Overview of FRTB and intuition behind the mathematics

This is an overview of the market risk capital computation under the proposed FRTB approach. While there is abundant literature focusing on models, very few focus on mathematical intuition. In the following sections, I will try to give an overview of the market risk computation framework and intuition behind using sensitivity analysis/Taylor series in estimating the market risk capital.

Market Risk FRTB

• Market risk capital under FRTB ?can be modelled either under the standardized approach or model based approach. ? Capital charge for mkt risk = max (capital under standardized, capital under advanced models)
• ?Capital charge for Market Risk under the model based approach =

? = (Capital for desk that qualify for Model-Based Approach) + (Profit and Loss Attribution (PLA) Add-On) + (Standardized Approach for Non-IMA Desks)

[PLAT test, Pnl Attribution, RFET test are explained below]

-???????? Unlike under the Basel III, in FTRB each desk needs to take approval from the regulator to qualify for IMA. The main change in terms of models is replacing VaR with ES.

-???????? Capital Charge for market risk per IMA at desk level : . Most common multiplier is 1.2 for ES, where it is was 3 or 4 Var.

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• Capital Charge for Market Risk under standardized approach = Capital Charge based on sensitivities (Delta, Gamma, Curvature) ??+

???????????????????????????????????????????????????????? Capital Charge for DRC +

????????????????????????????????????????????????????? ????????????? Capital Charge for Residual Add-on

• Capital charge for default risk. This captures the Jump-to-Default risk that is not covered in credit risk. This charge is zero for sovereign. It is simply a percentage of exposure to equity and debt instruments issued by non-sovereigns.

·???????? Capital charge for residual risk. It is catch all if a trade can’t be classified.

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The intuition behind using sensitivities for computing capital

Deriving risk weights (RW) from sensitivity analysis is appropriate because once you assess the expected change in the portfolio (read as risk), it is intuitive to allocate capital to mitigate the risk. The following explains the mathematics behind capital computation using the Taylor series

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Let us say a bank has only one trade vi ?(fixed income trade)

r1: represents the risk factor

?The above equation simply shows that once you have sensitivities (calculated from the pricing engine) and ?is available, it is possible to understand the change in your portfolio.

?Extrapolating this to multiple trades labelled i to n

Sensitivity can be calculated more precisely by aggregating across buckets and currencies

• ? Once you have trade level sensitivity change, you aggregate across the bucket (tenor represented by the first formula) and then aggregate across the currencies (the second formula below)

Key things to remember in FRTB

• Securitization products must be modelled through the Standardized approach. They are not allowed under the modelled approach.

• PnL Attribution: Attributing the PnL to the drivers. It is explaining the pnl to the risk factors that drive the portfolio.
• Desks that fail the PLAT test must be modelled through the Standardized approach.

-???????? FRTB is not applicable for instruments in the Banking Book

-???????? FRTB is applicable at desk level. You do not look at the trades right away. First several tests are done to assess if the desk qualifies for IMA or Standardized. Regulatory approval process is required for desk level (Rates, Forex, Currency, Commodity, Equity, Credit)

-???????? FRTB does not allow changes to the internal models without informing to the regulators.

• Actual PnL/Hypothetical Pnl:

-???????? Actual Pnl (APL): Actual pnl including the changes in the portfolio from yesterday to today. Real pnl from yesterday to today.

-???????? Hypothetical PnL : Assuming the today’s end of portfolio is same as the yesterday’s end of the day portfolio. You change the risk factors from yesterday to today, you calculate the PnL and compare against the VaR.

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• ? Backtesting test: It is comparing actual pnl (APL) is compared against the HPL

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In summary, at a desk level you are allowed higher exceptions because you have fewer trades that do not offset between each other.

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• PLAT (PnL Attribution test): Comparing the PnL derived from the finance with the Risk Theoretical PnL. Front office models give more weightage to the prevailing risk factors, where as risk gives more weightage to historical factors. Front office models do not care of the risk models as I lost money or made money today.

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For the Amber desks, PLAT add on applicable. PLAT Add on is 50% of the difference between the IMA Capital Charge and Standardized charge. You further multiply this by factor

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?Comparing distributions

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• Spearman correlations: It assess if vectors are closely correlated. First you rank order the points.
• K-S test???????????????????? : It assess if these RTPL and HPL are from the same distribution. The smaller the KS metric the best as the difference between the two distributions is smaller.?
• RFET (Risk Factor Eligibility Test) in IMA (Internal Models Approach) : The RFET acts as a filter to decide which instruments or risk factors are eligible for modeling under the IMA. It ensures only those with sufficient liquidity and observable data (e.g., 24 price points annually) can be included, while others are shifted to the standardized approach

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Currency Pair (USD/EUR): 50 prices → Passes (exceeds the 24 required).

Commodity: 10 prices → Fails (falls short of 24).

Stock: 30 prices → Passes (exceeds the 24 required).

Derivates: 24 real price observations are required over the past 12 months. ( 4observations/quarter)

Credit : 12 per annum/4 per quarter.

Overall, in the context of the Fundamental Review of the Trading Book (FRTB), Taylor series expansions are instrumental in approximating changes in the value of financial instruments based on their sensitivities to underlying risk factors. This method is particularly useful for calculating Value at Risk (VaR) and Expected Shortfall (ES) measures, which are central to FRTB's market risk framework. Understanding the intuition behind framework helps to get a better grasp of models.