In my previous post I provided details on the construct of market risk capital calculation under FRTB’s internal model method. The method is based on the calculation of the stressed expected shortfall for a trading desk. This post will provide details on the method prescribed under FRTB rules without spending too much time on the mathematical details of the expected shortfall measure.

### What is Expected Shortfall?

It is a one tailed statistic which measures the **expected loss** during an * n *day period conditional upon the loss greater than the

*percentile of the loss distribution. For example, 10 day expected shortfall (ES) at 99*

**p**^{th}^{th}percentile is the average amount that is lost over the 10 day period provided the loss is greater than the 99

^{th}percentile. ES provide an answer to the question that

*if portfolio lose money, what is the expected loss at a given confidence level*. From the practitioner’s point of view ES is the average of all the losses in the loss distribution which occur beyond a required confidence level. I will not discuss the statistical properties of this measure but it is worthwhile to note that ES exhibits sub-additive property i.e. the ES of two portfolios combined together will not be more than the summation of ES of individual portfolios. This is a desirable property from the portfolio risk management perspective which is not present in VaR measure and has always been considered as one of the shortcomings in using VaR for risk measurement purposes.

### Stressed Expected Shortfall

Under FRTB rules, stressed expected shortfall is to be calculated at 97.5^{th} percentile for each trading desk for the market risk capital calculation purposes. Following steps are to be followed:

- Expected shortfall, referred as
, is calculated for the most severe twelve month stressed period on a regulator approved**ES**_{R,S}*reduced set of risk factors*. Bank specified reduced set of risk factor must be able to explain a minimum of 75% of the expected shortfall calculated using a full set of risk factors, referred as, on a most recent twelve month period. This condition is tested over a previous twelve week period;**ES**_{F,C} - Expected shortfall, referred as
, is calculated for the reduced set of risk factor on a most recent twelve month period;**ES**_{R,C} - Stressed expected shortfall is then calculated as . The ratio between
and**ES**_{F,C }is floored at 1.**ES**_{F,C}

Next section will detail the method to calculate expected shortfall as per the FRTB rules.

### Expected Shortfall Method

For market risk capital calculation purposes, ES is to be calculated at 97.5^{th} percentile for each trading desk. Appropriate *liquidity horizon *is to be used for scaling up an ES from the base horizon of 10 days.

#### Liquidity Horizon

There are five liquidity horizons for the risk factors across all the asset classes, which are summarised in a table below:

j |
LH_{j} |
Risk Factor Categories |

1 | 10 | Interest Rate: USD, EUR, GBP, AUD, JPY, SEK, CAD and Bank’s domestic currencyEquity: Price (Large Cap)FX: Rate for specified pairs |

2 | 20 | Interest Rate: currencies not covered aboveCredit Spread: Sovereign (IG)Equity: Price (Small Cap), Volatility (Large Cap)FX: Rate for unspecified pairsCommodities: Energy & Carbon Emissions Trading Price, Precious Metals and Base Metals Prices |

3 | 40 | Credit Spread: Sovereign (HY), Corporate (IG)FX: Volatility, Other Risk Factors (like correlations) |

4 | 60 | Interest Rate: Volatility, Other Risk Factors (like correlations): Corporate (HY)Credit Spread Equity: Volatility (Small Cap), Other Risk Factors (like correlations)Commodity: Other Commodity Prices, Volatility (Energy & Carbon Emissions, Precious Metals and Base Metals) |

5 | 120 | Credit Spread: Volatility, Other Risk Factors (like correlations)Commodities: Volatility (Other Commodities) |

Expected Shortfall can not be scaled up from the horizon shorter than the base horizon.

#### Liquidity Horizon Adjusted ES

Following formula is used for calculating liquidity horizon adjusted ES:

In this equation:

*T*is set at 10 days as a base horizon;- is the expected shortfall of a portfolio
*P*at the base horizon*T*(i.e. 10 days) which is calculated by shocking all the risk factors of each position. For example, in case of a portfolio of EUR/USD FX Options all the positions will be exposed to EUR/USD spot, EUR/USD FX volatility surface, EUR and USD Interest Rate Curves. All of these risk factors will have to be shocked to calculate this parameter. In this example we assume the domestic currency of the portfolio is either EUR or USD. - is the expected shortfall of a portfolio
*P*at the base horizon*T*(i.e. 10 days) by shocking only those risk factors of each position whose liquidity horizon is*j*(as per the table above) and keeping other risk factors constant.

Let us take an example of an FX portfolio. From the table above we can see that FX risk factors in general will be subject to the liquidity horizons of 10 days, 20 days or 40 days. Eq. 1 will operate like this:-

- is calculated by shocking all the risk factors which have liquidity horizon of 10 days, 20 days or 40 days;
- is calculated by shocking all the risk factors which have liquidity horizon of 20 days or 40 days;
- is calculated by shocking all the risk factors which have liquidity horizon of 40 days. As there are no FX risk factors whose liquidity horizon is more than 40 days, ES will not be calculated for rest of the liquidity horizons.

Based on this, Eq. 1 for the FX portfolio can be written as

Clearly in the above equation, interest rate risk factors have not been considered which appear in the calculation of forward rates and discounting. Assuming all the risk factors are from the longest liquidity horizon, total expected shortfall ES scales up from the 10 day ES as per following scaling factors:

j |
LHj |
Scaling Factor |

1 | 10 | 1.00 |

2 | 20 | 1.41 |

3 | 40 | 2.00 |

4 | 60 | 2.45 |

5 | 120 | 3.46 |

Recall Eq. 6 from my previous post (Nuts & Bolts of FRTB – Internal Model),

In the above equation, all the components are calculated individually using Eq. 1.

### Number of Expected Shortfalls

As can be seen above, there are a number of nested expected shortfall calculations to be performed for calculating a capital number. But realistically how many such calculations are required? Assuming a bank’s trading portfolio is exposed to all the risk factors following ES calculations will be required:

- 5 for
,one for each liquidity horizon;**ES**_{Diversified} - 3 each for
,**ES**_{GIRR},**ES**_{FX}and**ES**_{Equity}, totalling to 12;**ES**_{Commodity} - 4 for
;**ES**_{CS}

The above totals to 21 expected shortfall calculations. However, these are to be calculated on a full set for current period * ES_{F,C}*, reduced set for the current period

*and reduced set for the stressed period*

**ES**_{R,C}*. So these 21 expected shortfall calculations are tripled to*

**ES**_{R,S}**63**. This highlights the computational demands of the market risk capital calculation method under FRTB.