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I know that for a swap for example, the swap rate is just what adopts equilibrium for both legs by no arbitrage, on the other hand for a FRA its just the same only with one period of time. Considering what most economist do which is saying that the forward rates are the expectations of the marke (at least here) why are these numbers (just values of equilibrium given today's spot rates) good predictors for anything?

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Strictly speaking, any risk-free interest rate can be composed into three components:

  1. The rate expectations component is the market's "true" expectation for future interest rate.
  2. A bond risk premium component: longer maturity bonds have higher duration risk than cash. Accordingly market participants will demand more compensation for taking on duration risk; i.e., they'll ask for a higher yield. This is why even if market expectation for rates for the next 100 years is completely flat, the yield curve will still typically be upward sloping.
  3. A convexity bias component: longer maturity bonds are more positively convex, providing more return advantages when rates increase or decrease. Because of this "convexity advantage," investors are willing to accept a lower yield (all else equal). This is why the long end of the yield curve sometimes dip down.

So simply put, forward rates do NOT represent market expectation of future interest rates – you must subtract bond risk premium and add back convexity bias.

It is also well known that forward rates frequently overstate subsequently realized interest rates. This is known as "forward rate bias."

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Forwards have never really been that accurate at predicting the future shape and yield of the the bonds that you're looking at. That being said the bootstrapping process represents real executable transactions. In the case of the swap if the forward curve upon swap execution is realized the returns on the swap will be zero for both legs.

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