The term structure is upward sloping and linear in the continuously compounded interest rate. i.e., the 1-year rate is 1.5%, the 2-year rate is 2%, the 3-year rate is 2.5% and so on. A pension fund needs to pay 10 million each in years 2 and 4 from now. 

(a) What is the present value of the fund's liabilities? Calculate the duration and convexity of the liabilities.  

(b) The fund's current assets are in cash, and the net worth is zero. To hedge against the interest rate risk the fund plans to invest all the cash in a single zero coupon bond. What is the maturity Th of this bond for the immunization strategy to work? Explain each step in your calculations. 

(c) Immediately after implementing the immunization strategy the term structure shifts down by 100 basis points but is still upwards sloping and linear. What is the new present value of the liabilities? What is the new present value of the position in the zero coupon bond? What is the new net worth under the immunization strategy described? 

(d) Alternatively, suppose that after implementing the immunization strategy the term structure flattens out at 2.98% This means that the continuously compounded rate is the same for all maturities. What is the new present value of the liabilities? What is the new present value of the position in the zero coupon bond? What is the new net worth under the immunization strategy described?