What is P&L attribution (P&L explain) and how does it relate to the risk-theoretical P&L in FRTB?

P&L attribution (also called P&L explain) is a critical model validation tool under the FRTB. It tests whether the risk model used for internal model approach (IMA) capital can actually explain the desk's real profits and losses.

The Three P&Ls

1. Actual P&L (APL)

The real daily profit or loss from the desk's trading activity. Includes all sources: market moves, new trades, fees, bid-ask spread capture, etc.

2. Hypothetical P&L (HPL)

The P&L that would have occurred if the portfolio had been held constant (no new trades). Strips out intraday trading effects. Used for VaR backtesting.

HPL = Full revaluation of yesterday's end-of-day positions using today's market data

3. Risk-Theoretical P&L (RTPL)

The P&L predicted by the desk's risk model (sensitivities). Computed by multiplying risk factors' actual moves by the model's sensitivities.

RTPL = SUM(delta_i x Delta_RF_i) + 0.5 x SUM(gamma_ij x Delta_RF_i x Delta_RF_j) + ...

Why Compare Them?

The FRTB P&L attribution test compares RTPL to HPL to validate that the risk model captures the desk's actual risk drivers.

If RTPL closely matches HPL: The model is good — it captures the right risk factors with the right sensitivities.

If they diverge significantly: The model is missing risk factors, using wrong sensitivities, or aggregating incorrectly. The desk may lose its IMA approval.

The FRTB P&L Attribution Tests

Two statistical tests are applied:

1. Spearman Rank Correlation Test

Measures whether HPL and RTPL move in the same direction on the same days.

• Green zone: correlation > 0.80
• Amber zone: 0.70 < correlation < 0.80
• Red zone: correlation < 0.70

2. Kolmogorov-Smirnov (KS) Test

Tests whether the distribution of P&L differences (HPL - RTPL) is centered at zero.

• Compares the empirical CDF of the ratio (HPL - RTPL) / sigma_HPL against a reference

Consequences of Failing

• Amber zone: The desk faces a capital surcharge but keeps IMA
• Red zone: The desk must fall back to the standardized approach (SA), which typically requires more capital
• Tests are run at the desk level, not the firm level — different desks can be on different approaches

Common Causes of P&L Explain Failure

1. Missing risk factors — the model doesn't include a factor the portfolio is exposed to
2. Cross-gamma effects — model uses only first-order sensitivities but the portfolio has significant convexity
3. Pricing model differences — front office uses one model, risk uses another
4. Stale marks — end-of-day prices don't reflect true market levels

For more on FRTB implementation, explore our FRM Part II course.