A fund manager has a bond portfolio worth $$25$ million. The $($Macaulay$)$ duration of the portfolio in September will be $5\mathrm{.}9$ years. The December $10-$year Treasury bond futures price is currently $94\mathrm{.}37($implying a yield of $5\mathrm{.}63\%$ p$.$a$.)$ with duration $7\mathrm{.}7$ years.

How should the portfolio manager immunise the portfolio against changes in interest rates $($i$.$e$.$ target zero duration$)$ over the next three months? Remember that the assumed coupon rate for the Treasury bond futures is $6\%$ p$.$a$.($semi$-$annual$)$ with a face value of $$100,000.$ The calculations provided previously was wrong