Assume a par value of $1,000. Caspian Sea plans to issue a 20.00 year, annual pay bond that has a coupon rate of 8.15%. If the yield to maturity for the bond is 7.56%, what will the price of the bond be?

Answer 1

To determine the price of the bond, we can use the present value formula for bonds, which is the sum of the present value of the coupon payments and the present value of the par value at maturity.

The price of the bond can be calculated using the following formula:

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

Where:

• \(P\) = price of the bond
• \(C\) = annual coupon payment
• \(r\) = yield to maturity (as a decimal)
• \(FV\) = face value of the bond
• \(n\) = number of years to maturity

Given the values:

• Face Value (\(FV\)) = $1,000
• Coupon Rate = 8.15%
• Yield to Maturity (\(r\)) = 7.56% or 0.0756 in decimal
• Number of years to maturity = 20

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

\[ C = \text{Coupon Rate} \times \text{Face Value} = 0.0815 \times 1000 = 81.50 \]

Now we can plug in the values:

\[ P = \sum_{t=1}^{20} \frac{81.50}{(1 + 0.0756)^t} + \frac{1000}{(1 + 0.0756)^{20}} \]

Calculate the Present Value of Coupons

The present value of an annuity formula can be used to simplify the calculation of the coupon payments:

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

Substituting in the values:

\[ PV_{\text{coupons}} = 81.50 \times \left( \frac{1 - (1 + 0.0756)^{-20}}{0.0756} \right) \]

First, calculate \((1 + 0.0756)^{-20}\):

\[ (1 + 0.0756)^{-20} = (1.0756)^{-20} \approx 0.28418 \]

Now substitute back in:

\[ PV_{\text{coupons}} = 81.50 \times \left( \frac{1 - 0.28418}{0.0756} \right) \] \[ PV_{\text{coupons}} = 81.50 \times \left( \frac{0.71582}{0.0756} \right) \approx 81.50 \times 9.4661 \approx 771.41 \]

Calculate the Present Value of the Face Value

Now we calculate the present value of the face value:

\[ PV_{\text{face value}} = \frac{1000}{(1 + 0.0756)^{20}} \approx \frac{1000}{(1.0756)^{20}} \]

Calculate \((1.0756)^{20}\):

\[ (1.0756)^{20} \approx 4.35199 \]

Now substituting back in:

\[ PV_{\text{face value}} = \frac{1000}{4.35199} \approx 229.56 \]

Total Price of the Bond

Finally, add the present value of the coupons and the present value of the face value together:

\[ P = PV_{\text{coupons}} + PV_{\text{face value}} = 771.41 + 229.56 \approx 1000.97 \]

Therefore, the price of the bond is approximately:

\[ \boxed{1000.97} \]