Expected Shortfall Basel III style: what is the idea?

Question

I would like to do a qualitative question about the Expected shortfall in the Basel 3 document.

First of all let me introduce few definitions.

Suppose to have a portfolio $P$ depending on a family of risk factors. Let $T$ be a time horizon (for the Basel document $T = 10 $ days).

Now I introduce a family of liquidity horizons usefull to classificate the risk factors:

$$ \begin{matrix} j & & LH_j \\ \hline & & \\ 1 & & 10 \ days \\ 2 & & 20 \ days \\ 3 & & 40 \ days\\ 4 & & 60 \ days\\ 5 & & 120 \ days \end{matrix} $$

Thanks to this definition we can introduce the families $Q(P,j)$ of the risk factors whose liquidity horizons are at least as long as $LH_j$.

Finally we define the following terms: \begin{align} ES_T(P) = & \mbox{ES for the horizon T wrt all the risk factors} \\ ES_T(P,j) = & \mbox{ES for the horizon T where all the risk factors NOT belonging} \\ & \mbox{to Q(P,j) are freezed} \end{align}

Now that we have done with definitions I can make my question:

The Basel document gives this definition of Expected shortfall that I can not understand from an economical point of view: $$ ES = \sqrt{\left(ES_T(P)\right)^2 + \sum_{j\geqslant 2} \left( ES_T(P,j) \sqrt{\frac{LH_j - LH_{j-1}}{T}} \right)^2} $$

The first term is just the right term...but it seems a good idea to introduce other terms in order to take in account of ES wrt a subset of risk factors.

Now: I really do not understand the presence of the terms $\frac{LH_j - LH_{j-1}}{T}$, in particular it seems like they have to do the job of a weighted terms but in general they are equal to the following values $(1, 2, 2, 6)$ so that I can' t understand what is the meaning or the purpose of such terms.

Thank you in advice for your help. Ciao!

Answer 1

I believe the document that @clarkmaio referred to is Minimum capital requirements for market risk and the issue described can be found on page 52.

As explained here:

The revised FRTB rules require ES to be calculated using a base liquidity horizon of 10-days and this ES to be scaled by mapping each risk factor to one of the risk categories below:

Meaning that a portfolio of:

• Minor currencies will have an ES of sqrt(20/10) or 1.41 times higher;
• Credit spread corporates (IG) will have an ES of sqrt(40/10) or 2 times higher;
• Interest rate options will have an ES of 2.45 times higher;
• Credit Spread volatility products will have an ES of 3.46 times higher;

These reflect the increased time to liquidate such positions in a time of market stress, resulting in a higher potential market loss.

Example 1:

• FX: volatility
• n = 40
• j = 3

$$ ES = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \sqrt{\frac{LH_2 - LH_1}{T}} \right)^2 + \left( ES_T(P,3) \sqrt{\frac{LH_3 - LH_2}{T}} \right)^2} = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \sqrt{\frac{20 - 10}{10}} \right)^2 + \left( ES_T(P,3) \sqrt{\frac{40 - 20}{10}} \right)^2} = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \right)^2 + 2 \times \left( ES_T(P,3) \right)^2} = \sqrt{4 \times \left(ES_T(P)\right)^2} = 2 \times ES_T(P) $$

Example 2:

• Credit spread: volatility
• n = 120
• j = 5

$$ ES = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \sqrt{\frac{LH_2 - LH_1}{T}} \right)^2 + \left( ES_T(P,3) \sqrt{\frac{LH_3 - LH_2}{T}} \right)^2 + \left( ES_T(P,4) \sqrt{\frac{LH_4 - LH_3}{T}} \right)^2 + \left( ES_T(P,5) \sqrt{\frac{LH_5 - LH_4}{T}} \right)^2} = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \right)^2 + 2 \times \left( ES_T(P,3) \right)^2 + 2 \times \left( ES_T(P,4) \right)^2 + 6 \times \left( ES_T(P,5) \right)^2} = \sqrt{12} \times ES_T(P) = 3.46 \times ES_T(P) $$