To calculate the price of the bond, we can use the present value formula for bonds, which takes into account the present value of the bond's future cash flows. The bond pays interest semi-annually, so we need to adjust the coupon rate and the yield to maturity accordingly.
Given Data:
- Par Value (Face Value) = $1,000
- Coupon Rate = 8.10% annually
- Yield to Maturity (YTM) = 7.52% annually
- Years to Maturity = 18.00 years
- Number of periods per year (n) = 2 (since it is semi-annual)
Step 1: Calculate the semi-annual coupon payment
\[ \text{Coupon Payment} = \frac{\text{Coupon Rate}}{n} \times \text{Par Value} = \frac{8.10%}{2} \times 1000 = 0.0405 \times 1000 = 40.50 \]
Step 2: Calculate the semi-annual yield (YTM)
\[ \text{YTM (semi-annual)} = \frac{7.52%}{2} = 0.0376 \]
Step 3: Determine the total number of periods
\[ \text{Total Periods} = \text{Years to Maturity} \times n = 18 \times 2 = 36 \]
Step 4: Calculate the Present Value of the Coupon Payments
The present value of an annuity formula is used here:
\[ PV_{\text{coupons}} = C \times \left(1 - (1 + r)^{-n}\right) / r \]
Where:
- \(C\) = Coupon Payment = $40.50
- \(r\) = semi-annual YTM = 0.0376
- \(n\) = total number of periods = 36
\[ PV_{\text{coupons}} = 40.50 \times \left(1 - (1 + 0.0376)^{-36}\right) / 0.0376 \]
Calculating \( (1 + 0.0376)^{-36} \):
\[ (1 + 0.0376)^{-36} \approx 0.2871 \]
Now substituting this back:
\[ PV_{\text{coupons}} = 40.50 \times \left(1 - 0.2871\right) / 0.0376 \]
\[ PV_{\text{coupons}} = 40.50 \times \left(0.7129\right) / 0.0376 \approx 40.50 \times 18.9906 \approx 770.08 \]
Step 5: Calculate the Present Value of the Par Value
The present value of the par value is calculated using the formula:
\[ PV_{\text{par}} = \frac{F}{(1 + r)^{n}} \] Where:
- \(F\) = Face Value = $1,000
- \(r\) = semi-annual YTM = 0.0376
- \(n\) = total number of periods = 36
\[ PV_{\text{par}} = \frac{1000}{(1 + 0.0376)^{36}} = \frac{1000}{(1.0376)^{36}} \]
Calculating \((1.0376)^{36}\):
\[ (1.0376)^{36} \approx 3.69484 \]
Now substituting this back:
\[ PV_{\text{par}} = \frac{1000}{3.69484} \approx 270.20 \]
Step 6: Calculate Total Present Value (Price of the Bond)
\[ \text{Price of the Bond} = PV_{\text{coupons}} + PV_{\text{par}} = 770.08 + 270.20 \approx 1040.28 \]
Conclusion:
The estimated price of the bond will be approximately $1,040.28.