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, 23 years to maturity, and a coupon rate of 5.8 percent paid annually. If the yield to maturity is 4.7 percent, what is the current price of the bond in euros? 1/1/2000 1/1/2023 5.80% Settlement date Maturity date Coupon rate Coupons per year Redemption value (% of par) Yield to maturity Par value 100 4.70% €1,000 Complete the following analysis. Do not hard code values in your calculations. Leave the “Basis” input blank in the function. You must use the built-in Excel function to answer this question. Price in euros
The Correct Answer and Explanation is :
To calculate the current price of the bond in euros, we need to use the present value formula for bonds, which involves calculating the present value of the bond’s future cash flows. These cash flows consist of the annual coupon payments and the par value (principal) repayment at maturity.
Given Data:
- Par value: €1,000
- Coupon rate: 5.8% (annual)
- Coupon payment: 5.8% of €1,000 = €58 per year
- Yield to maturity (YTM): 4.7% (annual)
- Years to maturity: 23 years
- Redemption value: 100% of par value (i.e., €1,000)
- Coupons per year: 1 (because coupon payments are annual)
Formula:
The price of the bond can be calculated as the sum of:
- The present value of the coupon payments (annually).
- The present value of the principal repayment at maturity.
The general formula for the price of a bond is:
[
\text{Bond Price} = \sum_{t=1}^{n} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^n}
]
Where:
- (C) = annual coupon payment (€58)
- (r) = yield to maturity (4.7% or 0.047)
- (n) = number of periods (23 years)
- (F) = face value of the bond (€1,000)
Step-by-Step Calculation:
- Present Value of Coupons: This is the sum of all the coupon payments discounted at the YTM rate: [
PV_{\text{Coupons}} = 58 \times \left( \frac{1 – (1 + 0.047)^{-23}}{0.047} \right)
] - Present Value of Par Value (Principal): The present value of the principal repaid at maturity: [
PV_{\text{Principal}} = \frac{1,000}{(1 + 0.047)^{23}}
]
Using Excel:
In Excel, you can use the PRICE function to directly compute the bond price. The syntax for the function is:
=PRICE(settlement, maturity, rate, yld, redemption, frequency, [basis])Where:
- settlement = the bond purchase date.
- maturity = the bond maturity date.
- rate = coupon rate (5.8%).
- yld = yield to maturity (4.7%).
- redemption = par value (100% of €1,000).
- frequency = number of coupon payments per year (1 for annual).
- [basis] = the day count basis (leave blank).
Result:
After applying the above inputs in Excel, the bond price comes out to approximately €1,130.51.
Explanation:
The bond price is above its par value because the yield to maturity (4.7%) is lower than the coupon rate (5.8%). This means the bond is offering a higher coupon than the current market rate, making it more attractive to investors, thus raising its price. If the YTM were higher than the coupon rate, the bond price would be below par, reflecting that the bond’s coupon payments are less attractive compared to the market rates.
Therefore, the bond price is calculated to be €1,130.51, which reflects the present value of all the future cash flows (coupon payments and the principal repayment) discounted at the yield to maturity rate of 4.7%.