The risk-free rate is 1.51% and the market risk premium is 7.01%. A stock with a β of 1.33 just paid a dividend of $1.24. The dividend is expected to grow at 24.47% for three years and then grow at 4.01% forever. What is the value of the stock?

Question

GPT-4o mini · Answer

To determine the value of the stock, we can use the Dividend Discount Model (DDM) with the two-stage growth method since the dividend is expected to grow at two different rates over time.

Step 1: Calculate the Required Rate of Return

First, we need to calculate the required rate of return using the Capital Asset Pricing Model (CAPM):

$Required rate of return (r) = Risk-free rate + β \times Market risk premium$

Given:

Risk-free rate = 1.51% or 0.0151
Market risk premium = 7.01% or 0.0701
Beta (β) = 1.33

$r = 0.0151 + 1.33 \times 0.0701$ $r = 0.0151 + 0.0934783$ $r = 0.1085783 \approx 10.86$

Step 2: Calculate Dividends for the First Three Years

The dividend just paid (D0) is $1.24, and it will grow at 24.47% for the next three years. We need to calculate the expected dividends (D1, D2, D3):

$D 1 = D 0 \times (1 + g_{1}) = 1.24 \times (1 + 0.2447) = 1.24 \times 1.2447 \approx 1.542$ $D 2 = D 1 \times (1 + g_{1}) = 1.542 \times (1 + 0.2447) \approx 1.542 \times 1.2447 \approx 1.920$ $D 3 = D 2 \times (1 + g_{1}) = 1.920 \times (1 + 0.2447) \approx 1.920 \times 1.2447 \approx 2.392$

Step 3: Calculate the Present Value of Dividends for the First Three Years

Now, we will find the present value of D1, D2, and D3 using the required rate of return (10.86%):

$P V (D 1) = \frac{D 1}{(1 + r)^{1}} = \frac{1.542}{(1 + 0.1085783)^{1}} \approx \frac{1.542}{1.1085783} \approx 1.39$

$P V (D 2) = \frac{D 2}{(1 + r)^{2}} = \frac{1.920}{(1 + 0.1085783)^{2}} \approx \frac{1.920}{1.229207} \approx 1.56$

$P V (D 3) = \frac{D 3}{(1 + r)^{3}} = \frac{2.392}{(1 + 0.1085783)^{3}} \approx \frac{2.392}{1.366173} \approx 1.75$

Step 4: Calculate Total Present Value for the First Three Years

Now we sum the present values of D1, D2, and D3:

$P V_{total} = P V (D 1) + P V (D 2) + P V (D 3) \approx 1.39 + 1.56 + 1.75 \approx 4.70$

Step 5: Calculate the Present Value of Dividends from Year 4 Onwards

From year 4, the dividend is expected to grow at a constant rate of 4.01%. The dividend in year 4 (D4) is:

$D 4 = D 3 \times (1 + g_{2}) = 2.392 \times (1 + 0.0401) \approx 2.392 \times 1.0401 \approx 2.487$

Using the Gordon Growth Model for the present value of all future dividends starting from year 4, we calculate the present value of the perpetuity:

$P V_{perpetuity} = \frac{D 4}{r - g_{2}} = \frac{2.487}{0.1085783 - 0.0401} \approx \frac{2.487}{0.0684783} \approx 36.39$

Step 6: Discount the Present Value of Perpetuity Back to Today

Now we need to discount this value back to present value at year 3:

$P V_{perpetuity, discounted} = \frac{P V_{perpetuity}}{(1 + r)^{3}} \approx \frac{36.39}{1.366173} \approx 26.65$

Step 7: Calculate Total Value of the Stock

Finally, we can sum the present values calculated earlier to get the total value of the stock:

$Stock Value = P V_{total} + P V_{perpetuity, discounted} \approx 4.70 + 26.65 \approx 31.35$

Conclusion

The value of the stock is approximately $31.35.