The Frenall Company just paid a common stock dividend of $4.00 per share. The required rate of return on Frenall stock is 18.4 percent. Due to a major restructuring of the company’s production process, Frenall’s dividends are expected to decline by 25 percent in Year 1, and 14 percent in Year 2. From that point on, the company’s dividends are expected to grow at a rate of 6.4 percent per year forever. Given these expectations, compute the current equilibrium market price of Frenall’s stock.
To compute the current equilibrium market price of Frenall's stock, we can use the Gordon growth model, also known as the Dividend Discount Model (DDM).
The formula for the Gordon growth model is:
\[ P_0 = \frac{D_0(1 + g)}{r - g} \]
Where:
- \( P_0 \) is the current equilibrium market price of the stock
- \( D_0 \) is the dividend just paid (i.e., the most recent dividend)
- \( g \) is the expected growth rate of dividends
- \( r \) is the required rate of return
Given data:
- \( D_0 \) (dividend just paid) = $4.00 per share
- \( r \) (required rate of return) = 18.4%
- \( g \) (expected dividend growth rate):
- Decline by 25% in Year 1, so \( g_1 = 1 - 0.25 = 0.75 \)
- Decline by 14% in Year 2, so \( g_2 = 1 - 0.14 = 0.86 \)
- From Year 3 onwards, grow at a rate of 6.4%, so \( g = 0.064 \)
Let's calculate the current equilibrium market price using the data provided:
Step 1: Calculate the expected dividends for the first three years:
- Year 1 dividend (D1): \( D_1 = D_0 \times g_1 = 4 \times 0.75 \)
- Year 2 dividend (D2): \( D_2 = D_1 \times g_2 \)
Step 2: Calculate the expected dividend from Year 3 onwards using the constant growth rate:
- Year 3 dividend (D3): \( D_3 = D_2 \times (1 + g) \)
Step 3: Calculate the present value of dividends:
- Present value of Year 1 dividend: \( PV_1 = \frac{D_1}{{(1 + r)}^1} \)
- Present value of Year 2 dividend: \( PV_2 = \frac{D_2}{{(1 + r)}^2} \)
- Present value of Year 3 dividend onwards: \( PV_3 = \frac{D_3}{{(r - g)}^3} \)
Step 4: Calculate the current equilibrium market price, \( P_0 \):
- \( P_0 = PV_1 + PV_2 + PV_3 \)
Let's now calculate the current equilibrium market price:
Step 1:
\( D1 = 4 \times 0.75 = 3.00 \)
\( D2 = 3.00 \times 0.86 = 2.58 \)
Step 2:
\( D3 = 2.58 \times (1 + 0.064) \)
Step 3:
\( PV_1 = \frac{3.00}{{(1 + 0.184)}^1} \)
\( PV_2 = \frac{2.58}{{(1 + 0.184)}^2} \)
\( PV_3 = \frac{D3}{{(0.184 - 0.064)}^3} \)
Step 4:
\( P_0 = PV_1 + PV_2 + PV_3 \)
By substituting the calculated values and performing the calculations, we can find the current equilibrium market price of Frenall's stock.