Does the regulatory feedback loop between PD and Asset Correlation under CRR imply a unique fixed-point MoC?

External question — Quantitative Finance Stack Exchange Author: Jan Original post: https://quant.stackexchange.com/questions/85862 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. In the regulatory credit risk framework (Articles 153 and 154 of the Capital Requirements Regulation, CRR), the asset correlation parameter $R$ is endogenously defined as a strictly decreasing function of the probability of default ( $PD$ ). When introducing a Margin of Conservatism Category C, $\mathrm{MoC}_{C}$ , to account for statistical uncertainty due to a limited historical observation period $T$ , a paradox arises: an elevated asset correlation increases the variance of the default rate ( $DR$ ), requiring a higher $\mathrm{MoC}_{C}$ . However, this higher conservative probability of default, $$ PD_{\mathrm{conservative}} = PD_{0}+\mathrm{MoC}_{C}, $$ endogenously compresses the regulatory asset correlation $R(PD)$ . The following animation illustrates the feedback loop between PD and Asset Correlation under CRR. I propose that these counteracting forces establish a self-stabilizing, equilibrium-seeking system, where the equilibrium correpsonds to a "regulatory MoC C". I formulate this interaction as a first-order fixed-point problem using Picard iteration to find the unique optimal $\mathrm{MoC}_{C}$ . Please note that there is a working paper with more details available: http://dx.doi.org/10.2139/ssrn.7559681 Mathematical Framework Let $PD_{0}$ be the best-estimate probability of default. Under the CRR, the asset correlation $R(PD)$ is given by $R(PD)=a_{1}\cdot\frac{1-\exp(a_{3}\cdot PD)}{1-\exp(a_{3})}+a_{2}\cdot\left(1-\frac{1-\exp(a_{3}\cdot PD)}{1-\exp(a_{3})}\right)$ Following Bluhm, Overbeck, and Wagner (2003), the variance of the annual default rate within the Vasicek model framework is $\operatorname{Var}(DR\mid PD)=\Phi_{2}\left(\begin{pmatrix}\Phi^{-1}(PD)\\\Phi^{-1}(PD)\end{pmatrix};\begin{pmatrix}1&R(PD)\\R(PD)&1\end{pmatrix}\right)-PD^{2}$ where $\Phi^{-1}(\cdot)$ denotes the inverse standard normal cumulative distribution function, and $\Phi_{2}(\cdot;\Sigma)$ denotes the bivariate standard normal cumulative distribution function with correlation matrix $\Sigma$ . The Iterative Update Scheme To incorporate $\mathrm{MoC}_{C}$ under a target confidence level $\beta$ over $T$ years, define the sequential updating rule for $k\geq 0$ as $PD_{k+1}=PD_{0}+\frac{\Phi^{-1}(\beta)}{\sqrt{T}}\sqrt{\operatorname{Var}(DR\mid PD_{k})}\equiv g(PD_{k})$ The initialization is $PD^{(0)}=PD_{0}$ By differentiating $g(PD)$ with respect to $PD$ , using Leibniz's rule for the bivariate integral, we obtain the system's trajectory multiplier: $\frac{\mathrm{d}g(PD)}{\mathrm{d}PD}=\frac{\Phi^{-1}(\beta)}{2\sqrt{T\cdot\operatorname{Var}(DR\mid PD)}}\cdot\left[2\Phi\left(\Phi^{-1}(PD)\sqrt{\frac{1-R(PD)}{1+R(PD)}}\right)+\phi_{2}\left(\Phi^{-1}(PD),\Phi^{-1}(PD);R(PD)\right)\frac{\mathrm{d}R(PD)}{\mathrm{d}PD}-2PD\right]$ where $\phi_{2}(x,y;\rho)$ denotes the bivariate standard normal density with correlation parameter $\rho$ . Numerical simulations in the relevant range from zero to one show that $\left|\frac{\mathrm{d}g(PD)}{\mathrm{d}PD}\right|<1$ satisfying the contraction-mapping conditions of the Banach fixed-point theorem. Thus, the sequence converges to a unique, asymptotically stable equilibrium $PD^{*}=g(PD^{*})$ , where the optimal margin is defined as $\mathrm{MoC}_{C}^{*}=PD^{*}-PD_{0}$ Questions for the Community Methodological appropriateness: Is it conceptually sound to utilize the macroeconomic, systemic asset correlation function $R(PD)$ defined by regulators to dynamically scale a micro-portfolio uncertainty buffer $\mathrm{MoC}_{C}$ ? Does anyone know whether EU auditors or regulators, including the EBA and ECB, accept fixed-point or attractor-state methods for quantifying Category C Margins of Conservatism under the IRB repair guidelines?