Daily P&L Explain

A trading desk's P&L system reports the book made $340,000 today. That number alone tells a risk manager almost nothing useful — was it because rates moved and the book was long duration, because volatility spiked and the book was long vega, or because of a single large position that happened to gap? A P&L explain (also called P&L attribution or a risk factor attribution) answers that by decomposing the day's total P&L into contributions from each risk factor the book is exposed to, so the number can be sanity-checked against known market moves rather than taken on faith.

Building the explain

For a position with sensitivity $\Delta$ to a risk factor, and that factor moving by $\delta x$, the first-order contribution to P&L is:

$$\text{P\&L}_{\text{explained}} \approx \sum_k \frac{\partial V}{\partial x_k} \times \delta x_k$$

In words: for every risk factor $k$ the book is exposed to (an interest rate, an equity index level, an implied volatility, a credit spread), multiply the book's sensitivity to that factor by how much the factor actually moved, then add up the contributions across all factors. For an options book this typically includes delta (spot move), gamma (a second-order correction for large spot moves), vega (implied vol move), and theta (pure time decay), because a linear delta-only explain badly misses P&L on days with large moves.

Worked example

A book has spot delta of $50,000 per 1% move in an index, gamma worth an extra $2,000 per 1% squared for a large move, and vega of $8,000 per one-point change in implied vol. Today the index rose 2%, and implied vol fell 1.5 points.

1. Delta P&L. $50{,}000 \times 2 = 100{,}000$, i.e. $100,000.
2. Gamma P&L. Second-order correction, using the move squared: $2{,}000 \times 2^2 = 2{,}000 \times 4 = 8{,}000$, i.e. $8,000.
3. Vega P&L. $8{,}000 \times (-1.5) = -12{,}000$, i.e. -$12,000, a loss because the book is long vega and vol fell.
4. Total explained. $100{,}000 + 8{,}000 - 12{,}000 = 96{,}000$, i.e. $96,000.

If the actual reported P&L for the day was $102,000, there's a $6,000 unexplained residual. That residual might come from intraday trading, a risk factor not modeled (like a name-specific idiosyncratic move), or a genuine error in position data, and finding out which is exactly the point of running the explain.

What this means in practice

Every sizeable trading desk runs a daily P&L explain, because it's the primary check that the risk system's sensitivities are actually correct — a persistently large unexplained residual means either the risk model is missing a factor the book is genuinely exposed to, or the position data feeding the risk system disagrees with what's actually booked, both of which are serious problems a firm needs to catch quickly. It's also the mechanism by which a risk manager can tell a trader "yesterday's gain was 90% a rates call and 10% skill," turning a single opaque number into a story a desk head can act on.