The yield to maturity of a $1 000 bond with a 7.1% coupon rate, semi-annual coupons, and two years to maturity is 8.4% APR, compounded semi-annually. What must its price be?The price of the bond is $ (Round to the nearest cent.)
Question
The yield to maturity of a (Round to the nearest cent.)
Solution
To calculate the price of the bond, we need to calculate the present value of the bond's cash flows, which are the semi-annual coupon payments and the face value of the bond at maturity.
Step 1: Calculate the semi-annual coupon payment The semi-annual coupon payment is half of the annual coupon payment. The annual coupon payment is the face value of the bond times the coupon rate. So, the semi-annual coupon payment = (7.1% / 2) * 35.5
Step 2: Calculate the semi-annual yield to maturity The semi-annual yield to maturity is half of the annual yield to maturity. So, the semi-annual yield to maturity = 8.4% / 2 = 4.2%
Step 3: Calculate the present value of the semi-annual coupon payments The present value of the semi-annual coupon payments is the sum of the present values of each individual coupon payment. This is a geometric series that can be summed up using the formula for the present value of an annuity: PV = C * [(1 - (1 + r)^-n) / r] where: C = semi-annual coupon payment = 35.5 * [(1 - (1 + 0.042)^-4) / 0.042] = $131.57
Step 4: Calculate the present value of the face value at maturity The present value of the face value at maturity is the face value discounted back to the present using the semi-annual yield to maturity. So, PV = 885.54
Step 5: Calculate the price of the bond The price of the bond is the sum of the present value of the semi-annual coupon payments and the present value of the face value at maturity. So, the price of the bond = 885.54 = $1,017.11
Therefore, the price of the bond must be approximately $1,017.11.
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