Graphical Designs is offering \( 10-10 \) preferred stock. The stock will pay anannual dividend of \( \$ 10 \) with the first dividend payment occurring 10 years from today. The required return on this stock is 4.60 percent. What is the price of the stock today? a. \( \$ 132.55 \) b. \( \$ 145.03 \) c. \( \$ 138.65 \) d. \( \$ 217.39 \) e. \( \$ 135.60 \)

To find the price of the preferred stock today, we can use the formula for the present value of a future cash flow. Since the first dividend payment occurs 10 years from today, we need to discount the future cash flows back to the present value. ### Known Conditions: - Annual dividend (\(D\)) = \$10 - Required return (\(r\)) = 4.60% = 0.046 - Time until first payment (\(t\)) = 10 years ### Steps to Calculate the Price of the Stock: 1. **Calculate the Present Value of the Dividend Payment:** The present value (\(PV\)) of a future cash flow can be calculated using the formula: \[ PV = \frac{D}{(1 + r)^t} \] where: - \(D\) is the dividend payment, - \(r\) is the required return, - \(t\) is the time in years until the payment. 2. **Substituting the Values:** \[ PV = \frac{10}{(1 + 0.046)^{10}} \] Now, let's calculate this value. Calculate the value by following steps: - step0: Calculate: \(\frac{10}{\left(1+0.046\right)^{10}}\) - step1: Add the numbers: \(\frac{10}{1.046^{10}}\) - step2: Convert the expressions: \(\frac{10}{\left(\frac{523}{500}\right)^{10}}\) - step3: Simplify the expression: \(\frac{10}{\frac{523^{10}}{500^{10}}}\) - step4: Rewrite the expression: \(\frac{10\times 500^{10}}{523^{10}}\) The present value of the dividend payment is approximately \( 6.38 \). 3. **Calculate the Present Value of All Future Dividends:** Since the preferred stock pays dividends indefinitely after the first payment, we can treat the dividends as a perpetuity starting from year 10. The present value of a perpetuity can be calculated using the formula: \[ PV_{\text{perpetuity}} = \frac{D}{r} \] where \(D\) is the annual dividend and \(r\) is the required return. 4. **Calculate the Present Value of the Perpetuity:** \[ PV_{\text{perpetuity}} = \frac{10}{0.046} \] 5. **Discount the Present Value of the Perpetuity Back to Today:** Since this perpetuity starts in year 10, we need to discount it back to today: \[ PV_{\text{today}} = \frac{PV_{\text{perpetuity}}}{(1 + r)^{10}} \] Now, let's calculate the present value of the perpetuity and then discount it back to today. Calculate the value by following steps: - step0: Calculate: \(\frac{10}{0.046}\) - step1: Convert the expressions: \(\frac{10}{\frac{23}{500}}\) - step2: Multiply by the reciprocal: \(10\times \frac{500}{23}\) - step3: Multiply: \(\frac{10\times 500}{23}\) - step4: Multiply: \(\frac{5000}{23}\) Calculate or simplify the expression \( (1 + 0.046)^10 \). Calculate the value by following steps: - step0: Calculate: \(\left(1+0.046\right)^{10}\) - step1: Add the numbers: \(1.046^{10}\) - step2: Convert the expressions: \(\left(\frac{523}{500}\right)^{10}\) - step3: Use the properties of exponents: \(\frac{523^{10}}{500^{10}}\) Calculate or simplify the expression \( 217.391304/(1.567895) \). Calculate the value by following steps: - step0: Calculate: \(\frac{217.391304}{1.567895}\) - step1: Convert the expressions: \(\frac{\frac{27173913}{125000}}{\frac{313579}{200000}}\) - step2: Multiply by the reciprocal: \(\frac{27173913}{125000}\times \frac{200000}{313579}\) - step3: Reduce the numbers: \(\frac{27173913}{5}\times \frac{8}{313579}\) - step4: Multiply the fractions: \(\frac{27173913\times 8}{5\times 313579}\) - step5: Multiply: \(\frac{217391304}{1567895}\) The present value of the perpetuity is approximately \( 217.39 \). Now, when we discount this value back to today, we find that: \[ PV_{\text{today}} \approx 138.65 \] ### Conclusion: The price of the stock today is approximately \( 138.65 \). Thus, the correct answer is: **c. \( \$ 138.65 \)**.