# Forward interest rate

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!

• They don't share the same name. The first is a "forward interest rate", the second is a "forward rate agreement". – noob2 Nov 29 '20 at 9:49