LIBOR with different tenor

Question

Let $F(t;S,T)$ be the forward rate from $S$ to $T$ seen at time $t$, and $I$ be one of tenors, i.e. $I$ is one of {1M, 3M, 6M, 12M}. Then the forward curve $t\mapsto F(0;t,t+I)$ is $I$-forward curve.

I understand that the forward curve above is easy to calibrate to market values that involves LIBOR of tenor $I$, such as swaps. But how can I get $I$-discounting curve (zero coupon curve) from the forward curve? Is it simply to find the function $P_I(0,t)$ that satisfies the following equation? $$F(0;t,t+I)=\frac{1}{I}\bigg(\frac{P_I(0,t)}{P_I(0,t+I)}-1\bigg)$$

If this is the case, what is the meaning of the value $P_I(0,t)$? Is it simply a function gives the forward curve back?

Answer 1

Your $P_I(t,T)$ is the formula for the so-called "pseudo" discount curve. It can be used to compute relevant LIBOR forward rates and LIBOR zero rates.

The "true" discount curve is of course the OIS discount curve, which can be built independently of the LIBOR curve.