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?

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    $\begingroup$ The aim is not to predict the future value of the rate, but to price e.g. options on $L(t,S,T)$ (such as caps and floors). a) The fact that $L(t,S,T)$ is a martingale under the T-forward measure, means that it's easier to use this measure for this pricing purpose. And b) the $P(t,S,T)$ is this measure's numéraire. $\endgroup$ – byouness May 23 '18 at 11:01

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