A stock just paid a dividend of $2.95. The dividend is expected to grow at 27.40% for five years and then grow at 4.80% thereafter. The required return on the stock is 12.42%. What is the value of the stock?

Answer 1

To calculate the value of the stock, we will use the Dividend Discount Model (DDM), which is particularly useful for valuing stocks that pay dividends that are expected to grow at a certain percentage.

The steps to calculate the value of the stock are as follows:

1. Calculate the dividends for the first 5 years where the dividend is expected to grow at 27.40%.
2. Calculate the value of the stock at the end of year 5 using the Gordon Growth Model (for the period after year 5 when the growth rate changes to 4.80%).
3. Calculate the present value of all future cash flows (dividends and the terminal value).

Step 1: Calculate the expected dividends for the first 5 years.

Given:

• D0 (the dividend just paid) = $2.95
• Growth rate for the first 5 years (g1) = 27.40% = 0.274
• Required return (r) = 12.42% = 0.1242

The expected dividends for the next 5 years are:

• D1 = D0(1 + g1) = 2.95 * (1 + 0.274)
• D2 = D1(1 + g1)
• D3 = D2(1 + g1)
• D4 = D3(1 + g1)
• D5 = D4(1 + g1)

Calculating each dividend:

1. \( D1 = 2.95 \times (1 + 0.274) = 2.95 \times 1.274 = 3.76 \)
2. \( D2 = 3.76 \times (1 + 0.274) = 3.76 \times 1.274 = 4.79 \)
3. \( D3 = 4.79 \times (1 + 0.274) = 4.79 \times 1.274 = 6.10 \)
4. \( D4 = 6.10 \times (1 + 0.274) = 6.10 \times 1.274 = 7.77 \)
5. \( D5 = 7.77 \times (1 + 0.274) = 7.77 \times 1.274 = 9.88 \)

Step 2: Calculate the terminal value at the end of year 5.

After year 5, the dividend will grow at a constant rate of 4.80%. We will use the Gordon Growth Model to find the terminal value (TV) at the end of year 5:

The formula for the terminal value is: \[ TV = \frac{D6}{r - g2} \] Where:

• \( g2 = 4.80% = 0.048 \)
• \( D6 = D5(1 + g2) = 9.88 \times (1 + 0.048) = 9.88 \times 1.048 = 10.35 \)

Now calculating the terminal value: \[ TV = \frac{10.35}{0.1242 - 0.048} = \frac{10.35}{0.0762} \approx 135.53 \]

Step 3: Calculate the present value of the expected dividends and the terminal value.

The present value of each dividend (PV) can be calculated using the formula: \[ PV = \frac{D}{(1 + r)^t} \] Where \( D \) is the expected dividend and \( t \) is the year.

Now, we can calculate the present value for each dividend and the terminal value:

1. \( PV(D1) = \frac{3.76}{(1 + 0.1242)^1} \approx \frac{3.76}{1.1242} \approx 3.34 \)
2. \( PV(D2) = \frac{4.79}{(1 + 0.1242)^2} \approx \frac{4.79}{1.2622} \approx 3.79 \)
3. \( PV(D3) = \frac{6.10}{(1 + 0.1242)^3} \approx \frac{6.10}{1.4190} \approx 4.29 \)
4. \( PV(D4) = \frac{7.77}{(1 + 0.1242)^4} \approx \frac{7.77}{1.5941} \approx 4.87 \)
5. \( PV(D5) = \frac{9.88}{(1 + 0.1242)^5} \approx \frac{9.88}{1.7886} \approx 5.52 \)
6. \( PV(TV) = \frac{135.53}{(1 + 0.1242)^5} \approx \frac{135.53}{1.7886} \approx 75.82 \)

Finally, summing up the present values:

\[ Value = PV(D1) + PV(D2) + PV(D3) + PV(D4) + PV(D5) + PV(TV) \]

Calculating:

\[ Value = 3.34 + 3.79 + 4.29 + 4.87 + 5.52 + 75.82 \approx 97.63 \]

Thus, the estimated value of the stock is approximately $97.63.