4.10.6 Advanced Extensions

Basic Prerequisites: 04 · Aggregate Loss & LDA and Basel & Regulation.

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1. Intuition & Practical Objective

Two extensions that take LDA into the real world:

1. Basel capital: the Standardised Measurement Approach (SMA). After decades of letting banks build their own op-risk models (the AMA), the Basel Committee decided the internal models were too complex, incomparable, and gaming-prone - and in 2016/2017 replaced all op-risk approaches (BIA, TSA, ASA, AMA) with a single non-model-based SMA: a financial-statement proxy (the Business Indicator) scaled by internal loss history. This page derives and runs the SMA formula so the regulatory machinery is concrete, not opaque.
2. Insurance / risk transfer. Because op risk cannot be hedged, the only market mitigant is insurance - per-event covers and, crucially, aggregate stop-loss covers. The page quantifies what insurance actually buys: per-event caps barely dent a frequency-driven tail, while an aggregate stop-loss directly cuts the 99.9% loss.

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2. Mathematical Ground Truth & Derivations

2.1 The SMA (final form, Basel III d424, 2017)

Step 1 - Business Indicator (BI): a three-year average proxy of op-risk exposure,

$$\text{BI}=\text{ILDC}+\text{SC}+\text{FC},$$

where (bars denote 3-year averages):

• $\text{ILDC}=\min [|\overline{\text{InterestIncome}-\text{InterestExpense}}|;2.25\%\cdot \overline{\text{InterestEarningAssets}}]+\overline{\text{DividendIncome}}$,
• $\text{SC}=\max (\overline{\text{OOI}};\overline{\text{OOE}})+\max (\overline{\text{FeeIncome}};\overline{\text{FeeExpense}})$,
• $\text{FC}=|\overline{{\text{NetP\&L}}_{\text{TradingBook}}}|+|\overline{{\text{NetP\&L}}_{\text{BankingBook}}}|$.

Step 2 - Business Indicator Component (BIC): piecewise-linear, progressively steeper in the BI (3 buckets, marginal coefficients ${\alpha }_i$):

Bucket	BI range (€bn)	BIC
1	$\le 1$	$0.12\cdot \text{BI}$
2	$1<\text{BI}\le 30$	$0.12\cdot 1+0.15(\text{BI}-1)$
3	$>30$	$0.12\cdot 1+0.15\cdot 29+0.18(\text{BI}-30)$

(The d424 worked example $BI=35\Rightarrow$ BIC $=0.12+4.35+0.90=5.37$ bn is reproduced exactly below.)

Step 3 - Loss Component (LC) and Internal Loss Multiplier (ILM):

$$\text{LC}=15\times (\text{average annual op losses over previous 10 years}),$$ $$\text{ILM}=\ln [e-1+(\frac{\text{LC}}{\text{BIC}}{)}^{0.8}].$$

The ILM equals 1 when $\text{LC}=\text{BIC}$; it is bounded below by $\ln (e-1)\approx 0.541$ and grows slowly (log) with loss experience, so banks with better loss records hold less capital and vice-versa.

Step 4 - capital:

$$\text{ORC}=\text{BIC}\cdot \text{ILM},\text{RWA}=12.5\cdot \text{ORC}.$$

For bucket-1 banks ($\text{BI}\le 1$bn) the ILM is set to 1 and ORC $=0.12\text{BI}$ - no loss data used.

2.2 Insurance / risk transfer on the aggregate loss

An aggregate stop-loss with attachment $A$ and limit $L$ makes the insurer pay $\min [\max (S-A,0),L-A]$, so the net annual loss is

$$S_{\text{net}}=S-\min [\max (S-A,0),L-A]=\{\begin{array}{ll}S,& S\le A\\ A,& A<S\le L\\ S-(L-A),& S>L.\end{array}$$

Because op-risk tail risk is aggregate (many events stacking in one year), risk transfer that targets the total (stop-loss) is far more effective than per-event caps - the code below shows a 26.5% VaR reduction from a stop-loss versus ~2% from a per-event cap.

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3. Computational Implementation - run the SMA and test risk transfer

A. SMA capital. Stdlib only; reproduces the d424 cross-check exactly.

Here LC (€0.9bn) < BIC (€1.32bn), so the ILM (0.898) is below 1 - this bank’s better-than-average loss record earns it less capital than the pure-BI baseline. The SMA is deliberately incentive-aligned.

B. Insurance: per-event cap vs aggregate stop-loss. Compare net VaR after each risk-transfer on the same compound-Poisson aggregate.

The aggregate stop-loss (attachment €1M) caps the 99.9% loss at €1M, a 26.5% reduction in VaR, while the expected loss barely moves (€606k→€603k). Compare the per-event cap of the same size in a companion exercise - it cuts VaR by only ~2% - because a clumpy Poisson tail is driven by many events stacking, not single giants. Design the transfer to the risk that drives the tail: aggregate, not per-event.

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4. Failure Modes & First-Principles Breakdowns

1. SMA trades risk-sensitivity for simplicity by design. The BI is a proxy (a high-fee, high-margin bank can be over-capitalised; the 2016 consultation fixed some of this with NIM caps and fee floors). It is non-model-based, so it cannot reflect idiosyncratic tail risk the way LDA tries to - accept SMA for regulatory capital and keep LDA for economic capital.
2. The ILM’s log growth under-rewards/over-penalises at the extremes. Because $\text{ILM}\to \ln (e-1)$ is bounded below and grows slowly, banks with very heavy losses can be under-capitalised relative to their true tail - a deliberate regulatory floor, not an estimate of risk.
3. Risk-transfer moral hazard. Insurance only lowers capital-relevant loss if the cover genuinely transfers tail risk (aggregate stop-loss). A per-event cap that doesn’t touch the aggregate tail is cosmetically “insurance” but structurally useless - precisely the failure demonstrated above.
4. 10-year loss data for LC is still scarce. The SMA Loss Component averages 10 years of annual losses; a decade is barely enough to see one 99.9% event, so the LC itself inherits the data-scarcity failure mode of 05 · Failure Modes - the SMA is simpler but not data-independent.

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5. References

• BCBS, Revised standardised measurement approach for operational risk (2016, d305)
• BCBS, Basel III: Finalising post-crisis reforms (2017, d424), op-risk section
• BCBS, Basel II (2006), ¶645–655
• Cruz, Peters & Shevchenko, Fundamental Aspects of Operational Risk and Insurance Analytics (2015, Wiley)
• Panjer, Operational Risk: Modeling Analytics (2006)

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6. Connected Graph Bridges

• Base: 04 · Aggregate Loss & LDA · Basel & Regulation
• Back: 05 · Failure Modes · Index Hub
• Sibling: Model Risk & Validation · Liquidity Risk & Funding (risk-transfer analogues)