a. Many years ago, Castles in the Sand Incorporated issued bonds at face value at a yield to maturity of 5.6%. Now, with 5 years left until the maturity of the bonds, the company has run into hard times and the yield to maturity on the bonds has increased to 13%. What is now the price of the bond? (Assume semiannual coupon payments.) Note: Do not round intermediate calculations. Round your answer to 2 decimal places.
b. Suppose that investors believe that Castles can make good on the promised coupon payments but that the company will go bankrupt when the bond matures and the principal comes due. The expectation is that investors will receive only 82% of face value at maturity. If they buy the bond today, what yield to maturity do they expect to receive? Note: Do not round intermediate calculations. Enter your answer as a percent rounded to 2 decimal places.
a. To calculate the price of the bond, we need to discount the future cash flows (coupon payments and face value) at the new yield to maturity rate of 13%.
First, let's calculate the coupon payment. Since the bonds have a semiannual coupon payment, we need to divide the annual coupon rate by 2. The annual coupon rate is 5.6% of the face value.
Coupon payment = (5.6% / 2) * Face value
Next, we need to calculate the present value of the coupon payments and the face value at the new yield to maturity rate of 13%.
Present value of coupon payments = Coupon payment / (1 + (Yield to maturity / 2)) + Coupon payment / (1 + (Yield to maturity / 2))^2 + ... + Coupon payment / (1 + (Yield to maturity / 2))^n
where n is the number of periods until maturity (in this case, 5 years).
Present value of face value = Face value / (1 + (Yield to maturity / 2))^n
Finally, we can calculate the price of the bond by summing the present value of coupon payments and the present value of face value.
Price of the bond = Present value of coupon payments + Present value of face value
b. If investors expect to receive only 82% of the face value at maturity, we can adjust the calculation by using this expected payment instead of the face value.
Present value of face value = Expected payment / (1 + (Yield to maturity / 2))^n
To find the yield to maturity that investors expect to receive, we need to solve for it in the equation:
Price of the bond = Present value of coupon payments + Present value of face value
Let's solve both equations to find the answers.
a. Price of the bond:
Coupon payment = (5.6% / 2) * Face value
Present value of coupon payments = Coupon payment / (1 + (13% / 2)) + Coupon payment / (1 + (13% / 2))^2 + Coupon payment / (1 + (13% / 2))^3 + Coupon payment / (1 + (13% / 2))^4 + Coupon payment / (1 + (13% / 2))^5
Present value of face value = Face value / (1 + (13% / 2))^5
Price of the bond = Present value of coupon payments + Present value of face value
b. Yield to maturity:
Present value of face value = Expected payment / (1 + (Yield to maturity / 2))^5
Price of the bond = Present value of coupon payments + Present value of face value
By solving these equations, we can find the price of the bond in scenario a and the yield to maturity in scenario b.