Products Market risk and margin

FRTB-SA

Fundamental Review of the Trading Book, standardised approach

Market risk capital with a derivation attached to every figure.

FRTB-SA sets bank capital for trading book market risk from sensitivities, default risk and residual risk add-ons. GraphAtlas runs the full sensitivities-based method, shows which correlation scenario binds, and allocates the charge to desks, risk factors and trades.

Try the calculation Request a walkthrough

Verification: 6 statements. 0 proved, 1 certified, 5 tested.

Regime: Basel Framework, MAR21 to MAR23
Set by: Basel Committee on Banking Supervision
European Commission and the European Supervisory Authorities
Built for: Market risk and regulatory reporting
Desk heads and capital management
Model validation
Version: 5.1

Try it

Run it on sample data

Edit any sensitivity and watch the charge, the binding scenario and each risk factor’s share move. Then see why the correlation tables need a floor.

Fig. 1 GIRR delta capital for an illustrative two-currency book

Verifying receipt

Capital charge, USD

36,584,378 US dollars

The charge is the highest of three correlation scenarios. Low correlation binds.

Correlations cut to 2ρ − 1 or 75% of ρ, whichever is larger.

Share of the Low scenario charge, by currency bucket and risk factor
Bucket and tenor	PV01	Weighted	Share	Share, drawn to scale
	K 34.32m		33,903,114
		918,000	637,702
		2,142,000	1,867,707
		5,952,000	5,776,397
		12,402,000	11,610,487
		8,136,000	6,981,665
		31,845,000	23,907,478
		−23,496,000	−12,325,828
		−8,019,000	−3,227,700
		−6,237,000	−1,998,301
		3,201,000	673,507
	K 5.98m		2,681,264
Charge, low scenario			36,584,378	Shares sum exactly
Shares sum exactly

Fig. 2 Are the prescribed GIRR correlations a valid correlation matrix?

Verifying receipt

The prescribed correlations, used as they are.

Correlation between tenors, medium scenario

Tenor	3m	6m	1y	2y	3y	5y	10y	15y	20y	30y
3m	100	97	91	81	72	57	40	40	40	40
6m	97	100	97	91	86	76	57	42	40	40
1y	91	97	100	97	94	89	76	66	57	42
2y	81	91	97	100	99	96	89	82	76	66
3y	72	86	94	99	100	98	93	89	84	76
5y	57	76	89	96	98	100	97	94	91	86
10y	40	57	76	89	93	97	100	99	97	94
15y	40	42	66	82	89	94	99	100	99	97
20y	40	40	57	76	84	91	97	99	100	99
30y	40	40	42	66	76	86	94	97	99	100

Marks an entry held at the 40% floor, in the medium scenario.

5y against 10y: exp(−3% × |5 − 10| ÷ 5) = 97.04%. That is above the 40% floor, so it stands.
Rule: Basel Framework, MAR21.46 and Table 2, MAR21.6. January 2019 text.

Eigenvalues, smallest first

of 10 below zero

A real correlation matrix has none, because no mix of risks can have a negative variance. The eigenvalues are computed here in floating point. The check further down is what carries the claim.

The ten eigenvalues of the matrix, smallest first
#	Value	Drawn to scale
1	−0.1650
2	−0.1063
3	0.0002
4	0.0068
5	0.0079
6	0.0114
7	0.0914
8	0.3000
9	1.7676
10	8.0859

A book that shows it

These ten weighted sensitivities, in currency units, give a quadratic form w^TRw below zero. A matrix with no negative eigenvalue can never do that.

3m: 761,242
6m: −82,082
1y: −1,000,000
2y: −340,798
3y: 73,905
5y: 586,548
10y: 753,976
15y: 237,253
20y: −166,172
30y: −829,740

w^TRw in floating point−559,882,732,895

The sum above is rounded at every step. The check below does it again in exact fractions, on the matrix as it is stored.

Charge of the book above under each scenario
What the rule does with this book	Bucket charge K	Capital
High scenario	floored at 0	0
Medium scenario the one this book breaks	floored at 0	0
Low scenario	83,247	83,247

The rule floors the quantity under the root at zero (MAR21.4(4)), so a bucket charge never turns imaginary. The capital is the largest of the three scenarios, so a scenario that floors to zero does not decide the result.

Inputs and outputs

What goes in. What comes out

The plate runs a slice of the engine. This is the whole shape of it.

What goes in

PV01 by tenor: The change in present value for a one basis point rise in the zero rate at each of the ten prescribed tenors, per currency.
Reporting currency: The currency the charge is stated in. Each currency bucket is converted before it is combined.

What comes out

Capital charge: The largest of the charges under the high, medium and low correlation scenarios, with the binding scenario named.
Attribution: Each risk factor’s exact share of the charge, in every scenario. The shares add up to the charge.
Correlation certificate: A check that the prescribed correlation matrices are, or are not, positive semidefinite, with a witness anyone can recompute.
Derivation: Every figure as a tree of steps, each with the paragraph it applies.
Receipt: A hash of the rule parameters, the inputs and the result. Anyone can recompute it.

What the plate covers

In the plate above

GIRR delta, the sensitivities-based method for interest rate risk, in all three correlation scenarios.
The exact share of the charge that belongs to each risk factor.
A check, in exact arithmetic, of whether the prescribed correlation matrices are valid correlation matrices.

In the licensed engine, not the plate

The other risk classes, and vega and curvature, which the licensed engine runs from your sensitivities.
The default risk charge and the residual risk add-on.
Aggregation by desk and by trade, and the reporting templates.
Amendments and consolidations after the January 2019 text.

Clause map

Every rule has an address

Each row names the clause, the function that carries it and the statements that check it. A test opens each file and confirms the function exists.

Rules of the regulation, the code that carries each one and the statements that check it
Reference	Rule	Carried by	Checked by
MAR21.42, Table 1	Risk weights by tenor, in basis points.	`GIRR_RISK_WEIGHT_BP` src/kernel/frtb.ts	Tested FR-04
MAR21.46, Table 2	Correlation between two tenors, from a decay formula with a floor of 40 per cent.	`baseTenorCorrelation` src/kernel/frtb.ts	Tested FR-04
MAR21.6	The high and low correlation scenarios, and the same rule across currencies.	`scenarioCorrelation` src/kernel/frtb.ts	Tested FR-04
MAR21.4(4)	The bucket charge is the root of the weighted sensitivities combined through the correlations, with the quantity under the root floored at zero.	`evaluateScenario` src/kernel/frtb.ts	Tested FR-05 Certified FR-03
MAR21.4(5) and MAR21.50	Buckets combine through the cross-currency correlation, with a clipped fallback when the quantity under the root is negative.	`girrDelta` src/kernel/frtb.ts	Tested FR-05
MAR21.7	The capital requirement is the largest of the charges under the three scenarios.	`girrDelta` src/kernel/frtb.ts	Tested FR-06 Tested FR-01 Tested FR-02

Verification

What we claim, and how it is checked

Every statement is registered, and each says which level of assurance it has earned. The ladder is explained on the verification page.

Certified

True for the case in hand, with evidence anyone can recheck.

FR-03

Each prescribed GIRR delta correlation matrix is indefinite, shown by an integer witness vector whose quadratic form, summed in exact arithmetic on the stored matrix, is negative by far more than rounding in the entries could explain.

tests/kernel/frtb.test.tssrc/kernel/frtb.tssrc/kernel/rational.ts

Tested

True on every input we tried.

FR-01

Risk factor shares add up to the capital charge in every correlation scenario, and each matches a numerical derivative of the charge.

tests/kernel/frtb.test.ts
FR-02

The capital charge is positively homogeneous of degree one and unchanged when every position flips sign.

tests/kernel/frtb.test.ts
FR-04

The risk weights, the decay parameter, the floor and the scenario rules reproduce every entry of the published correlation table and the worked example in the January 2019 text of MAR21.

tests/kernel/frtb.test.ts

Show the other 2 Show fewer

FR-05

A bucket whose quadratic form is negative is floored at zero, and the clipped fallback applies when the cross-bucket quantity under the root is negative.

tests/kernel/frtb.test.ts
FR-06

The capital charge is never negative and equals the largest of the three scenario charges.

tests/kernel/frtb.test.ts

Read the verification statement

Use it

The same call, anywhere

This sample is written from the input the plate starts with, so the code and the numbers cannot drift apart.

Call

import { girrDelta, girrReceiptPayload } from '@graphatlas/kernel/frtb-sa';
import { receipt } from '@graphatlas/kernel/receipt';

const book = {
  reportingCurrency: 'USD',
  buckets: [
    {
      currency: 'USD',
      pv01: {
        0.25: 5400, 0.5: 12600, 1: 37200, 2: 95400, 3: 67800, 5: 289500,
        10: -213600, 15: -72900, 20: -56700, 30: 29100,
      },
    },
    {
      currency: 'EUR',
      pv01: {
        0.25: -1800, 0.5: 8700, 1: 15300, 2: 28800, 3: -25200, 5: -115200,
        10: 158100, 15: -78300, 20: 42300, 30: 9900,
      },
    },
  ],
};

const result = girrDelta(book);
const hash = await receipt(girrReceiptPayload(book, result));

Returns

const out = {
  capital: 36584378.23,
  binding: 'low',
  charges: { high: 32150928.21, medium: 34439068.77, low: 36584378.23 },
  receipt: 'deab796a4e2b81f900a5e4b2f9113cd77ec62a794257f6b64dac904f231025e7',
};

The call and the numbers come from the engine that built this page, run on the same input as the plate above. PV01 is in US dollars per basis point. Values are rounded to two places for display. The receipt hashes the unrounded numbers.

Runs on

Browser
Edge
Node.js
Python
JVM
.NET
Excel
Command line
Container

Green surfaces are measured on every build of this site. Each must return the same receipt, bit for bit. The others are packaged targets that the build does not measure. How the platform works

Changes

When the rule moves, so does the engine

Publications that touch this regime, and the release that answered each.

15 Jul 2026

EU Takes effect

Reporting standards for market risk under the standardised approach

Shipped as FRTB-SA 5.1, 9 days later
19 May 2026

BCBS Clarification

Clarification on curvature risk for index options

Shipped as FRTB-SA 5.0, 8 days later

Open the full feed

Tenor	3m	6m	1y	2y	3y	5y	10y	15y	20y	30y
3m	100	97	91	81	72	57	40	40	40	40
6m	97	100	97	91	86	76	57	42	40	40
1y	91	97	100	97	94	89	76	66	57	42
2y	81	91	97	100	99	96	89	82	76	66
3y	72	86	94	99	100	98	93	89	84	76
5y	57	76	89	96	98	100	97	94	91	86
10y	40	57	76	89	93	97	100	99	97	94
15y	40	42	66	82	89	94	99	100	99	97
20y	40	40	57	76	84	91	97	99	100	99
30y	40	40	42	66	76	86	94	97	99	100

Tenor	3m	6m	1y	2y	3y	5y	10y	15y	20y	30y
3m	100	97	91	81	72	57	40	40	40	40
6m	97	100	97	91	86	76	57	42	40	40
1y	91	97	100	97	94	89	76	66	57	42
2y	81	91	97	100	99	96	89	82	76	66
3y	72	86	94	99	100	98	93	89	84	76
5y	57	76	89	96	98	100	97	94	91	86
10y	40	57	76	89	93	97	100	99	97	94
15y	40	42	66	82	89	94	99	100	99	97
20y	40	40	57	76	84	91	97	99	100	99
30y	40	40	42	66	76	86	94	97	99	100

FRTB-SA

What goes in

What comes out

What the plate covers

In the plate above

In the licensed engine, not the plate

Runs on

Reporting standards for market risk under the standardised approach

Clarification on curvature risk for index options

Bring us one book

Bring us
one book

Tenor	3m	6m	1y	2y	3y	5y	10y	15y	20y	30y
3m	100	97	91	81	72	57	40	40	40	40
6m	97	100	97	91	86	76	57	42	40	40
1y	91	97	100	97	94	89	76	66	57	42
2y	81	91	97	100	99	96	89	82	76	66
3y	72	86	94	99	100	98	93	89	84	76
5y	57	76	89	96	98	100	97	94	91	86
10y	40	57	76	89	93	97	100	99	97	94
15y	40	42	66	82	89	94	99	100	99	97
20y	40	40	57	76	84	91	97	99	100	99
30y	40	40	42	66	76	86	94	97	99	100