A stock just paid a dividend of \( \$ 5.05 \) and is expected to maintain a constant dividend growth rate of 4.2 percent indefinitely. If the current stock price is \( \$ 69 \), what is the required return on the stock?

To find the required return on the stock, we can use the Gordon Growth Model (also known as the Dividend Discount Model for a stock with constant growth). The formula is given by: \[ r = \frac{D_1}{P_0} + g \] Where: - \( r \) = required return - \( D_1 \) = dividend expected next year - \( P_0 \) = current stock price - \( g \) = growth rate of the dividend ### Step 1: Calculate \( D_1 \) The dividend expected next year (\( D_1 \)) can be calculated using the formula: \[ D_1 = D_0 \times (1 + g) \] Where: - \( D_0 = 5.05 \) (the most recent dividend) - \( g = 0.042 \) (the growth rate) ### Step 2: Substitute the values Now, we can calculate \( D_1 \): \[ D_1 = 5.05 \times (1 + 0.042) \] ### Step 3: Calculate \( r \) After calculating \( D_1 \), we can substitute \( D_1 \), \( P_0 \), and \( g \) into the required return formula to find \( r \). Let's perform these calculations. Calculate the value by following steps: - step0: Calculate: \(5.05\left(1+0.042\right)\) - step1: Add the numbers: \(5.05\times 1.042\) - step2: Multiply the numbers: \(5.2621\) Calculate or simplify the expression \( (D_1 / 69) + 0.042 \). Simplify the expression by following steps: - step0: Solution: \(\frac{D_{1}}{69}+0.042\) - step1: Convert the expressions: \(\frac{D_{1}}{69}+\frac{21}{500}\) - step2: Reduce fractions to a common denominator: \(\frac{D_{1}\times 500}{69\times 500}+\frac{21\times 69}{500\times 69}\) - step3: Multiply the numbers: \(\frac{D_{1}\times 500}{34500}+\frac{21\times 69}{500\times 69}\) - step4: Multiply the numbers: \(\frac{D_{1}\times 500}{34500}+\frac{21\times 69}{34500}\) - step5: Transform the expression: \(\frac{D_{1}\times 500+21\times 69}{34500}\) - step6: Reorder the terms: \(\frac{500D_{1}+21\times 69}{34500}\) - step7: Multiply the numbers: \(\frac{500D_{1}+1449}{34500}\) Calculate or simplify the expression \( (5.2621 / 69) + 0.042 \). Calculate the value by following steps: - step0: Calculate: \(\left(\frac{5.2621}{69}\right)+0.042\) - step1: Divide the terms: \(\frac{52621}{690000}+0.042\) - step2: Convert the expressions: \(\frac{52621}{690000}+\frac{21}{500}\) - step3: Reduce fractions to a common denominator: \(\frac{52621}{690000}+\frac{21\times 1380}{500\times 1380}\) - step4: Multiply the numbers: \(\frac{52621}{690000}+\frac{21\times 1380}{690000}\) - step5: Transform the expression: \(\frac{52621+21\times 1380}{690000}\) - step6: Multiply the numbers: \(\frac{52621+28980}{690000}\) - step7: Add the numbers: \(\frac{81601}{690000}\) ### Step 1: Calculate \( D_1 \) We calculated \( D_1 \) as follows: \[ D_1 = 5.05 \times (1 + 0.042) = 5.2621 \] ### Step 2: Calculate the Required Return \( r \) Now, we substitute \( D_1 \), \( P_0 \), and \( g \) into the required return formula: \[ r = \frac{D_1}{P_0} + g = \frac{5.2621}{69} + 0.042 \] Calculating this gives: \[ r \approx 0.118262 \] ### Final Result Thus, the required return on the stock is approximately: \[ r \approx 11.83\% \] This means that investors would require a return of about 11.83% on this stock given the current price and expected growth in dividends.