To calculate the price of a bond, we can use the present value formula for bonds, which considers the present value of both the annuity (the semi-annual coupon payments) and the lump sum (par value at maturity). Here’s the breakdown:

Given Information:

• Par Value (Face Value), $F = 1000$
• Coupon Rate, $C = 7.96\%$
• Yield to Maturity (YTM), $YTM = 8.10\%$
• Time to Maturity, $T = 7 \text{ years}$
• Payment Frequency = Semi-annual (2 times a year)

Since it’s a semi-annual bond:

• The coupon payment per period is: $\text{Coupon Payment} = \frac{1000 \times 7.96\%}{2} = 39.80$
• The number of periods is: $N = 7 \times 2 = 14 \text{ periods}$
• The yield per period is: $\text{Yield per period} = \frac{8.10\%}{2} = 4.05\%$

Step 1: Present Value of the Annuity (Coupon Payments)

The formula for the present value of an annuity is: $PV_{\text{coupons}} = \text{Coupon Payment} \times \frac{1 - (1 + r)^{-N}}{r}$ where:

• $r = 0.0405$ (yield per period)
• $N = 14$

Substituting values: $PV_{\text{coupons}} = 39.80 \times \frac{1 - (1 + 0.0405)^{-14}}{0.0405}$

Step 2: Present Value of the Par Value (Face Value at Maturity)

The present value of the par value is calculated as: $PV_{\text{face value}} = F \times (1 + r)^{-N}$ Substituting values: $PV_{\text{face value}} = 1000 \times (1 + 0.0405)^{-14}$

Step 3: Calculate Total Price of the Bond

The price of the bond is the sum of $PV_{\text{coupons}}$ and $PV_{\text{face value}}$: $\text{Bond Price} = PV_{\text{coupons}} + PV_{\text{face value}}$

Let's compute these values.The price of the bond is approximately $992.63.

Would you like further details or have any questions about this calculation?

Here are some related questions for further exploration:

1. How does changing the yield to maturity affect the bond price?
2. What would the bond price be if the coupon rate were higher?
3. How does the frequency of coupon payments impact the bond price?
4. What is the difference between yield to maturity and current yield?
5. How is bond duration calculated, and how does it affect bond pricing?

Tip: When the yield to maturity is higher than the coupon rate, the bond typically trades at a discount (below par value).