Backtesting and the P&L attribution test · RiskAtoms Academy

By the end you will be able to

1Apply desk-level backtesting at 99% and 97.5% and judge its limits with the binomial distribution.
2Run the bank-wide traffic light and read the multiplier m_c.
3Compute the P&L attribution test's Spearman correlation and KS statistic and place a desk in its zone.
4Decide which of RiskAtoms Bank plc's desks may use the internal model, and with what consequence.

Why it matters

Under the FRTB a bank does not get permission for a model; each desk earns it, every quarter. RiskAtoms Bank plc's four desks all pass backtesting. But the FX and commodities desk's risk model has no Brent factor, and its daily P&L from the model ranks the days with a Spearman correlation of only 0.34 against the real thing. It fails the P&L attribution test and falls back to the standardised approach. The Rates desk's simpler curve model lands in between, in the amber zone, and its cost appears in L4.3.08: Total market risk capital.

Core explanation

Two tests, desk by desk

L4.3.05: Internal models: expected shortfall and L4.3.06: Non-modellable risk factors and the internal DRC built the model. Before any desk may use it, the desk must pass two tests each quarter (MAR30.4): backtesting, which checks the VaR against what happened, and the P&L attribution test, which checks that the risk model sees the same P&L as the desk's pricing models. A separate bank-wide backtest sets the multiplier.

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 test period is the last 250 days of the history (days 500 to 749), which include the end of the stressed spell. VaR is a one-day historical simulation on the previous 250 days, recomputed with today's positions; L3.2.14: Backtesting backtested a delta-normal VaR on the same days, so its counts differ. Actual P&L is taken equal to hypothetical P&L.

Desk-level backtesting

Desk-level backtesting compares each desk's one-day VaR at both 99% and 97.5% with its P&L over the last 12 months. A desk with more than 12 exceptions at 99%, or more than 30 at 97.5%, must use the standardised approach (MAR32.19). A correct model would show 2.5 and 6.25 on average (M-C04: The binomial distribution), and would breach the limits with probabilities of 0.0002% and less than 0.0001%. The limits are loose on purpose: a model with a true 2% exception rate, twice what it claims, still passes the 99% limit 99.8% of the time (M-D04: Confidence intervals and hypothesis tests). They catch broken models; the bank-wide test and the multiplier deal with merely optimistic ones.

Desk	Exceptions at 99%	Exceptions at 97.5%	Within the limits (12, 30)
Equities	5	7	yes
Rates	2	6	yes
FX and commodities	2	4	yes
Credit	2	3	yes

The bank-wide traffic light

The zones of L3.2.14: Backtesting are still there: 99% one-day VaR, 250 days, green up to 4 exceptions, amber 5 to 9 (called yellow in L2.5.02: Return distributions), red 10 or more. Two things changed. Exceptions are counted against both actual and hypothetical P&L, and the larger count is used (MAR32.5). And the multiplier, now applied to a stressed ES, runs from 1.5 rather than 3 (MAR32.9; MAR33.42):

MAR32.9, Table 1
Exceptions	Zone	m_c
0	green	1.50
1	green	1.50
2	green	1.50
3	green	1.50
4	green	1.50
5	amber	1.70
6	amber	1.76
7	amber	1.83
8	amber	1.88
9	amber	1.92
10 or more	red	2.00

The bank-wide backtest covers the desks on the model. For the three model desks together there are 2 exceptions, on days 515 and 520: green, m_c = 1.5. The whole book, with the FX and commodities desk back in, has 1.

Daily P&L of Equities, Rates and Credit together, with the one-day 99% VaR drawn below zero. Red bars are exceptions.

The P&L attribution test

L3.2.08: P&L attribution and explain explained a day's P&L factor by factor. The P&L attribution test asks a narrower question: does the risk model see the same P&L as the front office? For each desk and day it compares the risk-theoretical P&L with the HPL over the last 250 days, using two statistics (M-D05: Rank correlation and distribution tests). The Spearman correlation checks that the two rank the days alike; the KS statistic checks that their distributions have the same shape. The zones (MAR32.42, Table 2):

KS thresholds 0.09 and 0.12 correspond to p-values of 0.264 and 0.055
Zone	Spearman	KS	Consequence
green	> 0.80 and	< 0.09	IMA
amber	otherwise	otherwise	IMA with a capital surcharge
red	< 0.70 or	> 0.12	standardised approach

The book's desks use different risk models, chosen to show each zone. Equities and Credit use a second-order Taylor expansion of full revaluation; Rates moves the 2-year rate by 0.643 times the 10-year (a one-factor curve); FX and commodities has no Brent factor, because the futures are a new product.

Desk	Spearman	KS	PLA zone	Backtesting	Uses
Equities	1.0000	0.008	green	passes	IMA
Rates	0.7580	0.096	amber	passes	IMA + surcharge
FX and commodities	0.3367	0.236	red	passes	SA
Credit	1.0000	0.008	green	passes	IMA

Reading the results

A red desk goes to the standardised approach and comes back only after it is green again and meets the backtesting requirements over the past 12 months; an amber desk keeps the model but pays a surcharge (MAR32.43–32.44). Whatever the mix of zones, at least 10% of the bank's aggregate market risk capital must sit on desks that pass both tests, or the bank loses the IMA altogether (MAR32.2).

The formal version

For a desk with HPL h_t and RTPL r_t over n = 250 days, rank differences d_t, empirical distribution functions F, and X the number of exceptions of a correct VaR at confidence c (MAR32.5–32.9, 32.18–32.19, 32.37–32.42):

\begin{aligned} X &\sim \text{Bin}(250,\ 1-c), \qquad \text{desk out if } X_{99} > 12 \ \text{or}\ X_{97.5} > 30 \\ m_c &= 1.5 + \text{add-on}(X_{\text{bank}}), \qquad X_{\text{bank}} = \max\big(X^{\text{actual}},\ X^{\text{hyp}}\big) \\ r_S &= \text{corr}\big(\text{rank}(h),\ \text{rank}(r)\big) = 1 - \frac{6\sum_t d_t^2}{n(n^2-1)} \quad \text{(no ties)} \\ \text{KS} &= \max_x \big|F_{h}(x) - F_{r}(x)\big| \\ \text{green} &: r_S > 0.80 \ \text{and}\ \text{KS} < 0.09; \qquad \text{red}: r_S < 0.70 \ \text{or}\ \text{KS} > 0.12 \end{aligned}

The add-on is 0 for up to 4 exceptions, then 0.20, 0.26, 0.33, 0.38, 0.42 and 0.50 for 10 or more (Table 1). Ties in the ranks take their average rank; the rank-difference formula then becomes approximate, and the correlation of the ranks is the definition.

Try it

Start on FX and commodities, in the red zone. Move the slider a step at a time to fix its model and note the first setting where it turns amber and the first where it turns green: KS and Spearman cross their thresholds at different points. KS can wobble on the way, rising for a step before it falls, because it is the single largest gap between the two distributions, not an average. Then look at Equities, where the two P&Ls are almost one line.

Try itThe P&L attribution test, desk by desk

Desk

Fix the model: move RTPL towards HPL0%

Risk model: No Brent factor in the model (new product).

Spearman0.3367

KS statistic0.236

Zonered

Green: Spearman above 0.80 and KS below 0.09. Red: Spearman below 0.70 or KS above 0.12. Amber: anything else (MAR32.42).

Each dot is a day. On the dashed line the model is exact.

HPLRTPL

The KS statistic is the largest vertical gap, in red.

250 days (days 500 to 749), £ thousands. HPL is today's desk revalued in full on each day's moves; RTPL is what the desk's risk model gives for the same moves. Illustrative.

Worked example

The Rates desk, test by test

Days 500 to 749, illustrative.

Is the Rates desk on the model?

Step 1 of 5

1
Desk backtesting
2 exceptions at 99% and 6 at 97.5%: within 12 and 30. Passed.

Regulatory lens

Basel: MAR32.1–32.45. Status as at 3 November 2026 as at 3 Nov 2026.

In the UK

The PRA splits the PLA amber zone in two. Yellow desks were green and have deteriorated within set thresholds; orange desks previously fell below the lowest threshold and may not use the model until they are green again (PS17/23, para 3.35). Yellow desks pay the capital surcharge; red and orange desks are treated as failing the test (PRA Rulebook, IMA Part, Article 325ba(6)). The IMA starts on 1 January 2028, and for its first year the PLA test is for monitoring only, so a failing desk does not automatically move to the standardised approach. CP9/26 (June 2026) proposes to extend that period from one year to three (paras 2.9–2.10). Not final as at 3 November 2026.

In the EU

CRR Article 325bf (backtesting) and 325bg (PLA), with the thresholds in Delegated Regulation (EU) 2022/2059. The EU also has four zones, green, yellow, orange and red, with Basel's Spearman and KS thresholds (EBA final draft RTS, EBA/RTS/2020/02, Article 9). Until 31 December 2029, Delegated Regulation (EU) 2026/1221 lets desks that fail the PLA test keep using the internal model, with the test for monitoring only (Article 495i).

In the US

The agencies' proposals of 19 March 2026 would move the models approach from VaR to expected shortfall with liquidity horizons, and with it the tests on each desk; 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 correct model is amber one year in nine

With 250 days at 99%, a perfectly calibrated VaR has five or more exceptions with probability 10.8%. About one year in nine, a bank with a correct model pays a higher multiplier by bad luck alone, the price of a test simple enough to apply everywhere.

Spark AtomOne desk alone would be amber

The Equities desk has 5 exceptions at 99%, all in the stressed spell: comfortably within its desk limit of 12, but enough to put it in the amber zone if it were the bank's only model desk. Together with Rates and Credit, whose P&L offsets some of its losses, the bank-wide count is 2.

Key takeaways

Each desk is backtested at 99% and 97.5% over 12 months; more than 12 or 30 exceptions sends it to the SA. All four of the bank's desks pass.
The bank-wide test keeps the green, amber and red zones, counts the larger of actual and hypothetical exceptions, and sets m_c from 1.5 to 2.0. The model desks have 2 exceptions: m_c 1.5.
The PLA test compares RTPL with HPL by Spearman correlation (ranks) and the KS statistic (shape) over 250 days.
Green: Spearman above 0.80 and KS below 0.09. Red: Spearman below 0.70 or KS above 0.12, and the desk goes to the SA. Amber in between, with a surcharge.
On the book Equities and Credit are green, Rates amber (0.758, 0.096) and FX and commodities red (0.337, 0.236).
The UK and the EU split amber into yellow and orange, and both make the PLA test monitoring-only for a time: the EU to 2029, the UK for the first year, or three as proposed in CP9/26.

Reaction

Check what stuck

Question 1 · objective 1

A desk's 97.5% VaR model is correct. How many exceptions should it expect in 250 days?

Question 2 · objective 1

A desk has 14 exceptions at 99% and 25 at 97.5% over the last 12 months. Its PLA test is green. What happens?

AIt leaves the IMA and uses the standardised approach: more than 12 exceptions at 99%BIt stays on the IMA: 25 is below the 97.5% limit of 30CIt stays on the IMA with a capital surchargeDOnly the bank-wide multiplier rises

Question 3 · objective 3

Over five days a desk's HPL is 1, 2, 3, 4, 5 and its RTPL is 2, 1, 4, 3, 5 (in £m). What is the Spearman correlation?

Question 4 · objective 3

HPL on four days is 1, 2, 3, 4 and RTPL is 2, 3, 4, 5. What is the KS statistic?

Question 5 · objective 2

The model desks together have 6 exceptions at 99% over 250 days. What is m_c?

Question 6 · objective 3

A desk's Spearman correlation is 0.85 and its KS statistic 0.10. Which PLA zone is it in?

AAmberBGreenCRedDIt depends on its backtesting

Glossary

Desk-level backtesting: Desk-level backtesting: each trading desk's one-day VaR at 99% and at 97.5% compared with its daily P&L over the last 12 months. More than 12 exceptions at 99%, or 30 at 97.5%, and the desk leaves the IMA (MAR32.18–32.19).
Bank-wide backtesting: Bank-wide backtesting: the one-day 99% VaR of all the model desks together against their daily P&L over 250 days. Its exceptions set the multiplier m_c (MAR32.5–32.11).
Hypothetical P&L (HPL): Hypothetical P&L (HPL): the P&L of yesterday's end-of-day positions from today's market moves, valued with the bank's own pricing models; no new trades, fees or commissions.
Actual P&L: Actual P&L: the desk's real daily P&L, including intraday trading and new trades but not fees and commissions.
Risk-theoretical P&L (RTPL): Risk-theoretical P&L (RTPL): the P&L the desk's risk model gives for the same day, using only its own risk factors and valuation methods.
P&L attribution test: P&L attribution test (PLA): a quarterly test of whether a desk's RTPL tracks its HPL, with the Spearman correlation and the KS statistic over the last 250 days (MAR32.34–32.44).
Spearman correlation: Spearman correlation: the correlation of the ranks of two series. It asks whether good and bad days are ordered alike, whatever their sizes.
KS statistic: Kolmogorov-Smirnov (KS) statistic: the largest vertical gap between the empirical distribution functions of two samples. It asks whether the two P&Ls have the same shape.
Amber zone: Amber zone (PLA): neither green nor red. The desk may stay on the IMA but adds a capital surcharge (MAR32.44, MAR33.43).

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, MAR32: Internal models approach: backtesting and profit and loss attribution test requirements, bis.org: the full test design, with its FAQs. Advanced.
European Banking Authority, Final draft RTS on back-testing and profit and loss attribution requirements (EBA/RTS/2020/02): a worked definition of each PLA metric and zone. Advanced.

Origins

Basel Committee on Banking Supervision, MAR32, paras 32.1–32.45, bis.org (read in the Swiss SIF copy).
Prudential Regulation Authority, PS17/23, December 2023, para 3.35; PRA Rulebook: Market Risk: Internal Model Approach (CRR) Part, Article 325ba, version from 1 January 2028; CP9/26, June 2026, paras 2.9–2.10.
European Banking Authority, final draft RTS EBA/RTS/2020/02, Articles 8–9.
Commission Delegated Regulation (EU) 2022/2059 (title and contents); Commission Delegated Regulation (EU) 2026/1221 of 4 June 2026, Article 495i.
Board of Governors of the Federal Reserve System, press release and fact sheet, 19 March 2026.

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