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I have the following question from Hull, problem 6.17:


On August 1 a portfolio manager has a bond portfolio worth $10 million. The duration of the portfolio in October will be 7.1 years. The December Treasury bond futures price is currently 91-12 and the cheapest-to-deliver bond will have a duration of 8.8 years at maturity. How should the portfolio manager immunize the portfolio against changes in interest rates over the next two months?


The solution is as follows:

The treasurer should short Treasury bond futures contract. If bond prices go down, this futures position will provide offsetting gains. The number of contracts that should be shorted is:

$$\begin{align} Number\ of\ Contracts& =\frac{$\ 10\,000\,000}{\bbox[yellow, 5px,border:2px solid red]{$\ 91\,375}}\times\frac{7.1\ \text{years}}{8.8\ \text{years}} \newline & = 88.30 \ \text{contracts}\newline \therefore Number\ of\ Contracts &\approx 88\ \text{contracts}\end{align}$$


Question:

How was the $91,375 calculated?

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The question reads "...price is currently 91-12..." where 91-12 is notated in futures quotes (pg. 2 of Treasury Futures). Coupon-bearing securities are frequently quoted in percent of par to the nearest 1/32nd of 1% of par. This means that: $$\begin{align}91\mbox{-}12 & = 91 + \frac{12}{32}\newline & = 91 + 0.375\newline & = 91.375 \newline \newline \therefore 91\mbox{-}12& =91.375\%\end{align}$$

To calculate the future contract price: $$\begin{align}Future\ Contract\ Price& = 91.375\%\times $\ 1\,000\,000\newline &=$\ 91\,375\end{align}$$

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