MR 18 – Fundamental Review of the Trading Book

FRTB, short for the Fundamental Review of the Trading Book, is the Basel Committee on Banking Supervision’s overhaul of market risk capital. It began with a consultative paper in May 2012. The Committee then collected industry comments, revised the proposal, and asked banks to run Quantitative Impact Studies (QISs), trial calculations showing how a proposed rule would change their capital. The final rules appeared in January 2019, with an agreed start date of January 2023, although many national regulators signalled a delay of a year or two.

What came before FRTB

Under Basel I, market risk capital rested on a value at risk (VaR) figure with a 10-day horizon and 99% confidence. It was a current measure, since each day’s figure drew on market behaviour over the recent past, usually one to four years. Basel II.5 added a stressed VaR, fed with market moves from a 250-day window of severe stress, and each bank searched its history for the 250-day window whose moves would hurt its present portfolio the most.

The main changes FRTB makes

FRTB keeps the stress idea but changes almost everything around it. The risk measure becomes stressed expected shortfall (ES) at 97.5% confidence rather than VaR at 99%. The single 10-day horizon gives way to liquidity horizons that run from 10 days to 120 days depending on how quickly a position could be exited. A detailed standardized approach sits alongside an internal models approach, and every bank must compute capital under the standardized approach even after it has been approved to use its own models. That requirement fits the Committee’s wider decision, announced in December 2017, that total required capital should be at least 72.5% of the standardized figure, a floor phased in over five years and fully in place by 2028. Since 2008 the Committee has steadily trusted internal models less and leaned on standardized numbers as a backstop.

Most FRTB calculations also happen desk by desk, and internal model approval is granted to individual desks, so a bank’s foreign exchange desk can run its own model while its equity desk is held to the standardized approach. Finally, FRTB redraws the boundary between the two books more firmly. Capital is computed so differently in each that banks had been placing instruments wherever the charge was smaller. The incremental risk charge of Basel II.5 had already made this regulatory arbitrage less rewarding, and FRTB goes further with sharper definitions.

VaR gives the loss a bank will not exceed with a chosen probability but says nothing about how bad losses are beyond that point. Expected shortfall fills the gap: it is the average loss given that the loss lands in the tail beyond the VaR. FRTB uses stressed ES at 97.5% confidence, so the tail average comes from market moves seen in a stressed period, just as stressed VaR did.

The two measures almost coincide for a normal distribution

The confidence level of 97.5% was not chosen at random. When losses are normally distributed with mean μ and standard deviation σ, the 99% VaR is μ + 2.326σ and the 97.5% ES is μ + 2.338σ, so the switch barely changes the number for a well-behaved, thin-tailed portfolio. Normal ES has a closed form. Let X be the confidence level and Y the standard normal value exceeded with probability 1 − X:

Because the normal density of the loss at the VaR point is f(VaR), the same result can be written as μ plus σ² times f(VaR) divided by (1 − X).

Why ES matters when tails are heavy

The near equality is a property of the normal case only. Real market losses tend to have fatter tails, and a fat tail drags the tail average far out even when the 99% quantile barely moves. The 97.5% ES can then sit well above the 99% VaR, so FRTB charges more capital to precisely the portfolios with the most severe extreme losses.

Basel I and Basel II.5 measured every position over 10 days, as if an illiquid corporate bond could be unwound as fast as a large cap stock. FRTB drops that assumption. It speaks of risk factors, the market variables a portfolio depends on, and of shocks, the changes in those factors under stressed conditions. Each shock is measured over a liquidity horizon reflecting how long exposure to that factor would take to exit in stressed markets. Five horizons exist: 10, 20, 40, 60 and 120 days.

The table below lists the allocation, grouped by asset class, with a final column showing the internal models category that each horizon maps to (category 1 for 10 days up to category 5 for 120 days). Where a range appears, the horizon depends on the currency concerned.

Two patterns stand out: volatilities usually get longer horizons than the matching prices, and lower credit quality or smaller size pushes a factor toward the long end. A large cap stock price gets 10 days, while credit spread volatility gets the full 120.

Standardized approach capital adds three pieces: a sensitivity based risk charge, a default risk charge and a residual risk add-on. The sensitivity piece uses seven risk classes, each matching a kind of trading desk. They cover equity, commodity, foreign exchange and general interest rate risk, plus three separate credit spread classes, and each class gets a delta, a vega and a curvature risk charge.

The aggregation formula

The delta charge for one risk class combines weighted sensitivities across all of its risk factors:

The bank supplies the sensitivities δ_i, while the Basel Committee fixes the risk weights W_i and the correlations ρ_ij. Correlations are uncertain, so the bank must rerun the calculation with every specified correlation multiplied by 1.25, by 1.00 and by 0.75, and then use the largest of the three results.

Defining the sensitivities

For equity prices, exchange rates and commodity prices, a delta measures the response to a proportional change in the factor. If a 1% rise in a commodity price would add $3,000 to a portfolio, the delta is 3,000/0.01 = 300,000. For interest rates and credit spreads, the delta measures the response to an absolute change. If a one basis point (0.0001) rise in a rate would cut the portfolio’s value by $200, the delta is −200/0.0001 = −2,000,000.

Where the risk weights come from

Take price factors first, where deltas are percentage sensitivities. Setting each W_i to the factor’s daily volatility would make equation 18.1 return the standard deviation of the portfolio’s daily value change, and using stressed daily volatilities would give the stressed daily standard deviation. Regulators scale the stressed daily volatility up further to build in the liquidity horizon and the confidence level. With a stressed daily volatility of 2% and a 20-day horizon, the weight might be 0.02 × √20 × 2.338 = 0.209, where 2.338 is the normal multiplier for a 97.5% ES. Rates and spreads work the same way with weights expressed in basis points, so a multiple of 1 would yield a one-day standard deviation.

Vega risk reuses the delta machinery, which works because a derivative’s value is often close to linear in volatility. In equation 18.1 the factors indexed by i and j become volatilities within the risk class, and each δ_i becomes a vega, the value change for a small move in volatility i, measured in percentage or absolute terms as the bank chooses. Now ρ_ij correlates moves in two volatilities, and W_i reflects the volatility of that volatility, its liquidity horizon and the confidence level.

No diversification across classes

FRTB allows no diversification between risk classes, nor between delta and vega inside a class. So the charge so far is simply the seven delta charges plus the seven vega charges, all added.

Treating each point on a curve as a risk factor

Rates, spreads and volatilities usually come as term structures defined by a set of points. A rate curve is often described by 10 zero-coupon rates: 3 and 6 months, then 1, 2, 3, 5, 10, 15, 20 and 30 years. Each vertex is its own risk factor in equation 18.1, and its delta comes from bumping that vertex alone by one basis point. The Basel Committee publishes a weight per vertex and correlations between vertices of the same curve.

Across two different curves, say US dollar and euro, FRTB simplifies: any point A on one and any point B on the other share one common correlation, whatever their maturities.

Delta and vega charges treat value as linear in each factor, which misses the gamma risk of option books. The curvature risk charge captures it. The bank moves factor i up and down by its risk weight W_i, fully revalues, and compares with the linear prediction of plus or minus W_iδ_i:

A positive figure means curvature made the outcome worse than delta predicted; if both are negative, the charge is zero.

Figure 2 is the profile of a short option position, with both revalued points below the delta line. A long option position is convex, both revalued points sit above the line, and its charge is zero.

Curves and aggregation

For a term structure, the whole curve shifts up and down in parallel by the largest W_i among its points, perhaps the three-month vertex’s weight for a rate curve, and the delta effect is removed vertex by vertex using each δ_i. Charges for different factors are then aggregated much as deltas are, with Committee correlations where diversification is allowed.

FRTB splits credit exposure to a counterparty into two risks that are capitalised separately. Changes in the counterparty’s credit spread are market risk and run through the delta, vega and curvature machinery already described. The possibility of an outright default, often called jump-to-default (JTD) risk, goes into its own default risk charge.

Default risk charge

The default risk charge multiplies each exposure by a loss given default (LGD) and by a default risk weight, and both inputs are set by the Basel Committee. Senior debt, for instance, has an LGD of 75%, and a counterparty rated A carries a default risk weight of 3%. Equity positions also attract a default charge, with an LGD of 100%, since a shareholder recovers little in a default. The rules also specify when long and short exposures may offset each other.

Residual risk add-on

Some risks slip through the sensitivity framework altogether. The residual risk add-on covers them, notably exotic options that cannot be written as combinations of plain vanilla options. It equals the notional amount of the transaction multiplied by a risk weight set by the Committee, and for exotic options that weight is 1%.

The simplified standardized approach

Large banks must use the full standardized approach. For smaller banks, a simplified version was proposed in a June 2017 consultation and kept in the final January 2019 rules; among other things it drops vega and gamma risk. That should ease adoption in places such as the United States, with many small banks doing mostly simple transactions.

Under the internal models approach, a bank estimates stressed ES at 97.5% confidence by a method it chooses, most likely historical simulation. Each risk factor goes into a category by liquidity horizon: category 1 for 10 days, 2 for 20 days, 3 for 40 days, 4 for 60 days and 5 for 120 days.

Overlapping 10-day changes

Basel I and Basel II.5 let banks scale a one-day result up to 10 days with a √10 multiplier. FRTB instead requires actual 10-day changes taken from a past stressed period. Econometricians would rather use non-overlapping windows for independent observations, but that would demand a very long history, so FRTB uses overlapping windows. Scenario one replays every factor’s percentage move from Day 0 to Day 10 of the stress window, scenario two the move from Day 1 to Day 11, and so on one day at a time. Neighbouring scenarios share nine days and so are dependent; the estimate stays unbiased, yet the effective sample shrinks and the result gets noisier. The Committee presumably judges 10-day changes more informative anyway, which holds if daily changes are dependent while successive 10-day changes are close to independent.

Five nested ES calculations

The bank computes ES₁ by applying 10-day changes to every risk factor. ES₂ applies them only to factors in categories 2 to 5, holding category 1 fixed. ES₃ shocks categories 3, 4 and 5; ES₄ shocks categories 4 and 5; and ES₅ shocks category 5 alone. These are combined as follows, where LH_j is the liquidity horizon of category j:

The logic builds up step by step. With only categories 1 and 2, all factors move over a first 10 days and category 2 factors move again over a second, independent 10 days, so an extended square root rule gives the square root of ES₁² + ES₂². Category 3 factors need 40 days, not 20, so they move for 20 more independent days, with their 20-day ES taken as √2 times the 10-day ES, adding 2ES₃² under the root. Continuing through categories 4 and 5 yields equation 18.2, the cascade approach, which works for VaR too.

Five ES calculations per desk add up: six desks need 5 × 6 = 30. The stressed period is the stretch of history most damaging to today’s portfolio, and the search must reach back to 2007.

When some risk factors lack history

Under Basel II.5, some current risk factors had no data for the candidate 250-day stress windows. FRTB lets the search for the stressed period use a subset of factors, provided it covers at least 75% of today’s risk factors. The resulting stressed ES is then grossed up by a ratio taken from the most recent 12 months, namely ES with all risk factors divided by ES with the subset. This step can double the workload from 30 to 60 ES calculations.

Partial ES and the weighted expected shortfall

ES is computed for the whole portfolio and for each of the six desks. A desk’s figure, called a partial expected shortfall, comes from shocking only that desk’s risk factors while the rest stay fixed. Because diversification across desks is lost, the partial figures add up to at least the whole portfolio ES, and in practice to more. FRTB blends the two views into a weighted expected shortfall, with λ set by the Basel Committee at 0.5:

Non-modelable risk factors and the capital formula

A risk factor is non-modelable if it has fewer than 24 observations in a year, or if more than a month passes between consecutive observations. Such factors are excluded from the ES model and capitalised through special stress test rules. Capital for day t is then:

The stressed ES behind capital is not backtested, for two reasons: ES is harder to backtest than VaR, and a stressed measure cannot be backtested at all, since its extreme data should not recur at the stress window’s frequency.

Desk level VaR backtest

Instead, each trading desk backtests a one-day VaR estimated from the most recent 12 months of data, at both 99% and 97.5% confidence. A desk with over 12 exceptions at 99%, or over 30 exceptions at 97.5%, loses internal model status and must calculate capital with the standardized approach until neither condition holds any longer. Regulators may add tests, such as a daily p-value, the probability of a profit below the actual profit or a loss above the actual loss; a sound model gives uniformly distributed p-values.

Profit and loss attribution

The profit and loss attribution test checks that the risk model’s pricing tracks reality. Each day the bank compares the actual profit or loss with the model’s prediction. Let U be the actual minus the model profit or loss on a day and V be the actual profit or loss. Two ratios are computed:

The actual profit or loss here is hypothetical, meaning the profit or loss the desk would have made with no trading during the day, so that intraday trading does not distort the comparison. If the ratios fall outside these ranges on four or more occasions within a 12-month period, the desk must move to the standardized approach.

Exposure to a company’s credit has two faces under FRTB. With credit spread risk, the worry is a change in the company’s spread that reprices instruments linked to it. With jump-to-default risk, the worry is an actual default by the company.

In the internal models approach, credit spread risk is treated like any other market risk. Its liquidity horizons run from 20 to 120 days depending on the type of spread, and credit spread volatility is assigned 120 days. Jump-to-default risk is handled the way default risk is handled in the banking book: the capital charge comes from a VaR with a one-year horizon and 99.9% confidence. In the standardized approach the same split appears as the sensitivity charges for spreads and the separate default risk charge.

Securitizations move to the standardized approach

Basel II.5 introduced the comprehensive risk measure (CRM) charge for securitization products, including collateralized debt obligations and asset-backed securities, and it let approved banks compute that charge with their own models. The Basel Committee concluded that this did not work well, because different banks produced widely different capital numbers for an identical portfolio. Under FRTB, securitizations must use the standardized approach.

Roughly, the trading book holds what the bank intends to trade and the banking book what it expects to hold to maturity. The first attracts market risk capital and the second credit risk capital, computed so differently that the choice of book matters. Banks exploited this in the past, often booking credit-dependent instruments as trading positions since the market risk charge there was lower than the credit risk charge would have been.

Stricter tests for the trading book

FRTB makes the boundary clearer and less a matter of judgment. A declared intent to trade is no longer enough. The bank must actually be able to trade the instrument and to manage its risks on a trading desk, and the instrument’s daily value changes should feed into equity and pose risks to solvency. FRTB also sets out, instrument type by instrument type, which book a position belongs in.

No switching later

The assignment is made when the instrument is first booked, and strict rules prevent moving it afterwards. A transfer is allowed only in extraordinary circumstances, with the examples given being the closure of a trading desk or a change in accounting standards for recognising fair value. Even then, any capital benefit created by the move is disallowed, which removes the incentive to shift positions to cut capital.