‘Even though most corporate bonds in the United States make coupon payments semiannually; bonds issued elsewhere often have annual coupon payments: Suppose German company issues bond with par value of €1,000,10 years to maturity, and a coupon rate of 6.8 percent paid annually: If the yield to maturity is 7.9 percent, what is the current price of the bond? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) Price’
The Correct Answer and Explanation is:
To calculate the current price of the bond, we use the present value formula for bonds, which adds the present value of the coupon payments and the present value of the face value at maturity.
Given:
- Par Value (Face Value) = €1,000
- Time to Maturity = 10 years
- Coupon Rate = 6.8% annually
- Yield to Maturity (YTM) = 7.9% annually
- Coupon Payment = 6.8% × €1,000 = €68 annually
Step-by-Step Calculation:
The price of the bond is:Price=∑t=11068(1+0.079)t+1000(1+0.079)10\text{Price} = \sum_{t=1}^{10} \frac{68}{(1 + 0.079)^t} + \frac{1000}{(1 + 0.079)^{10}}Price=t=1∑10(1+0.079)t68+(1+0.079)101000
We calculate each term:
Present Value of Coupons:
PVcoupons=68×(1−1(1+0.079)10)÷0.079PV_{\text{coupons}} = 68 \times \left(1 – \frac{1}{(1 + 0.079)^{10}}\right) \div 0.079PVcoupons=68×(1−(1+0.079)101)÷0.079PVcoupons=68×(1−12.09800576)÷0.079=68×0.523335÷0.079≈68×6.625≈450.50PV_{\text{coupons}} = 68 \times \left(1 – \frac{1}{2.09800576}\right) \div 0.079 = 68 \times 0.523335 \div 0.079 \approx 68 \times 6.625 \approx 450.50PVcoupons=68×(1−2.098005761)÷0.079=68×0.523335÷0.079≈68×6.625≈450.50
Present Value of Face Value:
PVface=1000(1+0.079)10=10002.09800576≈476.65PV_{\text{face}} = \frac{1000}{(1 + 0.079)^{10}} = \frac{1000}{2.09800576} \approx 476.65PVface=(1+0.079)101000=2.098005761000≈476.65
Total Price:
Price=PVcoupons+PVface=450.50+476.65=927.15\text{Price} = PV_{\text{coupons}} + PV_{\text{face}} = 450.50 + 476.65 = \boxed{927.15}Price=PVcoupons+PVface=450.50+476.65=927.15
Explanation
The value of a bond is the sum of the present values of its future cash flows, which include the periodic coupon payments and the repayment of the face value at maturity. Since this bond pays annual coupons and has 10 years to maturity, the investor will receive ten annual payments of €68 and a €1,000 lump sum in year ten.
The discount rate used is the yield to maturity, which reflects the return an investor would earn if they held the bond until maturity, assuming all coupon payments are reinvested at the same rate. In this case, the YTM is higher than the coupon rate, meaning the bond should be priced at a discount.
We discount each cash flow using the formula for the present value of an annuity for the coupons and the present value of a lump sum for the face value. The coupon stream is worth about €450.50, while the discounted face value is worth about €476.65. Adding these together gives the total present value of the bond as €927.15.
This price reflects a market condition where investors demand a higher return than the bond offers through its coupon, thus lowering its market price below par. The longer time to maturity also amplifies the effect of the discount rate, especially on the face value portion, due to compounding. Therefore, the bond is trading at a discount to par because of the higher required yield.
