To calculate the price of a bond, we can use the present value formula for the cash flows associated with the bond. The present value of a bond is the sum of the present value of its future coupon payments and the present value of the par value repaid at maturity.
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Bond Details:
- Par value (Face value, \( F \)) = $1,000
- Coupon rate = 16.00%
- Annual coupon payment (\( C \)) = Coupon rate × Par value = 0.16 × $1,000 = $160
- Years to maturity (\( N \)) = 20
- Yield to maturity (YTM, \( r \)) = 16.00% or 0.16
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Calculate the Present Value of Coupon Payments: The present value of the coupon payments is calculated using the formula for the present value of an annuity: \[ PV_c = C \times \left( \frac{1 - (1 + r)^{-N}}{r} \right) \] Substituting the values: \[ PV_c = 160 \times \left( \frac{1 - (1 + 0.16)^{-20}}{0.16} \right) \]
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Calculate the Present Value of the Par Value: The present value of the par value is calculated using the formula for the present value of a lump sum: \[ PV_F = \frac{F}{(1 + r)^N} \] Substituting the values: \[ PV_F = \frac{1,000}{(1 + 0.16)^{20}} \]
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Calculate the Total Present Value (Price of the Bond): The price of the bond \( P \) is the sum of the present values of the coupon payments and the par value: \[ P = PV_c + PV_F \]
Let's do the calculations step-by-step:
Step 1: Calculate Present Value of Coupon Payments
\[ PV_c = 160 \times \left( \frac{1 - (1 + 0.16)^{-20}}{0.16} \right) \]
Calculating \( (1 + 0.16)^{-20} \): \[ (1 + 0.16)^{-20} \approx (1.16)^{-20} \approx 0.0807 \] Therefore, \[ PV_c = 160 \times \left( \frac{1 - 0.0807}{0.16} \right) \approx 160 \times \left( \frac{0.9193}{0.16} \right) \approx 160 \times 5.7463 \approx 919.04 \]
Step 2: Calculate Present Value of Par Value
\[ PV_F = \frac{1,000}{(1 + 0.16)^{20}} \approx \frac{1,000}{(1.16)^{20}} \]
Calculating \( (1 + 0.16)^{20} \): \[ (1.16)^{20} \approx 6.883 \] Therefore, \[ PV_F = \frac{1,000}{6.883} \approx 145.66 \]
Step 3: Calculate the Price of the Bond
Now add the two present values together to find the price of the bond: \[ P = PV_c + PV_F \approx 919.04 + 145.66 \approx 1,064.70 \]
Final Answer
The price of the bond is approximately $1,064.70.