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Treasury futures contract has no carry, but what is its rolldown (if it exists)?

In the above answer to carry, @Helin mentioned "...bonds have expected rolldown returns that will flow through to futures..."

Does it mean that futures' rolldown is exactly the same as the rolldown of its CTD bond in spot space? Thanks.

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Roll in futures language is typically defined in terms of the futures curve, i.e. the difference between (successive) contracts, say RXM3 minus RXH3. You can find more information on this for STIR contracts in Jha, S. (2011), Interest Rate Markets: A Practical Approach to Fixed Income. I'd say this is primarily what people refer to as "futures rolldown".

Since bond futures are a bit different to most other futures contracts, you can make an argument that the rolldown of the CTD flows through to the contract itself. In the absence of a gross/net basis, the price of the future relates to CTD price via the conversion ratio: $P_{fut}=P_{ctd}\times C$. If the shape of the yield curve stays constant, the change in the CTD bond price caused by the passage of time will then result in a profit or loss on the future.

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It should be the same as the fwd bond. You should be able to measure the fwds roll down as fwd yield minus the aged spot bonds yield (assuming the spot curve doesn't change) at term.

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