To find the price of a bond, you can use the formula for the present value of the bond's cash flows, which includes both the coupon payments and the par value at maturity.
Given:
- Par value (Face value) = \( F = 1,000 \)
- Coupon rate = \( 15% \)
- Coupon payment (C) = \( \frac{15% \times 1,000}{2} = 75 \) (since it's semi-annual)
- Number of periods (N) = \( 5 \times 2 = 10 \) (since it's a 5-year bond with semi-annual payments)
- Yield to maturity (YTM) = \( 15% \) (since it matches the coupon rate, the yield per period is \( \frac{15%}{2} = 7.5% \))
Next, we will calculate the price of the bond using the present value of the cash flows from the coupon payments and the principal.
The formula for the price of the bond is:
\[ P = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right) + \frac{F}{(1 + r)^{N}} \]
Where:
- \( P \) = Price of the bond
- \( C \) = Coupon payment per period
- \( r \) = Yield per period (as a decimal)
- \( N \) = Total number of periods
- \( F \) = Face value of the bond
Substituting the given values:
- Calculate the coupon payment \( C = 75 \)
- Calculate \( r = \frac{15%}{2} = 0.075 \) (as a decimal)
- Calculate \( N = 10 \)
- Calculate \( F = 1,000 \)
Now, substitute into the formula:
\[ P = 75 \times \left( \frac{1 - (1 + 0.075)^{-10}}{0.075} \right) + \frac{1,000}{(1 + 0.075)^{10}} \]
Calculating each component:
- Calculate \( (1 + 0.075)^{-10} = (1.075)^{-10} \approx 0.5083 \)
- Calculate \( 1 - (1 + 0.075)^{-10} \approx 1 - 0.5083 = 0.4917 \)
- Therefore, \( \frac{1 - (1.075)^{-10}}{0.075} \approx \frac{0.4917}{0.075} \approx 6.5556 \)
- Calculate the present value of the coupon payments: \[ 75 \times 6.5556 \approx 491.67 \]
- Calculate \( (1 + 0.075)^{10} = (1.075)^{10} \approx 2.0610 \)
- Therefore, the present value of the face value: \[ \frac{1,000}{2.0610} \approx 485.00 \]
Now, sum the present values:
\[ P \approx 491.67 + 485.00 \approx 976.67 \]
Thus, the price of the bond is approximately $976.67.