Let $L(t, S, T)$ denote the forward rate from time S to T observed at time t, assuming t < S < T.
A lot of modelling work is centered around this rate, but how is this rate useful? How are we supposed to interpret it?
a. Under the real world measure, $E_t[L(S, S, T)] \neq L(t, S, T)$. So this rate is not a good predictor of L(S, S, T). In fact, the above-mentioned equation holds only under the T-forward measure, but how should we think about it if the equality is under the forward measure?
b. Let P(t,S, T) denote the zero coupon bond price implied from L(t, S, T). Again P(t, S, T) is not the expectation under the real work measure of P(S, S, T). What justifies using P(t, S, T) as a discount factor?