A stock just paid a dividend of $2.32. The dividend is expected to grow at 28.40% for five years and then grow at 4.63% thereafter. The required return on the stock is 13.79%. What is the value of the stock?

To find the value of the stock given the dividend growth model, we can split the calculation into two parts: the first part involves calculating the dividends for the first five years when the dividend grows at a high growth rate, and the second part calculates the terminal value beyond year five when the growth rate stabilizes.

Step 1: Calculate dividends for the first 5 years

The most recent dividend paid (D0) is $2.32. The expected growth rate for the first five years is 28.40% (or 0.2840).

1. Calculate D1: \[ D_1 = D_0 \times (1 + g) = 2.32 \times (1 + 0.2840) = 2.32 \times 1.284 = 2.98368 \]

2. Calculate D2: \[ D_2 = D_1 \times (1 + g) = 2.98368 \times (1 + 0.2840) = 2.98368 \times 1.284 = 3.82549 \]

3. Calculate D3: \[ D_3 = D_2 \times (1 + g) = 3.82549 \times (1 + 0.2840) = 3.82549 \times 1.284 = 4.91385 \]

4. Calculate D4: \[ D_4 = D_3 \times (1 + g) = 4.91385 \times (1 + 0.2840) = 4.91385 \times 1.284 = 6.30691 \]

5. Calculate D5: \[ D_5 = D_4 \times (1 + g) = 6.30691 \times (1 + 0.2840) = 6.30691 \times 1.284 = 8.06663 \]

Step 2: Calculate present value of the dividends for the first 5 years

The required return is 13.79% (or 0.1379). Now we discount each of these dividends back to the present value:

\[ PV(D_1) = \frac{D_1}{(1 + r)^1} = \frac{2.98368}{(1 + 0.1379)^1} = \frac{2.98368}{1.1379} \approx 2.6174 \]

\[ PV(D_2) = \frac{D_2}{(1 + r)^2} = \frac{3.82549}{(1 + 0.1379)^2} = \frac{3.82549}{1.29588} \approx 2.9520 \]

\[ PV(D_3) = \frac{D_3}{(1 + r)^3} = \frac{4.91385}{(1 + 0.1379)^3} = \frac{4.91385}{1.47119} \approx 3.3446 \]

\[ PV(D_4) = \frac{D_4}{(1 + r)^4} = \frac{6.30691}{(1 + 0.1379)^4} = \frac{6.30691}{1.6725} \approx 3.7692 \]

\[ PV(D_5) = \frac{D_5}{(1 + r)^5} = \frac{8.06663}{(1 + 0.1379)^5} = \frac{8.06663}{1.90268} \approx 4.2437 \]

Adding these present values together: \[ PV_{\text{first 5 years}} = 2.6174 + 2.9520 + 3.3446 + 3.7692 + 4.2437 \approx 16.9269 \]

Step 3: Calculate terminal value at the end of year 5

After the first 5 years, the dividend is expected to grow at a perpetuity growth rate of 4.63% (or 0.0463). The dividend at the end of year 5 (D6) is: \[ D_6 = D_5 \times (1 + g) = 8.06663 \times (1 + 0.0463) = 8.06663 \times 1.0463 \approx 8.437 \]

The terminal value (TV) at the end of year 5 can be calculated using the Gordon Growth Model: \[ TV = \frac{D_6}{r - g} = \frac{8.437}{0.1379 - 0.0463} = \frac{8.437}{0.0916} \approx 92.008 \]

Now, discount this back to the present value: \[ PV(TV) = \frac{TV}{(1 + r)^5} = \frac{92.008}{(1 + 0.1379)^5} = \frac{92.008}{1.90268} \approx 48.4266 \]

Step 4: Calculate the total present value

Finally, the total value of the stock (PV) is the sum of the present values of the first five years of dividends and the present value of the terminal value: \[ PV_{\text{total}} = PV_{\text{first 5 years}} + PV(TV) = 16.9269 + 48.4266 \approx 65.3535 \]

Conclusion

The value of the stock is approximately $65.35.