You have 1 share of stock A and 1 share of stock B. The total value of the two holdings is $373.68. Stock A is expected to pay annual dividends forever. The first dividend for stock A is expected to be $8.02 and all subsequent dividends are expected to increase by 3.74 percent per year forever. The expected return for stock A is 10.15 percent per year and the first dividend will be paid in 1 year. Stock B is expected to pay annual dividends forever. The first dividend for stock B is expected to be $10.39 and all subsequent dividends are expected to increase by X percent per year forever. The expected return for stock B is 7.71 percent per year and the first dividend will be paid in 1 year. What is X?

To determine $ X $, the growth rate for stock B’s dividends, we can use the **Gordon Growth Model (Dividend Discount Model)**. The formula for the value of a stock paying dividends that grow perpetually is:

\[
P = \frac{D_1}{r - g}
\]

where:  
- $ P $ = Current price of the stock  
- $ D_1 $ = Next period’s dividend (first dividend)  
- $ r $ = Expected return (discount rate)  
- $ g $ = Dividend growth rate  

We are given the following data for **two stocks (A and B)**:

### **Stock A:**
- Total value of holdings (A + B): $ 373.68 $
- Expected return $ r_A = 10.15\% = 0.1015 $
- First dividend $ D_{A,1} = 8.02 $
- Dividend growth rate $ g_A = 3.74\% = 0.0374 $

Using the Gordon Growth Model for stock A:

\[
P_A = \frac{8.02}{0.1015 - 0.0374} = \frac{8.02}{0.0641} \approx 125.12
\]

So, the current value of stock A is approximately $ 125.12 $.

---

### **Stock B:**
For stock B:
- Total value of holdings (A + B) = 373.68  
- Value of stock A = 125.12  
- Value of stock B $ P_B = 373.68 - 125.12 = 248.56 $  
- Expected return $ r_B = 7.71\% = 0.0771 $  
- First dividend $ D_{B,1} = 10.39 $  
- Growth rate of dividends = $ X $

Using the Gordon Growth Model for stock B:

\[
P_B = \frac{10.39}{0.0771 - X}
\]

Setting $ P_B = 248.56 $ and solving for $ X $:

\[
248.56 = \frac{10.39}{0.0771 - X}
\]

Rearrange:

\[
0.0771 - X = \frac{10.39}{248.56}
\]

\[
0.0771 - X \approx 0.0418
\]

\[
X \approx 0.0771 - 0.0418 = 0.0353 = 3.53\%
\]

---

### **Answer:**
The dividend growth rate $ X $ for stock B is approximately **3.53%**.

---

Would you like a further breakdown or explanation?  

Here are five related questions:
1. How does the Gordon Growth Model change if dividends do not grow perpetually?
2. What assumptions does the Gordon Growth Model rely on?
3. How sensitive is the stock price to changes in the expected return?
4. What if dividends grow at a variable rate instead of a constant rate?
5. How can we estimate the expected return if the dividend growth rate and price are known?

**Tip:** When expected returns are close to the dividend growth rate, the stock price becomes more sensitive to small changes in either variable.

Learn how to calculate the dividend growth rate for stock B given a portfolio value of $373.68, with stock A and stock B paying annual dividends that grow perpetually. This problem uses the Gordon Growth Model to find the growth rate X for stock B's dividends, given expected returns, initial dividends, and growth rates.

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