To calculate the price of a bond, we can use the present value formula for the cash flows generated by the bond. The bond price \( P \) can be calculated using the formula:
\[ P = \sum_{t=1}^{n} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^n} \]
where:
- \( C \) = coupon payment,
- \( y \) = yield to maturity (as a decimal),
- \( F \) = face value of the bond,
- \( n \) = number of periods until maturity.
Given:
- Par value \( F = 1,000 \)
- Coupon rate = 7.95% (annual), therefore semi-annual coupon payment \( C = \frac{0.0795 \times 1000}{2} = 39.75 \)
- Yield to maturity \( y = \frac{0.0846}{2} = 0.0423 \) (since it's semi-annual)
- Years to maturity = 9, so number of periods \( n = 9 \times 2 = 18 \)
Now we can substitute these values into the bond price formula:
- Calculate the present value of the coupon payments:
\[ PV_{\text{coupons}} = \sum_{t=1}^{18} \frac{39.75}{(1+0.0423)^t} \]
This can be computed using the formula for the present value of an annuity:
\[ PV_{\text{coupons}} = C \times \left( \frac{1 - (1+y)^{-n}}{y} \right) \] \[ PV_{\text{coupons}} = 39.75 \times \left( \frac{1 - (1+0.0423)^{-18}}{0.0423} \right) \]
Calculating the factors:
\[ (1+0.0423)^{-18} \approx (1.0423)^{-18} \approx 0.3842 \] \[ PV_{\text{coupons}} = 39.75 \times \left( \frac{1 - 0.3842}{0.0423} \right) \approx 39.75 \times \left( \frac{0.6158}{0.0423} \right) \approx 39.75 \times 14.5544 \approx 578.12 \]
- Calculate the present value of the face value:
\[ PV_{\text{face value}} = \frac{1000}{(1+y)^n} = \frac{1000}{(1+0.0423)^{18}} \approx \frac{1000}{(1.0423)^{18}} \approx \frac{1000}{1.9104} \approx 523.49 \]
- Add the two present values together to find the price of the bond:
\[ P = PV_{\text{coupons}} + PV_{\text{face value}} \approx 578.12 + 523.49 \approx 1101.61 \]
So, the price of the bond is approximately $1,101.61.
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