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I'm very confused about the terms regarding libor curves in general.

When people talk about libor curve, I picture it as a curve with different libor maturities (i.e. 1 week, 1 month, 3-month and 6-month etc.).

However I can't picture what a 3-month or 6-month libor curve look like, isn't that just a single value at that time? can someone please kindly explain?

And also when people talk about forecasting a libor, what do they mean? How is it connected with bootstrapping method?

Thanks so much in advance for helping!

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A 3 month libor curve is a set of forward rates for 3 month libor. Thus, the curve begins at where 3 month libor is today , and takes different values for each possible forward observation date. Loosely speaking, this curve represents where the market thinks 3 month libor will set in the future.

The analogous statement holds for 1mo libor , 6 month libor , etc.

A curve consisting of lending rates for various tenors 1 week, 1 month, 3 month etc is an interbank lending curve but this is not normally referred to as "the libor curve".

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There are a lot of intricacies involved, and I'll focus on high-level stuff.

Let's go back to the basics. If we have the 3-month LIBOR rate and the 6-month LIBOR rate, can we calculate the 3-month forward 3-month LIBOR rate? Before the financial crisis, the answer was typically assumed to be yes. The math is simple:

$$\left(1 + \frac{\text{3-month LIBOR}}{4}\right)\left(1 + \frac{\text{3-month forward 3-month LIBOR}}{4}\right) = 1 + \frac{\text{6-month LIBOR}}{2}.$$

You can apply similar math to calculate any forward rate, be it an $x$-year forward 3-month rate, or an $x$-year forward 12-month rate.

But going back to the formula above, it assumes that after the first three months, your counterparty will be there to roll your maturing loan into another three-month loan. During the financial crisis, rolling these short-term loans became a serious issue, and people woke up to realize that LIBOR was not as risk free as they thought. The implications are profound and extend beyond short-term loans. For longer maturity swaps, for example, people realized that discounting cash flows using LIBOR resulted in them holding insufficient collateral to cover losses.

So nowadays, we have to account for the fact that LIBORs of different tenors are not only NOT risk-free, but carry different degrees of risk. A three-month LIBOR and a six-month LIBOR no longer belong to the same curve, and the 3-month forward 3-month LIBOR cannot be imputed from them (there are in fact traded instruments, 3s6s basis, that tell us how much this risk is worth). Accordingly, we build 1-month LIBOR curves for $x$-year forward 1-month curve, 3-month LIBOR curve for $x$-year forward 3-month rate, etc. None of these curves are used to discount cash flows; for that, yet another curve, typically the OIS curve, is required.

See MULTIPLE CURVE CONSTRUCTION for a discussion of how modern yield curves are constructed.

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