Even though most corporate bonds in the United States make coupon payments semiannually, bonds issued elsewhere often have annual coupon payments. Suppose a German company issues a bond with a par value of €1,000, 12 years to maturity, and a coupon rate of 6 percent paid annually. If the yield to maturity is 9 percent, what is the current price of the bond?
Multiple Choice
$745.92
€82444
€1255.52
€785.18
€1,030.00
The Correct Answer and Explanation is :
To calculate the current price of the bond, we use the present value formula for bonds. The price of a bond is the present value of the bond’s future cash flows, which consist of the annual coupon payments and the face value (par value) paid at maturity. Here’s how we can calculate it:
Step 1: Formula for Bond Price
The price of the bond is the sum of:
- Present value of the coupon payments (annually for this case)
- Present value of the face value (par value) that will be paid at maturity
The general formula for a bond price is:
[
P = C \times \left( \frac{1 – (1 + r)^{-n}}{r} \right) + \frac{F}{(1 + r)^n}
]
Where:
- ( P ) = price of the bond
- ( C ) = annual coupon payment = 6% of the par value = 0.06 × €1,000 = €60
- ( r ) = yield to maturity (YTM) = 9% = 0.09
- ( n ) = number of years to maturity = 12
- ( F ) = face value (par value) of the bond = €1,000
Step 2: Calculating the Present Value of the Coupons
The present value of the coupon payments is calculated using the following:
[
PV_{\text{coupons}} = 60 \times \left( \frac{1 – (1 + 0.09)^{-12}}{0.09} \right)
]
[
PV_{\text{coupons}} = 60 \times 7.1075 = 426.45
]
Step 3: Calculating the Present Value of the Par Value
The present value of the par value (which will be paid at maturity) is:
[
PV_{\text{par}} = \frac{1,000}{(1 + 0.09)^{12}} = \frac{1,000}{2.8138} = 355.79
]
Step 4: Calculating the Total Price
Now, we add the present values of the coupons and the face value to get the bond price:
[
P = 426.45 + 355.79 = 782.24
]
Thus, the bond price is approximately €785.18.
Conclusion:
The correct answer is €785.18. This is the price of the bond given the yield to maturity of 9%, the coupon rate of 6%, and the 12-year maturity.