**PnL Explained** also called **P&L Explain**, **P&L Attribution** or **Profit and Loss Explained** is an income statement with commentary which Product control products and which traders, especially derivatives (swaps and options) use, that attributes or *explains* the daily fluctuation in the value of a portfolio of trades to the root causes of the changes.

P&L is the day-over-day change in the value of a portfolio of trades typically calculated using the following formula: PnL = Value today – Value from Prior Day

Report

A PnL Explained Report will usually contain one row per trade or group of trades and will have at a minimum these columns:

- Column 1: PnL — This is the PnL as calculated outside of the PnL Explained report
- Column 2: PnL Explained — This is the sum of the explanatory columns
- Column 3: PnL Unexplained — This is calculated as PnL – PnL Explained (i.e., Column 1 – Column 2)
- Column 4: Impact of Time — This is the PnL due to the change in time.
- Column 5: Impact of Prices — This is the PnL, i.e., the change in the value of a portfolio due to changes in commodity or equity/stock prices
- Column 6: Impact of Interest Rates — This is the PnL due to changes in interest rates
- Column 7: Impact of Volatility — This is the PnL due to changes in volatilities. Volatilities are used to value Option (finance) (i.e., calls and puts)
- Column 8: Impact of New Trades — PnL from trades done on the current day
- Column 9: Impact of Cancellation / Amendment – PnL from trades cancelled or changed on the current day

Methodologies

There are two methodologies for calculating Pnl Explained, the ‘sensitivities’ method and the ‘revaluation’ method.

**Sensitivities method**

The Sensitivities Method involves first calculating option sensitivities known as the Greeks because of the common practice of representing the sensitivities using Greek letters. For example, the delta of an option is the value an option changes due to a $1 move in the underlying commodity or equity/stock. To calculate ‘Impact of Prices’ the formula is: Impact of Prices = Option Delta * Price Move; so if the price moves $100 and the option’s delta is 0.05% then the ‘Impact of Prices’ is $0.05. To generalize, then, for example to yield curves:

Impact of Prices = Position Sensitivity * Move in the variable in question

**Revaluation method**

The Revaluation Method recalculates the value of a trade based on the current and the prior day’s prices. The formula for Impact of Prices using the Revaluation Method is

- Impact of Prices = (Trade Value using Today’s Prices) – (Trade Value using Prior Day’s Prices)

for some small value assets such as loose tools

- Depreciation = value at the beginning of the year (opening balance) + purchases in the year – value at the end of the year (closing balance)

PnL Unexplained

PnL unexplained is a critical metric that regulators and product control within a bank alike pay attention to.

PnL Attribution is used to test the hypothesis that the risk factors identified for a risky position are sufficient to materially explain the value change expected from the risky position;. Such that if position sensitivities to those risk factors are calculated, then the value change observed over a day can be attributed to the market price change of those risk factors, with the magnitude of the estimated as a sum product of the risk factor sensitivities and the corresponding daily risk factor price change.

Any residual P&L left unexplained (PnL Unexplained) would be expected to be small **IF** the identified risk factors are indeed sufficient to materially explain the expected value change of the position AND if the models used to calculate sensitivities to these risk factors are correct. PnL Unexplained is thus a critical metric that when large may highlight instances where the risk factors classified for a risky position are incomplete or the models used for sensitivities calculations are incorrect or inconsistent.^{[1]}

References

**^***“Why P&L Attribution? Or judging weathermen…” Acuity Derivatives. Retrieved 10 September 2012.*

Ofer Abarbanel is a 25 year securities lending broker and expert who has advised many Israeli regulators, among them the Israel Tax Authority, with respect to stock loans, repurchase agreements and credit derivatives. Founder of TBIL.co STATX Fund.