You currently have pension fund assets of $15 million in a bond portfolio with a Macaulay duration of 10.

Your liabilities are $2.7 million per year starting 30 years from today (i.e. at time 30) and lasting for 30 years (i.e. the last payment is at time 60). The yield curve is flat with spot rates constant at 4% for all maturities.

(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) What is the change in the funding status (assets less liabilities) of the fund if interest rates rise by 0.10%?

(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. The market has a 1-year zero-coupon bond and a 100-year zero-coupon bond. How would you allocate your assets to ensure that your pension fund is hedged against an interest rate change?