The IRB risk-weight formula · RiskAtoms Academy

By the end you will be able to

1Explain in words what the conditional PD at 99.9% is, and why capital covers it less expected loss.
2Compute the asset correlation and the maturity adjustment, and say why each falls as PD rises.
3Work out K, the risk weight and RWA for a corporate loan, and add them up across a book.
4Check the formula against CRE99 and explain why Basel 3.1 has no 1.06 scaling factor.

Why it matters

RiskAtoms Bank plc lends £1,926.6m to Bramwick Stores plc, a grade-6 retailer with a PD of 1.13%. The standardised approach weights it at 100%. The bank's IRB permission weights it at 85.65%, a number nobody looks up in a table: it comes out of one formula with a normal distribution, a logarithm and two exponentials in it. That formula sets £12,461.0m of the bank's RWA. This lesson takes it apart.

Core explanation

One bad year in a thousand

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.

The credit figures are the Level 3 credit book (Module 3.3): its twelve corporate borrowers and three mortgage segments, each scaled up to stand for a slice of the bank's £12bn corporate and £30bn mortgage books. Grades, PDs and LGDs are unchanged.

The idea: capital for the bad year

L3.3.10: Expected loss split credit losses into an expected part, covered by pricing and provisions, and an unexpected part, covered by capital. The IRB formula follows that split exactly: “Risk weighted assets are designed to address unexpected losses” (CRE31.1). Expected loss is compared with provisions instead (L4.2.07: Expected loss, provisions and the shortfall).

How bad a year? L3.3.11: Unexpected loss and default correlation built the one-factor model: every borrower's fortune is part economy and part its own luck. Fix the economy at a bad value and, in a large book, the share of borrowers that default is no longer random. It is the conditional PD, a probability conditional on the state of the economy (M-C12: Conditional probability and Bayes' rule). Basel picks the economy's 99.9th percentile, its worst year in a thousand (M-C09: Quantiles). For loans like Bramwick's:

The economy's worst year in…	Percentile	Default rate of Bramwick-like loans
2 (a median year)	50.0%	0.57%
10	90.0%	2.78%
100	99.0%	7.92%
1,000	99.9%	14.85%

The average year's rate is 1.13%; in the median year it is lower, because a few bad years pull the average up. In the 1-in-1,000 year it is 14.85%. Capital per £1, before the maturity adjustment, is LGD times the gap between the two: 40% × (14.85% − 1.13%) = 5.49%. The BCBS chose 99.9% so that a bank's losses should exceed its capital “on average once in a thousand years”, set high partly to protect against estimation errors in PD and LGD (explanatory note, section 5.1).

Only the intuition here

The theory behind treating each loan on its own, the asymptotic single risk factor model, and its limits (one factor, an infinitely fine-grained book) are in M6.4: Portfolio credit models. This lesson takes the formula as the rule.

The correlation: from 24% down to 12%

The bad-year default rate depends on how much borrowers move together, the asset correlation R. The bank does not estimate it: the rule sets it. For corporates, sovereigns and banks it slides from 24% at the lowest PDs to 12% at the highest, with a weight that falls exponentially in PD (M-A03: Exponents). The BCBS reasoning: “Asset correlations decrease with increasing PDs”, because a weak borrower's default owes more to its own problems, and larger firms are more tied to the economy (explanatory note, section 5.2). Two adjustments follow from the same logic:

Firm-size adjustment: a corporate with group sales under €50m gets up to 4 points off R (CRE31.8). Pellcourt Properties Ltd has sales of €40.25m, so its R is 20.16% instead of 21.02%, and its risk weight 34.4% instead of 36.1%.
Large financial institutions: R × 1.25 for regulated financial institutions with assets of USD 100bn or more, and for all unregulated ones (CRE31.7), added in Basel III after the 2008 crisis.

PD	R corporate	R, sales €5m	R, large FI	b(PD)	Maturity factor at M = 5
0.05%	23.70%	19.70%	29.63%	0.2861	3.005
0.10%	23.41%	19.41%	29.27%	0.2469	2.569
0.25%	22.59%	18.59%	28.24%	0.1996	2.139
0.50%	21.35%	17.35%	26.68%	0.1671	1.892
1.00%	19.28%	15.28%	24.10%	0.1375	1.693
2.00%	16.41%	12.41%	20.52%	0.1108	1.531
5.00%	12.99%	8.99%	16.23%	0.0799	1.363
10.00%	12.08%	8.08%	15.10%	0.0599	1.263
20.00%	12.00%	8.00%	15.00%	0.0427	1.183

Retail uses its own correlations: 15% for mortgages, 4% for qualifying revolving retail and 3% to 16% for other retail. They are in L4.2.06: IRB for retail and specialised lending.

The maturity adjustment

A five-year loan has more time than a one-year loan to be downgraded before it is repaid, so it needs more capital. The formula multiplies by a factor that grows with the effective maturity M, at a slope b(PD) built from the natural logarithm of PD (M-A04: Logarithms). The slope is larger for strong borrowers: a AA name has room to fall, while a B name is already close to default and its risk is mostly in the PD. The factor is built so that at M = 1 it is exactly 1, and F-IRB fixes M at 2.5 years (CRE32.44). For Bramwick:

M (years)	Maturity factor	K	Risk weight	RWA (£m)
1.0	1.0000	5.49%	68.62%	1,322.0
2.5	1.2482	6.85%	85.65%	1,650.1
5.0	1.6619	9.12%	114.03%	2,196.9

M = 1 yearM = 2.5 years (F-IRB)M = 5 years

The curves fan out most on the left: maturity matters most where PD is low. The PD axis is logarithmic. On an ordinary PD axis the curve would be concave, rising steeply at low PDs and then flattening, because the falling correlation offsets part of the rising PD: doubling PD from 1% to 2% multiplies the risk weight by only 1.24.

From K to RWA

K is capital per £1. L4.1.04: Risk-weighted assets showed why RWA are 12.5 times a capital charge: 12.5 is 1 ÷ 8%. So the risk weight is 12.5 × K, and RWA are that times EAD (CRE31.3, CRE31.5). Bramwick's 6.85% becomes a risk weight of 85.65%, RWA of £1,650.1m and a Pillar 1 minimum of £132.0m, against £1,926.6m of RWA under the standardised approach.

Checking it, and the 1.06 story

CRE31 prints the formula as images, so how do you know you have it right? The Basel Framework itself publishes illustrative risk weights in CRE99 Table 1. Every figure below is computed by this page; the largest gap to the published two decimals is 0.005 percentage points.

PD	CRE99: corporate, LGD 40%	Computed	× 1.06	CRE99: sales €5m	Computed, €5m	CRE99: mortgage, LGD 45%	Computed, mortgage
0.05%	17.47%	17.47%	18.52%	13.69%	13.69%	6.23%	6.23%
0.25%	43.97%	43.97%	46.61%	34.68%	34.68%	21.30%	21.30%
1.00%	82.06%	82.06%	86.98%	64.35%	64.35%	56.40%	56.40%
5.00%	133.20%	133.20%	141.20%	99.79%	99.79%	148.22%	148.22%
20.00%	211.76%	211.76%	224.47%	167.48%	167.48%	253.12%	253.12%

The “× 1.06” column does not match, and that is the point. Basel II added a scaling factor of 1.06 to IRB credit RWA, a calibration to keep overall capital near its old level. In May 2006, after its fifth quantitative impact study, the Committee decided to keep it (BCBS press release, 24 May 2006). The finalised Basel III standards dropped it: CRE31 has no such factor, and the CRE99 weights only come out without it. The PRA, consulting on the UK's rules, wrote that “The Basel 3.1 standards remove the 1.06 scaling factor on the basis that it is no longer necessary” (CP16/22, para 4.119). On this bank, putting it back would add £747.7m of RWA.

What moves an IRB risk weight?

The formal version

For a non-defaulted corporate, sovereign or bank exposure (CRE31.5 to CRE31.8; the formula is an image in CRE31 and was read in the BCBS explanatory note, sections 4.6 and 5.2, and in old UK CRR Article 153):

\begin{aligned} R &= 0.12 \cdot \frac{1-e^{-50\,\text{PD}}}{1-e^{-50}} + 0.24 \cdot \left(1-\frac{1-e^{-50\,\text{PD}}}{1-e^{-50}}\right) \;[-\,0.04\left(1-\tfrac{S-5}{45}\right)] \\ b &= \big(0.11852 - 0.05478 \cdot \ln \text{PD}\big)^2 \\ K &= \left[\text{LGD}\cdot N\!\left(\frac{G(\text{PD}) + \sqrt{R}\cdot G(0.999)}{\sqrt{1-R}}\right) - \text{PD}\cdot\text{LGD}\right] \cdot \frac{1 + (M-2.5)\,b}{1 - 1.5\,b} \\ \text{RW} &= 12.5 \cdot K, \qquad \text{RWA} = \text{RW} \cdot \text{EAD} \end{aligned}

$\text{PD}$ : one-year probability of default, at least 0.05% except for sovereigns (CRE32.4). $\text{LGD}$ : loss given default, a decimal. $M$ : effective maturity in years. $S$ : group sales in € millions, between 5 and 50; the bracketed term applies only to SMEs (CRE31.8). $\text{EAD}$ : exposure at default, in £.
$N$ : the standard normal distribution function (M-C06: The normal distribution); $G$ : its inverse, so $G(0.999) = 3.0902$ . In L3.3.11: Unexpected loss and default correlation these were $\Phi$ and $\Phi^{-1}$ .
The explanatory note writes the argument of N as $(1-R)^{-0.5}\,G(\text{PD}) + \left(\tfrac{R}{1-R}\right)^{0.5} G(0.999)$ : the same thing, since $x^{-0.5} = 1/\sqrt{x}$ (M-A03: Exponents).
Defaulted exposures use $K = \max(0,\ \text{LGD} - \text{BEEL})$ instead (CRE31.3; L4.2.07: Expected loss, provisions and the shortfall). There is no 1.06 anywhere.

Try it

Start at Bramwick. Slide PD down below 0.05% and watch the floor stop it. Set M to 1 and then 5. Switch to Large bank, then to Mortgage. Finally switch the rule set to the old UK CRR and see every weight grow by 6%.

Try itThe IRB risk weight, step by step

Asset class

PD (one year)1.13%

LGD40%

Effective maturity M, years2.50

Rule set

PD used, max(PD, 0.05%)1.130%

G(PD)−2.2801

Correlation R18.82%

Conditional PD at 99.9%14.85%

UL = LGD × (cond. PD − PD)5.489%

Maturity factor (b = 0.1326)1.2482

K6.852%

EL = PD × LGD (not in K)0.452%

Risk weight85.65%

RWA on £1,926.6m£1,650.1m

Raise M to 5 years and watch the strong borrowers' weights rise most; set it to 1 year and the maturity factor is exactly 1.

Starts at Bramwick Stores plc: PD 1.13%, LGD 40%, M 2.5, EAD £1,926.6m. Illustrative data, £ millions. Large bank sets LGD to 45%, the F-IRB senior unsecured LGD for financial institutions (CRE32.6), as well as R × 1.25; switching away restores 40%. The QRRE revolver PD floor (0.1%) is not applied here, and the old UK CRR choice changes only the 1.06 (its PD floor was 0.03%).

Worked example

Bramwick, then the whole IRB book

Bramwick Stores plc: PD 1.13%, senior unsecured, so LGD 40% under F-IRB (CRE32.6), M = 2.5, EAD £1,926.6m (drawn plus 40% of the undrawn line). £ millions.

From PD to RWA

Step 1 of 6

1
Correlation
$\begin{aligned} w &= \frac{1-e^{-50 \times 0.0113}}{1-e^{-50}} = 0.4316 \\ R &= 0.12 \times 0.4316 + 0.24 \times 0.5684 = 0.1882 \end{aligned}$

Corporate slice	PD	LGD	R	Cond. PD	K	RW	EAD	RWA
Thornvale Engineering Ltd	0.28%	35.0%	22.43%	6.89%	3.27%	40.9%	1,372.5	560.7
Bramwick Stores plc	1.13%	40.0%	18.82%	14.85%	6.85%	85.6%	1,926.6	1,650.1
Pellcourt Properties LtdSME: firm-size adjustment	0.57%	22.0%	20.16%	10.05%	2.75%	34.4%	3,034.8	1,043.2
Ostreth Energie GmbH	0.14%	40.0%	23.19%	4.34%	2.56%	32.0%	1,949.5	623.7
Larkhollow Freight Ltd	2.26%	34.3%	15.88%	20.01%	7.25%	90.6%	720.1	652.3
Mirefield Care Homes Ltd	1.13%	26.3%	18.82%	14.85%	4.51%	56.4%	1,109.1	625.0
Quorrel Systems BV	0.57%	40.0%	21.02%	10.52%	5.25%	65.6%	878.2	576.1
Hastemoor Construction Ltd	4.53%	40.0%	13.25%	27.11%	10.32%	129.0%	603.3	778.1
Vellacourt Hôtels SAS	2.26%	25.9%	15.88%	20.01%	5.48%	68.5%	992.3	679.4
Cindral Pharma Inc	0.07%	40.0%	23.59%	2.63%	1.71%	21.4%	1,317.3	281.4
Graybeck Plastics Ltd	11.31%	75.0%	12.04%	44.16%	26.92%	336.5%	298.6	1,005.0
Sallowmere Logistics GmbH	0.28%	32.9%	22.43%	6.89%	3.07%	38.4%	1,197.8	459.7
Corporates, F-IRB						58.0%	15,400.0	8,934.7

Mortgage segment	PD	LGD	R	Cond. PD	K	RW	EAD	RWA
LTV up to 60%	0.30%	5%	15%	4.63%	0.22%	2.7%	11,544.5	312.1
LTV 60–80%	0.60%	12%	15%	7.68%	0.85%	10.6%	13,119.6	1,394.1
LTV 80–95%	1.50%	20%	15%	14.56%	2.61%	32.6%	5,575.9	1,820.1
Mortgages, retail IRB						11.7%	30,240.0	3,526.3

Secured slices carry an F-IRB LGD below 40%: the collateral formula LGD* of CRE32.10 blends 40% on the unsecured part with 20% or 25% on the part covered after a 40% haircut. Graybeck's subordinated loan takes 75% (CRE32.7). The mortgages have no maturity factor (CRE31.13).

Regulatory lens

The formula is CRE31; its inputs and their floors are CRE32; how a bank may estimate them is CRE36. Statements on jurisdictions are as at 3 November 2026 as at 3 Nov 2026.

In the UK

Under the UK CRR in force until 31 December 2026 the risk weight ends “× 12,5 × 1,06” (Articles 153(1) and 154(1)). The PRA proposed removing the factor with Basel 3.1 (CP16/22, paras 4.118 to 4.119), and the UK's Basel 3.1 rules apply from 1 January 2027 (PS1/26, para 1.23). What else changes for UK IRB banks is in L4.2.11: Credit risk capital: what changes from 2027.

In the EU

CRR3 (Regulation (EU) 2024/1623) applies from 1 January 2025 and drops the 1.06 from the IRB risk weight, raises the PD floor from 0.03% to 0.05%, and confines exposures to institutions and to corporates with sales above €500m to F-IRB (EBA statement on CRR3 and credit risk, 17 July 2024; KPMG summary of CRR3, 2024).

In the US

The US advanced approaches still multiply credit RWA by 1.06: “Credit-risk-weighted assets means 1.06 multiplied by the sum of” the wholesale, retail and other parts (12 CFR 217.2). On 19 March 2026 the agencies proposed removing the advanced approaches from the capital framework; comments closed on 18 June 2026 (Federal Reserve fact sheet, 19 March 2026). As far as we could find, no final rule had been published by 3 November 2026.

Spark Atoms

Spark AtomA 6% top-up that outlived its reason

The 1.06 was a calibration, not a model: a multiplier to bring IRB capital back towards the old level. In May 2006 the fifth impact study showed minimum capital for large banks falling by 6.8% on average, yet the Committee kept 1.06, pointing to a more benign economy during the fourth and fifth studies than during the third and to downturn LGDs still being worked on (BCBS press release, 24 May 2006). It took the 2017 reforms to remove it.

Spark AtomWhy 99.9% and not 99%?

At 99% Bramwick-like loans would need capital for a default rate of 7.92%; at 99.9% it is 14.85%. The explanatory note gives the reasons for the higher level: Tier 2 cannot absorb losses as Tier 1 does, and the high level guards against errors in the banks' own PD and LGD estimates and in the model itself (section 5.1).

Key takeaways

K covers the loss in the economy's worst year in a thousand less expected loss: LGD × (conditional PD − PD), times a maturity factor.
The correlation is set by the rule: 24% to 12% for corporates as PD rises, less for SMEs, × 1.25 for large financial institutions.
The maturity factor is 1 at one year and rises with M, most for low PDs. F-IRB uses 2.5 years: 1.248 for Bramwick.
RW = 12.5 × K. Bramwick: K 6.85%, RW 85.65%, RWA £1,650.1m; the IRB book £12,461.0m.
CRE99's illustrative weights check the formula. Basel 3.1 has no 1.06; the old UK CRR and the US advanced approaches do.

Reaction

Check what stuck

Question 1 · objective 2

What is the asset correlation R for a corporate with PD 1% and sales above €50m, in %?

%

Question 2 · objective 2

Why does the corporate correlation fall from 24% to 12% as PD rises?

AWeaker borrowers default more for reasons of their own, so the economy explains less of their riskBSupervisors wanted to reward banks for lending to weak borrowersCHigher-PD loans are usually shorter, so they need less capitalDThe correlation is estimated from each bank's own defaults

Question 3 · objective 1

A mortgage pool has PD 2% and the mortgage correlation of 15%. What is its default rate in the 1-in-1,000 year (the conditional PD at 99.9%), in %?

%

Question 4 · objective 3

An IRB exposure has K = 5% and EAD of £400m. What are its RWA, in £m?

£m

Question 5 · objective 4

Under Basel 3.1, what happens to the 1.06 scaling factor?

AIt is gone: RW = 12.5 × K, and the CRE99 illustrative risk weights only match without itBIt now applies to standardised RWA as wellCIt is replaced by a factor of 1.25 on every exposureDIt stays for corporates and is removed for retail

Glossary

K (IRB capital requirement): K: the IRB capital requirement per £1 of exposure at default. The loss in a 1-in-1,000 year less expected loss, times the maturity adjustment.
Conditional PD: Conditional PD: the default rate of a large pool of similar borrowers when the economy is at a given bad percentile, here its 99.9th worst.
Asset correlation: Asset correlation R: how strongly borrowers' fortunes move with one shared economic factor. Set by the rule, not estimated by the bank.
Firm-size adjustment: Firm-size adjustment: lowers the correlation of corporates with group sales under €50m by up to 4 percentage points (CRE31.8).
Maturity slope b(PD): Maturity slope b(PD): how fast capital rises with each extra year of effective maturity. Larger for low PDs.
Effective maturity: Effective maturity M: the cash-flow-weighted remaining life of the exposure, floored at one year and capped at five under A-IRB; 2.5 years under F-IRB.
ASRF model: Asymptotic single risk factor model: the one-factor model in a portfolio so large and fine-grained that only the shared factor matters. It lets each loan's capital be computed on its own.
1.06 scaling factor: The 1.06 scaling factor: a multiplier on IRB credit RWA added in Basel II in 2004 to keep capital close to its old level. Basel 3.1 has none.

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, An Explanatory Note on the Basel II IRB Risk Weight Functions, July 2005: every piece of the formula, with the reasons for the constants. Intermediate.
M6.4: Portfolio credit models: the asymptotic single risk factor model and its limits. Advanced.

Origins

Basel Committee on Banking Supervision, CRE31: IRB approach: risk weight functions, paras 31.1–31.16, and CRE32: IRB approach: risk components, paras 32.4, 32.6–32.7, 32.10 and 32.44, bis.org.
Basel Committee on Banking Supervision, CRE99: Application guidance, Table 1, bis.org.
Basel Committee on Banking Supervision, An Explanatory Note on the Basel II IRB Risk Weight Functions, July 2005, sections 4.6, 4.7, 5.1–5.3.
Basel Committee on Banking Supervision, “Basel Committee maintains calibration of Basel II Framework”, press release, 24 May 2006.
Regulation (EU) No 575/2013 as it applies in the UK, Articles 153 and 154, legislation.gov.uk.
European Banking Authority, Statement on the application of CRR 3 in relation to credit risk, 17 July 2024.
KPMG AG Wirtschaftsprüfungsgesellschaft, EU-Bankenpaket und CRR III: Diese regulatorischen Neuerungen erwarten Institute im IRB-Ansatz, 2024.
Prudential Regulation Authority, CP16/22, chapter 4, paras 4.118–4.119, November 2022; and PS1/26, para 1.23, January 2026.
12 CFR 217.2; Board of Governors of the Federal Reserve System, Fact sheet on the capital proposals, 19 March 2026.

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