**After completing this reading you should be able to:**

- Describe the changes to the Basel framework for calculating market risk capital under the Fundamental Review of the Trading Book (FRTB) and the motivations for these changes.
- Compare the various liquidity horizons proposed by the FRTB for different asset classes and explain how a bank can calculate its expected shortfall using the various horizons.
- Explain the FRTB revisions to Basel regulations in the following areas:
- Classification of positions in the trading book compared to the banking book
- Backtesting, profit and loss attribution, credit risk and securitizations

## Changes to the Basel Framework for Calculating Market Risk Capital under the Fundamental Review of the Trading Book (FRTB)

The Fundamental Review of the Trading Book is an international standard that sets out rules regarding the amount of capital that banks must hold to protect themselves against **market risk**. These rules have been designed to **replace** a series of patches introduced to remove **deficiencies** in the framework that was in place in the run-up to the 2007-2009 financial crisis. The new rules set a higher bar for banks with respect to risk measurement and management. The review process is spearheaded by the **Basel Committee on Banking Supervision.**

A comprehensive consultative document proposing major revisions to the calculation of regulatory capital for market risk was published in May 2012. After that, the Basel Committee on Banking Supervision invited banks to scrutinize the proposals and assess their applicability. The final version of the rules was published in January 2016. Guidelines released initially required compliance with the new rules by the end of 2016, but this was pushed forward to December 2022 following another round of consultations with banks and other stakeholders. It is important for risk managers to understand the nature of the proposed changes and the new calculation methodology.

### Initial Requirements

Under Basel I regulations, banks were required to calculate market risk capital based on a **value at risk** calculated for a **10-day horizon** with a **99% confidence interval**. This process generated a very “current” VaR because the 10-day horizon incorporated a recent period of time, typically one to four years.

Basel II.5 regulations stipulated an additional requirement. Banks had to add a stressed VaR measure to the current value captured with the 10-day VaR. But unlike the VaR, the stressed VaR used a 250-day period. The purpose of the stressed VaR was to measure the behavior of market variables during a 250-day period of stressed market conditions. A **period of stress** was defined as a period where the observed movements in market variables would lead to significant financial stress for the current portfolio. In that regard, banks enjoyed a bit of freedom because they were allowed to go back through time and self-select a 250-day window of time that would have triggered significant financial difficulty for their current portfolio.

### Motivation for changes

After consultations, it was felt that the 10-day VaR was really not the best measurement for a bank’s true risk. As we know, the VaR simply answers the question, “How bad can things get?” The VaR tells us, with a given level of confidence, that losses will not exceed a certain threshold. But there lies the problem because the VaR is **indifferent** to the losses that can occur beyond this threshold. In fact, it has been argued that because of this indifference, portfolios with identical VaR can carry different risk. A further problem identified has much to do with the fact that the VaR is an **incoherent** measure; that the risk of a portfolio can be larger than the sum of the individual positions that make up the portfolio.

Suitability of the 10-day VaR (not just the VaR itself) was also questioned.

Although the 10-day VaR is undoubtedly useful for day-to-day internal risk management purposes, it has been argued that it does not meet the objective of prudential regulation which seeks to ensure that banks have sufficient capital to survive low probability, or “tail”, events. Further weaknesses identified in the 10-day VaR include:

- Its inability to adequately capture credit risk;
- Its inability to capture market liquidity risk;
- Its inability to incentivize banks to monitor tail risk
- To a limited extent, the inadequate capture of basis risk.

### Changes

FRTB proposes the abandonment of the 10-day VaR at 99% confidence in favor of the **expected shortfall** with a 97.5% **confidence interval.** The move is informed by the fact that the expected shortfall helps overcome some of the weaknesses identified in VaR. In particular, the ES does not ignore tail losses. It seeks to answer the question: “If things get bad, what is the estimated loss on the bank’s P&L?”

Perhaps an example will help you get the true picture here. Let’s consider a $100 million bond portfolio with a 2% probability of default. The table below details the portfolio’s default schedule.

$$ \textbf{Table 1 – Example Default Schedule for } $\textbf{100 Million Bond Portfolio} $$

$$ \begin{array}{c|c|c} \textbf{Confidence level} & \textbf{Default} & \textbf{Loss} \\ \hline {95\%} & \text{No} & {$0} \\ \hline {96\%} & \text{No} & {$0} \\ \hline {97\%} & \text{No} & {$0} \\ \hline {98\%} & \text{No} & {$0} \\ \hline {99\%} & \text{Yes} & {$\text{100 million}} \\ \hline {99.9\%} & \text{Yes} & {$100 \text{ million}} \\ \end{array} $$

At the 95% confidence, notice that the expected loss is zero and therefore solely relying on the 95% VaR would imply a loss of $0. But would we end up with a similar value if we used the expected shortfall as our preferred risk measure?

By definition, the expected shortfall, at \(\alpha\) confidence, measures the potential dollar loss conditional on the loss exceeding the VaR at α level of confidence. In three out of the 5 loss assessment levels beyond the 95% level, the expected loss is $0, but in the remaining two cases, the expected loss is $100 million due to default. That means 40% of the tail risk would yield a loss, so the expected shortfall is $40 million (i.e., 40% × $100 million). So while the 95% VaR is $0, the 95% ES is $40 million. Clearly, this presents a very different risk perspective than using the VaR alone.

It is important to note that for a normal distribution a mean of \(\mu\) and standard deviation of \(\sigma\), the 10-day VaR at the 99% confidence level yields approximately the same result as the expected shortfall at the 97.5 confidence level. Precisely, the 99% VaR is given by \(\mu+2.326\sigma\), and the 97.5% expected shortfall is given by \(\mu+2.338\sigma\). For non-normal distributions, however, the two measures are not equivalent. When the loss distribution has a heavier tail than a normal distribution, the 97.5% ES can be considerably greater than the 99% VaR.

Under FRTB proposals banks no longer have to combine the 10-day VaR and the 250-day stressed VaR risk measures. Instead, they are required to calculate capital based exclusively on expected shortfall using a 250-day stressed period. However, banks have retained the freedom to self-select a 250-day window of exceptionally difficult financial stress.

## Regulatory Capital under the Standardized Approach

Under the standardized approach, the capital requirement is the simple sum of three components:

- Risk charges under the sensitivities based method
- A default risk charge, and
- A residual risk add-on

**Component 1: Risk charges under the sensitivities based method**

Under the first component, seven risk classes (corresponding to trading desks) are defined:

- General interest rate risk,
- Foreign exchange risk,
- Commodity risk,
- Equity risk,
- Credit spread risk – non securitization,
- Credit spread risk – securitization,
- Credit spread risk – securitization correlation trading portfolio

For each of these classes, a delta risk charge, vega risk charge, and curvature risk charge are calculated.

The delta risk charge takes into account the weight and weighted sensitivities (deltas) of each of the above risk classes, as well as the correlation between risk classes:

$$ \text{Delta risk charge} =\sum_{\text i} \sum_{\text j} \rho_{\text {ij}} \sigma_{\text i} \sigma_{\text j} {\text w}_{\text i} {\text w}_{\text j} $$

While the risk weights \(\text w_{\text i}\) and correlations \(\rho_{\text {ij}}\) are determined by the Basel Committee, the weighted sensitivities, \(\delta_{\text i}\), are determined by individual banks. Let’s say, for example, that a 1% increase in the price of a commodity, say, gold increases the value of the portfolio by $5,000. In this case, the delta will be 5,000/0.01 = 500,000.

**Component 2: Default Risk Charge**

The default risk charge is intended to capture jump-to-default-risk, i.e., the loss that would be suffered by the investor if the issuer of the bond or equity were to default. The default risk charge is calculated for every existing instrument separately and is a function of the face (notional) amount, the market value of the instrument and the Loss Given Default (LGD). To determine the default risk charge, there are three steps involved:

**Step 1: Gross JTD risk positions (Gross JTD)**

JTD loss amounts for each instrument subject to default risk are determined.

**Step 2: Net JTD risk positions (Net JTD)**

All long and short positions featuring the same obligor are offset to produce net short or net long amounts.

For offsetting to occur, the short exposure must have the same or lower seniority relative to the long exposure. For example, a long exposure in a bond may offset a short exposure in equity, but a short exposure in a bond cannot offset a long exposure in equity.

**Step 3: Discounting**

The net short exposures are discounted by a hedge benefit ratio.

**Step 4: Weighting**

Default risk weights are applied to arrive at the capital charge

Both the LGD and the risk weights are specified by the Basel Committee. For example, the LGD for senior debt is specified as 75% and the default risk for a counterparty rated A is 3%.

**Component 3: Residual Risk Add-on**

The residual risk add-on is to be calculated for all instruments bearing residual risk, i.e., instruments whose total risk cannot be captured via the delta/vega/curvature approach.

Instruments with a residual risk add-on are those that are subject to vega or curvature risk capital charges in the trading book but have pay-offs that cannot be written or perfectly replicated as a finite linear combination of vanilla options. To calculate the add-on, the notional amount of the transaction is multiplied by a risk weight that is specified by the Basel Committee. The risk weight of exotic options is 1%.

### A Simplified Approach

It is worth noting that the standardized approach applies to large systematically important banks. In June 2017, the Basel committee put forward a consultative document outlining a simplified version of the standardized approach for use by small banks (banks with a low concentration of trading book activity, or those with insufficient infrastructure to successfully implement the sensitivities-based method).

The simplified version makes a number of significant simplifications. Most notably, it removes capital requirements for vega and curvature risks, simplifies the calculation of basis risk, and reduces the correlation scenarios to be applied in associated calculations.

## Comparing the Various Liquidity Horizons Proposed by the FRTB for Different Asset Classes

In broad terms, market liquidity refers to the capacity to offset or extinguish (eliminate) a risk position, over a short time period, at current market prices. According to FRTB proposals, market liquidity across trading positions should be based on the concept of liquidity horizons. Under FRTB, the term liquidity horizon represents “the time required to sell a financial instrument or hedge all its material risks, in a stressed market, without materially affecting market prices.”

Researchers and the Basel Committee on Supervision agree that a uniform 10-day liquidity horizon as originally proposed in Basel regulations is inappropriate because it is nearly impossible for banks to exit all risk positions within a 10-day period due to market illiquidity. As such, five different liquidity horizons have been proposed: **10 days, 20 days, 60 days, 120 days,** and **250 days**. For example, the calculation of regulatory capital for a 120-day horizon (essentially 6 months’ worth of trading days) is intended to shield a bank from significant risks while waiting six months to recover from underlying price volatility.

The new proposals specify a liquidity horizon for each major risk factor. For example,

- Investment grade sovereign credit spreads are assigned a 20-day horizon;
- Structured products credit spreads are assigned a 250-day horizon
- Precious metals are assigned a 20-day horizon

The full list of risk factors and the corresponding liquidity horizons is given below.

$$ \textbf{Table 2 – Risk Factor Liquidity Horizons} $$

$$ \begin{array}{c|c} \textbf{Risk Factor Horizon} & \textbf{(days)} \\ \hline \text{Interest rate (dependent on currency)} & {10-60} \\ \hline \text{Interest rate volatility} & {60} \\ \hline \text{Credit spread: sovereign, investment grade} & {20} \\ \hline \text{Credit spread: sovereign, non-investment grade} & {40} \\ \hline \text{Credit spread: corporate, investment grade} & {40} \\ \hline \text{Credit spread: corporate, non-investment grade} & {60} \\ \hline \text{Credit spread: other} & {120} \\ \hline \text{Credit spread volatility} & {120} \\ \hline \text{Equity price: large cap} & {10} \\ \hline \text{Equity price: small cap} & {20} \\ \hline \text{Equity price: large cap volatility} & {20} \\ \hline \text{Equity price: small cap volatility} & {60} \\ \hline \text{Equity: other} & {60} \\ \hline \text{Foreign exchange rate (dependent on currency)} & {10-40} \\ \hline \text{Foreign exchange volatility} & {40} \\ \hline \text{Energy price} & {20} \\ \hline \text{Precious metal price} & {20} \\ \hline \text{Other commodities price} & {60} \\ \hline \text{Energy price volatility} & {60} \\ \hline \text{Precious metal volatility} & {60} \\ \hline \text{Other commodities price volatility} & {120} \\ \hline \text{Commodity (other)} & {120} \\ \end{array} $$

### Calculating the Expected Shortfall Using the Internal Models Approach

In line with the proposals under FRTB, the internal models approach (IMA) requires banks to estimate stressed ES with a 97.5% confidence.

Once risk factors have been allocated liquidity horizons as indicated in the table above, they are then put into 5 categories as follows:

- Category 1: risk factors with 10-day horizons.
- Category 2: risk factors with 20-day horizons.
- Category 3: risk factors with 60-day horizons.
- Category 4: risk factors with 120-day horizons.
- Category 5: risk factors with 250-day horizons.

The use of these five categories is informed by the fact that risk factor shocks might not be correlated across liquidity horizons. Under the IMA, the expected shortfall is measured over a base horizon of 10 days. The expected shortfall is measured through five successive shocks to the categories in a nested pairing scheme. First, banks calculate ES when 10-day changes are made to all risk factors, with the resulting ES denoted \(\text{ES}_1\). Next, they are required to calculate ES when 10-day shocks are made to all risk factors in categories 2 and above but holding category 1 constant (The resulting value is denoted \(\text{ES}_2\)). The process continues with ES calculation when 10-day changes are made to all risk factors in categories 3 and above but holding categories 1 and 2 constant (The resulting value is denoted \(\text{ES}_3\)). They are then required to calculate ES when 10-day changes are made to all risk factors in categories 4 and 5 with risk factors in categories 1,2, and 3 being kept constant. (the resulting value is denoted \(\text{ES}_4\)). Finally, 10-day changes are made to risk factors in category 5 (the resulting value is denoted \(\text{ES}_5\)).

$$ \textbf{Table 3 – Expected Shortfall Using the Internal Models Approach} $$

$$ \begin{array}{c|c|c} \textbf{Shocks} & \textbf{Held constant} & \bf{\text{ES}_{\text j}} \\ \hline \text{All risk factors} & \text{None} & {\text{ES}_1} \\ \hline \text{Categories 2,3,4,5} & \text{Category 1} & {\text{ES}_2} \\ \hline \text{Categories 3,4,5} & \text{Categories 1, 2} & {\text{ES}_3} \\ \hline \text{Categories 4,5} & \text{Categories 1, 2, 3} & {\text{ES}_4} \\ \hline \text{Category 5} & \text{Categories 1, 2, 3, 4} & {\text{ES}_5} \\ \end{array} $$

The liquidity-adjusted ES is then determined as follows:

$$ \sqrt{ \text{ES}_1^2+\sum_{\text j=2}^5 \left(\text{ES}_{\text j} \sqrt{\cfrac {\text{LH}_{\text j}-\text{LH}_{\text j-1}}{10}}\right)^2 } $$

Where \(\text{LH}_{\text j}\) is the liquidity horizon for category j.

Pending formal approval of the internal models approach, banks are required to continue applying the **revised standardized approach**, RSA. Under the RSA, risks (assets) are grouped into “buckets,” which are created based on the concept of liquidity horizons.

The standardized risk measure for each bucket is then calculated using the following formula:

$$ \sum_{\text i} {\text w}_{\text i}^2 {\text v}_{\text i}^2+2\sum_{\text i} \sum_{\text j < {\text i}} \rho_{\text {ij}} {\text w}_{\text i} {\text w}_{\text j} {\text v}_{\text i} {\text v}_{\text j} $$

Where:

\(\text v_{\text i}\) = value of the i^{th} risk factor (i^{th} instrument)

\(\text w_{\text i}\) = Regulatory risk weight of risk factor i (determined by the Basel Committee)

\(\rho_{\text {ij}}\) = the correlation between risk factors I and j (determined by the Basel committee)

As seen above, the basic concept of a bucket’s risk measure is to multiply the instrument’s market value by a risk weight. A bank’s regulatory capital is then determined as the aggregate of the standardized risk measures for all the buckets.

## FRTB Revisions to Basel Regulations in the Classification of Positions

One of the issues extensively addressed in FRTB has much to do with regulatory modifications with respect to the trading book and banking book. Historically, the trading book consists of instruments that the bank intends to trade, while the banking book consists of instruments that are expected to be held to maturity at historical cost. Instruments in the banking book are subject to credit risk capital whereas those in the trading book are subject to market risk capital.

The two categories of capital are calculated in quite different ways, a situation that has been blamed for regulatory arbitrage in years past (whereby firms capitalize on loopholes in **requirements** in order to circumvent unfavorable **capital needs**). For example, some banks are on record for holding credit-dependent instruments in the trading book because by so doing, they are subject to less regulatory capital than they would be if they had been placed in the banking book.

FRTB has attempted to make the distinction between the banking book and the trading book clearer and less subjective. To be allocated to the trading book, the bank must demonstrate more than an intent to trade. Precisely, some two criteria must be met:

- The bank must be able to trade the asset, and physically manage the associated risks of the underlying asset on the trading desk.
- The day-to-day price fluctuations must affect the bank’s equity position and pose a risk to bank solvency.

If these two conditions are met, the bank is allowed to allocate the asset to the trading book.

In addition, FRTB proposes further restrictions on reclassification.

Once an asset has been acquired and initially assigned to either the trading book or the banking book, it cannot be reclassified except for extraordinary circumstances. The goal is to avoid a situation where banks reclassify assets if such a move results in more favorable capital requirements. Good examples of “extraordinary circumstances” would be a firm-wide shift in accounting practices or a total closure of the trading desk.

## FRTB Revisions to Backtesting, Profit and Loss Attribution, Credit Risk and Securitizations

### Credit Risk

In an attempt to further mitigate regulatory arbitrage, FRTB distinguishes two types of credit risk exposure to a company:

*Credit spread risk*is the risk that the company’s credit spread will change, causing the mark-to-market value of the instrument to change.*Jump-to-default risk*is the risk that there will be a default by the company.

Credit spread risk can be addressed by calculating the expected shortfall.

Jump to default risk is subject to an **incremental default risk** (IDR) charge. All risk assets that are exposed to the risk of default – including equities – are subject to IDR calculation. The IDR charge is calculated based on a 99.9% VaR with a one-year time horizon.

### Profit And Loss (P&L) Attribution & Back-testing

Under FRTB, banks can either use their own internal models or a standardized approach to calculate capital. For a desk to qualify for the **internal models approach**, it must pass two tests: a profit and loss attribution test and a back-test.

Under the profit and loss attribution test, banks are required to compare the actual profit or loss in a day with that predicted by their models. Two ratios must be established:

$$ \text{mean ratio}=\cfrac {\text{mean of U}}{\text{standard deviation of V}} $$

$$ \text{variance ratio}=\cfrac {\text{variance of U}}{\text{variance of V}} $$

where:

U = the difference between the actual and model profit/loss in a day, and

V = the actual profit/ loss in a day.

The mean ratio is considered acceptable if it lies between -10% and 10%, while the variance ratio must be less than 20%. To use the internal models approach, there must not be **four** or more situations in a 12-month period where the ratios are outside these ranges. Otherwise, the desk must use the standardized approach for determining capital.

FRTB does not recommend back-testing of the stressed ES measures that are used to calculate capital under the internal models approach for two reasons. First, it is more difficult to back-test ES than VaR. Second, it is fairly difficult to back test stressed ES measures because it is statistically very unlikely for stressed scenarios to occur in the future with the same frequency. For these reasons, FRTB recommends back-testing the VaR.

What happens is that the hypothetical P&L and Actual P&L are both compared to the 1-day holding period VaR over the past 12 months. Both 99% and 97.5% confidence levels are used. Use of the internal models approach is conditioned on an experience of not more than 12 exceptions at 99% or not more than 30 exceptions at 97.5% in the most recent 12 month period. Otherwise, all positions must be capitalized using the standardized approach.

### Securitization

Basel II.5 introduced the Comprehensive Risk Measure (CRM) charge to cover the risks in securitized assets such as asset-backed securities and collateralized debt obligations. Under Basel II.5 guidelines, a bank (with regulatory approval) is free to use its own models to determine the CRM charge. Through the FRTB, the Basel Committee is of the view that this is unsatisfactory because it results in too much variation in the capital charges calculated by different banks for the same portfolio. Therefore, FRTB requires banks to use the standardized approach for securitizations.

## Question

Billow Bank has sometimes been finding it difficult to search for past stressed periods using all market variables. This fact can be attributed to shortage of historical data for some of the variables. How does the Fundamental Review of the Trading Book (FRTB) deal with this challenge?

- The FRTB can allow the stressed period computations to be based on a subset of market variables and the results scaled up by the ratio of the current Expected Shortfall using the full set of risk factors to the current Expected Shortfall measure using the reduced set of factors.
- One-day changes in market variables are typically considered for the computation of a one-day VaR, which is then multiplied by the square root of 10 to determine an estimate of the 10-day VaR.
- A shock equal to the change between Day 1 and Day 11 for the equity price would be considered by the second trial, the same way it would consider a shock equal to the change between Day 1 and Day 121 for the credit spread and so on.
- All the above

The correct answer is **A**.

To easily search for past stressed periods using all market variables, the calculation of stressed period can be based on a subset of market variables and the results scaled up by the ratio of the current Expected Shortfall using the full set of risk factors to the current Expected Shortfall measure using the reduced set of factors.

Options **B** and **C** do not deal with the challenge of searching for past stressed periods using market variables.