A pension fund has just paid some of its liabilities, and as a result of this transaction the fund is no longer fully immunized. The fund manager decides that instead of changing the portfolio, the firm should hedge its position using a futures contract on a Treasury bond. The fund manager wants to hedge against parallel changes to the spot rate curve. Use the following set of information to determine the numerical values of the hedging position:

- Yearly spot rate sequence: .05, .053, .056, .058, .06, .061.

- Liabilities: \(\$ 1\) million in 1 year, \$2 million in 2 years, and \(\$ 1\) million in 3 years.

- Current bond portfolio: \(\$ 4.253\) million in par value of zero-coupon bonds maturing in 2 years. (Use the continuous-time formulas for discounting: \(e^{-r t}\).)

- The hedge is to be constructed using futures contracts on zero-coupon bonds maturing in 6 years, with a contract delivery date in 1 year.