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I have some confusion regarding forward interest rate. It seems that there are two notions of forward interest rate:

  1. The interest rate $f$ agreed at time $t$ for investment over a future time period between $T_1$ and $T_2$. And this interest satisfies $$e^{r_{T_1}T_1}e^{f(T_2-T_1)}=e^{r_{T_2}T_2}$$
  2. The price at time $t\leq T$ for a contract that pays the spot (short) rate $R(T)$ at time $T$.

It seems to me that they are different concepts and just happen to share the same name. I am wondering if I am missing something obvious and whether there is some deeper connection between the two notions? Thanks for any help!

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    $\begingroup$ They don't share the same name. The first is a "forward interest rate", the second is a "forward rate agreement". $\endgroup$ – noob2 Nov 29 '20 at 9:49
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You are right that they measure different things. As do the "forward" for 3m rates in 3 years time versus the same 5 years time. These are the same kind of different "forwards". As are the "forwards" for 2 year rates in 3 years time versus 1 year rates in 4 years time.

The rates markets doesn't find this problematic or confusing, because it has a simple notation that transparently covers all eventualities. The "AyBy" rate is the rate from A year's time for the next B years.

So if your horizon is, say, 5 years, then your (2) above is simply the 5y3m contract/swap. Your (1) above might be the 1y4y, 2y3y, 3y2y, 4y1y etc. rate, depending on when you want your rate in 5 years time to finish.

The concession that rates markets will sometimes make for those who don't know their 2y3m rates from their 1y1y rates is to call the former (your 2) "the short-term forward strip" or "STIR" and the latter (your 1) "forward-starting rates".

hope this helps to clarify.

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