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Not only in the U.S., but I'd venture to say in all developed countries, the short end of the curve is controlled not by the market, but by the central bank. Only further out, the curve is controlled by the market, with the forwards being the market's view on what the shorter-term rates will be in the future.

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In discrete and 2-year setting, the question can be formulated to: given $R_1<R_2$, show $R_1<c<R_2$ from $$\frac{c}{1+R_1}+\frac{1+c}{(1+R_2)^2}=1.$$ This can be proven by contradiction: 1) assume $c<=R_1$, then $$1=\frac{c}{1+R_1}+\frac{1+c}{(1+R_2)^2}<\frac{R_1}{1+R_1}+\frac{1+R_1}{(1+R_1)^2}=1$$ 2) assume $c>=R_2$, then 1=\frac{c}{...

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Just to test understanding of the reasoning: $\frac{c}{1+R_{01}}+\frac{1+c}{\left(1+R_{01}\right)\left(1+R_{12}\right)}=1$ Multiply through by 1+c: $c\frac{1+c}{1+R_{01}}+\frac{\left(1+c\right)^2}{\left(1+R_{01}\right)\left(1+R_{12}\right)}=1+c$ Now $\frac{1+c}{1+R_{01}}>1$ as per the reasoning in the previous answer because the curve is upward ...

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This question came to me as well, the reasoning in the solution is not convincing, neither is the solution given by Magic. In Magic's answer, the reasoning "$c_2$ is some average of $R_1$ and $R_{12}$, then lower than $R_2$" is incorrect since $R_{12}$ is bigger than $R_2$ in upward sloping. We can prove it rigorously in the 2-year setting as below. In ...

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Here is a nice survey of how this model and its alternatives are used by the central banks: https://www.bis.org/publ/bppdf/bispap25a.pdf

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