Derivative of the Basel Risk-Weight Function with Respect to MoC C

External question — Quantitative Finance Stack Exchange Author: Jan Original post: https://quant.stackexchange.com/questions/85861 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. I am examining the differentiation of the Basel/CRR risk-weight function with respect to the Margin of Conservatism Category C (MoC C). More specifically, if the conservative PD changes because of MoC C, should the resulting change in the regulatory asset correlation also be included when differentiating the risk weight? I derive the following expression and obtain numerical agreement with a finite-difference approximation over a PD range of 0.03%–5%. I would appreciate confirmation that the analytical derivation and its application to the grade-level aggregation are correct. References Tasche, D. (2010), “Estimating discriminatory power and PD curves when the number of defaults is small.” Wosnitza, J. H. (2026), “Quantification of margin of conservatism category C: Correlations and quantification levels,” Journal of Credit Risk, 22(3), pp. 51–73. DOI: 10.21314/JCR.2026.005 Derivation We differentiate the risk-weight function with respect to Margin of Conservatism Category C. Define $$ p:=PD, \qquad m:=\mathrm{MoC}, \qquad q:=\widetilde{PD}=\text{Conservative PD}. $$ Apart from certain factors, the risk-weight function is $$ RW(q)= LGD \left[ \Phi\left( \frac{ \Phi^{-1}(q) + \sqrt{R(q)}\cdot\Phi^{-1}(\alpha) }{ \sqrt{1-R(q)} } \right) -q \right]. $$ For convenience, define $$ z(q) := \frac{ \Phi^{-1}(q) + \sqrt{R(q)}\cdot\Phi^{-1}(\alpha) }{ \sqrt{1-R(q)} }. $$ Thus, $$ RW(q)=LGD\cdot[\Phi(z(q))-q]. $$ Because q affects the risk weight directly and also indirectly through the asset-correlation parameter $R(q)$ , the chain rule gives $$ \frac{dRW(q)}{dm}= \left[ \frac{\partial RW(q)}{\partial q} + \frac{\partial RW(q)}{\partial R} \frac{dR(q)}{dq} \right] \frac{dq}{dm}. \tag{1} $$ (A) Derivative of the conservative PD with respect to MoC C Following Tasche (2010) and Wosnitza (2026), the best-estimate probability of default is $$ p= LRADR\cdot \frac{dF_D(s)}{dF_{\mathrm{All}}(s)}. $$ The conservative probability of default is defined as $$ q= \left(LRADR+\beta m\right) \frac{dF_D(s)}{dF_{\mathrm{All}}(s)}, $$ where $\beta$ is a scaling factor. Using the definition of p, we obtain $$ q= \left(LRADR+\beta m\right) \frac{p}{LRADR}. $$ Therefore, $$ q= p+\beta m\frac{p}{LRADR}. $$ Differentiating with respect to m gives $$ \frac{dq}{dm}= \beta\frac{p}{LRADR}. \tag{A} $$ Equivalently, since $$ \frac{dF_D(s)}{dF_{\mathrm{All}}(s)}= \frac{q}{LRADR+\beta m}, $$ we can write $$ \frac{dq}{dm}= \beta\frac{q}{LRADR+\beta m}. $$ At m=0, we have q=p, and therefore $$ \left.\frac{dq}{dm}\right|_{m=0}= \beta\frac{p}{LRADR}. $$ Substituting this result into equation (1) yields $$ \frac{dRW(q)}{dm}= \left[ \frac{\partial RW(q)}{\partial q} + \frac{\partial RW(q)}{\partial R} \frac{dR(q)}{dq} \right] \beta\frac{p}{LRADR}. $$ (B) Derivative of the asset-correlation parameter Articles 153 and 154 of the CRR define the asset-correlation parameter as a function of the probability of default: $$ R(q)= a_1 \frac{1-\exp(a_3q)} {1-\exp(a_3)} + a_2 \left[ 1- \frac{1-\exp(a_3q)} {1-\exp(a_3)} \right]. $$ Expanding this expression gives $$ R(q)= \frac{a_1-a_2}{1-\exp(a_3)} + a_2 + \exp(a_3q) \frac{a_2-a_1}{1-\exp(a_3)}. $$ Therefore, $$ \frac{dR(q)}{dq}= a_3\exp(a_3q) \frac{a_2-a_1}{1-\exp(a_3)}. \tag{B} $$ Since $a_3<0$ , $a_2>a_1$ , and $1-\exp(a_3)>0$ , it follows that $$ \frac{dR(q)}{dq}<0. $$ Substituting equation (B) into equation (1), we obtain $$ \begin{aligned} \frac{dRW(q)}{dm} ={}& \Bigg[ \frac{\partial RW(q)}{\partial q} + \frac{\partial RW(q)}{\partial R} a_3\exp(a_3q) \frac{a_2-a_1}{1-\exp(a_3)} \Bigg] \beta\frac{p}{LRADR}. \end{aligned} \tag{2} $$ Equivalently, factoring out (LGD), $$ \begin{aligned} \frac{dRW(q)}{dm} ={}& \Bigg[ \frac{1}{LGD} \frac{\partial RW(q)}{\partial q} + \frac{1}{LGD} \frac{\partial RW(q)}{\partial R} a_3\exp(a_3q) \frac{a_2-a_1}{1-\exp(a_3)} \Bigg] \ \cdot LGD\cdot\beta\frac{p}{LRADR}. \end{aligned} \tag{3} $$ (C) Derivative with respect to the asset-correlation parameter The derivative of $z(q)$ with respect to R is $$ \frac{\partial z(q)}{\partial R}= \frac{ \Phi^{-1}(\alpha) \left[ \sqrt{\frac{1-R(q)}{R(q)}} + \sqrt{\frac{R(q)}{1-R(q)}} \right] + \frac{\Phi^{-1}(q)}{\sqrt{1-R(q)}} }{ 2[1-R(q)] }. $$ Since $$ RW(q)=LGD[\Phi(z(q))-q], $$ we have $$ \frac{1}{LGD} \frac{\partial RW(q)}{\partial R}= \phi(z(q)) \frac{\partial z(q)}{\partial R}. $$ Consequently, $$ \boxed{ \frac{1}{LGD} \frac{\partial RW(q)}{\partial R}= \phi(z(q)) \frac{ \Phi^{-1}(\alpha) \left[ \sqrt{\frac{1-R(q)}{R(q)}} + \sqrt{\frac{R(q)}{1-R(q)}} \right] + \frac{\Phi^{-1}(q)}{\sqrt{1-R(q)}} }{ 2[1-R(q)] }. } \tag{C} $$ (D) Derivative with respect to the conservative PD Holding $R(q)$ fixed, the direct partial derivative of the risk weight with respect to q is $$ \frac{1}{LGD} \frac{\partial RW(q)}{\partial q}= \phi(z(q)) \frac{1} {\sqrt{1-R(q)}\cdot \phi\left(\Phi^{-1}(q)\right)} -1. \tag{D} $$ Substituting equations (B), (C), and (D) into equation (3), we obtain $$ \begin{aligned} \frac{dRW(q)}{dm} ={}& \Bigg[ \phi(z(q)) \frac{1} {\sqrt{1-R(q)}\cdot \phi\left(\Phi^{-1}(q)\right)} -1 \ &\quad+ \phi(z(q)) \frac{ \Phi^{-1}(\alpha) \left[ \sqrt{\frac{1-R(q)}{R(q)}} + \sqrt{\frac{R(q)}{1-R(q)}} \right] + \frac{\Phi^{-1}(q)}{\sqrt{1-R(q)}} }{ 2[1-R(q)] } \ &\quad\quad\times a_3\exp(a_3q) \frac{a_2-a_1}{1-\exp(a_3)} \Bigg] LGD\cdot\beta\frac{p}{LRADR}. \end{aligned} \tag{4} $$ At (m=0), we have (q=p). Therefore, $$ \begin{aligned} \left.\frac{dRW}{dm}\right|_{m=0} ={}& \Bigg[ \phi(z(p)) \frac{1} {\sqrt{1-R(p)}\cdot \phi\left(\Phi^{-1}(p)\right)} -1+\phi(z(p)) \frac{ \Phi^{-1}(\alpha) \left[ \sqrt{\frac{1-R(p)}{R(p)}} + \sqrt{\frac{R(p)}{1-R(p)}} \right] + \frac{\Phi^{-1}(p)}{\sqrt{1-R(p)}} }{ 2[1-R(p)] } \ &\quad\quad\times a_3\exp(a_3p) \frac{a_2-a_1}{1-\exp(a_3)} \Bigg] LGD\cdot\beta\frac{p}{LRADR}. \end{aligned} \tag{5} $$ PD = transpose(0.03/100 : 0.01/100 : 5/100); LGD = 1; % a1 = 0.03; a2 = 0.16; a3 = -35; % R = @(PD) a1 * (1 - exp(a3*PD)) / (1 - exp(a3)) + a2 * (1 - (1 - exp(a3*PD))/(1 - exp(a3))); A = @(PD) (norminv(PD) + sqrt(R(PD)) * norminv(0.999)) ./ sqrt(1 - R(PD)); RW = @(PD) (normcdf(A(PD)) - PD) * LGD; % R_Default = (mvncdf(norminv(PD) * ones(1,2), [0, 0], [1, R; R, 1]) - PD^2) / (PD * (1-PD)); % h = 10^-6; NumDiff = (RW(PD+h) - RW(PD))/h; MyDiff = (normpdf(A(PD)) .* 1./sqrt(1 - R(PD)) .* 1./normpdf(norminv(PD)) - 1 + normpdf(A(PD)) .* (norminv(0.999) * (sqrt((1 - R(PD)) ./ R(PD)) + sqrt(R(PD) ./ (1 - R(PD)))) + norminv(PD) ./ sqrt(1 - R(PD))) ./ (2*(1-R(PD))) * a3 .* exp(a3 * PD) * (a2 - a1) / (1 - exp(a3))) * LGD * 1 ; % display(NumDiff) display(MyDiff) % figure hold on plot(100*PD, NumDiff, '-', LineWidth=4, Color="k") plot(100*PD, MyDiff, '--', LineWidth=2, Color=ones(1,3) * 0.85) hold off % xlabel("Probability of Default", 'FontSize', 16) xticks(1:5) xtickformat('percentage') ax = gca; ax.FontSize = 14; ylabel('$\frac{ d\mathrm{RW(PD)}}{dPD}$', ... 'Interpreter', 'latex', 'FontSize', 16); ```