To calculate the price of a bond, we can use the present value formula for both the coupon payments and the face value at maturity.
Given:
- Par value (Face value) = $1,000
- Coupon rate = 9.00%
- Yield to maturity (YTM) = 9.00%
- Time to maturity = 20 years
- Payment frequency = semi-annual
First, we'll calculate the semi-annual coupon payment: \[ \text{Coupon Payment} = \left( \text{Coupon Rate} \times \text{Par Value} \right) / \text{Number of Payments per Year} \] \[ \text{Coupon Payment} = \left( 0.09 \times 1,000 \right) / 2 = 45 \]
Next, we calculate the total number of payments: \[ \text{Total Payments} = \text{Years to Maturity} \times \text{Payments per Year} = 20 \times 2 = 40 \]
Now, since the yield to maturity (YTM) is equal to the coupon rate, the price of the bond can also be determined easily. When the YTM equals the coupon rate, the bond will be priced at par value. Thus:
\[ \text{Bond Price} = \text{Par Value} = 1,000 \]
To confirm formulaically, the bond price can be calculated using the present value of the coupon payments and the present value of the face value at maturity:
- Present Value of Coupons: \[ PV_{\text{coupons}} = C \left( \frac{1 - (1 + r)^{-n}}{r} \right) \] Where:
- \(C\) = Coupon Payment = $45
- \(r\) = Yield per period = \(0.09 / 2 = 0.045\)
- \(n\) = Total number of payments = 40
Calculating this part: \[ PV_{\text{coupons}} = 45 \left( \frac{1 - (1 + 0.045)^{-40}}{0.045} \right) \] \[ PV_{\text{coupons}} = 45 \left( \frac{1 - (1.045)^{-40}}{0.045} \right) \]
Calculating \( (1.045)^{-40} \): \[ (1.045)^{-40} \approx 0.12924 \] Then: \[ PV_{\text{coupons}} = 45 \left( \frac{1 - 0.12924}{0.045} \right) \] \[ PV_{\text{coupons}} = 45 \left( \frac{0.87076}{0.045} \right) \approx 45 \times 19.263 \approx 868.84 \]
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Present Value of Face Value: \[ PV_{\text{face value}} = \frac{F}{(1 + r)^{n}} = \frac{1,000}{(1 + 0.045)^{40}} \] Calculating \( (1.045)^{40} \): \[ (1.045)^{40} \approx 6.758 \] Thus: \[ PV_{\text{face value}} = \frac{1,000}{6.758} \approx 148.308 \]
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Total Bond Price: \[ \text{Bond Price} = PV_{\text{coupons}} + PV_{\text{face value}} \approx 868.84 + 148.308 \approx 1,017.148 \]
Given that the YTM equals the coupon rate, the result should be $1,000 since it has to sell at par. So in the context of financial principles:
Final Price of the Bond: \( $1,000 \).
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