Converting Treasury Futures Quotes into Contract Dollar Value
Summary
The document explains how to convert a Treasury bond futures quote written in points and thirty-seconds into a percentage of par, then into a dollar contract value. In its example, the quote 91-12 means 91 plus twelve thirty-seconds, or 91.375 percent of par. Applying that percentage to a contract’s stated $100,000 face value yields a dollar value of $91,375.
This conversion supplies the contract-value input in a duration hedge calculation for a bond portfolio. The example’s hedge calls for short Treasury futures, with the number of contracts adjusted by the portfolio’s duration relative to the cheapest-to-deliver bond’s duration. The note answers a narrow quotation-convention question; it does not derive the full hedge formula or discuss basis risk, conversion factors, or changes in duration. Its arithmetic relies on the contract face amount and quote convention specified in the example.
Key ideas
- Treasury futures prices may be quoted in points and thirty-seconds of a point.
- A quote of 91-12 converts to 91.375 percent of par.
- Multiplying the quoted percentage by the contract face amount gives its dollar value.
- Duration ratios can then be used with contract value to size a futures hedge.
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Full text
# How to calculate a future contracts price?
# How to calculate a future contracts price?
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?
## Answer by 3kstc (score 0, accepted)
https://quant.stackexchange.com/a/37132
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: $$\begin{align}Future\ Contract\ Price& = 91.375\%\times $\ 1\,000\,000\newline &=$\ 91\,375\end{align}$$Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.