The primary function of a risk factor is the quantification of market risk associated with a market variable such as a yc or a vol surface. RFs are used in 1) no-arbitrage pricing models and 2) market risk models. NAPMs use RFs to calculate daily changes in the risk-neutral $valuation of an OTC derivative. MRMs use historical trends in the RFs to calculate market risk $capital as buffer against adverse rate movements that lead to NAPM $losses.
FRTB is the regulation that specifies the rules for calculating MR $capital. It allows two types of MRMs: 1) SA, and 2) IMA
There are three principles that apply to both types. The first is that MR $capital should be based on rate “shocks” that capture NAPM daily worst-case $losses. The second is that those shocks should be derived from historical observable market rate movements. And the third is that they should be independent of bank-specific MR modelling logic.
In the case of SA, regulators ensure that this third principle is adhered to by providing the shocks directly to banks. SA, unlike IMA, does not allow banks to calculate their own shocks. The SA RF shocks are called risk weights. They are shocks of reg RFs. Reg RFs are the observable RFs shown in 3) in the diagram below abstracted using interpolation and deterministic techniques to the reg-defined “vertices” shown in 1).
In 2) below, x is the vector of data points associated with the bank’s fitted RFs. V(x) is the NAPM valuation of an OTC. V(x) is calculated using bank-proprietary models, given the x inputs. Daily changes in V(x)=Daily P&L. These internal RF objects (x=x1, x2, … xn) are the model parameters and settings and the resulting model-derived curves and surfaces. Bank-specific fitting techniques are used to fit these RFs so they match the shapes of observable RFs.
Despite principle three above, SA MRMs are still exposed to bank-specific model-risk. The SA-provided RF shocks need to be applied to reg RF sensitivities dV/dR – shown under 1) below – whose creation require internal quant logic.
The first step in this internal quant logic is the creation of sensitivities for the fitted RFs. These are created by bumping x, the vector of fitted RFs, by standardised amounts such as 1bp or 1% and re-calculating V(x). The result is $quantities such as PV01, deltas and vegas – also known as risk sensitivities (RS) or Greeks. These RS’s are stored in a Jacobian matrix dV/dx. The next step is the RF mapping. RF mapping is the process of mapping the internal RS’s (the dV/dx’s) to the reg RS buckets in dV/dR. The mapping process uses a Jacobian matrix of RF mapping weights dx/dR whose constituents are created by bumping the reg RFs by 1bps or 1%, re-building the fitted curves and observing their movement.
Once the RF mapping weights are calculated, they are multiplied – using the calculus chain rule – by the internal RS’s to create the reg RS’s that the reg RF weights are applied to – to create the MR $capital.
