Consider a forward rate agreement on LIBOR (say), which starts 2 months from now, expires after 3 months and has strike $K$, and is based on $3M$ LIBOR -- $FRA_{2\times 5}$. Now the present value of this contract is,
\begin{equation} \frac{\alpha\cdot (L_{3m}(2m) - K)}{1 + \alpha\cdot L_{3m}(2m)} \end{equation}
where $\alpha$ is the relevant day count fraction, $L_{3m}(2m)$ refers to the 3 month LIBOR rate in 2 months time (assume unit notional for ease).
I see the term $1 + \alpha\cdot L_{3m}(2m)$ in the above equation as a discount rate applied to the pay out $L_{3m}(2m) - K$.
Why do we apply the $L_{3m}(2m)$ rate for the discounting? Why not apply a short term money market rate or any other rate for that matter -- I guess I am also asking how we choose the rate that we apply for the discounting?