The sensitivities-based method: delta and vega · RiskAtoms Academy

By the end you will be able to

1Turn a position into sensitivities to the rule's risk factors, then into weighted sensitivities, in the right units.
2Combine weighted sensitivities into a bucket charge K_b and buckets into a class charge with γ, including the alternative S_b.
3Run the three correlation scenarios and say which binds and why.
4Rebuild RiskAtoms Bank's £33.41m SBM by class and scenario, and explain why its four desks need more on their own.

Why it matters

RiskAtoms Bank's trading desk carries nine weighted sensitivities. Added up without regard to sign, they come to £42.22m. The rule's charge is £33.41m, the £417.7m of RWA in L4.1.04: Risk-weighted assets. The difference is correlation: how the rule lets a short in 10-year rates offset a long in 2-year rates, and a US index position add to a UK one. This lesson shows where each pound of that £33.41m comes from.

Core explanation

From sensitivities to capital

Illustrative dataRiskAtoms Bank plc's capital book

The same made-up UK bank as Level 3: £67bn of assets and £4bn of equity at 31 December 2025, in £ millions. Module 4.1 adds its regulatory layer: own funds, deductions, risk-weighted assets by risk type, buffers, the leverage exposure and MREL, computed from the Level 3 credit, trading, derivatives and operational risk books. Every figure is illustrative, not any bank's data.

Pro forma. The book applies Basel 3.1, the 2017 finalisation of Basel III, to the 31 December 2025 balance sheet. On that date the bank still reported under the old UK CRR. The UK applies Basel 3.1 from 1 January 2027, and the internal model approach for market risk from 1 January 2028 (PRA PS1/26, paras 1.23 and 1.24). So the book shows the bank as it would look under the new rules.

Market risk in Module 4.3. The positions are the Level 3 trading desk (M3.2: Market risk): eight positions in four books, Equities, Rates, FX and commodities, and Credit, valued by full revaluation on eight risk factors. The client derivatives are the four netting sets of M3.4: Counterparty credit risk. The £417.7m of market risk RWA in L4.1.04: Risk-weighted assets is the sensitivities-based method only: it leaves out the default risk charge and the residual risk add-on, which this module adds. No position has been changed; where a lesson adds one to teach a rule, it is labelled illustrative.

Risk classes, risk factors, sensitivities

The sensitivities-based method starts where L3.2.02: Risk factors and position mapping left off: map every position to risk factors. The rule fixes the list. It has seven risk classes: general interest rate risk (GIRR), three kinds of credit spread risk (CSR: non-securitisations, securitisations, and the correlation trading portfolio), equity, commodity and FX (MAR21.1). Within each, the risk factors are prescribed:

Risk class	Delta risk factors
GIRR	Each currency's risk-free curve at ten tenors: 3 and 6 months, 1, 2, 3, 5, 10, 15, 20 and 30 years (MAR21.8)
CSR non-securitisation	Each issuer's credit spread curves, bond and CDS, at 6 months and 1, 3, 5 and 10 years (MAR21.9)
Equity	Each share's spot price and each share's repo rate (MAR21.12)
Commodity	Each commodity's spot price, by time to maturity and delivery location (MAR21.13)
FX	Each currency's exchange rate against the reporting currency, here sterling (MAR21.14)

A sensitivity is a partial derivative (M-B02: Partial derivatives), measured by a bump and scaled to a unit move: for rates and spreads, the value change for a 1bp rise divided by 0.0001 (MAR21.19–21.20); for equity, commodity and FX, the change for a 1% rise divided by 0.01 (MAR21.21, 21.23–21.24). So the swap's PV01 of £56,583 per bp at 2 years becomes a sensitivity of £565.8m, and the UK index futures' £60m of exposure is a sensitivity of £60m. The units are large on purpose: the risk weight then scales them to a stressed move. Vega uses its own measure: vega times implied volatility (MAR21.25).

Buckets and risk weights

Risk factors are grouped into buckets: one bucket per currency for GIRR, by credit quality and sector for CSR (MAR21.51), by market cap, economy and sector for equity, by commodity group, and by currency for FX. Each risk factor gets a risk weight, and the weighted sensitivity is WS = RW × s. For the bank:

Class	Bucket	Risk factor	s (£m)	RW	WS (£m)
GIRR	GBP	GBP 2y	351.43	0.919%	3.231
GIRR	GBP	GBP 10y	−924.89	0.778%	−7.194
CSR	1	UK government, 10y	−796.25	0.5%	−3.981
CSR	4	Corporate bonds (BBB industrials), 5y	−343.03	3%	−10.291
Equity	12	UK index spot	40.73	15%	6.110
Equity	12	US index spot	30.00	15%	4.500
Commodity	2	Brent	12.31	35%	4.308
FX	USD	USD/GBP	10.00	10.607%	1.061
Equity vega	12	UK index vol, 0.5y	1.99	77.782%	1.544

GIRR: 1.3% at 2 years and 1.1% at 10 years (MAR21.42), each divided by √2 because sterling is one of the specified currencies (MAR21.44). The 2-year line is the swap plus the bonds' share of 2-year rate risk; the 10-year line is the gilt plus the bonds' share.
CSR: the corporate bonds are IG industrials, bucket 4 at 3%; the gilt is an IG sovereign, bucket 1 at 0.5% (MAR21.53). The gilt's yield moves with the 10-year rate in the Level 3 book, so its rate PV01 stands in for its spread CS01.
Equity: both indices are large-cap advanced-economy indices, bucket 12 at 15% (MAR21.77). The US basket is treated as the index (beta 1).
FX: the risk factor is the value of a dollar in pounds. The basket is long £30m of dollars, the forward sells £20m, so the net is £10m. USD/GBP is a specified pair: 15% ÷ √2 (MAR21.87–21.88).
Commodity: Brent is liquid energy, bucket 2 at 35% (MAR21.82). Vega: the puts' vega of £124,091 per vol point times 100 times the 16% implied volatility is £1.985m, weighted at 55% × √(20/10) = 77.78% for a large-cap index with a 20-day liquidity horizon (MAR21.92, Table 13).

Inside a bucket: K_b

Weighted sensitivities in one bucket combine exactly like the volatility of a portfolio in L3.2.09: Parametric VaR for a portfolio: square them, add the cross terms weighted by the correlation, take the square root. In matrix form the radicand is the quadratic form $WS^{\top} \rho\, WS$ (M-E02: Matrix multiplication and quadratic forms). The result is the bucket charge K_b. The rule sets the correlations (M-C10: Covariance and correlation): between GIRR tenors e^{−3%×|T₁−T₂|/min(T₁,T₂)}, at least 40% (MAR21.46), so 88.69% between 2 and 10 years; 80% between two indices in bucket 12 (MAR21.78); for CSR, 35% between names, 65% between tenors and 99.9% between bond and CDS curves, multiplied together (MAR21.54).

Across buckets: γ and S_b

A risk class with several buckets combines their K_b, plus a cross term for each pair: the cross-bucket correlation γ times the two buckets' net sums S_b. The bank's CSR has two buckets, the gilt (1) and the bonds (4), with γ = 20% (MAR21.57). Using the net sums can push the radicand below zero when buckets hedge each other strongly. Then the rule switches to an alternative: each S_b is clipped to lie between −K_b and +K_b, and the sum is recomputed (MAR21.4(5)(b)). It never bites on the bank's book; the formal version shows a case where it does.

Three correlation scenarios

Correlations are estimates, and they move in a crisis. So the rule does not trust one set: it runs the whole calculation with the set correlations (medium), with every correlation raised by a quarter and capped at 100% (high), and with every correlation lowered to max(2ρ − 1, 0.75ρ) (low), and the bank holds the highest of the three totals (MAR21.6–21.7). These are the correlation scenarios. A raised correlation hurts positions that add up; a lowered one hurts positions that hedge each other.

Scenario	GIRR 2y–10y ρ	Equity UK–US ρ	CSR γ(1, 4)
Medium (as set)	88.69%	80.00%	20.00%
High: × 1.25, cap 1	100.00%	100.00%	25.00%
Low	77.38%	60.00%	15.00%

Which scenario hurts which book?

The formal version

For each risk class and each of delta and vega, with WS_k = RW_k × s_k (MAR21.4; the formulas are images in MAR21 and were read as text in the PRA's Market Risk: Advanced Standardised Approach (CRR) Part, Articles 325f and 325h):

\begin{aligned} K_b &= \sqrt{\max\Big(0,\ \sum_k WS_k^2 + \sum_k \sum_{l \ne k} \rho_{kl}\, WS_k\, WS_l\Big)} \\ \text{Charge}_{\text{class}} &= \sqrt{\sum_b K_b^2 + \sum_b \sum_{c \ne b} \gamma_{bc}\, S_b\, S_c}, \qquad S_b = \sum_k WS_k \\ \text{if the radicand} < 0:\quad S_b &= \max\big(\min(\textstyle\sum_k WS_k,\ K_b),\ -K_b\big) \\ \rho^{\text{high}} &= \min(1.25\,\rho,\ 1), \qquad \rho^{\text{low}} = \max(2\rho - 1,\ 0.75\,\rho) \quad (\text{and the same for } \gamma) \\ \text{SBM} &= \max_{\text{medium, high, low}} \ \sum_{\text{classes}} \big(\text{Delta} + \text{Vega} + \text{Curvature}\big) \end{aligned}

$s_k$ : the net sensitivity to risk factor k across all positions; $RW_k$ : its risk weight; $\rho_{kl}$ : the correlation of two factors in bucket b; $\gamma_{bc}$ : the correlation of buckets b and c.
Vega uses the same aggregation with vega risk weights $RW = \min\big(55\% \times \sqrt{LH/10},\ 100\%\big)$ , LH being the liquidity horizon of MAR21.92 Table 13. Curvature has its own formula (L4.3.03: The curvature charge).
The alternative S_b, on a made-up pair of equity index buckets 12 and 13 (γ = 75%): bucket 12 holds two uncorrelated names of +5 each, bucket 13 two of −5. Then $K_{12} = K_{13} = \sqrt{50} = 7.071$ , S = ±10, and $50 + 50 + 2 \times 0.75 \times 10 \times (-10) = -50$ . Clipping S to ±7.071 gives $50 + 50 - 2 \times 0.75 \times 50 = 25$ , a charge of 5.

Try it

Start with the GIRR pair and watch low bind. Flip the sign of WS₂ with the button, which keeps its exact size, and see high take over. Load the equity pair and drag ρ to 100%: the charge becomes the plain sum, and all three scenarios give the same charge. For CSR, two buckets of one sensitivity each, the same algebra runs with γ.

Try itTwo sensitivities, three scenarios

Load one of the bank's pairs

WS₁: GBP 2y (£m)£3.23m

WS₂: GBP 10y (£m)−£7.19m

Correlation ρ (medium scenario)88.69%

Scenario	ρ used	Radicand	Charge
Medium	88.69%	20.965	£4.579m
High	100.00%	15.709	£3.963m
Low (binds)	77.38%	26.221	£5.121m

Medium

£4.58m

High

£3.96m

Low

£5.12m

|WS₁| + |WS₂|

£10.42m

Charge that counts£5.121m

The two offset each other, so the hedge is worth least when the correlation is lowest: the low scenario binds.

One bucket: K_b = √(WS₁² + WS₂² + 2ρ WS₁WS₂), floored at zero inside the root. High multiplies the correlation by 1.25, capped at 100%; low takes max(2ρ − 1, 0.75ρ). In the full SBM the scenario is chosen once for the whole bank, not pair by pair.

Worked example

RiskAtoms Bank's SBM, class by class

£ millions. The high scenario is shown, since it binds; the table after the steps has all three.

Delta and vega, high scenario

Step 1 of 6

1
GIRR
$\begin{aligned} WS_{2y} &= 351.43 \times \tfrac{1.3\%}{\sqrt 2} = 3.231, \qquad WS_{10y} = -924.89 \times \tfrac{1.1\%}{\sqrt 2} = -7.194 \\ \rho^{\text{high}} &= \min(1.25 \times 0.8869,\ 1) = 1 \\ K_{\text{GBP}} &= \sqrt{3.231^2 + -7.194^2 + 2 \times 1 \times 3.231 \times (-7.194)} = 3.963 \end{aligned}$

Scenario	GIRR	CSR	Eq. delta	Eq. vega	Commodity	FX	Total
Medium	4.579	11.753	10.078	1.544	4.308	1.061	33.323
High (binds)	3.963	11.926	10.610	1.544	4.308	1.061	33.412
Low	5.121	11.578	9.517	1.544	4.308	1.061	33.128

Desk by desk

Run the same calculation on each desk alone, each in its own binding scenario, and the charges are larger than the bank's in total:

Equities (high)

£15.34m

Rates (low)

£7.93m

FX and commodities (medium)

£6.43m

Credit (high)

£13.26m

Sum of the desks

£42.95m

The bank (high)

£33.41m

The £9.54m gap is offsetting across desks. The equities desk alone is long dollars through the US basket, and the FX desk alone is short dollars through the forward; together the net is £10m. The credit desk's bonds lose value when rates rise, as the gilt does, while the swap gains; on its own the rates desk is long 2-year and short 10-year rate risk, which is why low binds for it. The rules allow these offsets within a risk class across the whole bank, whatever desk the positions sit on.

Regulatory lens

The SBM is MAR21; the parameters checked here are MAR21.4–21.8, 21.19–21.25, 21.42–21.57, 21.72–21.92. Statements on jurisdictions are as at 3 November 2026 as at 3 Nov 2026.

In the UK

The PRA's Market Risk: Advanced Standardised Approach (CRR) Part applies from 1 January 2027 (PS1/26, para 1.23). Its Article 325f sets K_b and the cross-bucket formula, with the same alternative S_b, and Article 325h the three scenarios, with low as max(2ρ − 100%, 75%ρ) and the highest of the three taken.

In the EU

CRR3 carries the same method, but the EU has postponed it twice; from 1 January 2027 a delegated act applies temporary adjustments to the end of 2029, including a 0.9 multiplier on the SBM (Delegated Regulation (EU) 2026/1221, Article 495s). L4.3.11: Market risk and CVA capital: what changes from 2027 has the detail.

In the US

The US market risk rule in force (12 CFR 217, subpart F) has no sensitivities-based method. The agencies' proposals of 19 March 2026 would “introduce a standardized approach for market risk to serve as a credible fallback approach” (Federal Reserve fact sheet); comments closed on 18 June 2026, and as far as we could find no final rule had been published by 3 November 2026.

Spark Atoms

Spark AtomWhy √2 for sterling?

Only the risk weights of a short list of currencies may be divided by √2: EUR, USD, GBP, AUD, JPY, SEK and CAD, plus the bank's own reporting currency (MAR21.44). For RiskAtoms Bank that relief takes its GIRR weighted sensitivities down by about 29%, since 1/√2 = 0.707.

Spark AtomThe charge is a function of nine numbers

Once the sensitivities are known, the whole £33.41m depends on nothing else: no history, no simulation, no model. That is the point of a standardised approach. Two banks with the same sensitivities get the same charge, which is what makes it usable as a fall-back and as a benchmark for internal models (L4.3.08: Total market risk capital).

Key takeaways

The SBM maps positions to prescribed risk factors in seven risk classes, measures sensitivities by bumping, and weights each by a set risk weight: WS = RW × s.
Within a bucket, K_b = √(WSᵀρWS), floored at zero; across buckets, √(ΣK_b² + ΣΣγS_bS_c), with S_b clipped to ±K_b if that would go negative.
Everything runs three times, with correlations as set, raised by 25% and lowered; the highest total counts. Offsetting positions bind in low, adding ones in high.
RiskAtoms Bank: GIRR £3.96m, CSR £11.93m, equity £12.15m, FX £1.06m, commodity £4.31m: £33.41m in the high scenario, RWA £417.7m.
Desk by desk the charges add up to £42.95m: offsets between desks are worth £9.54m.

Reaction

Check what stuck

Question 1 · objective 1

A sterling swap book has PV01 £20,000 per bp at the 5-year tenor. The GIRR weight at 5 years is 1.1%, and GBP is a specified currency. What is its weighted sensitivity, in £m?

£m

Question 2 · objective 2

Two weighted sensitivities of +6 and +4 (£m) sit in one bucket with a correlation of 80%. What is K_b in the medium scenario, in £m?

£m

Question 3 · objective 3

The bank's GIRR bucket has +3.23 at 2 years and −7.19 at 10 years. Which scenario gives the largest GIRR charge on its own?

ALow: the two offset, so the hedge is worth least when the correlation is lowestBHigh: the correlation is raised, so every charge goes upCMedium: the set correlations always give the largest chargeDAll three give the same charge

Question 4 · objective 4

The four desks' SBMs, each worked out alone, add up to £42.95m. The bank's SBM is £33.41m. Why the gap?

AThe desks hedge each other: the equity desk's dollars against the FX forward, the bonds' rate risk against the swapBDesks are charged at a higher risk weight than the bankCThe bank-level SBM uses the medium scenario and the desks use the high oneDCurvature is only charged desk by desk

Question 5 · objective 1

An index option book has vega £50,000 per vol point at 20% implied volatility, on a large-cap index (weight 77.78%). What is its weighted vega sensitivity, in £m?

£m

Glossary

Sensitivities-based method: Sensitivities-based method (SBM): the main part of the FRTB standardised approach. Delta, vega and curvature charges computed from sensitivities with set risk weights and correlations.
Risk factor: Risk factor: a market variable the rule prescribes, such as the GBP rate at the 5-year tenor, one issuer's credit spread at one tenor, or one share's spot price.
Sensitivity: Sensitivity s_k: the change in value for a small bump in one risk factor, scaled to a unit move. A partial derivative, measured by revaluing.
Bucket: Bucket: a group of risk factors that the rule treats as close relatives, such as one currency's rate curve or IG industrials' credit spreads.
Weighted sensitivity: Weighted sensitivity WS_k: the sensitivity times its risk weight, the rule's stressed move for that risk factor. It is a loss-sized number in currency.
Bucket charge K_b: Bucket charge K_b: the weighted sensitivities in a bucket combined with the rule's correlations, like the volatility of a portfolio.
Cross-bucket correlation γ: Cross-bucket correlation γ: how the net positions of two buckets in one risk class offset or add.
Correlation scenarios: Correlation scenarios: the SBM is computed with the set correlations (medium), with them raised by 25% (high) and with them lowered (low). The highest total counts.
Vega sensitivity: Vega sensitivity: for an option, vega times implied volatility (MAR21.25), so a 1.00 relative move in the volatility. Weighted at up to 100%.

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, MAR21: Standardised approach: sensitivities-based method: the risk weight and correlation tables for every class, including the buckets the bank does not use. Advanced.
L3.2.09: Parametric VaR for a portfolio: the same quadratic form, used for parametric VaR. Intermediate.

Origins

Basel Committee on Banking Supervision, MAR21: Standardised approach: sensitivities-based method, paras 21.1, 21.4–21.9, 21.12–21.14, 21.19–21.25, 21.42–21.57, 21.72–21.92, bis.org.
Prudential Regulation Authority, PRA Rulebook, Market Risk: Advanced Standardised Approach (CRR) Part, Articles 325f and 325h, version from 1 January 2027; PS1/26, para 1.23, January 2026.
Commission Delegated Regulation (EU) 2026/1221, Article 495s, Official Journal of the European Union, 2026.
Board of Governors of the Federal Reserve System, Fact sheet on the capital proposals, 19 March 2026.

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