Question

Assume a par value of $1,000. Caspian Sea plans to issue a 9.00 year, semi-annual pay bond that has a coupon rate of 7.95%. If the yield to maturity for the bond is 8.46%, what will the price of the bond be?

Answer 1

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:

1. 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:

\[
(1+0.0423)^{-18} \approx (1.0423)^{-18} \approx 0.3842
\]
\[
PV_{\text{coupons}} = 39.75 \times \left( \frac{1 - 0.3842}{0.0423} \right) \approx 39.75 \times \left( \frac{0.6158}{0.0423} \right) \approx 39.75 \times 14.5544 \approx 578.12
\]

2. Calculate the present value of the face value:

\[
PV_{\text{face value}} = \frac{1000}{(1+y)^n} = \frac{1000}{(1+0.0423)^{18}} \approx \frac{1000}{(1.0423)^{18}} \approx \frac{1000}{1.9104} \approx 523.49
\]

3. 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**.