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# Why do multiple LIBOR curves exist?

## A summary of what I found without going too much into the technical details.

May 13 · 5 min read Photo by Jon Cellier on Unsplash

# Basic Information on the LIBOR

The LIBOR was previously known as the British Bankers’ Association (BBA) LIBOR. Now that the responsibility for its administration was transferred to the Intercontinental Exchange (ICE), it is also known as the ICE LIBOR. The LIBOR is published every London business day for 5 currencies (USD, GBP, EUR, CHF and JPY) for 7 different tenors (O/N, 1W, 1M, 2M, 3M, 6M and 12M).

# My First Encounter with the Topic Implied forward rate equation

This is the relationship between the implied forward rate between two points in time (T0 and T1) as of time t, in terms of discount factors.

In my answer, I mentioned something along the lines of using the 3-month LIBOR spot rate and 6-month LIBOR spot rate to compute the 3-month LIBOR rate 3 month forward.

In today’s situation, this would not be correct and we will explore the reason below. A more appropriate example would be the use of US Treasury rates, where the lines of reasoning would hold and the relationship would look something like this: Relationship between spot and forward US Treasury rates

# The LIBOR Curves

## Pricing Instruments using Curves

Below is an example of how the Treasury yield curve would look. These curves are often upward-sloping, though they could be flat or downward-sloping (in rarer cases). By Ldecola — Own work, CC BY-SA 4.0, https://commons.wikimedia.org/w/index.php?curid=69078386

Recognising that the rates used for discounting cash flows are unlikely to be constant, here comes the concept of pricing using a curve. Intuitively, the interest rates used for discounting are obtained by matching the tenors of the cash flows with the x-axis of the curve.

The Treasury yield curve shown above would also be called a spot curve because the plotted rates are known as spot rates (eg. 5Y interest rate today). Another type of curve that is often used is called the forward curve, which plots the forward rates.

## Reasons for Multiple LIBOR Curves

Before the 2008 financial crisis, the LIBOR was assumed to be risk-free. This means that we could obtain the 3M LIBOR 3 month forward rate from the 3M and 6M LIBOR spot rates. Visually, this would be how the relationship would look (similar to the one I used for US Treasury rates): Relationship between the LIBOR spot and forward rates

The relationship assumes that counterparties can first take up a 3M loan, then roll the maturing loan 3 months later into a new 3M loan to earn a return equivalent to taking on a 6M loan. However, rolling these loans became a problem during the financial crisis when firms had issues with paying back the notional on the maturing loans, which pointed to the presence of credit risk in these LIBOR rates.

In fact, the LIBOR rates of different tenors carry different levels of risks. Therefore, the the different LIBOR rates (1M, 3M, 6M etc.) no longer belong to the same curve and each tenor has its own curve.

## The use of LIBOR as a Discounting Rate

After it was shown that LIBOR rates were not risk-free, the cash flows of a swap started to use Overnight Indexed Swap (OIS) discounting (using discount factors derived from the OIS curve). The pricing of a vanilla interest rate swap would thereafter require at least 2 different curves.

# Ending Words…

If you are interested, other relevant topics would include the LIBOR reform (which was due in part to the LIBOR scandal), swap pricing using the multi-curve framework, just to name a few.

If you have any questions regarding the article or spotted any mistakes, please feel free to drop me a message. This article represents the knowledge that I gained while digging through the internet (articles, books, academic papers) for information.