# Basic LIBOR curve question

I'm new to the quant finance and have a very basic question about LIBOR curve.

LIBOR is published every day for 4 different tenors (1M, 3M, 6M, 1Y), and each rate means how much annual interest should be paid when leading banks borrow money from another.

In my understanding, there should be a unique LIBOR yield curve, in which 1M, 3M, 6M, 1Y point values are the same as the quoted value above.

But it doesn't seem to be the case. There's a LIBOR curve for each 4 different tenors. Given this, what does the value of 1M LIBOR curve at 1Y point?

And, when you model LIBOR using short rate model, you're modelling the unique LIBOR short rate, not the LIBOR of each tenor separately. Correct?

Thx!

Libor rates include credit risk. It is riskier to make a 6m loan than two 3m loan. So the 6M Libor curve is not the same as the 3M one. Their difference is the basis spread.

When using a short rate model, you are modelling one curve. As a first approximation, you can deduce the other curves by adding a deterministic basis spread.

• You are claiming that the curve is always upward sloping. I dont think so. – emcor Jun 11 '15 at 21:44
• I don't think I claimed anything of the sort. – AFK Jun 11 '15 at 22:26
• @emcor AFK is right. Th common assumption before the crisis is $(1 + \text{3-month LIBOR} / 4) \times(1 + \text{3-month forward 3-month LIBOR}/4) = 1 + \text{6-month LIBOR}/2$. This of course assumes LIBOR is risk-free, which turned out to be untrue. Post the crisis, the 1-month, 3-month, 6-month, 12-month, etc. curves must all be built separately, all the while accounting for the non-zero bases amongst them. – Helin Jun 11 '15 at 23:52
• +1 I was wrong ; ) – emcor Jun 12 '15 at 9:12

You wrote Given this, what does the value of 1M LIBOR curve at 1Y point represent?

It is a real number X such that:

The following deal can be agreed today in the swap market: You will pay me the amount X (fixed in advance) one year from now, and in return I agree to pay you one year from now the amount Y equal to the 1 Month Libor Rate published at that time. Note that X is known today while Y is unknown and will only be known later, it could turn out to be greater or smaller than X.

I found this Mercurio paper (PDF) helpful and accessible.

The first fifteen pages or so provide a nice background on why multiple curves are used (not sure if outdated, though). Mercurio first motivates the use of multiple curves with an example of what seems to be an arbitrage opportunity. We're given 3m LIBOR, 6m LIBOR, and a 3$\times$3m FRA on a certain date such thatthe implied 3$\times$3m forward LIBOR rate exceeds the FRA, indicating a possible arbitrage opportunity of buying a 6m bond, selling a 3m bond, then entering into a payer FRA in 3m time. However, if we allow for the possibility that:

1. our counterparty could default at some point within the 6m window
2. liquidity could dry up between now and in 3m time

then this strategy may not be an aribitrage opportunity. He then explains in Section 2.3:

Such rates, in fact, become compatible with each other as soon as credit and liquidity risks are taken into account. However, instead of explicitly modeling credit and liquidity effects, practitioners seem to deal with the above discrepancies by segmenting market rates, labeling them differently according to their application period. This results in the construction of different zero-coupon curves, one for each possible rate length considered. One of this curves, or any version obtained by mixing “inhomogeneous rates”, is then elected to act as the discount curve.