## The risk transition in LIBOR transition

This post continue on the "transformers" series related to LIBOR discontinuation and follows our quant perspective on IBOR fallback. In this episode, we discuss the transition or transformation through time of the risk of a fixed portfolio. The explanation is done using the graph in Figure 1. If the meaning and content of the graph are obvious to you, then there is no need to read further; if this is not the case, you may want to spend a little bit of time reading.

Figure 1: A cryptic graph to be explained later.

### Single period swap

We start with the simplest portfolio, composed of a single swap on a single period. The date of the analysis is 30-Aug-2018 as in the previous episodes. The swap has a start date in 12 months and a 3-month tenor on USD-LIBOR-3M. The notional is 100m.

We look at the risk through the glasses of PV01. We compute the market quotes bucketed PV01 with respect to each tenors and then sum them by curve (OIS and LIBOR3M). This gives us two numbers for each date. Like mentioned in the previous episodes, those numbers have to be taken with a pinch of salt as they are obtained by adding sensitivities to different market realities (market quotes from different instruments with different conventions). They are enough for the qualitative analysis we perform, but may not be perfect for all purposes.

We first look at the trade risk in absence of LIBOR discontinuation. The risk is composed of the risk to the LIBOR fixing for roughly 2,500 USD/bp (100m/10,000/4) and a very small discounting amount from the fact that the swap is not ATM. The Y axis of the graph represent the PV01 in K USD/bp for the LIBOR and the OIS curves. The X axis is the date on which the risk is computed. To avoid complicating the picture, we have used the rate as of the first date and computed the implied forward curves for each day in the following year. The risk are computed with those forward curves. If we had used the actual market curves for each day, there would be on top of the changes described here some small ups and downs due to market fluctuations.

Figure 2: Risk transition for a one period swap in absence of discontinuation.

We now introduce the Announcement Date and the Discontinuation Date. We suppose that the announcement is 30-Dec-2018 and the actual discontinuation is 28-Feb-2018. Those dates do not affect our previous risk graph but we reported the dates for visualization facility.

Now we introduce a fallback option, starting with the OIS Benchmark option. The reason to start with that one, even if this is not in the ISDA consultation, is that this is the one the closest to the actual LIBOR in term of risk profile.

The big change happens on the announcement date. The only fixing in our swap is after the discontinuation date, it is then replaced by a fixing to the OIS benchmark. In term of risk, the OIS benchmark is on the Discounting/Overnight/OIS curve. On that date, the risk jumps from the LIBOR curve (dashed light blue) to the OIS curve (dashed dark blue). Then nothing spectacular happens to the risk up to the fixing date. On that date the risk decrease dramatically when the rate is known, leaving only a residual small OIS risk (coming from the difference between the fixing and the fixed rate of the trade) which disappears completely at maturity.

Note also that it is possible that the OIS fixing and LIBOR fixing dates will be slightly different because of non-good business days. For example USD-LIBOR is fixing according to the London calendar but SOFR according to the US Government Securities calendar (and obviously this is not yet defined for the OIS Benchmark financial fiction we use here).

Figure 3. Risk transition for a one period swap. OIS Benchmark option added.

Once the profile of one option is understood we can add the other three. The LIBOR profile will be the same for all options. It goes from something before the announcement date, and that something is the same for all options, to nothing. We do not repeat that part to avoid overloading the graph.

The other options included are Spot Overnight, Compounding Setting in Advance and Compounding Setting in Arrears. For the Spot Overnight, the fixing is also on one date, so the profile is very similar around the fixing date to the OIS Benchmark. The total PV01 risk between the announcement date and the fixing date is quite similar to the previous one. As discussed in a previous episode, this is not true when looking at the tenors/buckets level. For the Compounding Setting in Advance, the risk start to decrease three months before the actual fixing date. The fixing is obtained by compounding the rates over the three-month period preceding the fixing. So each day that is passing a small piece of the rate is know and there is no risk anymore on it. Each day the risk is decreasing slightly. Finally for the Compounding Setting in Arrears, the risk is roughly constant up to the start of the theoretical deposit underlying the fixing and slightly decrease up to the maturity date of the same fixing. This is a translated version of the previous description.
Figure 4. Risk transition for a one period swap. Legacy and all fallback options.

### Multi-periods swap

We now change the underlying instrument to a two-year swap starting in three months. The announcement date is 30-Dec-2018 and the discontinuation date is 30-Sep-2019. The swap has 8 3-month periods. The announcement date is in the first period and the discontinuation date is in the fourth period.

The profile in the absence of discontinuation is a standard profile with a small discounting risk and a LIBOR risk that steps down at each fixing date (yellow).

When we introduce the fallback, on the announcement date, the LIBOR risk of all the fixing after the discontinuation date (4) are transferred to the OIS curve. The light blue dashed line drops on the announcement date by the equivalent of 4 fixings risk and the OIS risk jumps in the opposite direction. The 3 fixings that are between the announcement date and the discontinuation date are not affected by the fallback, this is why they is still three quarterly drops on the LIBOR light blue line.

Figure 5: The risk profile for the OIS and LIBOR curves for a two-year swap. Legacy swap and all fallback options.

The three options propose a slightly different profile around the fixings. The OIS Benchmark and Spot Overnight are similar to the original LIBOR with risk drops. The Compounding options have the risk that linearly decrease (actually not completely linearly, but by a discrete drop of the same amount each day); they differ on when this decrease starts: on the fixing date or one original index tenor before.

No risk is lost in transition, most risks are transformed. In the end all risks die.

Don't fallback, step forward!