# Expected Shortfall Basel III style: what is the idea?

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.

• Good question. Could you specify a bit more the concept of "liquidity horizon"? So what does it mean to say that "risk factor $X$ has a liquidity horizon of at least $LH_j$"? – Daneel Olivaw Jun 6 '17 at 17:40
• Can you give a more precise reference to "the Basel 3 document" that you are referring to ? (For those who find the BCBS documentation hard to navigate). – noob2 Jun 6 '17 at 19:26

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:

$$ES = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \sqrt{\frac{LH_2 - LH_1}{T}} \right)^2 + \left( ES_T(P,3) \sqrt{\frac{LH_3 - LH_2}{T}} \right)^2 + \left( ES_T(P,4) \sqrt{\frac{LH_4 - LH_3}{T}} \right)^2 + \left( ES_T(P,5) \sqrt{\frac{LH_5 - LH_4}{T}} \right)^2} = \sqrt{\left(ES_T(P)\right)^2 + \left( ES_T(P,2) \right)^2 + 2 \times \left( ES_T(P,3) \right)^2 + 2 \times \left( ES_T(P,4) \right)^2 + 6 \times \left( ES_T(P,5) \right)^2} = \sqrt{12} \times ES_T(P) = 3.46 \times ES_T(P)$$
• First of all thank you for your answer! Now the reason of that term start to ble clear! I have just a question about the examples you gave me: infact it seems like you are computing the term $\frac{LH_j}{T}$...i.e. you are not doing the subtraction with the liquidity horizon of the previous family (and I must say that it would have more sense with the term $\frac{LH_j}{T}$). Thank you in advice! – clarkmaio Jun 7 '17 at 7:05
• For example for Minor Currencies: what I expect from the formula is to have the term: $$\frac{LH_2 - LH_1}{T} = \frac{20 - 10}{10} = 1$$ – clarkmaio Jun 7 '17 at 7:37
• Wait: pay attention! in general you have that $$ES_T(P) \not = ES_T(P,j)$$ since the second one has some constant risk factors. So that if, for example, you compute the ES by Monte Carlo method you have to use two different models for simulations. This means that $$ES_T(P)^2 + ES_T(P,2)^2 + 2 \times ES_T(P,3)^2 \not = 4 \times ES_T(P)$$ – clarkmaio Jun 7 '17 at 10:42