CRR Asset Correlation Feedback and a Fixed-Point MoC Method

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


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












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?