Allocating Basel Granularity Adjustment Across Exposures

# How to calculate contributions to the granularity adjustment


# How to calculate contributions to the granularity adjustment












In the Basel pillar 2 framework a granularity adjustment is introduced. While the capital requirements in pillar 1 do not take concetrations into account, this is meant to be covered with this adjustment in pillar 2. In Gordy and Lütkebohmert: Granularity Adjustment for Regulatory Capital Assessment a simplified adjustment is proposed (formula 17): $$ GA = \frac{1}{2 K^*} \sum_{i=1}^n s_i^2 C_i \left(\delta (K_i + R_i) + K_i \right), $$ where $n$ is the number of assets, $s_i = \frac{A_i}{\sum_{j=1}^n A_j}$ is the fraction of exposure $A_i$ over the total exposure;$C_i$ depends on $LGD_i$, $R_i$ is the expected loss $R_i = PD_i \cdot E[LGD_i]$, $\delta$ is a regulatory constant and $K_i$ is the capital requirment for exposure $i$. Finally, $K^*$ is the weighted average capital requirement: $$ K^* = \sum_{j=1}^n s_i K_i. $$

Thus, $GA$ takes into account risk via $PD$ and $LGD$ as well as concentration via $s_i$.

How can we attribute this number to the individual assets?

A straight forward expression would be the summand in the above sum: $$ GA_i = \frac{1}{2 K^*} s_i^2 C_i \left(\delta (K_i + R_i) + K_i \right). $$

- However, by definition all these contributions are positive, and one could expect negative contributions in certain scenarios (e.g. if concentration is reduced by adding a position of small risk and small weight).

- If I did the calculations right, then the above expression for $GA_i$ is not the derivative of $GA$ w.r.t. $s_i$ multiplied by $s_i$ as the Euler allocation would indicate: $$ s_i \cdot \frac{d GA(s)}{d s_i} \neq GA_i. $$ Thus the allocation by $GA_i$ is not an "Euler-allocation".

The questions I would like to discuss:

- Is there any literature on attribution of the granulrarity adjustment? I have not found any.

- Although the definition of $GA_i$ above comes very natural due to the functional form of $GA$ it seems to be counter-intuitive. Which alternatives are there?

EDIT: I try to illustrate the derivative in the following. Set $d_i = C_i \left(\delta (K_i + R_i) + K_i \right)$ so that we can write the above formula more compactly: $$ GA = \frac{1}{2 K^*} \sum_{i=1}^n s_i^2 d_i, $$ where $K^* = \sum_{i=1}^n s_i K_i$. Then my manual calculations yield (we need the product rule, as $K^*$ is a function of $s_i$): $$ \frac{d GA(s)}{d s_i} = \frac{s_i d_i K^* - K_i/2 \cdot \sum_{i=1}^n s_i^2 d_i} { (K^*)^2}. $$ Using the same notation for the summand above we would compare this (with the factor $s_i$ to pass from the marginal contribution to the allocation) to: $$ GA_i = \frac{1}{2 K^*} s_i^2 d_i. $$ Let us write the expression above with the weight $s_i$: $$ s_i \cdot \frac{d GA(s)}{d s_i} = \frac{s_i^2 d_i K^* - s_i \cdot K_i/2 \cdot \sum_{i=1}^n s_i^2 d_i} { (K^*)^2}, $$ which can be written as: $$ s_i \cdot \frac{d GA(s)}{d s_i} = \frac{s_i^2 d_i}{K^*} - s_i\frac{K_i/2 \cdot \sum_{i=1}^n s_i^2 d_i} { (K^*)^2}. $$ Using the above, the Euler-like attribution does not equal the allocation by the summands.