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Why do rising interest rates sometimes lead to rising futures prices, and sometimes lead to falling futures prices?
You don't understand the quotation method of interest rate futures;

The quotation of futures interest rate is something like this: (1- interest rate) * 100. Interest rate rises, quotation falls, interest rate falls, quotation rises.

And when you buy and sell, you buy and sell according to the interest rate quotation. That is, your buying and selling price is negatively related to the interest rate.

Specifically, let's look at short-term treasury bonds futures in the United States.