To calculate the price of the bond, we can use the present value formula for bonds, which is the sum of the present value of future cash flows (coupon payments) and the present value of the par value (face value) at maturity.
Given Data:
- Par Value (Face Value) = \( P = 1,000 \)
- Coupon Rate = \( 7.80% \)
- Yield to Maturity (YTM) = \( 8.36% \)
- Time to Maturity = \( 18.00 \) years
- Number of Payments per Year = \( 2 \) (since it’s a semi-annual bond)
Step 1: Calculate the Coupon Payment
The coupon payment can be calculated as follows:
\[ C = \frac{{\text{{Coupon Rate}}}}{2} \times P = \frac{0.078}{2} \times 1,000 = 39 \]
Step 2: Calculate the Number of Periods
The total number of periods (semi-annual payments) can be calculated as follows:
\[ n = 18 \times 2 = 36 \]
Step 3: Calculate the Semi-Annual Yield
The semi-annual yield is calculated as follows:
\[ \text{{Semi-Annual YTM}} = \frac{8.36%}{2} = 4.18% = 0.0418 \]
Step 4: Calculate the Present Value of the Coupon Payments
The present value of the coupon payments can be calculated using the formula for the present value of an annuity:
\[ PV_{\text{coupons}} = C \times \left(1 - (1 + r)^{-n}\right) / r \]
Substituting values:
\[ PV_{\text{coupons}} = 39 \times \left(1 - (1 + 0.0418)^{-36}\right) / 0.0418 \]
Calculating further:
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Calculate \((1 + 0.0418)^{-36}\): \[ (1 + 0.0418)^{-36} \approx (1.0418)^{-36} \approx 0.198212 \]
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Substitute back into the PV formula: \[ PV_{\text{coupons}} = 39 \times \left(1 - 0.198212\right) / 0.0418 \] \[ PV_{\text{coupons}} = 39 \times (0.801788) / 0.0418 \approx 39 \times 19.184396 = 748.195 \]
Step 5: Calculate the Present Value of the Par Value
The present value of the par value can be calculated as follows:
\[ PV_{\text{par value}} = P \times (1 + r)^{-n} = 1,000 \times (1.0418)^{-36} = 1,000 \times 0.198212 \approx 198.212 \]
Step 6: Calculate the Bond Price
Now, we can find the price of the bond by summing the present values of the coupon payments and the present value of the par value:
\[ \text{Bond Price} = PV_{\text{coupons}} + PV_{\text{par value}} \approx 748.195 + 198.212 \approx 946.407 \]
Thus, the price of the bond is approximately $946.41.
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