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The risk-free rate is 1.49% and the market risk premium is 5.67%. A stock with a β of 1.78 just paid a dividend of $2.79. The dividend is expected to grow at 21.40% for five years and then grow at 4.50% forever. What is the value of the stock?

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To determine the value of the stock, we can use the Gordon Growth Model, also known as the Dividend Discount Model (DDM). The stock's value is determined by calculating the present value of the expected future dividends. The procedure involves two phases of growth:

  1. High growth phase for the first five years
  2. Stable growth phase thereafter

Given Data:

  • Risk-Free Rate (\( r_f \)) = 1.49%
  • Market Risk Premium = 5.67%
  • Beta (β) = 1.78
  • Dividend just paid (D₀) = $2.79
  • Growth rate for the first 5 years (g₁) = 21.40%
  • Growth rate thereafter (g₂) = 4.50%

Step 1: Calculate the Required Rate of Return

Using the Capital Asset Pricing Model (CAPM): \[ r = r_f + \beta \times \text{Market Risk Premium \] Substituting the values: \[ r = 1.49% + 1.78 \times 5.67% \] \[ r = 1.49% + 10.0866% \] \[ r \approx 11.5766% \]

Step 2: Calculate Dividends for the First Five Years

We calculate the expected dividends over the first five years while incorporating the growth rate of 21.40%.

  • Year 1: \( D_1 = D_0 \times (1 + g_1) = 2.79 \times (1 + 0.214) = 2.79 \times 1.214 = 3.39006 \)
  • Year 2: \( D_2 = D_1 \times (1 + g_1) = 3.39006 \times 1.214 = 4.1193063 \)
  • Year 3: \( D_3 = D_2 \times (1 + g_1) = 4.1193063 \times 1.214 = 5.001345387 \)
  • Year 4: \( D_4 = D_3 \times (1 + g_1) = 5.001345387 \times 1.214 = 6.072348295 \)
  • Year 5: \( D_5 = D_4 \times (1 + g_1) = 6.072348295 \times 1.214 = 7.3681636238 \)

Step 3: Present Value of the First Five Years' Dividends

\[ PV = \frac{D_1}{(1 + r)^1} + \frac{D_2}{(1 + r)^2} + \frac{D_3}{(1 + r)^3} + \frac{D_4}{(1 + r)^4} + \frac{D_5}{(1 + r)^5} \] Calculating the present values:

  • \( PV(D_1) = \frac{3.39006}{(1 + 0.115766)^1} = \frac{3.39006}{1.115766} \approx 3.042556129 \)
  • \( PV(D_2) = \frac{4.1193063}{(1 + 0.115766)^2} = \frac{4.1193063}{1.24502792} \approx 3.308731218 \)
  • \( PV(D_3) = \frac{5.001345387}{(1 + 0.115766)^3} = \frac{5.001345387}{1.39104412} \approx 3.597892139 \)
  • \( PV(D_4) = \frac{6.072348295}{(1 + 0.115766)^4} = \frac{6.072348295}{1.55240713} \approx 3.911063258 \)
  • \( PV(D_5) = \frac{7.3681636238}{(1 + 0.115766)^5} = \frac{7.3681636238}{1.72236569} \approx 4.276153510 \)

Now summing these present values: \[ PV_{\text{first 5 years}} = 3.042556129 + 3.308731218 + 3.597892139 + 3.911063258 + 4.276153510 \approx 17.13639625 \]

Step 4: Calculate Terminal Value at Year 5

After year 5, the stock will grow at a constant rate of 4.50%. The terminal value (TV) at the end of year 5 can be calculated as follows: \[ TV = \frac{D_6}{(r - g_2)} \] Where: \[ D_6 = D_5 \times (1 + g_2) = 7.3681636238 \times (1 + 0.045) \approx 7.3681636238 \times 1.045 = 7.7058171 \] Thus, \[ TV = \frac{7.7058171}{(0.115766 - 0.045)} = \frac{7.7058171}{0.070766} \approx 108.352522 \]

Step 5: Present Value of Terminal Value

The present value of the terminal value is obtained by discounting it back to present value: \[ PV_{\text{TV}} = \frac{TV}{(1 + r)^5} = \frac{108.352522}{(1.115766)^5} \approx \frac{108.352522}{1.72236569} \approx 62.84502046 \]

Step 6: Total Present Value (Value of Stock)

Finally, the total present value (price of the stock) is: \[ \text{Stock Price} = PV_{\text{first 5 years}} + PV_{\text{TV}} \approx 17.13639625 + 62.84502046 \approx 79.98141671 \]

Thus, the estimated value of the stock is approximately: \[ \boxed{79.98} \]