FRTB IMA - Part 3: Move from VaR to ES and IMA Capital Charge Calculation

When VaR (Value at Risk) first emerged as a measure of market risk in early 1990s (thanks to JP Morgan) - it was widely accepted as an extremely useful metric. You see - big hot shot executive don't like mathematical jargons - they want to see a brief summary, preferably in common English language that can be used as an actionable material. Something that reads like :

We can say with 97% confidence, that the bank/portfolio will not have a loss of more than 2 million dollars in next one day.

VaR basically says exactly this. It does not overcomplicate things by introducing mathematical jargons, symbols, graphs, metrics that are complex to decipher. However, while it does make life easier and simpler on most days, it has drawbacks. You see - back in 2008 financial crisis even Lehman Brothers on a random weekday had a VaR report saying there is 99% confidence that the bank won't face a loss of more than X billion dollars in next 1 to 10 days.

But hey, what about the 1%. If the 1% does happen, can the loss be as big as... ... ... as the bank becomes bankrupt? For Lehman Brothers, it indeed did. Many banks suffered losses above and beyond what their VaR report said.

There were plethora of reasons. Basel attempted a stop gap solution with stressed VaR in Basel 2.5 regulations but what emerged to be a better metric was 'Expected Shortfall' which is mandated in FRTB IMA (Internal Models Approach). It is better than VaR in many ways including but not limited to the fact that it addresses sub-additivity (a property of being a good risk measure which VaR does not have).

Expected Shortfall as a risk measure addresses various issues.

Besides the change in risk metric (from VaR to ES): FRTB has also changed the associated liquidity horizon. In VaR, the liquidity horizon was fixed (i.e. 10 days) despite banks having some products that are far from being liquid. This is where FRTB introduced varying liquidity horizons (LH) from 10 days to 120 days depending upon the underlying risk factor. LH is however capped at the maturity of related instrument.

Enough theory though. Let us crunch some formulas and numbers now. You see the overall purpose of doing all this is to arrive at capital charge applicable for market risk, right?

So what's the overall capital charge

It is sum of three components

• The aggregated charge associated with Unapproved desk (under SA) denoted by 'Cu' in the equation.
• The aggregated charge associated with approved desk (under IMA). This part here involves ES calculation. This is what this article will focus on in detail. It is denoted by 'Ca' in the equation
• Default Risk Charge (DRC). Note that DRC for IMA is not same as DRC for SA. I shall cover this in detail in next article.

I have discussed first part in my earlier series of articles for 'standardised approach'. Let me discuss the second element in right hand side of the equation in this article.

Aggregated Charge associated with Approved Desks (under IMA): Ca

The aggregated capital charge under IMA is the max of either

"IMCC + SES calculated" OR "MultiplicationFactor*AvgIMCC + AvgSES"

Understanding IMCC

The aggregate capital charge for modellable risk factors, MRF (denoted as IMCC) is based on weighted average of constrained and unconstrained expected shortfall charges.

Understanding UnConstrained ES [IMCC(C)]

In the current system, the Value at Risk (VaR) is calculated based on the market movements of the previous year. However, this approach has been proven to be insufficient during turbulent times, like after the Lehman Brothers' default. To address this, capital requirements now include another measure called stressed Value at Risk (SVaR), which considers periods of stress.

Under the new internal models approach called FRTB, both VaR and SVaR are replaced with a single risk measure based on expected loss. This measure is calibrated to the most severe stress period between 2005 and now, typically one year long. However, obtaining high-quality data for such a long period is often not feasible for all risk factors.

To address this data limitation, a simplified treatment is allowed. A long data history is only required for a subset of risk factors that explain at least 75% of the total profit and loss variance. For all other risk factors, only data from the previous year is needed and considered.

Okay, but how do we calculate ES itself in Constrained/Unconstrained ES

Calculating ES involves recursive interplay with liquidity horizons split into 5 major categories.

Understanding Constrained ES [IMCC(Ci)]

ES charge should be also calculate within each asset class (FX, Interest Rate, Equity, Commodity, Credit Spread).

Understanding SES

For risk factors that can not be considered as modellable a Stressed Expected Shortfall (SES) measure should be calculated. Recall assessment of modellability in previous article.

Each non-modellable risk factor (NMRF) needs to be accounted for, by using a stressed scenario, that is at least as cautious as the expected shortfall calibration used for modelled risks (MRFs).

When calculating losses, we should 97.5% confidence level and take into account a period of extreme stress for the specific risk involved. For each risk that can't be easily predicted (since we need to figure out how long it takes to convert an asset into cash during this stressful period). We can determine this by looking at the longest time between two price observations in the past year or using the predetermined time for that particular risk.

The total regulatory capital measure for K risk factors in desks that are eligible for modeling but are considered unmodellable should be determined.

Idiosyncratic credit spread risk factors are specific factors that only impact certain assets. When dealing with non-modellable risk factors resulting from idiosyncratic credit spread risk, banks can use the same stress scenario observations. They can assume zero correlation when combining gains and losses.

Going back to original formula.

You will notice, there is still this term called 'mc' (i.e. multiplication factor) and average of IMCC and SES we have not talked about.

What's this multiplication factor ('mc')

The multiplication factor, which is either set at 1.5 or determined by regulatory authorities, reflects the assessment of the bank's risk management system. Additionally, banks need to add a "plus" to this factor, which is linked to the model's performance after the fact. This introduces a built-in incentive to ensure the model's accuracy. The value of the "plus" can range from 0 to 0.5 and depends on how well the bank's daily Value at Risk (VaR) at the 99th percentile aligns with actual observations using all risk factors.

How do we estimate this multiplication factor if we must?

• Bank’s 1-day static VaR measure at the 99th percentile is compared with desk’s one-day Actual P&L or Hypothetical P&L.
• An exception occurs when either the actual or hypothetical loss of the firm-wide trading book registered in a day of the backtesting period is higher than the corresponding daily risk measure given by the model.
• Multiplication Factor (mc) is then defined based on the number of exceptions generated during VaR 99% backtesting over the most recent 250 days

And what about IMCC and SES average?

It is the average of previous 60 days IMCC and SES that needs to be calculated by the bank quarterly.

Can we have all of this in a spreadsheet?

Each of above is also calculated in several broken parts and cannot be summarised just using words (which you will forget) : so I plan to put all this into perspective using spreadsheet that can show how each of them is calculated (mathematically) while summarising them contextually in brief text (which will also be availed in spreadsheet). I shall share that in comment section in one week or so. This spreadsheet will cover

• How constrained and unconstrained ES is calculated under IMA.
• How ES (Expected Shortfall) is calculated across liquidity horizons
• What set of risk factors are used (reduced set and full set) and when.
• Stressed scenario requirements (for SES)
• DRC for IMA
• How is multiplication factor (mc) estimated

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