A stock just paid a dividend of $1.02. The dividend is expected to grow at 26.77% for three years and then grow at 4.48% thereafter. The required return on the stock is 12.74%. What is the value of the stock?

Question

GPT-4o mini · Answer

To find the value of the stock, we can use the Dividend Discount Model (DDM) which accounts for both the short-term high growth and the long-term stable growth phases of dividends. We will calculate the expected dividends for the first three years when the growth is 26.77%, and then we will use the Gordon Growth Model for the dividends in perpetuity after the third year.

Step 1: Calculate the expected dividends for the first three years

Let $D_{0} = 1.02$ (the dividend just paid).

Using the growth rate $g_{1} = 26.77$ for the first three years, we can calculate:

Year 1 Dividend $D_{1}$ : $D_{1} = D_{0} \times (1 + g_{1}) = 1.02 \times (1 + 0.2677)$ $D_{1} \approx 1.02 \times 1.2677 \approx 1.294$
Year 2 Dividend $D_{2}$ : $D_{2} = D_{1} \times (1 + g_{1})$ $D_{2} \approx 1.294 \times 1.2677 \approx 1.64$
Year 3 Dividend $D_{3}$ : $D_{3} = D_{2} \times (1 + g_{1})$ $D_{3} \approx 1.64 \times 1.2677 \approx 2.08$

Step 2: Calculate the dividend for Year 4

The growth rate changes to $g_{2} = 4.48$ from Year 4 onward.

Year 4 Dividend $D_{4}$ : $D_{4} = D_{3} \times (1 + g_{2})$ $D_{4} \approx 2.08 \times (1 + 0.0448)$ $D_{4} \approx 2.08 \times 1.0448 \approx 2.18$

Step 3: Calculate the present value of the dividends for the first three years

We discount each of the dividends by the required return $r = 12.74$ :

Present Value of $D_{1}$ : $P V (D_{1}) = \frac{D_{1}}{(1 + r)^{1}} = \frac{1.294}{(1 + 0.1274)^{1}} \approx \frac{1.294}{1.1274} \approx 1.1477$
Present Value of $D_{2}$ : $P V (D_{2}) = \frac{D_{2}}{(1 + r)^{2}} = \frac{1.64}{(1 + 0.1274)^{2}} \approx \frac{1.64}{1.2718} \approx 1.2876$
Present Value of $D_{3}$ : $P V (D_{3}) = \frac{D_{3}}{(1 + r)^{3}} = \frac{2.08}{(1 + 0.1274)^{3}} \approx \frac{2.08}{1.4328} \approx 1.4535$

Step 4: Calculate the present value of the dividends from Year 4 onward

To find the value of the stock from Year 4 onward, we can use the Gordon Growth Model: $P_{3} = \frac{D_{4}}{r - g_{2}} = \frac{2.18}{0.1274 - 0.0448} = \frac{2.18}{0.0826} \approx 26.37$ Now we need to discount this back to present value at Year 3: $P V (P_{3}) = \frac{P_{3}}{(1 + r)^{3}} = \frac{26.37}{1.4328} \approx 18.42$

Step 5: Add all present values together to find the stock price

$P V_{T o t a l} = P V (D_{1}) + P V (D_{2}) + P V (D_{3}) + P V (P_{3})$ $P V_{T o t a l} \approx 1.1477 + 1.2876 + 1.4535 + 18.42 \approx 22.31$

Therefore, the value of the stock is approximately $22.31.