To calculate the price of the bonds issued by Trek Star Productions, we can use the present value formula for bonds, which consists of the present value of future cash flows from the coupon payments and the present value of the face value at maturity.
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Identify the variables:
- Face Value (FV): $1,000.00
- Annual Coupon Payment (C): $102.75
- Number of Years to Maturity (N): 14 years
- Market Interest Rate (r): 12.00% or 0.12
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Calculate the Present Value of Coupon Payments: The present value of the coupon payments can be calculated using the formula for the present value of an annuity:
\[ PV_{\text{coupons}} = C \times \left(\frac{1 - (1 + r)^{-N}}{r}\right) \]
Plugging in the values:
\[ PV_{\text{coupons}} = 102.75 \times \left(\frac{1 - (1 + 0.12)^{-14}}{0.12}\right) \]
Calculating \((1 + r)^{-N}\):
\[ (1 + 0.12)^{-14} \approx (1.12)^{-14} \approx 0.1396 \]
Now calculate the full expression for the Present Value of Coupons:
\[ PV_{\text{coupons}} = 102.75 \times \left(\frac{1 - 0.1396}{0.12}\right) = 102.75 \times \left(\frac{0.8604}{0.12}\right) \approx 102.75 \times 7.170 \]
\[ PV_{\text{coupons}} \approx 735.05 \]
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Calculate the Present Value of the Face Value: The present value of the face value can be calculated using the formula:
\[ PV_{\text{face}} = FV \times (1 + r)^{-N} \]
Plugging in the values:
\[ PV_{\text{face}} = 1,000 \times (1 + 0.12)^{-14} \approx 1,000 \times 0.1396 \approx 139.6 \]
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Calculate the Total Present Value (Price of the Bond): The price of the bond is the sum of the present values of the coupons and the present value of the face value:
\[ Price = PV_{\text{coupons}} + PV_{\text{face}} \approx 735.05 + 139.6 \approx 874.65 \]
Therefore, the price of the bonds is approximately $874.65.
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