Fundamental Review of the Trading Book: formula sheet

Key formulas

Basel II/2.5 internal model measure

VaR at 99% confidence, 10-day horizon (plus stressed VaR, IRC, CRM under Basel 2.5)

The old approach. Several separate charges added together.

FRTB internal model measure

Expected shortfall at 97.5% confidence, calibrated to a period of stress

ES averages losses beyond the cutoff, so it captures tail severity. It replaces VaR.

Liquidity horizons in FRTB

Risk-factor horizons of 10, 20, 40, 60 or 120 days

Less liquid risk factors get longer horizons. Replaces one flat 10-day horizon.

Normal-distribution check

ES at 97.5% ≈ VaR at 99% for a normal distribution

Used to explain why the confidence level moved. Not true for fat-tailed losses.

Default assignment rule

Instrument on the mandatory banking book list → banking book, whatever the intent; otherwise trading intent or hedge of a trading position → trading book; otherwise → banking book

The mandatory banking book list covers unlisted equity, real estate, retail and SME credit, and equity investments in funds that cannot be looked through or lack daily prices. Trading intent cannot override it, and there is no exception route. Presumptive trading book list items go to the trading book unless the supervisor approves an exception.

Reclassification add-on

Add-on = max(0, Capital before switch − Capital after switch)

Applies only when a switch reduces total capital. It is the capital difference measured at the time of the switch, held constant (not recomputed) and it runs off as the relevant positions mature. If the switch does not reduce capital, the add-on is zero.

Reclassification trigger

Switch only if extraordinary circumstances AND supervisor approval AND public disclosure

All three conditions are needed. Changes in market conditions or in accounting or tax treatment alone do not qualify.

Weighted sensitivity

WS_k = RW_k × s_k

s_k is the net sensitivity to risk factor k (for example PV01 for GIRR). RW_k is the prescribed risk weight.

Within-bucket aggregation

K_b = √( max(0, Σ WS_k² + Σ Σ ρ_kl × WS_k × WS_l) ), for k ≠ l

Used for delta and vega. ρ_kl is the prescribed correlation between risk factors in the same bucket.

Across-bucket aggregation

Charge = √( Σ K_b² + Σ Σ γ_bc × S_b × S_c ), for b ≠ c

S_b is the sum of weighted sensitivities in bucket b (K_b is the cap/floor in the alternative specification). γ_bc is the cross-bucket correlation.

Three correlation scenarios

High: ρ × 1.25, capped at 100% | Medium: ρ | Low: max(2ρ − 100%, 75% × ρ)

Apply to both ρ and γ. Compute capital under each scenario and use the highest result.

GIRR tenor correlation

ρ = max( e^(−θ × |T_k − T_l| ÷ min(T_k, T_l)), 40% ), with θ = 3%

Between different tenors of the same curve in the same currency. Cross-currency bucket correlation is 50%.

Vega risk weight

RW = min( RW_σ × √LH ÷ √10, 100% )

LH is the liquidity horizon of the risk class, in days. Vega sensitivity = vega × implied volatility.

Curvature risk position

CVR_up = −[ V(x + shock) − V(x) − RW_curv × s ]; CVR_down = −[ V(x − shock) − V(x) + RW_curv × s ]

s is the delta sensitivity of the instrument. A positive CVR is a loss. Within-bucket curvature aggregation uses ρ squared. Curvature applies to instruments with optionality.

Jump-to-default (JTD)

Long: max(LGD × notional + P&L, 0) | Short: min(LGD × notional + P&L, 0), with notional negative for shorts

Prescribed LGD: 75% senior debt, 100% non-senior and equity, 25% covered bonds.

DRC bucket charge

DRC_b = max(0, Σ RW × net long JTD − WtS × Σ RW × |net short JTD|)

WtS = Σ net long JTD ÷ (Σ net long JTD + Σ |net short JTD|). Netting only across the same obligor. Bucket charges are summed with no diversification.

Residual risk add-on

RRAO = 1.0% × gross notional (exotic underlyings) + 0.1% × gross notional (other residual risks)

Applies to instruments with exotic underlyings or other residual risks, such as gap, correlation or behavioural risks.

Total SA capital

Capital = SBM + DRC + RRAO

SBM is the highest of the three correlation scenario results.

Expected shortfall at 97.5%

ES(97.5%) = average of losses beyond the 97.5th percentile loss

Worst 2.5% tail. Basel uses 97.5% ES instead of 99% VaR.

Liquidity-horizon scaled ES

ES = √[ ES(P, T)² + Σ over j≥2 ( ES(P, Qj) × √((LHj − LHj−1) ÷ T) )² ]

T = 10 days. ES(P, Qj) uses only factors with horizon at least LHj. LH values: 10, 20, 40, 60, 120 days.

Stressed calibration

ES = ES(R,S) × max[ ES(F,C) ÷ ES(R,C), 1 ]

R = reduced factor set, F = full set, S = stress period, C = current period. Each ES term is the liquidity-adjusted ES computed on that factor set and period. The ratio is floored at 1. The reduced set must explain at least 75% of the full-set ES.

Internally modelled capital charge (IMCC)

IMCC = 0.5 × unconstrained ES + 0.5 × Σ risk-class constrained ES

ES terms are over IMA-approved desks only (desks that pass backtesting and P&L attribution).

IMA capital

Capital = max[ IMCC(t−1) + SES(t−1), mc × IMCC(avg) + SES(avg) ] + DRC

(t−1) = previous day's figure. avg = 60-day average. mc is the backtesting-based multiplier, at least 1.5. SES is the stress scenario capital requirement for NMRFs. DRC is the separate default risk charge. Total IMA capital is also subject to the output floor.

Risk factor eligibility test (RFET)

Modellable if: at least 24 real price observations in the last 12 months, with no 90-day period holding fewer than 4, OR at least 100 observations in the last 12 months

Observations must be real prices, and an observation counts at most once per day. The 24-observation route also needs at least 4 observations in every 90-day period. The 100-observation route has no 90-day condition. Fail both routes and the factor is an NMRF.

NMRF capital charge

SES is calibrated to a loss at least as large as the ES calibration (97.5%) over the stress period, with a liquidity horizon of at least 20 days, or the factor's own horizon if longer

NMRFs are charged outside the ES model. Aggregation depends on the factor type and the assumed correlations.

Hypothetical vs risk-theoretical P&L

PLA compares HPL (front office) with RTPL (risk model) using Spearman correlation and KS distance

HPL and RTPL are compared daily over the same 250 days, with RTPL using only the risk factors in the model.

PLA zones: Spearman correlation

Green: ρ > 0.80 | Amber: 0.70 ≤ ρ ≤ 0.80 | Red: ρ < 0.70

Higher correlation is better.

PLA zones: KS distance

Green: KS < 0.09 | Amber: 0.09 ≤ KS ≤ 0.12 | Red: KS > 0.12

Lower distance is better. Desk zone is the worse of the two metrics.

Desk-level backtesting limit

Desk fails if exceptions over last 250 days: > 12 at 99% VaR or > 30 at 97.5% VaR

Uses both actual and hypothetical P&L for exception counting.

Bank-level capital multiplier (99% VaR exceptions in 250 days)

Green: 0–4 exceptions, multiplier add-on 0 | Amber: 5–9, add-on 0.20 to 0.42 | Red: 10 or more, add-on 0.50

Under FRTB the base multiplier is 1.5, so the total ranges from 1.50 (green) to 2.00 (red). Amber add-ons by count: 5 = 0.20, 6 = 0.26, 7 = 0.33, 8 = 0.38, 9 = 0.42. Do not mix this with the older Basel market risk traffic light scale, which has a base of 3.0, add-ons from 0.40 to 1.00 and a total range of 3.0 to 4.0. Check which framework the question uses.

Gross JTD for a long position

JTD(long) = max(LGD × Notional + P&L, 0)

P&L = market value − notional. Prescribed LGD: 100% for equity and non-senior debt, 75% for senior debt, 25% for covered bonds.

Gross JTD for a short position

JTD(short) = min(−LGD × |Notional| + P&L, 0)

Shorts give negative JTD (a gain on default). Net long and short JTD to the same obligor before applying risk weights.

DRC risk weights (non-securitization)

AAA 0.5%; AA 2%; A 3%; BBB 6%; BB 15%; B 30%; CCC 50%; unrated 15%; defaulted 100%

Weights are default probabilities by credit quality. Memorise the order and the unrated figure.

Hedge benefit ratio

WtS = ΣNet JTD(long) ÷ (ΣNet JTD(long) + Σ|Net JTD(short)|)

Computed within each bucket. A higher ratio means shorts get less credit.

Bucket DRC

DRC(bucket) = max[ Σ RW × Net JTD(long) − WtS × Σ RW × |Net JTD(short)| ; 0 ]

Total DRC is the sum of bucket charges. There is no diversification across buckets.

BA-CVA counterparty term

SCVA(c) = (1 ÷ α) × RW(c) × Σ [M × EAD × DF] over netting sets, with α = 1.4

DF = 1 for banks using SA-CCR for exposure. For IMM banks DF = (1 − e^(−0.05M)) ÷ (0.05M). RW depends on sector and investment-grade or high-yield status.

Reduced BA-CVA

K(reduced) = √[ (ρ × ΣSCVA(c))² + (1 − ρ²) × ΣSCVA(c)² ], ρ = 50%

Sums are over counterparties. No hedge recognition in the reduced version. The full version adds hedge recognition for eligible hedges.

SA-CVA structure

Capital = m(CVA) × (K delta + K vega)

Sensitivity-based, built on counterparty credit spread and exposure risk factors. It needs supervisory approval. The multiplier m(CVA) is set by the supervisor, default 1.

Quick revision

• FRTB replaces VaR with Expected Shortfall to capture tail risk beyond a single quantile.
• Basel 2.5 added stressed VaR, the incremental risk charge and the comprehensive risk measure as patches.
• The trading book boundary is based on trading intent and tighter rules limit moving positions between books.
• The Standardized Approach is sensitivities-based and applies to all banks.
• The Internal Models Approach is approved desk by desk, not for the whole bank at once.
• Liquidity horizons vary by risk factor, replacing a single flat horizon.
• Risk factors that lack enough real price observations are non-modellable and get a separate stress-based charge.
• Desks must pass P&L attribution and backtesting to keep using internal models.
• A desk that fails the tests falls back to the Standardized Approach.
• CVA risk has its own capital framework, separate from the market risk charge.
• Securitization positions have separate treatment and are not simply handled with internal models.
• Always state the approach first, then the measure, then the interpretation.

Common mistakes

• Saying Basel 2.5 was the full redesign of market risk rules. Fix: Basel 2.5 was a quick 2009 patch that added charges. FRTB was the fundamental redesign.
• Stating that FRTB uses 99% VaR. Fix: FRTB uses ES at 97.5%. VaR at 99% belongs to Basel II and 2.5.
• Saying the bank may freely reclassify when its business strategy changes. Fix: Remember that reclassification is allowed only in extraordinary circumstances, with supervisor approval and public disclosure.
• Treating the presumptive list as absolute with no exceptions. Fix: The list is a rebuttable presumption. A bank can seek an exception with supervisory approval and must bear the burden of proof.
• Taking the average of the three correlation scenarios, or using only the medium scenario. Fix: Compute capital under each scenario and take the highest. The charge is conservative by design.
• Applying weights before netting sensitivities, or netting across different tenors or buckets before weighting. Fix: Net sensitivities only for the same risk factor. Then weight. Then use correlations to give partial offset across different factors.
• Saying FRTB uses 99% VaR with a longer horizon. Fix: Remember that IMA uses 97.5% ES. Under normality it is roughly equal to 99% VaR, but it captures tail size.
• Using the 97.5th percentile loss (the VaR) as the ES, or averaging the wrong observations. Fix: ES is the mean of losses beyond the cutoff. It is always at least as large as VaR at the same confidence.
• Taking the better of the two PLA metrics as the desk result. Fix: The desk takes the worse zone of Spearman and KS.
• Adding NMRFs into the Expected Shortfall model. Fix: NMRFs are excluded from ES and charged separately under the stress scenario capital charge.

Exam tips

• Memorise the Basel 2.5 additions: stressed VaR, IRC and CRM, with what each covers.
• Know the numbers: 99% VaR old, 97.5% ES new, horizons 10 to 120 days.
• Questions often ask why, not what. Link each flaw to its FRTB response.
• Watch for options that attribute old features to FRTB, or the reverse.
• Remember the boundary and regulatory arbitrage issue. It is easy to skip but often tested.
• Questions often hinge on one word: extraordinary, presumptive, approval or disclosure. Read for it.
• If an option says the bank may reclassify at its own discretion, it is almost certainly wrong.
• Know both directions: the add-on only matters when the switch reduces capital.