To calculate the price of a bond, you can use the present value formula for bonds, which takes into account the present value of the bond's coupon payments and the present value of the par value at maturity. Specifically, the price of the bond can be calculated using the following formula:
\[ P = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) + \frac{F}{(1 + r)^{n}} \]
Where:
- \( P \) = Price of the bond
- \( C \) = Coupon payment (semi-annual)
- \( r \) = Yield to maturity per period (semi-annual)
- \( n \) = Total number of periods (semi-annual)
- \( F \) = Face value of the bond
Step 1: Calculate the semi-annual coupon payment (\(C\))
The coupon payment is calculated as follows: \[ C = \text{Coupon Rate} \times \text{Par Value} \times \text{Period} \] The bond has a coupon rate of 8.07%, and since it pays semi-annually, each payment will be: \[ C = 0.0807 \times 1000 \times \frac{1}{2} = 40.35 \]
Step 2: Determine the semi-annual yield to maturity (\(r\))
The yield to maturity is given as 7.53% annually. Since the bond pays semi-annually, we divide it by 2: \[ r = \frac{7.53%}{2} = 0.03765 \]
Step 3: Determine the total number of periods (\(n\))
The bond matures in 6.00 years, and with semi-annual payments, the total number of periods is: \[ n = 6 \times 2 = 12 \]
Step 4: Calculate the price of the bond (\(P\))
Now we can plug the values into the bond price formula: \[ P = 40.35 \times \left( \frac{1 - (1 + 0.03765)^{-12}}{0.03765} \right) + \frac{1000}{(1 + 0.03765)^{12}} \]
Calculating each term:
- Calculate the present value of coupon payments: \[ \text{PV of Coupons} = 40.35 \times \left( \frac{1 - (1 + 0.03765)^{-12}}{0.03765} \right) \]
- Calculate the present value of face value: \[ \text{PV of Face Value} = \frac{1000}{(1 + 0.03765)^{12}} \]
Let's calculate each separately.
1. Calculate the present value of the coupon payments:
\[ 1 + 0.03765 \approx 1.03765 \] \[ (1.03765)^{-12} \approx 0.63573 \quad (\text{Using calculator}) \] So, \[ \frac{1 - 0.63573}{0.03765} \approx \frac{0.36427}{0.03765} \approx 9.6835 \] Then, \[ \text{PV of Coupons} \approx 40.35 \times 9.6835 \approx 390.82 \]
2. Calculate the present value of the face value:
\[ (1.03765)^{12} \approx 1.49725 \] So, \[ \text{PV of Face Value} \approx \frac{1000}{1.49725} \approx 667.37 \]
Final Price Calculation: \[ P \approx 390.82 + 667.37 \approx 1058.19 \]
Therefore, the price of the bond is approximately $1058.19.
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