- What impact does a higher long-term growth rate have on the stock's value?
- How do inflation and real interest rates affect the required return on equity?
Tip: Always ensure that growth rates in the dividend-discount model are less than the cost of capital for the formula to make sense mathematically.
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Math Problem Analysis
Mathematical Concepts
Dividend-Discount Model
Present Value of Dividends
Two-Stage Growth Model
Formulas
D2 = D1 × (1 + g1)
D3 = D2 × (1 + g1)
Terminal Value at Year 2 = D3 / (r - g2)
Present value of D1 = D1 / (1 + r)
Present value of D2 = D2 / (1 + r)^2
Theorems
Gordon Growth Model
Time Value of Money
Suitable Grade Level
College level - Finance
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