Assume a par value of $1,000. Caspian Sea plans to issue a 11.00 year, annual pay bond that has a coupon rate of 8.08%. If the yield to maturity for the bond is 7.53%, 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 bond's cash flows, which consist of the annual coupon payments and the par value at maturity. The formula for the price of the bond is:

\[ P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N} \]

Where:

• \( P \) = price of the bond
• \( C \) = annual coupon payment (\( C = \text{coupon rate} \times \text{par value} \))
• \( r \) = yield to maturity (as a decimal)
• \( F \) = par value of the bond
• \( N \) = number of years until maturity

Given:

• Par value \( F = 1000 \)
• Coupon rate = 8.08%
• Yield to maturity \( r = 7.53% \)
• Years until maturity \( N = 11 \)

First, calculate the annual coupon payment \( C \):

\[ C = 0.0808 \times 1000 = 80.80 \]

Next, substitute the values into the bond price formula:

\[ P = \sum_{t=1}^{11} \frac{80.80}{(1 + 0.0753)^t} + \frac{1000}{(1 + 0.0753)^{11}} \]

This can be separated into two parts:

1. Present value of the coupon payments:

\[ PV_{\text{coupons}} = \sum_{t=1}^{11} \frac{80.80}{(1 + 0.0753)^t} \]

2. Present value of the par value:

\[ PV_{\text{par}} = \frac{1000}{(1 + 0.0753)^{11}} \]

Calculation Part 1: PV of Coupons

Using the formula for the present value of an annuity:

\[ PV_{\text{annuity}} = C \times \left(1 - \frac{1}{(1 + r)^N}\right) \div r \]

Substituting the values we have:

\[ PV_{\text{coupons}} = 80.80 \times \left(1 - \frac{1}{(1 + 0.0753)^{11}}\right) \div 0.0753 \]

Calculating:

1. Calculate \((1 + r)^N\):

\[ (1 + 0.0753)^{11} \approx 2.2758 \]

2. Calculate the present value of the coupon payments:

\[ PV_{\text{coupons}} = 80.80 \times \left(1 - \frac{1}{2.2758}\right) \div 0.0753 \] \[ PV_{\text{coupons}} = 80.80 \times \left(1 - 0.4391\right) \div 0.0753 \approx 80.80 \times 0.5609 \div 0.0753 \approx 80.80 \times 7.4494 \approx 601.73 \]

Calculation Part 2: PV of Par Value

\[ PV_{\text{par}} = \frac{1000}{(1 + 0.0753)^{11}} \approx \frac{1000}{2.2758} \approx 439.28 \]

Total Price of the Bond

Now, add the present values of the coupons and the par value:

\[ P = PV_{\text{coupons}} + PV_{\text{par}} \approx 601.73 + 439.28 \approx 1040.99 \]

Thus, the price of the bond is approximately $1,040.99.