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

Photo by Jon Cellier on Unsplash

The London InterBank Offered Rate (LIBOR) is an interest rate at which banks charge each other in the interbank market. It is a primary benchmark for short-term interest rates around the world.

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).

I haven’t been too familiar with the intricacies of the LIBOR. What brought my attention to this topic was an interview question asking me to explain this equation:

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

Before going into the section on LIBOR curves, it is important to understand what are curves and their uses in pricing instruments. If you are familiar with the concept of curves, feel free to skip the first subsection.

## Pricing Instruments using Curves

At the basic level, when we price an instrument with multiple cash flows (eg. fixed rate coupon bonds), we discount each cash flow by a constant interest rate. In the real world, the 1Y spot rate would unlikely be the same as the 10Y spot rate (due to reasons such as liquidity premium).

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

Half a year since the interview question, I learnt that the LIBOR has separate curves for each tenor. Some common ones included the 1M, 3M and 6M LIBOR curves. One use of these LIBOR curves is to project LIBOR forward rates to value cash flows tied to floating rates (in for eg. an interest rate swap).

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

Prior to the 2008 financial crisis, LIBOR-based instruments such as interest rate swaps were valued using a single curve methodology. This means that future cash flows were determined by projecting LIBOR rates (forward curves) and these cash flows were discounted by the same set of LIBOR curves.

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.

Lapses in the valuation of financial instruments surfaced during the 2008 financial crisis, which led to a larger focus on the risk of traded products over the years (from an investment bank’s perspective). The changes to LIBOR I mentioned in the article is just a portion of the topic of LIBOR.

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.

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