Gugenheim, Incorporated, has a bond outstanding with a coupon rate of 6.9 percent and annual payments. The yield to maturity is 8.1 percent and the bond matures in 25 years. What is the market price if the bond has a par value of \( \$ 2,000 \) ?

To find the market price of the bond, we can use the present value formula for bonds, which discounts the future coupon payments and the par value at maturity back to the present. The formula is: \[ P = \sum \left( \frac{C}{(1 + r)^t} \right) + \frac{F}{(1 + r)^N} \] Where: - \( P \) = price of the bond - \( C \) = annual coupon payment = \( 0.069 \times 2000 = 138 \) - \( r \) = yield to maturity (as a decimal) = 0.081 - \( F \) = par value of the bond = 2000 - \( N \) = number of years to maturity = 25 Now, calculating the present value of the coupon payments: \[ P_{coupons} = \sum \left( \frac{138}{(1 + 0.081)^t} \right) \text{ for } t = 1 \text{ to } 25 \] This is a geometric series. The present value of the annuity can be calculated as: \[ P_{coupons} = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right) = 138 \times \left( \frac{1 - (1 + 0.081)^{-25}}{0.081} \right) \] Calculating this gives: \[ P_{coupons} \approx 138 \times 11.1572 \approx 1536.68 \] Now calculating the present value of the par value: \[ P_{par} = \frac{2000}{(1 + 0.081)^{25}} \approx \frac{2000}{5.8785} \approx 340.39 \] Finally, adding both present values together: \[ P = P_{coupons} + P_{par} \approx 1536.68 + 340.39 \approx 1877.07 \] Thus, the market price of the bond is approximately \( \$ 1,877.07 \).