The correct answer is A (6.9%).

To derive the ROE, we use the formula for the forward P/E ratio: $\text{P/E}_1 = \frac{p}{r - g}$ where:

• $p$ = dividend payout ratio = $1 - \text{retention rate}$ = $1 - b$
• $r$ = required rate of return
• $g$ = dividend growth rate = $b \times \text{ROE}$

Rearranging the formula to solve for ROE: $\text{ROE} = \frac{\left( \frac{1 - b}{\text{P/E}_1} \right) - r}{-b}$ Substituting the given values: $\text{ROE} = \frac{\left( \frac{1 - 0.45}{8} \right) - 0.10}{-0.45} \approx 6.9\%$

Option B (8.0%) is incorrect because it mistakenly interchanges the retention rate and dividend payout ratio in the calculation.

Option C (9.7%) is incorrect because it erroneously uses the retention rate instead of the dividend payout ratio in the numerator of the formula.