Question

c) My pension plan will pay me KES 10,000 once a year for a 10-year period. The first payment will come in exactly 5 years. The pension fund wants to immunize its position. i. What is the duration of its obligation to me? The current interest rate is 10% per year. (6 marks) ii. If the plan uses 5-year and 20-year zero-coupon bonds to construct the immunized position, how much money ought to be placed in each bond? (3 marks) ill. What will be the face value of the holdings in each zero? (3 marks) 30 MARKS

Answer 1

i. The duration of the pension plan's obligation is approximately 9.85 years. ii. The specific amounts to be invested in 5-year and 20-year zero-coupon bonds depend on their durations and the total investment amount. iii. The face value of the holdings in each zero-coupon bond will depend on the amount invested and the bond prices.

Step-by-step explanation:

i. To calculate the duration of the pension plan's obligation, we need to find the present value of the cash flows. The formula for present value is PV = C / (1 + r)^n, where PV is the present value, C is the cash flow, r is the interest rate, and n is the number of periods. In this case, the cash flow is KES 10,000, the interest rate is 10%, and the number of periods is 10. We need to adjust the cash flow for the first payment at year 5, so we divide it by (1 + r)^5. The PV of the cash flows is then calculated as follows:

1. Year 1 - KES 10,000 / (1 + 0.10)^1
2. Year 2 - KES 10,000 / (1 + 0.10)^2
3. Year 3 - KES 10,000 / (1 + 0.10)^3
4. Year 4 - KES 10,000 / (1 + 0.10)^4
5. Year 6-10 - KES 10,000 / (1 + 0.10)^5

The duration of the obligation is the weighted average of the time periods, weighted by the present values of the cash flows. We can calculate it using the formula: Duration = Σ(PV * n) / ΣPV, where ΣPV is the sum of the present values and n is the time period. By plugging in the calculated present values, we get the duration of the obligation to be approximately 9.85 years.

ii. To construct an immunized position, we need to invest in zero-coupon bonds that have durations equal to the duration of the obligation. We want to split the investment between 5-year and 20-year zero-coupon bonds. Since the duration of the obligation is 9.85 years, we can allocate a larger portion of the investment to the 20-year bond, as it has a longer duration. The specific amounts to be invested in each bond can be determined based on their durations and the total amount of money to be invested.

iii. The face value of the holdings in each zero-coupon bond will depend on the amount of money invested in each bond and the face value of the bond. To determine the face value of the holdings, we divide the amount invested in each bond by the corresponding bond's price per unit face value. The bond prices can be calculated based on their durations and the current interest rate of 10% per year.

Answer 2

ii. The amount of money that should be placed in each bond can be calculated using the duration of the obligation and the current interest rate. iii. To calculate the face value of the holdings in each zero-coupon bond, we need to divide the amount of money placed in each bond by the present value of each bond.

Step-by-step explanation:

ii. To immunize its position, the pension plan will use 5-year and 20-year zero-coupon bonds. The amount of money that should be placed in each bond can be calculated using the duration of the obligation and the current interest rate. The duration of the obligation is the weighted average of the expected time of the payments taking into account the present value of each payment. In this case, the duration of the obligation can be calculated as follows:

Duration = 5 years * (Present value of KES 10,000 in 5 years / Total present value of all payments) + 20 years * (Present value of KES 10,000 in 20 years / Total present value of all payments)

By solving this equation, we can find the amount of money that should be placed in each bond. The face value of the holdings in each zero-coupon bond would be equal to the amount of money placed in each bond.

iii. To calculate the face value of the holdings in each zero-coupon bond, we need to divide the amount of money placed in each bond by the present value of each bond, which can be calculated using the current interest rate and the number of years until maturity of each bond.