A stock just paid a dividend of $1.21. The dividend is expected to grow at 25.59% for three years and then grow at 4.63% thereafter. The required return on the stock is 10.49%. What is the value of the stock?

To find the value of the stock given the expected future dividends, we can use the Dividend Discount Model (DDM). The stock's value is calculated by predicting the future dividends for a certain period, discounting them to the present value, and then considering the stock's value at the end of that period as a perpetuity.

1. Calculate future dividends for the first 3 years:

The stock just paid a dividend of \( D_0 = 1.21 \).

Year 1:

\[ D_1 = D_0 \times (1 + g_1) = 1.21 \times (1 + 0.2559) = 1.21 \times 1.2559 \approx 1.519 \]

Year 2:

\[ D_2 = D_1 \times (1 + g_1) = 1.519 \times (1 + 0.2559) \approx 1.519 \times 1.2559 \approx 1.907 \]

Year 3:

\[ D_3 = D_2 \times (1 + g_1) = 1.907 \times (1 + 0.2559) \approx 1.907 \times 1.2559 \approx 2.395 \]

2. Calculate the dividend for Year 4 (where the growth changes):

From year 4 onward, the dividend will grow at \( g_2 = 4.63% \).

\[ D_4 = D_3 \times (1 + g_2) = 2.395 \times (1 + 0.0463) = 2.395 \times 1.0463 \approx 2.509 \]

3. Calculate the present value of dividends for the first 3 years:

Now we need to discount the dividends for years 1 to 3 back to the present using the required return of \( r = 10.49% \).

\[ PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{1.519}{(1 + 0.1049)} = \frac{1.519}{1.1049} \approx 1.376 \] \[ PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{1.907}{(1 + 0.1049)^2} = \frac{1.907}{1.22095} \approx 1.564 \] \[ PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{2.395}{(1 + 0.1049)^3} = \frac{2.395}{1.34846} \approx 1.773 \]

4. Sum the present value of the dividends for the first 3 years:

\[ \text{Total PV (Years 1-3)} = PV(D_1) + PV(D_2) + PV(D_3) \approx 1.376 + 1.564 + 1.773 \approx 4.713 \]

5. Calculate the present value of all dividends from Year 4 onwards (perpetuity):

Starting from Year 4, the dividends will grow perpetually at \( g_2 = 4.63%. \)

The value at Year 3, which is the present value of all future dividends from Year 4 onward, can be calculated using the formula for a growing perpetuity:

\[ P_3 = \frac{D_4}{r - g_2} \] \[ P_3 = \frac{2.509}{0.1049 - 0.0463} = \frac{2.509}{0.0586} \approx 42.851 \]

6. Now discount \( P_3 \) back to present value:

\[ PV(P_3) = \frac{P_3}{(1+r)^3} = \frac{42.851}{1.34846} \approx 31.76 \]

7. Calculate the total present value of the stock:

Finally, we add the present values of the first three years of dividends and the present value of the perpetuity:

\[ \text{Stock Value} = \text{Total PV (Years 1-3)} + PV(P_3) \approx 4.713 + 31.76 \approx 36.473 \]

Conclusion:

The value of the stock is approximately $36.47.