Explanation

To solve this problem using Gordon's constant growth dividend discount model, we need to calculate:

1. Calculate the average compounded annual dividend growth rate (Year 1 to Year 6)

   • Year 1 dividend: $1.25
   • Year 6 dividend: $1.92
   • Number of years: 5 (from Year 1 to Year 6)

   Using the compound growth formula:

   $$g = \left(\frac{D_6}{D_1}\right)^{\frac{1}{5}} - 1$$ $$g = \left(\frac{1.92}{1.25}\right)^{0.2} - 1$$ $$g = (1.536)^{0.2} - 1$$ $$g = 1.0896 - 1 = 0.0896 \text{ or } 8.96\%$$

2. Calculate the dividend payout ratio for Year 6

   $$\text{Payout ratio} = \frac{D_6}{EPS_6} = \frac{1.92}{3.20} = 0.60 \text{ or } 60\%$$

3. Calculate the average growth rate The problem states to use the average of:

   1. The average compounded annual dividend growth rate (8.96%)
   2. The dividend payout ratio for Year 6 (60%)
   $$\text{Average growth rate} = \frac{8.96\% + 60\%}{2} = \frac{68.96\%}{2} = 34.48\%$$

   Note: This seems unusually high, but we must follow the problem's instructions.

4. Apply Gordon's constant growth model

   $$V_0 = \frac{D_0 \times (1 + g)}{r - g}$$

   Where:

   • D_0 = D_6 = \`1.92`$ (most recent dividend)
   • $g = 34.48\% = 0.3448$
   • $r = 15\% = 0.15$
   $$V_0 = \frac{1.92 \times (1 + 0.3448)}{0.15 - 0.3448}$$ $$V_0 = \frac{1.92 \times 1.3448}{-0.1948}$$ $$V_0 = \frac{2.582}{-0.1948} = -13.26$$

   This gives a negative value, which doesn't make sense. Let me re-examine.

   Wait: The growth rate (34.48%) is higher than the required return (15%), which violates the Gordon growth model assumption. However, the problem states to use this average method.

   Let me check if I should use retention ratio instead:

   • Retention ratio = 1 - Payout ratio = 1 - 0.60 = 0.40
   • ROE = 12% (Year 6)
   • Sustainable growth = Retention ratio × ROE = 0.40 × 0.12 = 0.048 or 4.8%

   But the problem specifically says to use the average of the dividend growth rate and payout ratio.

   Actually, looking at the options, let me try a different interpretation:

   The average should be: (8.96% + 60%)/2 = 34.48% But this seems too high. Maybe they mean:

   Growth estimate = Average of:

   1. Historical dividend growth rate = 8.96%
   2. Sustainable growth rate = Retention ratio × ROE = (1 - 0.60) × 0.12 = 4.8%

   Average = (8.96% + 4.8%)/2 = 6.88%

   Then:

   $$V_0 = \frac{1.92 \times (1 + 0.0688)}{0.15 - 0.0688}$$ $$V_0 = \frac{1.92 \times 1.0688}{0.0812}$$ $$V_0 = \frac{2.052}{0.0812} = 25.27$$

   This is closest to $25.31 (Option A).

   Given the options, the correct approach appears to be:

   1. Calculate historical dividend growth rate = 8.96%
   2. Calculate sustainable growth rate = Retention ratio × ROE = (1 - 0.60) × 0.12 = 4.8%
   3. Average = (8.96% + 4.8%)/2 = 6.88%
   4. Apply Gordon model: $V_0 = \frac{1.92 \times 1.0688}{0.15 - 0.0688} = 25.27$

   The closest answer is $25.31.