# Good references on PNL explain?

Can anyone share good references for how PNL explain should be calculated and presented for the best use of a derivatives trading desk?

• darbyshire "Pricing and Trading Interest Rate Derivatives" has a reasonable amount of relevant material on this topic, whilst not specifically being a book dedicated to this exact task.
– Attack68
Nov 15 '20 at 18:33

I'm not aware of any great reference. However Peter Nash Effective product control: controlling for trading desks. Wiley (2018) chapter 10 Review of Mark-to-Market P&L is a good start.

I wrote some notes here that I hope may help.

You should have risk-theoretical P&L (RTPL - Taylor sereis approximation of the P&L) for all positions. For the positions that are marked to model, you should have Brute Force - both "Cumulative" and "Independent". (It is possible to do bruce force P&L explain for positions with observable price, but it's harder and less useful.)

• Hi Dimitri, I have a method of attributing theoretical PnL (or the change in any function between two points) into an additive series of single effects. Bivariate example: $f(x_1,y_1)-f(x_0,y_0) = e_x + e_y$, with $e_x = \frac{1}{2}\left(f(x_1,y_1)-f(x_0,y_1) + f(x_1,y_0)-f(x_0,y_0)\right)$ and likewise for $e_y$. This method is extendable to more components of course. It attributes cross-effects evenly on the single components. It can be used with effects like decay, rates, FX, credits etc and does not require sequential shifting. Would that work for you? I have a prototype in R at hand. Nov 14 '20 at 7:41
• Thanks!!! I'm always curious to see what people do, even when I have no immediate use for it. The Nash book is from a product control point of view, not mathematical. I I'm thinking of editing my old answer I cited and adding more details, particularly on RTPL, and on practical euristics that would help someone trying to build a P&L explain from ground up, a common task. I don't know any place that explains it to my liking. Certainly en.wikipedia.org/wiki/PnL_Explained Wikipedia article is less than great. Nov 14 '20 at 14:48

References

1. https://www.bis.org/publ/bcbs265.pdf This one is directly used by banks for programs such as FRTB.

2. https://assets.kpmg/content/dam/kpmg/xx/pdf/2018/10/frtb-white-paper-july-2018.pdf. This one describes it from a P&L variance ratio point of view.

3. https://en.wikipedia.org/wiki/PnL_Explained. Basic summary of P&L attribution.

Summary

The purpose of the P&L explanation (or Volcker P&L attribution) is to test how well your risk factors explain your actual P&L and hence the overall logic and consistency of the model.

$$P\&L_{unexplained} = P\&L_{model} - P\&L_{risk \ factors}$$

$$P\&L_{risk \ factors}$$, the "explained" portion of P&L, is estimated using the Greeks/sensitivities of the risk factors (sum of first order sensitivities with respect to individual risk factors multiplied by risk factor shifts). First order sensitivities (i.e. delta) use forward differences while second order sensitivities (i.e. gamma) use central differencing and are typically used for futures options.

$$P\&L_{model}=P\&L_{comprehensive} - P\&L_{new \ positions} - P\&L_{other}$$

is actual model P&L calculated from the price of a position on two consecutive days, where

$$P\&L_{comprehensive}=NPV_{T}-NPV_{T-1}-CASH-CVAHedges$$ and $$P\&L_{new \ positions}=P\&L_{new \ position} + P\&L_{trade \ event}$$ $$P\&L_{trade \ event}$$ is NPV changes from notional changes in existing positions and $$P\&L_{other}$$ are finance adjustments.

The $$P\&L_{unexplained}$$ thus compares the difference between the model P&L and the P&L of the risk factors used to explain the price movements. A modeler would like to expect that $$P\&L_{risk \ factors}$$ explains more than 90% of P&L. In other words, you would like to minimize the portion of P&L that is unexplained by the risk factors used in the model which are supposed to capture the effects of actual P&L experienced by the position.

• Thanks for the summary. I am aware of the basics; what I am really looking for is a reference like a textbook or academic paper that discusses more nuances. Nov 14 '20 at 14:37
• I have updated my answer to include references. The first link is the one banks implement due to regulatory requirements. More papers can be found from BIS website. Also the summary/formula section I provided is directly used in practice too in model testing of derivative products. Nov 14 '20 at 18:24