P&L attribution and explain · RiskAtoms Academy

By the end you will be able to

1Split a desk's daily P&L into risk factors with its sensitivities (delta, gamma, vega, theta).
2Build a revaluation-based explain, and say what each method leaves unexplained and why.
3Read the unexplained P&L as a diagnostic: missing factors, wrong sensitivities, big moves, bad data.
4Tell actual, hypothetical and risk-theoretical P&L apart, and say which one a risk model is judged on.

Why it matters

Every evening the desk head signs off a number: today the Markets desk lost £2,355k. The next question from the risk manager is always “why?”. If the answer adds up, the risk system understands the book. If it does not, something is wrong: a sensitivity is missing, a price is stale, or the book holds a risk nobody is measuring. The P&L explain is how desks find out, every day.

Core explanation

Where did today's money come from?

Illustrative dataRiskAtoms Bank plc's Markets desk

One made-up trading book on one made-up valuation day, in pounds: eight positions in four books, eight risk factors and 750 days of simulated factor moves. The levels (UK index 8,000, GBP/USD 1.30, Brent $80 and so on) are round illustrative numbers, not market data. The same book runs through every lesson in Module 3.2, so the figures agree from lesson to lesson.

We explain real days from the book's 750-day history. The P&L is the hypothetical P&L: today's eight positions, revalued in full with that day's factor moves and one day of time decay on the puts (−£10,531 a day). No new trades, fees or funding.

Three P&Ls

A desk has more than one P&L, and the explain has to say which it is explaining. Actual P&L is what finance reports: market moves plus everything the traders did during the day. Hypothetical P&L freezes yesterday's positions and applies today's market. Risk-theoretical P&L is what the risk model says those positions should have made, using only its own factors and pricing. The gap between hypothetical and risk-theoretical P&L measures how well the risk model sees the book.

Explain with sensitivities

L2.5.04: Sensitivities estimated P&L as sensitivity × move, and L3.2.06: Option Greeks across a book added up the Greeks across the book. A sensitivity-based explain does exactly that, factor by factor, every day. Take day 730, an ordinary loss of £933k: the US index fell 1.65%, the 10-year rate rose 2.2bp, and the other factors moved a little.

UK index−£70k

US index−£490k

2y rate+£43k

10y rate−£203k

GBP/USD−£33k

Brent−£105k

UK vol−£24k

IG spread−£43k

Time (theta)−£11k

Unexplained+£2k

Actual P&L−£933k

The US basket and the gilt explain most of the loss. Only £1,746 is left unexplained, 0.19% of the day. On a day like this the Greeks know the book.

A jump day

Day 486 is different: equities fell 6%, rates dropped 18 to 22bp, UK implied vol jumped 7 points and spreads widened 25bp. The book lost £2,355k. With deltas alone the explain misses £898k, because the puts gain far more than their delta says on a 6% fall. Adding the gamma terms brings the unexplained down to £92k.

Factor	Delta	Gamma	Delta + gamma	Revalued alone
UK index	−£2,444k	+£632k	−£1,812k	−£1,761k
US index	−£1,500k	+£38k	−£1,462k	−£1,463k
2y rate	−£633k	−£1k	−£634k	−£634k
10y rate	+£2,035k	+£19k	+£2,054k	+£2,054k
GBP/USD	+£150k	+£1k	+£151k	+£151k
Brent	−£862k	+£30k	−£831k	−£832k
UK vol	+£869k	+£82k	+£950k	+£928k
IG spread	−£858k	+£6k	−£852k	−£852k
Time (theta)	−£11k		−£11k	−£11k
Unexplained	+£898k		+£92k	+£65k
Actual P&L			−£2,355k	−£2,355k

Explain by revaluation

The last column is a revaluation-based explain: move one factor at a time, reprice the book in full, and record the change. It catches every effect of a factor on its own, however non-linear: the UK index line is −£1,761k, against −£1,812k from delta and gamma. What it cannot catch is the interaction of factors that moved together. The puts are worth more when spot falls and vol rises than the two moves separately suggest (their vanna), and the US basket's pound value depends on the index and the exchange rate together. That leaves £65k.

The waterfall: everything explained, nothing settled

A third method moves the factors one after another and keeps each move in place: the waterfall. The steps add up to the P&L exactly, so nothing is unexplained. But the shares depend on the order. On day 486, moving spot before vol gives the UK index −£1,761k and UK vol +£1,042k; moving vol first gives vol +£928k and the index −£1,647k. The cross effect has been handed to whichever factor came second.

Reading the unexplained

Across all 750 days, the typical size (root mean square) of the unexplained P&L is £115k with deltas alone, £22.1k with deltas and gammas and £23.6k by revaluation, against £972k for the P&L itself. Most of it comes from the 60 stressed days: there the sensitivity residual averages £67k, against £11.7k on calm days. Big moves are where the Taylor series runs out. A desk sets a tolerance, say more than 10% of the day's P&L and more than £25k. With deltas and gammas the book breaches it on 8 days of 750, 4 of them in the stressed spell; with deltas alone, on 104.

A residual that is large once, on a jump day, is expected. A residual that is large every day, or that tracks one factor's move, is a warning. If the risk system had lost the vol factor, the explain of day 486 would leave £1,042k unexplained, and the VaR would be blind to vega.

Sort itWhat does this unexplained P&L point to?

Sort each pattern by its likeliest cause.

The residual is near zero on calm days and large on the three biggest market days.
Every day the residual is about −0.9 × the move in the euro swap rate, which the desk trades but the risk system does not map.
A £400,000 residual on one quiet day, reversing the next day.
The residual grows steadily over a week as an option nears expiry.
A trade booked at the wrong notional shows up as unexplained P&L until it is amended.

The formal version

Write the book's value as a function of the factor levels and time, $V(x_1, \dots, x_8, t)$ . Over one day the factors move by $\Delta x_i$ and time by $\Delta t$ . A second-order Taylor expansion (M-B03: Taylor expansion) in several variables gives

\begin{aligned} \Delta V &= V(x + \Delta x,\, t + \Delta t) - V(x, t) \\ &= \underbrace{\sum_i \delta_i\,\Delta x_i}_{\text{deltas}} + \underbrace{\tfrac12 \sum_i \Gamma_{ii}\,(\Delta x_i)^2}_{\text{gammas}} + \underbrace{\sum_{i<j} \Gamma_{ij}\,\Delta x_i\,\Delta x_j}_{\text{cross gammas}} + \underbrace{\theta\,\Delta t}_{\text{time}} + R_3 \\[4pt] \delta_i &= \frac{\partial V}{\partial x_i}, \qquad \Gamma_{ij} = \frac{\partial^2 V}{\partial x_i\,\partial x_j}, \qquad \theta = \frac{\partial V}{\partial t} \end{aligned}

$R_3$ holds third and higher-order terms. Each method keeps a different part:

\begin{aligned} \text{Sensitivity explain:}\quad U_{\text{sens}} &= \Delta V - \sum_i \delta_i\Delta x_i - \tfrac12\sum_i \Gamma_{ii}(\Delta x_i)^2 - \theta\,\Delta t = \sum_{i<j}\Gamma_{ij}\Delta x_i\Delta x_j + R_3 \\ \text{Revaluation explain:}\quad U_{\text{reval}} &= \Delta V - \sum_i \big[V(x + \Delta x_i e_i) - V(x)\big] - \theta\,\Delta t = \text{cross effects only (all orders)} \\ \text{Waterfall:}\quad \Delta V &= \sum_{k=1}^{8} \big[V(x + \Delta x_{(1)} + \dots + \Delta x_{(k)}) - V(x + \Delta x_{(1)} + \dots + \Delta x_{(k-1)})\big] + \text{time} \end{aligned}

$e_i$ moves factor $i$ alone. The waterfall is a telescoping sum, so it is exact whatever the order, but the order decides which factor gets each cross effect.

The puts on day 486. Put through the expansion term by term, with the UK index falling 6% ( $\Delta x = -0.06$ ) and vol rising 7 points:

Term	Formula	P&L
Delta	δ Δx	+£1,156k
Gamma	½ Γ (Δx)²	+£524k
Vega	ν Δσ	+£869k
Volga	½ ∂²V/∂σ² (Δσ)²	+£82k
Vanna	∂²V/∂x∂σ Δx Δσ	+£402k
Theta	θ × 1 day	−£11k
Sum of terms		+£3,021k
Full revaluation		+£2,757k
Third and higher order		−£264k

Even the full second-order expansion is out by £264k on the puts: a 6% fall with a 7 point vol jump is too far for a two-term Taylor series. That is why a revaluation explain is the better tool on big days, and why L3.2.10: Historical simulation in practice and L3.2.11: Monte Carlo VaR revalue in full rather than trusting sensitivities.

Try it

Pick a day and a method. Start on day 486 and switch from deltas only to revaluation. Then tick the box to lose the vol factor and watch the unexplained line across the 750 days jump on the stressed days.

Try itExplain a day on the desk

Day486

Jump to

Explain method

The risk system has lost the UK vol factor (a mapping error)

UK index−£1,812k

US index−£1,462k

2y rate−£634k

10y rate+£2,054k

GBP/USD+£151k

Brent−£831k

UK vol+£950k

IG spread−£852k

Time (theta)−£11k

Unexplained+£92k

Actual P&L−£2,355k

Hypothetical P&L−£2,354.8k

Explained−£2,447.0k

Unexplained+£92.2k

Share of |P&L|3.9%

Within tolerance.

Unexplained P&L on every day with this method. Days outside the tolerance (more than 10% of the day's P&L and more than £25k): 8 of 750. The shaded band is the stressed spell.

Worked example

Day 205: the surprise rate rise

Explaining a rates shock

Step 1 of 5

1
The day
Day 205 is a surprise policy move in calm times: the 2-year swap rate rose 25bp and the 10-year 15bp, the UK index fell 0.8%, GBP/USD rose 1.2%, UK vol rose 0.6 points and the spread widened 3bp. Hypothetical P&L: −£1.087m.

Regulatory lens · looking ahead

Under Basel's FRTB, a desk that wants to use an internal model must pass a P&L attribution test each quarter: its risk-theoretical P&L is compared with its hypothetical P&L using a Spearman correlation and a Kolmogorov-Smirnov test, and desks that fail move to the standardised approach. M4.3: Market risk capital (FRTB) and CVA covers the test. Backtesting, which compares VaR with both hypothetical and actual P&L, is L3.2.14: Backtesting.

Spark Atoms

Spark AtomWhen the hedges were not real

In January 2008 Société Générale announced a loss of about €4.9bn from unwinding futures positions built up by one trader, Jérôme Kerviel, who had hidden them behind fictitious offsetting trades. On paper his book looked almost flat. Checks that tie each day's P&L back to real, confirmed positions and their risks are one of the controls that catch this.

Spark AtomWhy theta is not free money

The puts lose about £10,531 a day to time. That is the price of their gamma: on day 486 the gamma and vanna terms alone were worth £926k. For an option holder, theta paid on quiet days buys convexity on wild ones.

Key takeaways

A P&L explain splits each day's P&L into risk factors, time and other causes, and shows what is left over.
Sensitivity-based explain uses delta × move, ½ gamma × move² and theta: a Taylor expansion. It is good for small moves.
Revaluation-based explain moves one factor at a time; what it leaves is the cross effect of factors moving together.
A waterfall explains everything but its factor shares depend on the order.
Large residuals on jump days are expected; persistent residuals that track one factor point to a missing or mis-mapped risk.
Hypothetical P&L (fixed positions) is what a risk model should be judged against; actual P&L adds trading, fees and adjustments.

Reaction

Check what stuck

Glossary

P&L explain (attribution): A daily report that splits a desk's P&L into the parts due to each risk factor, time, new trades and other causes, and shows what is left unexplained.
Actual P&L: The desk's reported P&L for the day: market moves, new and amended trades, intraday trading, fees, commissions and valuation adjustments. (The actual P&L used in Basel backtesting leaves out fees and commissions.)
Hypothetical P&L: The P&L the end-of-day positions of yesterday would have made from today's market moves, with no new trades, fees or commissions.
Risk-theoretical P&L: Risk-theoretical P&L: the P&L the risk model predicts from its own risk factors and pricing for the day's moves.
Sensitivity-based explain: Explain built from sensitivities: delta × move for each factor, plus gamma terms and theta.
Revaluation-based explain: Explain built by revaluing the positions in full with one factor moved at a time.
Waterfall explain: Explain built by moving the factors one after another, keeping earlier moves in place. It explains everything, but each factor's share depends on the order.
Unexplained P&L: Actual P&L minus everything the explain attributes to a cause.
Theta: The change in an option's value from one day passing, with the market unchanged. Usually a cost for an option buyer.
Vanna: The cross sensitivity of an option to spot and volatility together: how its delta changes when volatility moves.

Deep Dives & Origins

Deep Dives

J. C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021, the chapter on the Greek letters (Taylor series expansions). Intermediate.
C. Alexander, Market Risk Analysis, Volume IV: Value-at-Risk Models, Wiley, 2008, on risk factor VaR and option portfolios. Advanced.

Origins

Basel Committee on Banking Supervision, Minimum capital requirements for market risk, BIS, January 2019 (P&L attribution test, MAR32).
Basel Committee on Banking Supervision, Supervisory framework for the use of “backtesting” in conjunction with the internal models approach to market risk capital requirements, BIS, January 1996.

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