You work in the treasury department of a large international company which seeks to borrow EUR $200$ million from a German bank for six months from Wednesday $20$ December $2023.$ Today is Monday $20$ November $2023.$ The credit rating of your company means that it borrows at $3-$month EURIBOR plus $113$ bp$.$

You plan to borrow successively twice at $3$ month EURIBOR plus $113$ bp$,$ while hedging possible changes in the EURIBOR rate by using two positions on the $3-$month EURIBOR futures contracts traded on Eurex, maturing respectively in December $2023,$ the FEU$3$Z$23$ contract, and in March $2024,$ the FEU$3$H$24$ contract. Simultaneous positions in consecutive futures contracts on the same underlying, here the $3-$month EURIBOR, is called a futures strip.

You take today a position in $200$ contracts on FEU$3$Z$23$ at the bid$-$ask quote $95\mathrm{.}90095\mathrm{.}905$ and simultaneously a position in N contracts on FEU$3$H$24$ at the bid$-$ask quote $96\mathrm{.}13096\mathrm{.}135.$ You must decide whether to buy or sell the contracts and you must also choose the optimal number N of FEU$3$H$24$ contracts to use. N is a positive integer if you buy the FEU$3$H$24$ contract and a negative integer if you sell the FEU$3$H$24$ contract.

The FEU$3$Z$23$ and FEU$3$H$24$ contracts mature respectively on Monday $18$ December $2023$ and Monday $18$ March $2024.$ We assume two equiprobable scenarios for the future $3-$month EURIBOR. In the Low scenario, the $3-$month EURIBOR is $4\mathrm{.}0000\%$ on Monday $18$ December $2023$ and $3\mathrm{.}7700\%$ on Monday $18$ March $2024.$ In the High scenario, the $3-$month EURIBOR is $4\mathrm{.}2000\%$ on Monday $18$ December $2023$ and $3\mathrm{.}9700\%$ on Monday $18$ March $2024.$

The trade date of the first $3-$month loan is Monday $18$ December $2023$ with settlement two business days later on Wednesday $20$ December $2023$ and maturity on Wednesday $20$ March $2024.$ The settlement of the FEU$3$Z$23$ contract is Wednesday $20$ December $2023$ based on the $3-$month EURIBOR rate fixed on Monday $18$ December $2023.$ The actual amount you borrow on Monday $18$ December $2023$ takes into account the profit or loss on the FEU$3$Z$23$ contract so as to net EUR $200$ million needed by your company. To be clear, you borrow less in case you receive a profit on your futures trade, and more otherwise. Weeglect any financing cost on your margin account with your broker during the life of the futures contract when computing the profit or loss on the futures contract and we assume that you will manage to answer any margin call during the life of the contract.

The trade date of the second $3-$month loan is Monday $18$ March $2024.$ This second loan settles on Wednesday $20$ March $2024$ based on the $3-$month EURIBOR rate fixed on Monday $18$ March $2024.$ The cash you obtain from this second loan at its settlement on Wednesday $20$ March $2024,$ together with the profit or loss from your position on the FEU$3$H$24$ contract, is used to repay exactly the amount you borrowed on the first loan. We neglect again any financing cost on your margin account during the life of the second futures contract and we assume that you will manage to answer any margin call during the life of this contract. The amounts of all financial transactions are rounded to the nearest Euro.

Question $3.$The quality of your hedge is measured by the absolute value of the difference in the cash amount your company repays at maturity of the second loan on Thursday $20$ June $2024$ under the two scenarios. The lower the value of this absolute difference the higher the quality of the hedge. Which value N do you select so as to maximize the quality of your hedge?