To determine the price of the bond, we can use the present value formula for bonds. The price of a bond is the present value of its future cash flows, which consist of the annual coupon payments and the par value at maturity.
Given:
- Par Value (FV) = $1,000
- Coupon Rate = 7.95%
- Coupon Payment (C) = Par Value * Coupon Rate = $1,000 * 7.95% = $79.50
- Yield to Maturity (YTM) = 8.45%
- Number of Years to Maturity (N) = 13.00 years
Step 1: Calculate the present value of the annual coupon payments
The present value of the coupon payments can be calculated using the formula for an annuity:
\[ PV_{\text{coupons}} = C \times \left(1 - \frac{1}{(1 + r)^N}\right) / r \]
where:
- \( C \) = annual coupon payment = $79.50
- \( r \) = yield to maturity / 100 = 8.45% / 100 = 0.0845
- \( N \) = number of years = 13
Calculating the present value of coupons:
\[ PV_{\text{coupons}} = 79.50 \times \left(1 - \frac{1}{(1 + 0.0845)^{13}}\right) / 0.0845 \]
Calculating \( (1 + 0.0845)^{13} \):
\[ (1.0845)^{13} \approx 2.491 \]
So then \( \frac{1}{(1.0845)^{13}} \approx 0.401 \).
Now substituting back into the formula:
\[ PV_{\text{coupons}} = 79.50 \times \left(1 - 0.401\right) / 0.0845 \approx 79.50 \times (0.599) / 0.0845 \]
\[ PV_{\text{coupons}} = 79.50 \times 7.0808 \approx 563.87 \]
Step 2: Calculate the present value of the par value
The present value of the par value is calculated as:
\[ PV_{\text{par}} = \frac{FV}{(1 + r)^N} \]
Substituting in the values:
\[ PV_{\text{par}} = \frac{1000}{(1 + 0.0845)^{13}} \approx \frac{1000}{2.491} \]
Calculating \( PV_{\text{par}} \):
\[ PV_{\text{par}} \approx 401.24 \]
Step 3: Calculate the total price of the bond
The total price of the bond is the sum of the present value of the coupon payments and the present value of the par value:
\[ \text{Price of the bond} = PV_{\text{coupons}} + PV_{\text{par}} \] \[ \text{Price of the bond} \approx 563.87 + 401.24 \approx 965.11 \]
Therefore, the price of the bond will be approximately $965.11.
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