P&L Attribution

People in risk management and the middle office use P&L attribution when they check which risks produced the daily P&L, and by how much.

Why split P&L by risk factor

The examination manual of the US Federal Reserve Board ("the manual" below) says this work is to be carried out by a group independent from the traders¹. The manual says that, as of its April 2003 section, this work was the responsibility of the risk management function or the middle office in most larger institutions.

P&L attribution shows how much of daily P&L came from which risk. Trading P&L arises from several factors. A financial institution therefore ties the P&L of each transaction to its underlying source and tries to understand why value changed. Examples of sources are capturing the bid/offer spread, the profit at origination of a transaction, and proprietary position taking. Some desks also split daily P&L by the risks they actually manage, such as delta, gamma, theta, rho, and vega².

Being able to show accurately where daily P&L came from is one of the key controls for checking that trading valuations are reasonable. Measuring P&L and tying it to its sources also helps confirm that risks are measured accurately and monitored². The calculation of daily P&L itself and the verification of valuations are covered in Daily Valuation and P&L Sign-Off. As a term, it is the same work as P&L explain.

How an attribution is built

The part of P&L that can be tied to risk factors is called explained P&L, and the gap between it and actual P&L is called unexplained P&L³.

$$\text{unexplained P\&L} = \text{actual P\&L} - (\text{sum of the contributions of each source})$$

Consider a hypothetical example in which a desk that holds many options is followed over two days. Because the manual gives no method for calculating each source's contribution, the example sets a simple convention that can be calculated by hand: the contribution is the previous day's sensitivity times the day's move in the risk factor.

Valuations, P&L, and the price of the underlying asset are in yen, and volatility is in %. The sensitivities at the end of the previous day are as follows.

• Delta is 200,000, a gain of 200,000 yen when the underlying asset rises by 1 yen.
• Gamma is −40,000.
• Vega is 300,000, a gain of 300,000 yen when volatility rises by 1 percentage point.
• Theta is −150,000, a loss of 150,000 yen in one day.

The gamma contribution is set as gamma times the square of the move, divided by 2.

Suppose this desk investigates the cause whenever the size of unexplained P&L (positive or negative) exceeds 100,000 yen. On day 1, the gain from delta is offset by the losses from gamma, vega, and theta, so explained P&L is −30,000 yen and unexplained P&L is −10,000 yen, within the threshold. On day 2, the price moves by 10 yen and the delta gain of 2,000,000 yen is offset by the gamma loss of 2,000,000 yen, but actual P&L is −350,000 yen, so unexplained P&L is −200,000 yen, above the threshold. This table alone does not settle whether the gap comes from a rough approximation or from a valuation error. That is why the cause is investigated. The limits of sensitivities are covered in Sensitivities: DV01, Vega and Key Rate Sensitivities.

Investigating unexplained P&L

When unexplained P&L is large or an unusual pattern of attribution appears, the valuation process may be flawed, and reported income may be overstated or understated. The position may also carry risks that are not adequately captured and measured². Depending on the result of the investigation, the valuation model may be changed, or the choice of valuation inputs, such as volatilities and correlations, may be challenged³.

The attribution process should be standardized and consistently applied across business units¹. The amount used as the threshold for investigation varies among institutions, and there are both cases in which a threshold amount is set and cases in which the risk management unit judges case by case³. Where the first-day profit of a new transaction is allocated among the sales, origination, and proprietary trading desks, and valuation reserves, the reasonableness of the allocation should also be reviewed independently³.

How it is used in regulation (the P&L attribution test)

Japan's capital adequacy notice (the Financial Services Agency standards under Article 14-2 of the Banking Act, hereafter "the Notice") requires banks that measure market risk capital under the internal models approach of the FRTB to carry out the P&L attribution test.

The test compares the front-office model with the internal models approach, to check that simplifying the valuation in the internal models approach has not produced a material difference. For each trading desk, it compares two P&Ls every day⁴.

• Risk-theoretical P&L: the P&L calculated with the desk's risk management model⁵.
• Hypothetical P&L: the daily P&L that arises when the positions held at the end of the previous day are revalued with the market data at the end of the current day. It excludes fees, intraday trades, and new or modified transactions⁵.

The risk factors of risk-theoretical P&L include all risk factors, whether or not they can be modeled⁶. Hypothetical P&L and actual P&L (the P&L produced in the process of calculating daily P&L) are calculated with the same valuation model. Risk-theoretical P&L is calculated with the desk's risk management model, so it uses a different model. The test compares risk-theoretical P&L with hypothetical P&L and does not use actual P&L. In backtesting, VaR is compared with both hypothetical P&L and actual P&L⁷.

The test has two indicators, both calculated from a time series of the latest 250 business days. One is the Spearman rank correlation indicator, which looks at the rank correlation, and the other is the Kolmogorov-Smirnov (KS) test indicator, which looks at how close the distributions are⁸. The result falls into green, amber, or red, as in the table below⁹.

Figure description

The figure checks the red condition first and the green criteria next. A desk that meets neither is amber.

Apply the flow in Figure 1 to three desks of Bank A. Consider a hypothetical example in which the indicators are calculated from the latest 250 business days.

Both indicators are checked. At the boundary values, a Spearman of exactly 0.80 is not "above 0.80", and a KS of exactly 0.09 is not "below 0.09", so in neither case is the desk green. Whether a desk that is not green is amber or red depends on the red condition: Spearman below 0.70 or KS above 0.12. Meeting either one makes it red. A Spearman of exactly 0.70 is not "below 0.70", so it does not meet the red condition on the Spearman side.

Deep diveCalculating a rank correlation by hand from 5 days of P&L

Consider a hypothetical example in which the Spearman indicator is calculated from 5 days of P&L, using the formula in the Notice (the covariance of the rank data divided by the product of the two standard deviations). The actual test uses 250 business days.

Day	Hypothetical P&L	Rank	Risk-theoretical P&L	Rank
1	−4	1	−3	1
2	2	3	1	3
3	−1	2	−2	2
4	6	5	4	4
5	3	4	5	5

The mean of the ranks is 3 for both. The products of the deviations from the mean are 4, 0, 1, 2 and 2, from day 1 in order. Their sum, 9, divided by 5 gives a covariance of 1.8. The mean of the squared deviations is (4 + 0 + 1 + 4 + 1) ÷ 5 = 2 for both sets of ranks, so both standard deviations are √2. The Spearman rank correlation indicator is 1.8 ÷ (√2 × √2) = 0.9⁸. It falls below 1 because the ranks of day 4 and day 5 are swapped.

When the result is amber or red, it is reported to the Commissioner of the Financial Services Agency, with the cause and the plan to deal with it¹⁰. A desk whose backtesting exceptions over the latest 12 months exceed 12 at the 99% confidence level, or 30 at 97.5%, also uses the standardized approach¹¹. To return from red to the internal models approach, the desk must be green in a repeated test. The number of backtesting exceptions over the latest 12 months must also not exceed these numbers⁹.

Relevant regulations

The substance of the rules is in Articles 275-4 to 275-9 of the Notice (the P&L attribution test) and in the definitions of P&L in Article 1. The Notice requires banks that use the internal models approach to carry out the desk-level P&L attribution test every quarter and update the classification of results¹². The January 2019 overview by the Financial Services Agency and the Bank of Japan also gives the values of the test indicators: for the Spearman rank correlation indicator and the KS test indicator, 0.80 and 0.09 for the amber zone and 0.70 and 0.12 for the red zone¹³.

Notes

1. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual", Section 2100.1, "Financial Performance" (April 2003), page 9 (PDF page 150), the second paragraph of "INCOME ATTRIBUTION". ↩ ↩²

2. The same manual, Section 2100.1, pages 8–9 (PDF pages 149–150), the opening of "INCOME ATTRIBUTION" and "Explained Profits and Losses". ↩ ↩² ↩³

3. The same manual, Section 2100.1, page 9 (PDF page 150), "Unexplained Profits and Losses" and the end of "Explained Profits and Losses". ↩ ↩² ↩³ ↩⁴

4. Financial Services Agency, "銀行法第十四条の二の規定に基づき、銀行がその保有する資産等に照らし自己資本の充実の状況が適当であるかどうかを判断するための基準" (Standards for determining whether a bank's capital adequacy is appropriate in light of the assets it holds, under Article 14-2 of the Banking Act, our translation; FSA Notice No. 19 of 2006, consolidated version updated on August 24, 2026), Article 275-4, paragraphs 1 and 3 (pages 462–463) (in Japanese). ↩

5. The same Notice, Article 1, items 91 (P&L attribution test), 92 (hypothetical P&L), 94 (risk-theoretical P&L), and 95 (actual P&L) (pages 24–25). ↩ ↩²

6. The same Notice, Article 275-5, paragraphs 1, 3 and 7 (page 463). Paragraph 7 provides that the same valuation model is used to calculate hypothetical P&L and actual P&L. ↩

7. The same Notice, Article 275-3, paragraph 1 (page 461). ↩

8. The same Notice, Article 275-7, paragraphs 1 to 4 (page 465). ↩ ↩²

9. The same Notice, Article 275-8 (pages 465–466). ↩ ↩²

10. The same Notice, Article 275-9, items 1 to 3 (pages 466–467). ↩

11. The same Notice, Article 275-3, paragraph 4 (page 462). It provides that a desk uses the standardized approach if its backtesting results show more than 12 exceptions at the 99% confidence level, or more than 30 at the 97.5% confidence level, over the latest 12 months. ↩

12. The same Notice, Article 272-6, item 10 (b) (3) (page 443). It sets the procedure of carrying out backtesting and the P&L attribution test every quarter and updating the classification. ↩

13. Financial Services Agency and Bank of Japan, "「マーケット・リスクの最低所要自己資本」の概要（2019 年 1 月）" (Overview of "Minimum capital requirements for market risk", January 2019, our translation), page 14 (PDF page 15), "IMA：2016 年の規則文書からの変更点" (IMA: changes from the 2016 rules text), "PLA テスト" (PLA test) (in Japanese). ↩