Topical Issues on Risks in Financial Markets.


Nuts & Bolts of FRTB – Internal Model

So far I have covered three components of the sensitivity based approach (SBA) of the standardised method to calculate market risk capital. These components are – Delta Risk Charge, Vega Risk Charge and Curvature Risk Charge. To determine total market risk capital required, banks are also required to calculate Default Risk Charge to capture jump-to-default risk for positions in non-securitised portfolio, securitised portfolio (with no correlations) and securitised portfolio (with correlations). In addition, banks may also have to calculate a residual risk add-on. I will cover these elements of the FRTB later. At this stage I consider it to be appropriate to introduce the internal model approach to calculate market risk capital.

Overview of Capital Calculation Approach

As noted in my introductory post on FRTB, banks will have to seek internal model approval from regulators at the trading desk level. All the desks which do not have approval to use internal model will be required to calculate capital using standardised method. Irrespective of the approval to use internal model, banks must calculate the standardised capital charge for each trading desk at least on a monthly basis. As per para 184 of the official rule, key objectives for this requirement are:

  1. use standardised capital charge for the trading desk which fail the eligibility criteria to use internal model, perhaps due to backtesting failure;
  2. utilise standardised capital charge numbers for the purposes of benchmarking banks with its peers within or across jurisdictions and;
  3. potentially for the model calibration purposes.

Picture below shows the different components of the standardised and the internal model approach and inter-relationships between two approaches. This demonstrates that standardised method is required to be calculated for all the trading desks but it is used only for those desks which do not either have a regulatory approval to use internal model or which become ineligible to use internal model potentially because of failing backtesting.  Mathematically,  aggregate capital charge for market risk can be written as

ACC = C_A + DRC + C_U  - Eq. 1

where CA is aggregate capital charge for eligible trading desks. This include capital charges for both modellable and non-modellable risk factors;

  • DRC is default risk charge ; and
  • CU is standardised capital charge for the unapproved/ineligible trading desks.

Capital Model Approach

My earlier posts covered the Sensitivities Based Approach of the standardised method. Default risk charge which is common to both the approaches will be covered later. Next section will cover the equation which is to be used to calculate market risk capital.

Capital Charge Equation

On a daily basis a bank must calculate and meet its capital requirement C_A which is higher of its

  1. previous day’s aggregate market risk capital charge; and
  2. average aggregate market risk capital charge over previous sixty business days

Aggregate market risk capital charge is calculated as per following equation –

C_A = max(IMCC_{t-1}+SES_{t-1};(m_c+\delta)*IMCC_{avg}+SES_{avg})   - Eq. 2


  • IMCC is the expected shortfall for the trading desk based on the modellable risk factors;
  • IMCC_{t-1} represent previous day’s expected shortfall for the trading desk;
  • IMCC_{avg} represent average expected shortfall for the trading desk over previous 60 days;
  • SES represent capital charge for the non-modellable risk factors which are capitalised using stress scenarios;
  • m_c represent a multiplication factor which is set at 1.5;
  • \delta is a factor to be set by regulators which can range between 0 and 0.5 depending upon bank’s risk management system and backtesting performance of the internal model to calculate VaR at 99 percentile on the full set of risk factors;

Aggregate capital charge for the modellable risk factors (IMCC) is calculated as the weighted average of the constrained and unconstrained expected shortfall charges. Mathematically,

IMCC = \rho*(IMCC(C)) + (1-\rho)*(\sum_{i=1}^RIMCC(C_i))  - Eq. 3

In this equation, \rho is set at 0.5 and is defined as the relative weight to the bank’s internal model;

  • IMCC(C) is a diversified stressed expected shortfall which is calculated as ES_{R,S}*\frac{ES_{F,C}}{ES_{R,C}}  - Eq. 4
  • IMCC(C_i) is a non-diversified (also called as constrained) stressed expected shortfall which is calculated by shocking risk factors relevant to one risk class and keeping other risk factors unchanged. Formula to calculate non-diversified stressed expected shortfall for a risk class i is ES_{R,S,i}*\frac{ES_{F,C,i}}{ES_{R,C,i}}  - Eq. 5

It may be easier to remember the above equation as

IMCC = 0.5*(ES_{Diversified} + ES_{GIRR} + ES_{CS} +ES_{Equity}+ ES_{FX} + ES_{Commodity})  - Eq. 6

where ES is to be calculated as per the equations Eq. 4 and Eq. 5 . These equations will  be detailed in my next post. The stylised example below shows capital charge under FRTB is higher than the fully diversified scenario but less than what it would be under standardised method. The simplified assumption above is that Expected Shortfall for each risk class is equal to the capital charge calculated using sensitivity based approach.

Risk Class Capital
Expected Shortfall for GIRR (ESGIRR) 20
Expected Shortfall for FX (ESFX) 20
Expected Shortfall for Equity (ESEquity) 20
Expected Shortfall for Commodity (ESCommodity) 20
Fully Diversified Expected Shortfall (ESDiversified) 50
Standardised Charge 80

In my next post I will provide details on the calculation of stressed expected shortfall for the market risk capital charge purposes.