In this transformer's blog installment, we look at a single trade present value and delta risk.

The trade we look at is a plain vanilla payer USD Fixed v LIBOR 3M starting in 6M and with a tenor of 5Y. The coupon is 1% and the notional 100m. The trade is largely in-the-money (the forward rate is 2.95%). As SOFR-indexed instruments are not liquid yet, we have used EFFR curves as a proxy.

### Delta ladder (Bucketed PV01 - Key rate duration)

We first look at the delta risk (par rate sensitivities) implied by the different options (without convexity adjustment at this stage). The delta ladder is as expected, with most of the risk on the LIBOR curve, short the rate on the 6M and long the rates at 5Y6M (between 5Y and 7Y with most of it on the 5Y). As the trade is largely in-the-money, there is also a non-negligible delta on the OIS curve.

The total for each column has to be taken with pinch of salt. The individual numbers are not with respect to the same instruments and the conventions are not the same for all of them.

Delta ladder for the legacy USD-LIBOR-3M trade |

Then we move to the different options. We start with an adjustment spread of 0 and then will add the spread later.

The first option we look at is the

*OIS benchmark*. This is not part of the ISDA list (part of our quant perspective proposal). We start with it because this is the closest to the IBOR trade is term of dates and structure. The adjusted RFR is also a term rate and the term has, by construction, exactly the same dates as the theoretical deposit underlying the LIBOR.

The delta is entirely moved from the LIBOR curve to the OIS curve. On the tenor side, it is very similar, short on the 6M and long between 5Y and 7Y.

Delta ladder for the option OIS Benchmark |

*Compounded Setting In Arrears*. As discussed in details in our perspective, this option will either break the coherency of a single fixing and/or not be achievable in practice. Nevertheless we include it here as we deal only with vanilla swaps where it is achievable. For the vanilla swaps, if we ignore the detail of non-good business days, the valuation and risk are the same as for the OIS benchmark option as proved in the Section 5.2 of our perspective. Not surprising, the risk in this case is almost the same as in the previous one.

**Delta ladder for the option**

*Compounding Setting In Arrears*The next option we look at is the ISDA proposed

*Spot Overnight*. This is the replacement of a term benchmark with an overnight benchmark with the same fixing date but a very different underlying period (and economic reality). From a delta ladder perspective, we see that most of the risk as been translated by three months backward.

**Delta ladder for the option**

*Spot Overnight*If we take the first period, with a LIBOR risk from 2BD+6M to 2BD+9M, it is now an overnight risk from 2BD+6M-2BD to 2BD+6M-1BD. A delta appears on the 3M because the forward rate of the first fixing is now impacted by the 3M rate and the 6M rate. If we were to look at the sensitivity of a 6Mx3M swap (one period LIBOR-3M swap starting in 6M), the risk from the forward is negative on 3M, positive on 6M and there is a little bit of discounting on the 9M.

What the delta ladder does not show is that most of the sensitivity is concentrated on few dates, one every 3 months. If the delta ladder had daily points we would see spikes at those dates. If such an option was selected, the delta ladder as presented here would become almost obsolete for risk management and other views of the risk would be required.

The last option we look at is the ISDA proposed

*Compounding Setting in Advance*. The rate is for a period of the same tenor as the LIBOR and the rate is overnight rates compounded, but the period is the period before the fixing instead of the period after in the LIBOR case. The impact on the delta is that the sensitivities related to the forward rates are move backward by three-months in the ladder.

**Delta ladder for the option**

*Compounding Setting In Advance*Note that the total delta is not very different from one option to the other.

### Present value and spread adjustment

Up to now, we have described only the risk (delta) for a given fixed adjustment spread of 0. What is happening when the spread is modified? In the next figure, we have displayed the impact of modifying the spread from 0 to 50 bps. The result is rather unspectacular. The change of present value is linear in the spread and the different options generate rather similar present values.

**Profile of present value for different levels of spread.**

At the level of a single swap the present values do not look very different and the value transfer is coming more from the spread choice than from the adjusted RFR option choice. When we will look at a large portfolio level in the next blog, we will see that those small differences add-up and can lead to significant impacts.

- Fallback transformers - Introduction
- Fallback transformers - Present value and delta
- Fallback transformers - Portfolio valuation
- Fallback transformers - Forward discontinuation
- Fallback transformers - Convexity adjustments
- Fallback transformers - magnified view on risk

Don't fallback, step forward!

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