Non-modellable risk factors and the internal DRC · RiskAtoms Academy

By the end you will be able to

1Apply the risk factor eligibility test to a record of real prices.
2Compute the stress scenario capital of non-modellable risk factors and aggregate it with MAR33.17.
3Describe the internal default risk charge (99.9%, one year, two types of systematic factors, PD floor) and read it off a simulation.
4Compare the bank's internal DRC with its standardised DRC and explain the gap.

Why it matters

A model is only as good as its data. RiskAtoms Bank plc's Credit desk has a price for bond issuer 3's spread 30 times in the last year, but none in the 90 days from day 157. Under the FRTB that spread may not go into the expected shortfall model at all; it is charged by a stress scenario instead. And ES, a ten-day measure, cannot see an issuer default. That needs a separate one-year, 99.9% model: on this desk it says £12.0m, about twice the standardised default risk charge of £6.14m.

Core explanation

What the ES model may not see

L4.3.05: Internal models: expected shortfall built the ES for risk factors the bank may model. This lesson covers the two pieces that sit next to it in the internal models approach: the factors the bank may not model, and the risk of default.

Illustrative dataRiskAtoms Bank plc's trading desk under the internal models approach

The same made-up Markets desk as Module 3.2: eight positions on four desks (Equities, Rates, FX and commodities, Credit), eight risk factors and 750 days of seeded history with a stressed spell, every position revalued in full. Figures are in £ millions at 31 December 2025.

Every internal-model figure is illustrative. The UK internal model approach for market risk starts on 1 January 2028 (PRA PS1/26, para 1.24), and the bank has no model permission: its capital is the standardised approach of the earlier lessons. The pages show what the bank's models would give if it had one.

The five observation records and the three stress shocks are invented for the lesson; the desk's eight-factor history has no data on these finer factors. The default simulation uses the obligors of L4.3.04: Default risk charge and residual risk add-on.

The risk factor eligibility test

L3.2.02: Risk factors and position mapping said that a bank using internal models must show each risk factor is backed by enough real prices. The risk factor eligibility test makes that a count. A real price is a trade or a committed quote, not a valuation. Each quarter, a factor passes if it has (a) at least 24 real prices a year with no 90-day period in the last 12 months holding fewer than 4, or (b) at least 100 in the last 12 months. At most one price a day counts (MAR31.13). Basel counts the 24 over the period used to calibrate the ES model; the book counts them over the last 12 months. Part (a) asks for prices spread through the year, not bunched.

A finer model of the desk than Level 3's eight factors would need factors like these five. Their records over the last year:

Shaded on the strip: each factor's emptiest 90 days
Risk factor	Quoted	Prices	Fewest in 90 days	RFET
UK index vol, 3m at the money	every weekday	261	64	passes (b)
GBP/USD basis, 1y	every 14 days	27	6	passes (a)
Bond issuer 3's spread	every 6 days to day 160, then every 40	30	0	fails
UK index skew, 3m	every 18 days	21	5	fails
UK index dividends, 2y	every 30 days	13	3	fails

The at-the-money volatility is quoted every weekday and passes (b). The cross-currency basis is quoted only fortnightly, but evenly, so 27 prices with at least 6 in every 90 days pass (a). The other 3 fail: the issuer spread for its silence, the skew and the dividends for too few prices. They become non-modellable risk factors.

Stress scenario capital

An NMRF leaves the ES model, and the bank must instead find a stress scenario for it at least as prudent as the ES calibration, over the greater of its liquidity horizon and 20 days (MAR33.16). If it cannot find one, it must use the maximum possible loss. On the book each NMRF's stress loss is its sensitivity times an assumed shock:

NMRF	Kind	Sensitivity	Shock	Stress loss
Bond issuer 3's spread basis to the IG curve	idiosyncratic credit	£4,288 per bp	60 bp	£0.257m
UK index vol skew at the puts' strike	other	£124,091 per vol point	4 vol points	£0.496m
UK index dividend expectations	other	£1,500 per bp	50 bp of dividend yield	£0.075m

The losses are then added with rules for their correlation (MAR33.17). Idiosyncratic credit spread and equity NMRFs, one issuer's own news, are taken to be independent: their losses add in quadrature. All the others aggregate with ρ = 0.6. On the book, issuer 3's spread stands alone at £0.257m; the skew and the dividends combine to £0.528m. SES is £0.785m, against £0.829m if the three were simply added and £0.564m if all were independent.

Sort itModellable or not?

For each record of real prices in the last 12 months, does the risk factor pass the RFET?

120 prices, all in March
26 prices, one every two weeks
40 prices, none in the last four months
Daily dealer valuations, no trades or committed quotes
23 prices, evenly spread

The internal default risk charge

L3.2.07: Credit spread risk separated credit spread risk from default risk. A ten-day ES at 97.5% will rarely see an issuer default, so the internal models approach adds an internal DRC, the counterpart of the standardised DRC of L4.3.04: Default risk charge and residual risk add-on. Its rules (MAR33.18–33.39):

ExploreThe internal DRC, rule by rule

Click a box to see what happens there.

Measure

A VaR at 99.9% of default losses over one year, computed weekly. Capital is the larger of the latest figure and its 12-week average (MAR33.20, 33.22).

The bank's model is the one-factor model of L3.3.11: Unexpected loss and default correlation with a second factor: each obligor's asset value loads 0.4 on a global factor and 0.3 on its region (UK or US), so two UK corporates have a correlation of 25% and a UK and a US one 16%. An obligor defaults when its asset value falls below Φ⁻¹(PD) (M-C06: The normal distribution). PDs are the one-year default column of the Level 3 rating matrix; the gilt's 0.02% is lifted to the floor of 0.03%.

Bond losses are market value less 40% recovery on face; share losses are the JTD of L4.3.04
Obligors	Number	Rating	PD	Loss if each defaults
Corporate bond issuers	8	BBB	0.20%	£6.00m
UK government (the gilt)	1	AA	0.03%	£58.80m
US basket names	5	A	0.05%	£6.00m
UK index names	10	A	0.05%	£2.18m

With 100,000 paths and a fixed seed, 2.18% of years see any default at all. The 99.9% VaR is the 100th worst loss (M-C09: Quantiles): £12.0m. 54 paths lose exactly that, 38 of them two bond issuers. Beyond it lie 46 worse paths, 28 of them a gilt default of £58.8m: far rarer than 1 in 1,000, so outside the VaR, but they lift the ES at 99.9% to £26.14m.

Explore the tail

Simulation

Confidence level

Loadings 0.4 global, 0.3 regional (correlation 25% within a region).

VaR at 99.9%£12.00m

ES at 99.9%£26.14m

Paths with any default2.18%

Standardised DRC, whole desk£6.14m

The worst of the 100,000 paths, grouped by what defaulted
What defaulted (worst paths first)	Loss	Paths
1 bond + 1 gilt	£64.80m	4
1 UK index share + 1 gilt	£60.98m	1
1 gilt	£58.80m	23
2 bonds + 2 US shares	£24.00m	1
3 bonds + 1 UK index share	£20.18m	1
1 bond + 6 UK index shares	£19.10m	1
3 bonds	£18.00m	4
2 bonds + 2 UK index shares	£16.37m	1

The internal DRC is about twice the standardised one on this desk. The SA weights each jump-to-default by a rating-based risk weight, 6% for a BBB name. The model asks what a 1-in-1,000 year looks like for a handful of large names, and the answer is two of them defaulting. Concentration, not the PDs, drives it. Switch the explorer to independent defaults and the 99.9% VaR halves to one default.

The formal version

RFET for a risk factor with n real prices in the last 12 months and n₉₀ the fewest in any 90-day period (MAR31.13). SES from stress losses: I the idiosyncratic credit spread NMRFs, J the idiosyncratic equity NMRFs, K the rest (MAR33.17). The default model with loadings a_g, a_r, standard normal factors Z_g, Z_r and ε_i, LGD-weighted loss L_i and the indicator 1 (MAR33.20–33.24):

\begin{aligned} \text{modellable} &\iff \big(n \ge 24 \ \text{and}\ n_{90} \ge 4\big)\ \text{or}\ n \ge 100 \\ \text{SES} &= \sqrt{\sum_{i\in I}\text{ISES}_{i}^2} + \sqrt{\sum_{j\in J}\text{ISES}_{j}^2} + \sqrt{\Big(\rho\sum_{k\in K}\text{SES}_{k}\Big)^2 + (1-\rho^2)\sum_{k\in K}\text{SES}_{k}^2}, \quad \rho = 0.6 \\ X_i &= a_g Z_g + a_r Z_r + \sqrt{1 - a_g^2 - a_r^2}\;\varepsilon_i, \qquad \text{default if } X_i < \Phi^{-1}\big(\max(\text{PD}_i,\ 0.03\%)\big) \\ L &= \sum_i L_i\,\mathbf{1}\big\{X_i < \Phi^{-1}\big(\max(\text{PD}_i,\ 0.03\%)\big)\big\}, \qquad \text{DRC} = \text{VaR}_{99.9\%}(L) \\ \text{capital} &= \max\big(\text{DRC}_{\text{latest}},\ \overline{\text{DRC}}_{12\text{ weeks}}\big) \end{aligned}

Each stress loss is taken over max(LH, 20 days). The SES form was read as text in the PRA's draft rules (CP9/26, appendix 2, Article 325bk(13)); the Basel MAR33.17 states the same zero correlation for I and J and ρ = 0.6 for the rest. The two-factor Gaussian form is the bank's choice: the rule asks only for two types of systematic factor.

Try it

Quote a factor every 14 days and it passes; open a quiet spell of 100 days and watch it fail, whichever minimum you choose. Close the spell and quote every 18 days instead: 21 real prices fail the minimum of 24. Switch the minimum to 16, the PRA's CP9/26 proposal for factors with a liquidity horizon over 20 days, and it passes. Then raise the skew shock and set ρ to 1: SES becomes a simple sum.

Try itIs it modellable, and what does it cost if not?

1. The risk factor eligibility test

A real price every14 days

Quiet spell from day 1500 days

Minimum count for (a)

Real prices in the year27

Fewest in any 90 days (shaded)6

Resultpasses (a)

2. Stress scenario capital for the book's three NMRFs

Issuer 3's spread: stress shock60.0 bp

UK index skew: stress shock4.0 vol points

UK index dividends: stress shock50.0 bp of dividend yield

Correlation ρ for the non-idiosyncratic NMRFs0.60

Issuer 3's spread (idiosyncratic credit)£0.257m

UK index skew (other)£0.496m

UK index dividends (other)£0.075m

Idiosyncratic credit, at zero correlation£0.257m

Others, at ρ£0.528m

SES£0.785m

Each stress loss is |sensitivity × shock|, the shock taken over max(liquidity horizon, 20 days). The shocks are assumed: the desk has no history for these factors. Illustrative, £ millions.

Worked example

SES and the internal DRC on the desk

£ millions, illustrative.

From records to capital

Step 1 of 7

1
RFET
Issuer 3's spread: 30 prices, but 0 in the 90 days from day 157. It fails (a), and 30 < 100 fails (b). The skew (21 prices) and the dividends (13) fail on count. Three NMRFs.

Regulatory lens

Basel: MAR31.12–31.16 (RFET), MAR33.16–33.17 (SES) and MAR33.18–33.39 (internal DRC). Status as at 3 November 2026 as at 3 Nov 2026.

In the UK

From 1 January 2028 (PS1/26, para 1.24), in the PRA Rulebook's Market Risk: Internal Model Approach (CRR) Part. The PRA does not let banks model central government and central bank exposures in the IMA DRC; they go to the standardised DRC, while the rest of the IMA still applies (PS17/23, para 3.41). On the book that gives £8.18m + £1.48m = £9.66m instead of £12.0m. Where no stress scenario can be found, the PRA replaces Basel's maximum possible loss with the firm's own judgement, floored at the largest loss historically observed for that NMRF (paras 3.48–3.50).

CP9/26 (June 2026) proposes, for 2028: 16 real prices instead of 24 for factors with a liquidity horizon over 20 days (para 2.14); a new “type 1” NMRF that fails the count but stays in the ES model with a zero-correlation add-on; dropping the idiosyncratic split; the NMRF stress period aligned with the ES one; and a monthly rather than daily calculation (paras 2.21–2.22). Not final as at 3 November 2026.

In the EU

The same tests are in CRR Articles 325be (modellability) and 325bk (stress scenario risk measure). Until 31 December 2029, Delegated Regulation (EU) 2026/1221 lets a factor with as few as two verifiable prices be treated as modellable, with a liquidity horizon set from 250 divided by its number of prices (Article 495j); pro-rates the count for new issues (495k); and sets to zero the PDs of issuers that would have a 0% standardised risk weight in the internal DRC (495l).

In the US

The 19 March 2026 proposals would bring an expected shortfall-based models approach; they were not final as at 3 November 2026 (L4.3.11: Market risk and CVA capital: what changes from 2027).

Spark Atoms

Spark AtomA lumpy tail, a steady quantile

The desk's default losses come in steps of £2.18m and £6m, so the simulated 99.9% VaR does not wander with the seed: with seeds 1, 2, 3 it is £12.0m, £12.0m, £12.0m again. A smooth portfolio of many small names would show more simulation noise, which is why L3.2.11: Monte Carlo VaR asked banks to document paths and seeds.

Spark AtomOnce in 3,333 years

The PD floor of 0.03% says that even the safest borrower defaults once in about 3,333 years. That is still beyond a 1-in-1,000 VaR, which is why the gilt never sets the internal DRC on this book but dominates its ES.

Key takeaways

A risk factor may be modelled only with enough real prices: 24 in a year with no 90-day gap below four, or 100 (MAR31.13).
Factors that fail become NMRFs, each charged by a stress scenario over max(LH, 20 days). On the book three fail, and SES is £0.785m.
SES adds idiosyncratic credit and equity losses at zero correlation and the rest at ρ = 0.6.
The internal DRC is a 99.9% one-year VaR of default losses from a model with two types of systematic factors, PDs floored at 0.03%: £12.0m on the book against the SA's £6.14m.
The UK keeps sovereigns out of the internal DRC; the EU's temporary rules to 2029 relax the RFET and zero the DRC on 0%-weighted issuers.
Every figure here is illustrative: the UK IMA starts on 1 January 2028.

Reaction

Check what stuck

Question 1 · objective 1

A risk factor has 30 real prices in the last 12 months: one every week for the first seven months, then none for five months. Is it modellable?

ANo: it has more than 24 prices, but some 90-day period holds fewer than fourBYes: 30 is more than 24CYes: the test only looks at the last 90 daysDNo: it needs 100 prices

Question 2 · objective 2

Two non-idiosyncratic NMRFs have stress losses of £3m and £4m. With ρ = 0.6, what is their aggregate stress scenario capital, in £m?

£m

Question 3 · objective 2

If the same two NMRFs were idiosyncratic credit spreads, what would the stress scenario capital be, in £m?

£m

Question 4 · objective 3

A bank simulates 50,000 one-year default paths. Which worst loss is its 99.9% VaR (counting the worst as 1)?

Question 5 · objective 4

Why is the bank's internal DRC (£12.0m) about twice its standardised DRC (£6.14m)?

AThe model's 1-in-1,000-year loss on a few large names is two £6m defaults, while the SA weights each name's JTD by a default risk weight such as 6% for BBBBThe internal DRC uses ES, which is always larger than VaRCThe internal DRC has no PD floorDThe SA leaves out equity positions

Glossary

Real price: Real price: a price from an actual transaction by the bank, a verifiable arm's-length trade between others, or a committed quote, possibly collected by a vendor. Valuations and collateral reconciliations do not count (MAR31.12).
Risk factor eligibility test (RFET): Risk factor eligibility test (RFET): at least 24 real prices a year (counted over the ES model's calibration period) with no 90-day period in the last 12 months holding fewer than four, or at least 100 in the last 12 months; at most one a day; checked quarterly (MAR31.13).
Non-modellable risk factor (NMRF): Non-modellable risk factor (NMRF): a risk factor that fails the RFET. It leaves the ES model and gets its own stress scenario capital.
Stress scenario capital (SES): Stress scenario capital (SES): the capital for NMRFs, built from one stress loss per factor and aggregated by MAR33.17.
Idiosyncratic NMRF: Idiosyncratic NMRF: a credit spread or equity factor driven by one issuer's own news, which can be shown to have no systematic part. These aggregate with zero correlation.
Internal default risk charge: Internal default risk charge: a 99.9% one-year VaR of default losses on the desk's credit and equity positions, from a model with two types of systematic factors (MAR33.18–33.39).
Systematic factor: Systematic factor: a shared driver of many obligors' credit quality, such as the global economy or a region. Each obligor's asset value loads on it, which makes defaults move together.
PD floor: PD floor: the IMA DRC may not use a one-year PD below 0.03% for any obligor (MAR33.24).

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, Minimum capital requirements for market risk, January 2019 (d457): the revised RFET and the internal DRC. Advanced.
Prudential Regulation Authority, CP9/26: Basel 3.1: Adjustments to the internal model approach (IMA) for market risk, June 2026, chapter 2: the UK's proposed changes to the RFET and NMRFs. Intermediate.

Origins

Basel Committee on Banking Supervision, MAR31, paras 31.12–31.16, and MAR33, paras 33.16–33.39, bis.org (read in the Swiss SIF copies).
Prudential Regulation Authority, PS17/23, December 2023, paras 3.35–3.50; PS1/26, January 2026, para 1.24; CP9/26, June 2026, paras 2.12–2.23, and its appendix 2 (draft PRA Rulebook instrument), Article 325bk.
Commission Delegated Regulation (EU) 2026/1221 of 4 June 2026, Articles 495j–495l.
Board of Governors of the Federal Reserve System, press release and fact sheet, 19 March 2026.

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