2. You manage a pension fund, and your liabilities consist of two payments as follows: Your currenlty have a total of $18 million in cash to cover those Time 10 years 30 years Payment $20 million $30 million liabilities. The term structure is currently flat at 5%, effective annual rate. (a) Compute the present value of your liabilities. Do you have enough assets to cover those liabilities? (b) Compute the Macaulay duration and the modified duration of your liabilities. (c) Compute an approximate change in the present value of your liabili- ties, using duration, when interest rates fall by 0.25%. (d) You are worried that a drop in interest rates will increase the value of your liabilities. To hedge against this exposure, you would like to invest your cash in a portfolio of bonds such that any change in the value of your liabilities is exactly compensated by a change in the value of your assets. Suppose that you can invest in a 1-year zero-coupon bond and in a 20-year zero-coupon bond. How would you allocate your assets to ensure that your pension fund is hedged against an interest rate change