Assume a par value of $1,000. Caspian Sea plans to issue a 7.00 year, semi-annual pay bond that has a coupon rate of 8.16%. If the yield to maturity for the bond is 7.74%, 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 bonds, which takes into account the present value of the coupon payments and the present value of the face value at maturity. Given that the bond has semi-annual payments, we need to adjust the coupon rate, the yield to maturity (YTM), and the number of periods accordingly.

Given:

• Par value (Face value) = $1,000
• Coupon rate = 8.16%
• Time to maturity = 7.00 years
• Yield to maturity = 7.74%
• Payment frequency = Semi-annual

Calculations:

1. Calculate the coupon payment: \[ \text{Coupon payment} = \text{Par Value} \times \left(\frac{\text{Coupon Rate}}{2}\right) = 1000 \times \left(\frac{8.16%}{2}\right) = 1000 \times 0.0408 = 40.80 \]

2. Calculate the number of periods (n): \[ n = \text{Years to Maturity} \times 2 = 7 \times 2 = 14 \]

3. Calculate the semi-annual yield (r): \[ r = \frac{\text{Yield to Maturity}}{2} = \frac{7.74%}{2} = 3.87% = 0.0387 \]

4. Calculate the present value of the coupon payments: \[ \text{PV of Coupons} = C \times \left(\frac{1 - (1 + r)^{-n}}{r}\right) \] Where \( C \) is the coupon payment. \[ \text{PV of Coupons} = 40.80 \times \left(\frac{1 - (1 + 0.0387)^{-14}}{0.0387}\right) \]

   Now we calculate \( (1 + 0.0387)^{-14} \): \[ (1 + 0.0387)^{-14} = (1.0387)^{-14} \approx 0.6080 \] Therefore, \[ 1 - (1 + 0.0387)^{-14} \approx 1 - 0.6080 = 0.3920 \]

   Now plug it back into the formula: \[ \text{PV of Coupons} = 40.80 \times \left(\frac{0.3920}{0.0387}\right) \approx 40.80 \times 10.14 \approx 414.03 \]

5. Calculate the present value of the par value: \[ \text{PV of Par Value} = \frac{\text{Par Value}}{(1 + r)^{n}} = \frac{1000}{(1 + 0.0387)^{14}} \] \[ (1 + 0.0387)^{14} \approx 1.66 \] Therefore, \[ \text{PV of Par Value} = \frac{1000}{1.66} \approx 602.41 \]

6. Add the present value of the coupon payments and the present value of the face value: \[ \text{Price of the Bond} = \text{PV of Coupons} + \text{PV of Par Value} \approx 414.03 + 602.41 \approx 1016.44 \]

Thus, the price of the bond will be approximately $1,016.44.