Derek bought a semi-annual pay bond issued by Caspian sea exactly 8.00 years ago from today. Right after receiving his coupon payment today, he will be selling the bond for a need for quick cash. The bond had 30 years to maturity at issue. The yield to maturity for the bond is 3.75%. If the coupon rate of the bond is 5.8.96%,at what price should he sell the bond today?

To determine the price at which Derek can sell his bond today, we need to calculate the present value of the bond's future cash flows using the bond's yield to maturity (YTM).

Given Data:

• Coupon Rate: 5.896%
• Yield to Maturity (YTM): 3.75%
• Years to Maturity at Issue: 30 years
• Years Held: 8 years
• Years Remaining to Maturity: \(30 - 8 = 22\) years
• Frequency of Payments: Semi-annual (twice a year)

Step 1: Calculate the Coupon Payment

The coupon payment \(C\) is calculated as follows: \[ C = \left( \text{Coupon Rate} \times \text{Face Value} \right) / \text{Payments per Year} \] Assuming the Face Value of the bond is \(1000\) (the typical face value):

\[ C = \left( 0.05896 \times 1000 \right) / 2 = 29.48 \]

Step 2: Calculate the Number of Periods Remaining

The number of periods remaining until the bond matures is given by: \[ \text{Number of Periods} = 2 \times \text{Years Remaining} \] \[ = 2 \times 22 = 44 , \text{periods} \]

Step 3: Market Rate per Period

The market (or yield) rate per period is: \[ \text{Market Rate per Period} = \text{YTM} / \text{Payments per Year} = 0.0375 / 2 = 0.01875 \]

Step 4: Calculate the Present Value of the Cash Flows

The present value of the coupon payments can be calculated using the present value of an annuity formula: \[ PV(\text{Coupons}) = C \times \left(1 - (1 + r)^{-n} \right) / r \] Where:

• \(C = 29.48\)
• \(r = 0.01875\)
• \(n = 44\)

Calculating this step-by-step:

1. Calculate \( (1 + r)^{-n} \): \[ (1 + 0.01875)^{-44} \approx 0.36815 \]

2. Substitute into the formula: \[ PV(\text{Coupons}) = 29.48 \times \left(1 - 0.36815\right) / 0.01875 \] \[ PV(\text{Coupons}) \approx 29.48 \times (0.63185) / 0.01875 \approx 29.48 \times 33.68533 \approx 994.74 \]

Step 5: Calculate the Present Value of Face Value

The present value of the face value is calculated as follows: \[ PV(\text{Face Value}) = \frac{\text{Face Value}}{(1 + r)^n} \] Substituting: \[ PV(\text{Face Value}) = \frac{1000}{(1 + 0.01875)^{44}} \approx \frac{1000}{2.1388} \approx 467.793 \]

Step 6: Total Present Value (Bond Price)

Now sum up the two present values calculated: \[ P = PV(\text{Coupons}) + PV(\text{Face Value}) \] \[ P \approx 994.74 + 467.793 \approx 1462.53 \]

Conclusion

Derek should sell the bond for approximately $1462.53 today.