When working on Murex rate curves, one quickly faces the problem of curve relationships and what is called as rate propagation. Once understood, it seems very simple. But before you get to that “Eureka” point, it might be confusing. So let’s dig into it and hopefully bring you some “Eureka” moment!

First of all, rate propagation only makes sense when you are working with multiple curves in the same currency. The rate propagation determines how are the other curves going to move when one curve moves. The setting sits under the rates general settings and can take 3 different values:

– Keep market quotes constant (KMQC)

– Keep zero rates constant (KZCC)

– Keep market quotes constant/Impact sensitivities

The first one and the 3rd one have the same results when perturbing one curve but the 3rd one tries to show the sensitivities due to other curves perturbation. Effectively you should hesitate between the second and the third one, as the first one does not show you the right sensitivities.

In the mode KMQC, the rate curves are recalibrated after each perturbation. Let’s take the following example:

USD DISC

USD 3M

USD 6M

USD DISC has no dependency on other curves. It can self calibrate.

USD 3M depends on USD DISC to calibrate its swap pillars as they are estimated on USD 3M but discounted on USD DISC. USD 6M depends on the 2 other curves as it contains basis swaps estimated on both 3M and 6M curves and discounted on DISC curve.

## Rate propagation mode : KZCC

In the mode KZCC, you basically assumes that if any rate changes, then the zero rates of the other curves do not change. So if your USD DISC curve changes, then the zero coupon rates of both USD 3M and USD 6M will remain the same. It means that the market rates of the USD 3M and 6M curves will change. Your ZC rate for the curves does not change, so the estimated rates will remain the same. But your discounting rate has changed (USD DISC has changed), so you need to change the fixed leg rate (aka your market quote) or your margin (basis swap market quotes) so that the NPV of the swaps remains 0.

## Rate propagation mode : KMQC

In this mode, you assume that the market quotes remain the same when one curve is perturbed. So your ZC rates should be recomputed. Let’s see why! Using the same example as above and perturbing the USD DISC curve will yield the following:

– USD 3M ZC rates will change

– USD 6M ZC rates will change

When you’ve changed your USD DISC curve, the discounting rates will change. So in the case of your IRS in the 3M curve, the discounting rate will change, the fixed leg rate remains constant (Keep Market Quote Constant!), so your fixed leg has a different NPV. As such, you need to modify your estimation rate (USD 3m ZC) to reach a NPV of 0.

Similarly for the 6M curve, the 3M leg will have changed NPV, the 6M leg has different discounting rates, so you need to adjust the 6m estimation rates to keep a NPV of 0, so the 6M zero rates will change.

STOP there. There are more complexities that you can lay on top of these propagation modes but the above will always stay true: you need to maintain a NPV of 0 in every instrument in your curves and that’s the only way to do it.

1 more question:

## I can’t reproduce the same behavior with manual shifting, why is it so?

What I wrote above is true and what should happen BUT sometimes the variation in values are actually quite small or 0. For instance, in KZCC, if your USD 3M curve is quite flat, then you won’t see much difference after the change in the discounting rates. Why? All flows fall on the same date for the fixed and floating leg. So you can sum both flows before applying the discount factor. If your estimation curve is flat, then prior to the shift of the discount rate, your sum of the 2 flows was already close to 0. As such, the impact of the discount factor is limited.

Rate relationships have changed a lot in the more recent versions with bugs and enhancements. So if there is something you can’t explain check with someone with more experience or a more recent version if you can.

Questions and comments are more than welcome!