A 5% coupon bond with 9 years to maturity has a yield-to-maturity of 7%. Assuming the coupons are paid semi-annually and the principal amount is equal to 100, what is the price of the bond?
To calculate the price of a bond, we need to determine the present value of its cash flows, which consist of the periodic coupon payments and the final repayment of the principal amount.
Let's break down the steps to calculate the price of the bond:
Step 1: Determine the total number of periods.
Since the coupons are paid semi-annually and the bond has 9 years to maturity, there will be 18 semi-annual periods (9 years * 2).
Step 2: Calculate the periodic coupon payment.
Since the bond has a 5% coupon rate and a principal amount of 100, the coupon payment per period will be 5% of 100, which is 5.
Step 3: Determine the present value of the periodic coupon payments.
To calculate the present value of the coupon payments, we need to discount each semi-annual payment at the yield-to-maturity. Since the yield-to-maturity is given as an annual rate, we need to divide it by 2 to get the semi-annual yield.
Using the formula for present value of an annuity:
PV = C * ((1 - (1 + r)^(-n)) / r)
where PV is the present value, C is the coupon payment, r is the semi-annual yield, and n is the total number of periods.
Using the given values, we can calculate:
PV_coupon = 5 * ((1 - (1 + 0.07/2)^(-18)) / (0.07/2))
Step 4: Determine the present value of the principal repayment.
At maturity, the bond will repay the principal amount of 100. To calculate its present value, we discount it back to the present using the semi-annual yield. Since this payment occurs at the end of the final period, we don't need to use the annuity formula.
PV_principal = 100 / (1 + 0.07/2)^18
Step 5: Calculate the price of the bond.
The price of the bond is the sum of the present values of the coupon payments and the principal repayment.
Price = PV_coupon + PV_principal
Now, let's perform the calculations:
PV_coupon = 5 * ((1 - (1 + 0.07/2)^(-18)) / (0.07/2))
PV_principal = 100 / (1 + 0.07/2)^18
Price = PV_coupon + PV_principal
After performing the calculations, we can find the price of the bond.