Hedging Bond Portfolio Duration with Treasury Futures
Summary
The document examines how to hedge a government bond portfolio against interest-rate changes with Treasury futures. It sets out a portfolio’s market value and modified duration, the deliverable bond’s duration, contract value, and a six-month rate, then questions how the lecture notes derive futures duration and the hedge ratio. The central issue is distinguishing the sensitivity of the futures price from the value of a futures position over the contract horizon.
The author asks for an interpretation of the duration formula and a first-principles explanation of how a short futures position offsets losses when rates rise. The numerical setup illustrates duration matching, but the document does not supply a resolution or empirical test. Its value is in identifying timing and notation as key sources of confusion; the proposed hedge also depends on the deliverable bond and the stated rate assumptions, so it should not be treated as a complete account of basis or contract risk.
Key ideas
- Duration matching can estimate the number of Treasury futures needed to hedge a bond portfolio’s rate exposure.
- The document questions whether the futures duration formula refers to a futures price or a discounted contract value.
- A hedge explanation should connect rate changes to the payoff of a short futures position.
- The example highlights that timing conventions and contract details affect the duration calculation.
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Full text
# Interest rate hedging using treasury futures – timing and duration
# Interest rate hedging using treasury futures – timing and duration
I'm pondering over the following (rather standard) problem:
> We have \$10 million invested in government bonds and are concerned with highly volatile interest rate over the next six months. We want to use the 6-month T-bond futures to protect the value of the portfolio. Duration of the bond portfolio is 6.80 years. Current futures price is 93 2/32 (for face value of \$100). The T-bond to be delivered has a duration of 9.20 years. Each contract delivers \$100,000 face value of bonds. Futures price for the total contract is \$93,062.50. 6-month interest rate is 4%.
Source: p. 15 of these lecture notes [pdf] by J. Wang
My understanding of the premise: If the interest rate increases by $\Delta y$, the value of the bonds portfolio changes (up to a linear approximation) by $$\frac{dP}{dy}\Delta y = -P\left(-\frac{1}{P}\frac{dP}{dy}\right)\Delta y = -P \times MD_P \times \Delta y = -10,000,000 \times 6.80 \times \Delta y.$$ We want to offset this by selling treasury futures.
The author now writes the following:
> Duration of the futures contract: $H = B(1+y)^{1/2}$. $$MD_H = MD_B + (1+y)^{-1}(1/2) = 9.20 + (1+04)^{-1}(0.5) = 9.68$$ Match duration: $$(\# \text{ of contracts})(93,062.50)(9.68) = (10,000,000)(6.80)$$
The expression for the (modified) duration is not what one obtains if applying the standard formula $MD_H = -\frac{1}{H}\frac{dH}{dy}$ to the expression $H = B(1+y)^{1/2}$. However it is obtained if applying the formula to the expression $B(1+y)^{-1/2}$, which could be interpreted as the present value of a bond worth $B$ in 6 months – but that's hardly the futures price. I think the issue comes down to the author being sloppy with timing and notation, and not separating the futures price from the value of holding a futures contract.
Would someone care to explain the author's thinking here? To what expression are we applying the formula for the modified duration, and why?
Ideally, I would also like a heuristic argument along the lines of "When the interest rate increases by $\Delta y$, the value of being short in the futures contract changes by ..." in order to provide some intuition for the solution.
Thanks!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.