Risk Theoretical PnL

One of the areas that is causing some concern and confusion is the idea of the Risk Theoretical PnL that is quite prominent in the FRTB regulatory paper from the BIS.

The idea behind this is better alignment of risk models to front office pricing models and consequently how instruments are marked by the trading desk. This will also serve as an extra layer to validate risk models along with backtesting models. Failure of backtesting or failure of the risk theoretical PnL to capture the risk factors adequately (that is to say the hypothetical PnL is consistently not explained correctly by the Risk Theoretical PnL) will lead to trading desks being moved from the internal models approach (IMA) to the Standardised Approach (SA). The disadvantage of this is of course much higher capital requirements which would be pretty detrimental to the bank.

Risk Theoretical PnL

Risk Theoretical PnL – Sample Spreadsheet

On the face of it the Risk Theoretical PnL is a fairly easy concept to understand. However implementation comes with challenges owing to the complexities of derivatives instruments and the inadequacy of a lot of systems in banks. You can download this spreadsheet to see what the approach should be broadly. In this spreadsheet we take a simple vanilla option position and look at its evolution over a year. Then we calculate the daily P&L and also the Risk Theoretical PnL based on a taylor series expansion of the greeks.

This approach works very well for simple derivatives (or linear) products that are explained by the main greeks from the model. However this become more complicated with larger market moves and more exotic products. Add to this the fact that a lot of risk models will assume proxies and simplify the underlying market data structure which could make it difficult for the risk model to capture the true risk.

Risk Theoretical PnL – Shortcomings

Depending on the desk in a banking institution they may have different yield curve setups – some desks which focus on the short end may have very granular setup in the short end, while other may be more granular in the long end of the curve. The point being that there are multiple curve structures across the bank. The risk function may then have to standardise the risk buckets in the curve to be able to have a standard measure of the risk. While this is convenient the granularity of the curve may be lost and as a result the risk model may no longer explain the PnL moves very well. This of course means that the Risk theoretical PnL may not be very good compared to the hypothetical PnL and as a result there is a danger that the risk model will be rejected and a given desk gets put on the SA.

The same type of problem also occurs with regard to using proxies in the equity or the credit world. These proxies may not be able to capture the Risk theoretical PnL adequately and again the desks will run the risk of having their risk models rejected and being placed on the standardised approach.

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