Yellow Lake Fashion stock is expected to pay a dividend of $7.78 in 1 year and a dividend of $6.56 in 2 years. The stock is currently priced at $177.7 and is expected to be priced at $202.42 in 1 year and at $X in 2 years. The stock’s dividends are paid annually and the next dividend is expected in 1 year. What is X?

We can solve this problem using the concept of **stock pricing** based on **expected dividends** and **future stock prices**. The stock's price today is the present value (PV) of all future cash flows, which include dividends and the future stock price. 

### Step-by-step solution:

1. **Given Information:**
   - Current stock price, $ P_0 = 177.7 $
   - Dividend in 1 year, $ D_1 = 7.78 $
   - Dividend in 2 years, $ D_2 = 6.56 $
   - Stock price in 1 year, $ P_1 = 202.42 $
   - Stock price in 2 years, $ P_2 = X $ (This is what we need to find)

2. **Time 1 (after 1 year):**
   The price of the stock in 1 year will be the present value of dividends and the stock price in 2 years:
   \[
   P_1 = \frac{D_2 + P_2}{1 + r}
   \]
   where $ r $ is the discount rate (the rate of return required by investors).

3. **Time 0 (today):**
   The price of the stock today is based on the expected dividend in 1 year and the price of the stock in 1 year:
   \[
   P_0 = \frac{D_1 + P_1}{1 + r}
   \]
   We can use this equation to find the discount rate $ r $.

### Step 1: Find the discount rate $ r $

Using the given information at time 0:
\[
177.7 = \frac{7.78 + 202.42}{1 + r}
\]

Solving for $ r $:
\[
177.7(1 + r) = 7.78 + 202.42
\]
\[
177.7(1 + r) = 210.2
\]
\[
1 + r = \frac{210.2}{177.7} = 1.1832
\]
\[
r = 1.1832 - 1 = 0.1832 \quad \text{or} \quad 18.32\%
\]

### Step 2: Find $ X $ (the stock price in 2 years)

Now that we know the discount rate $ r = 18.32\% $, we can use the equation for the stock price in 1 year to find $ P_2 $ (or $ X $):

\[
202.42 = \frac{6.56 + X}{1 + 0.1832}
\]
\[
202.42 = \frac{6.56 + X}{1.1832}
\]
\[
202.42 \times 1.1832 = 6.56 + X
\]
\[
239.41 = 6.56 + X
\]
\[
X = 239.41 - 6.56 = 232.85
\]

### Final Answer:
The stock price in 2 years, $ X $, is **$232.85**.

Would you like more details or have any questions?

Here are five relative questions for you to explore further:
1. How is the discount rate used in valuing future stock prices?
2. What assumptions are made in the dividend discount model?
3. How does the expected future stock price affect current valuations?
4. How does changing the discount rate affect the calculated stock price?
5. Why is it important for investors to estimate future stock prices?

**Tip:** Always double-check the assumptions regarding future dividends and stock prices, as they significantly impact valuation models.

Solve for the future stock price of Yellow Lake Fashion in 2 years using dividends, the current stock price, and the expected stock price in 1 year. This problem applies the Dividend Discount Model and involves calculating the discount rate and future stock price. Ideal for finance students.

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