What is P&L attribution, and how does the risk-theoretical P&L compare to actual P&L?

P&L attribution decomposes the daily profit or loss of a trading desk into components that can be explained by risk factor movements. It's both a risk management tool and a model validation technique.

The Basic Framework:

Actual P&L: What the desk actually earned or lost (from the accounting system)

Risk-Theoretical P&L (RTPL): What the risk model PREDICTS the desk should have earned/lost, based on:

• The desk's risk sensitivities (Greeks, durations, betas)
• Observed market moves in risk factors

RTPL = Sum of [Sensitivity_i x Delta(Risk Factor_i)]

For a simple equity portfolio:

RTPL = Delta x Delta(S) + 0.5 x Gamma x Delta(S)^2 + Vega x Delta(sigma) + Theta x Delta(t)

Example — Pinnacle Securities equity options desk:

Sources of the Unexplained P&L:

1. Missing risk factors: The model doesn't capture all drivers (e.g., skew, term structure)
2. Cross-gamma: Interaction effects between risk factors
3. Intraday trading: Risk sensitivities change as the desk trades during the day
4. Bid-ask capture: Actual trading at better prices than mid-market
5. Model approximation: Greeks are local linear/quadratic approximations
6. New deal P&L: Trades booked after risk was computed

Why Unexplained P&L Matters:

• Model validation: Large, persistent unexplained P&L suggests the risk model is missing important factors
• Basel FRTB P&L Attribution Test: Desks must pass a statistical test comparing RTPL to actual P&L. Failing desks are moved to the standardized approach (higher capital)
• The Spearman correlation between RTPL and actual P&L and the Kolmogorov-Smirnov test on the difference are used as metrics

FRM Key Points:

• Unexplained P&L should be small relative to actual P&L (typically < 10-15%)
• Systematic bias in RTPL indicates a model deficiency
• P&L attribution is forward-looking model validation (unlike backtesting, which is backward-looking)
• FRTB requires desks to pass P&L attribution tests to use internal models

Study P&L attribution and model validation in our FRM Part II course.