To calculate the price of a bond, you can use the present value formula for both the coupon payments and the par value at maturity.
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Calculate the semi-annual coupon payment:
\[ \text{Coupon payment} = \frac{\text{Coupon rate} \times \text{Par value}}{2} \]
\[ \text{Coupon payment} = \frac{0.0783 \times 1000}{2} = 39.15 \]
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Determine the total number of periods (n):
Since the bond has a maturity of 16.00 years and pays semi-annually:
\[ n = 16 \times 2 = 32 \text{ periods} \]
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Determine the yield per period (r):
The yield to maturity is annual, so for semi-annual:
\[ r = \frac{0.0824}{2} = 0.0412 \]
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Calculate the present value of the coupon payments:
The formula for the present value of an annuity (the coupon payments) is:
\[ PV_{\text{coupons}} = C \times \left(1 - (1 + r)^{-n}\right) / r \]
Where \(C\) is the coupon payment and \(r\) is the yield per period.
\[ PV_{\text{coupons}} = 39.15 \times \left(1 - (1 + 0.0412)^{-32}\right) / 0.0412 \]
Calculating the present value of coupons:
\[ PV_{\text{coupons}} = 39.15 \times \left(1 - (1.0412)^{-32}\right) / 0.0412 \]
First, calculate \( (1 + 0.0412)^{-32} \):
\[ (1.0412)^{-32} \approx 0.2633 \]
Then plug in the values:
\[ PV_{\text{coupons}} = 39.15 \times \left(1 - 0.2633\right) / 0.0412 \]
\[ = 39.15 \times 0.7367 / 0.0412 \approx 39.15 \times 17.87 \approx 699.34 \]
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Calculate the present value of the par value:
The formula for the present value of the par value (the face value at maturity) is:
\[ PV_{\text{par}} = \frac{F}{(1 + r)^n} \]
Where \(F\) is the face value.
\[ PV_{\text{par}} = \frac{1000}{(1.0412)^{32}} \]
Calculate \( (1.0412)^{32} \):
\[ (1.0412)^{32} \approx 3.5935 \]
Then:
\[ PV_{\text{par}} = \frac{1000}{3.5935} \approx 278.19 \]
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Calculate the total price of the bond:
\[ \text{Bond Price} = PV_{\text{coupons}} + PV_{\text{par}} = 699.34 + 278.19 \approx 977.53 \]
Therefore, the estimated price of the bond is approximately $977.53.