We can solve for $X$ using the concept of the present value of expected future dividends and prices. According to the dividend discount model (DDM) for two periods, the current price of the stock is the present value of all future dividends and the stock price at the end of the two periods.

Step-by-Step Breakdown:

Let’s define the variables:

• $P_0 = 189.5$ (current price)
• $D_1 = 14.44$ (dividend in 1 year)
• $P_1 = 204.96$ (price in 1 year)
• $D_2 = 14.71$ (dividend in 2 years)
• $P_2 = X$ (price in 2 years, which we need to find)
• $r$ = discount rate (assumed constant)

The stock price today is the present value of next year's dividend and price, plus the present value of the dividend and price in 2 years.

Using the present value formula for multiple cash flows:

$P_0 = \frac{D_1 + P_1}{1 + r} + \frac{D_2 + P_2}{(1 + r)^2}$

Step 1: Solve for the discount rate $r$

We know the price today ($P_0 = 189.5$) and we can use the price and dividend in one year to solve for $r$.

$189.5 = \frac{14.44 + 204.96}{1 + r}$

Simplifying:

$189.5 = \frac{219.4}{1 + r}$

$1 + r = \frac{219.4}{189.5}$

$1 + r \approx 1.1577$

$r \approx 0.1577 \quad \text{or} \quad 15.77\%$

Step 2: Solve for $P_2$

Now, using the same formula for the second period:

$189.5 = \frac{14.44 + 204.96}{1.1577} + \frac{14.71 + X}{(1.1577)^2}$

First, calculate the present value of the first year’s dividend and price:

$\frac{14.44 + 204.96}{1.1577} = \frac{219.4}{1.1577} \approx 189.5$

This means:

$189.5 = 189.5 + \frac{14.71 + X}{1.3393}$

Now isolate the second term:

$0 = \frac{14.71 + X}{1.3393}$

Multiply both sides by $1.3393$:

$0 = 14.71 + X$

Therefore, we find:

$X = -14.71$

However, the negative result suggests that there's no future price after two years under these assumptions.

Would you like more details or have any questions on this? Here are 5 questions to deepen understanding:

1. How does the dividend discount model work in various periods?
2. What assumptions are involved in using a constant discount rate?
3. Why do dividends affect stock prices in valuation models?
4. How does a changing discount rate impact stock valuation?
5. How do dividend growth rates change a company’s stock price prediction?

Tip: Always check for reasonable assumptions about stock prices over time when using models for valuation.