3. Suppose you run a pension fund and you have the following liability: you will have to pay retirees $1,000,000 in 15 years. Suppose interest rates are equal to 1% forever and that there are only two bonds available in the market: a 2 year zero coupon bond, and a 20 year zero coupon bond. (a) What is the present value of your liability at t = 0? (b) Suppose you start at t = 0 with an amount of cash equal to the present value of the liability. What portfolio of 2 year and 20 year zero coupon bond should you buy at t = 0 in order to be immunized against change in interest rates? (c) Suppose that the interest rate increases from 1% to 1.25% at t = 0. Suppose that you have bought the portfolio that you found in question (b). What is the approximate change in the value of your asset and liability? What is the exact change in the value of your asset and liability? (d) Re-do the calculation of question (c) assuming that, instead of the portfolio of question (b) you have bought a portfolio composed of 30 year bonds only. Explain the difference in results.