Suppose that it is February $20$ and a treasurer realizes that on July $17$ the company will have to issue $$5$ million of commercial paper with a maturity of $180$ days. If the paper were issued today, the company would realize $$4,820,000.$

$($In other words, the company would receive $$4,820,000$ for its paper and have to redeem it at $$5,000,000$ in $180$ days$$ time.$)$

The cheapest to deliver bond has a duration of $7\mathrm{.}5$ and the portfolio a duration of $15.$