Final answer:
To compute the price of both bonds, use the present value formula. Set up the portfolio to match the duration of the liability. If interest rates fall to 8% without rebalancing, calculate the realized yield. Find the duration of the portfolio after the drop in interest rates. To rebalance, adjust the weights of the bonds.
Step-by-step explanation:
a) To compute the price of both bonds, we can use the present value formula. The price of a bond is the sum of all the discounted future cash flows. Let's calculate the price of the 2-year zero-coupon bond (2YR) first. The bond will pay $100 at the end of Year 2. The discount rate is 10%, so the present value is $100 / (1+0.10)^2 = $83.33. Now, let's calculate the price of the 4-year zero-coupon bond (4YR). The bond will pay $100 at the end of Year 4. The present value is $100 / (1+0.10)^4 = $67.55.
b) To immunize the portfolio by duration matching, we need to find the weights of the bonds that will result in a portfolio duration of three years. The duration of the 2-year bond is 2 years, and the duration of the 4-year bond is 4 years. Let x represent the weight of the 2-year bond. The weight of the 4-year bond would then be 1 - x. To match the duration of the portfolio to three years, we can set up the equation 2x + 4(1-x) = 3. Solving for x, we get x = 1/2. Therefore, we need to invest half of the portfolio in the 2-year bond and half in the 4-year bond.
c) After the interest rates fall to 8%, the price of the bonds will increase. If we do not rebalance our portfolio, the realized yield after three years will be the difference between the final portfolio value and the initial investment, divided by the initial investment. The final portfolio value can be calculated by taking the price of the 2-year bond at 8% interest rate and multiplying it by the weight (1/2), and adding it to the price of the 4-year bond at 8% interest rate and multiplying it by the weight (1/2). The initial investment is the sum of the prices of the bonds at the current interest rate. We can calculate the realized yield using these values.
d) The duration of the portfolio after the drop in interest rates without rebalancing can be calculated by taking the weighted average of the durations of the individual bonds. Since we have equal weights for both bonds, the duration of the portfolio will be the average of the two durations. We can calculate the duration using the formula: duration = (2 x duration of 2-year bond + 4 x duration of 4-year bond) / 2.
e) To rebalance the portfolio, we need to adjust the weights of the two bonds to maintain a duration of three years. Since the interest rates have fallen to 8%, the new prices of the bonds will be different. We can calculate the new prices using the present value formula and then recalculate the weights of the bonds to match the desired duration of three years.