P&L Attribution

Before describing attribution it is necessary to define the P&L types that appear as reference values and outputs.

MTD and YTD P&L are computed as:

\[\text{MTD P\&L} = PV_\text{now} - \text{Monthly Adjustments} - \text{Brokerage Adjustments}\] \[\text{YTD P\&L} = PV_\text{now} - \text{YTD Adjustments} - \text{Brokerage Adjustments}\]

The fundamental decomposition of daily P&L is:

\[\underbrace{D_1 M_1 T_1 - D_0 M_0 T_0}_{\text{Total P\&L}} = \underbrace{(D_1 M_1 T_1 - D_0 M_1 T_1)}_{\text{Trade Activity}} + \underbrace{(D_0 M_1 T_0 - D_0 M_0 T_0)}_{\text{Market Data}} + \underbrace{(D_0 M_1 T_1 - D_0 M_1 T_0)}_{\text{Time}}\]

Where:

• \(D_0\), \(D_1\): Trade population at Day 0 and Day 1
• \(M_0\), \(M_1\): Market data (spots, vols, rates) at Day 0 and Day 1
• \(T_0\), \(T_1\): Valuation dates at Day 0 and Day 1

The three components are mutually exclusive and exhaustive:

                                  ┌─ NEW
             ┌─ TRADE ACTIVITY ──┤─ AMEND / RESTRUCTURE
             │                   │─ CANCEL
             │                   │─ EXPIRED
             │                   │─ EXERCISED
             │                   │─ FIXINGS
             │                   │─ FUNDING ROLLS
TOTAL P&L ───┤                   │─ CONTRA REVENUE (Brokerage)
             │                   └─ CVA / RESERVES
             │
             ├─ MARKET DATA ─────┬─ FX SPOT
             │                   ├─ ATM VOL
             │                   ├─ RISK REVERSAL
             │                   ├─ STRANGLE
             │                   ├─ IR PROJECTION
             │                   ├─ IR DISCOUNT
             │                   └─ CROSS EFFECTS
             │
             └─ TIME (THETA)

A fourth category, Unexplained P&L, captures any residual. A well-implemented attribution should drive this to zero. Its presence indicates missing or misclassified attribution components.

The full formula including cross effects:

\[\text{Market Movements} = (\text{ATM} + \text{RR} + \text{STR} + \text{FX Spot} + \text{IR Discount} + \text{IR Projection}) + \text{Cross Effects}\] \[\text{Cross Effects} = \text{Bump All} - (\text{ATM} + \text{RR} + \text{STR} + \text{FX Spot} + \text{IR Discount} + \text{IR Projection})\]

Bump and Reset is the standard methodology for market data attribution. Each component is computed by bumping only that component from Day 0 to Day 1 while holding all others at Day 0 levels. The baseline is always Day 0.

Bump and Reset ensures all components add to total market movements (up to cross effects). It does not capture interaction terms between components; those appear in the Cross Effects bucket.

In Bump and Run, bumps are applied cumulatively in a defined order. The \(\Delta P\&L\) at each step is measured against the previous step, not against Day 0.

Bump and Run is complementary to Bump and Reset. It is better for detecting cross-effects (a large step 2 increment implies a strong ATM×Spot interaction). Bump and Reset is better for attributing absolute magnitudes to individual components.

Trade Activity explains the P&L contribution from changes to the trade population between Day 0 and Day 1. It uses Day 1 market data for all sub-components so that trade changes and market data changes do not interact.

\[\text{Trade Activity} = D_1 M_1 T_1 - D_0 M_1 T_1\]

Sub-buckets (additive):

\[\text{Trade Activity} = \text{New} + \text{Amended} + \text{Cancelled} + \text{Expired} + \text{Exercised} + \text{Fixings} + \text{Funding Rolls} + \text{Contra Revenue} + \text{CVA} + \text{Reserves}\]

The market data component captures the P&L from changes in market rates and prices between Day 0 and Day 1, holding the trade population and date constant.

\[\text{Market Data} = D_0 M_1 T_0 - D_0 M_0 T_0\]

Sub-buckets (Bump and Reset methodology):

The time component captures the P&L resulting purely from the passage of one business day, holding market data and trade population constant.

Under Bump and Reset (Theta First): \[\Theta = NPV(D_0, M_0, T_1) - NPV(D_0, M_0, T_0)\]

Under Bump and Run (Theta Last): \[\Theta = NPV(D_0, M_1, T_1) - NPV(D_0, M_1, T_0)\]

Determining the correct deal population for an explain requires care:

• From a Market Data perspective, only Day 0 trades matter.
• From a Trade Activity perspective, all trade changes must be captured: trades that only exist on Day 0 (matured, cancelled, expired), trades that only exist on Day 1 (new), and trades that exist on both days.
• Settled positions must be included. A forward that matured on Day 1 appears as a cash Nostro position in the ledger on Day 1; if the cash position is omitted from trade activity, a spurious unexplained P&L appears.

1. Trade cut: Snapshot of the trade population.
2. Market data cut: Snapshot of market data (spots, forwards, vols, rates).
3. AM Flash / PM Flash: Preliminary P&L. Finance generates an exception report comparing traders' expected P&L against the Finance P&L. Traders sign off exceptions.
4. Rate sign-off: Finance formally approves the market data cut.
5. Full P&L execution: The official Flash is produced with all exceptions and trader comments attached and published.
6. Diff analysis: Comparison of AM Flash vs EOD Flash; exceptions reviewed. Final sign-off can only be completed once the total P&L difference is below a configured threshold.
7. EOD start: Full Actual P&L begins.

P&L is held in its natural currency (the denomination of each cash flow). At EOD, all amounts are consolidated into the branch functional currency (the domestic currency of the branch, e.g. GBP for a London branch).

Any P&L from spot conversion of non-domestic P&L is recorded as a P&L Adjustment (FX Adjustment). Traders are expected to hedge this FX exposure.

The aggregation hierarchy:

Trade → Structure → Book → Sub-Portfolio → Portfolio → Branch → Firm