Suppose g = 0 for t = 1 to 3, and then g is a constant 6%, Do = $2.00 and k is 13%. What is P.?
The correct answer and explanation is:
To find the price of a stock (P₀) under a non-constant growth model, we can use the dividend discount model (DDM), adjusting for variable growth. Here’s the given information:
- g = 0% from t = 1 to 3, then g = 6% starting from t = 4.
- D₀ = $2.00
- k (required rate of return) = 13%
Step 1: Find Dividends for Years 1–3 (Zero Growth)
Since g = 0% for the first 3 years:
- D₁ = D₂ = D₃ = $2.00
Step 2: Find D₄ (Growth Starts at t = 4)
From year 4 onward, dividend grows at 6%:
- D₄ = D₃ × (1 + g) = 2.00 × 1.06 = $2.12
Step 3: Find the Price at Year 3 (P₃)
From year 4 onward, growth is constant, so we can use the Gordon Growth Model to find P₃, the present value of all future dividends starting at year 4: P3=D4k−g=2.120.13−0.06=2.120.07=30.29P_3 = \frac{D_4}{k – g} = \frac{2.12}{0.13 – 0.06} = \frac{2.12}{0.07} = 30.29
Step 4: Find the Present Value (P₀)
Now discount D₁, D₂, D₃, and P₃ to the present: P0=2.00(1.13)1+2.00(1.13)2+2.00+30.29(1.13)3P_0 = \frac{2.00}{(1.13)^1} + \frac{2.00}{(1.13)^2} + \frac{2.00 + 30.29}{(1.13)^3} P0=2.001.13+2.001.2769+32.291.443P_0 = \frac{2.00}{1.13} + \frac{2.00}{1.2769} + \frac{32.29}{1.443} P0=1.7699+1.5661+22.37=25.71P_0 = 1.7699 + 1.5661 + 22.37 = \boxed{25.71}
Explanation (300 words)
This problem requires using a multi-stage Dividend Discount Model (DDM). Initially, the dividend does not grow for the first three years, remaining at $2.00 annually. After that, it begins to grow perpetually at 6%, creating a hybrid scenario: no-growth followed by constant growth.
We begin by identifying the dividends for each of the first three years. Since growth is zero, each dividend from t = 1 to t = 3 is simply $2.00.
Starting in year 4, growth begins at 6%. Thus, we calculate the year 4 dividend as D₄ = $2.00 × 1.06 = $2.12.
To find the value of all future dividends from t = 4 onward, we use the Gordon Growth Model, which assumes dividends grow at a constant rate indefinitely. This gives us P₃, the present value of all future dividends at t = 3, using: P3=D4k−gP_3 = \frac{D_4}{k – g}
The final step is to discount all cash flows—D₁, D₂, D₃, and P₃—back to the present using the required rate of return (13%). The result, P₀ = $25.71, represents the intrinsic value of the stock based on its expected future dividends.
✅ Final Answer: $25.71