4.9.2 Capital & RWA

Basic Prerequisites: 01 · From Zero and Credit Risk & the Merton Model.

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1. Intuition & Practical Objective

Capital regulation is one equation dressed in a thousand pages: capital ÷ risk-weighted assets ≥ a threshold. This page makes the equation concrete. The objective is to know exactly how each side is built - what counts as capital (the numerator, a quality ladder from CET1 down to Tier 2) and how exposures become risk-weighted assets (the denominator, credit + market + operational) - so that every later rule (FRTB, IRB, the output floor) is just a change to one of these two pieces.

The denominator is where the action is. RWA is a common currency for risk: a $100 sovereign bond at a 0% weight contributes nothing; a $100 unrated corporate loan at 100% contributes $100; a $100 retail mortgage at a 35% weight contributes $35. Capital is then divided by this sum. Two banks holding identical assets can need different capital purely because the rules weight them differently - and two banks with identical risk can report different RWA if one uses a model and the other does not. Understanding that gap is the whole point of the post-crisis reform.

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2. Mathematical Ground Truth & Derivations

The capital ratio. With total RWA over risk categories $c\in \{\text{credit, market, operational}\}$,

$$\mathrm{RWA}=\sum _c{\mathrm{RWA}}_c=\sum _c\sum _{i\in c}{E}_{i}r{w}_i,$$

and the headline ratios are

$$\text{CET1 ratio}=\frac{CET1}{\mathrm{RWA}},\text{Tier 1 ratio}=\frac{CET1+AT1}{\mathrm{RWA}},\text{Total ratio}=\frac{T1+T2}{\mathrm{RWA}},$$

with minima $4.5\%$, $6.0\%$, $8.0\%$ respectively (BCBS 2010). The $8\%$ rule is equivalent to $\mathrm{RWA}=12.5\times$ capital, which is why $12.5=\frac{1}{8}$ appears everywhere downstream.

The capital-quality ladder.

• CET1 (Common Equity Tier 1): common shares, share premium, retained earnings, plus disclosed reserves; fully loss-absorbing, the binding constraint.
• AT1 (Additional Tier 1): perpetual subordinated instruments with loss-absorption triggers (e.g. conversion below a CET1 trigger).
• Tier 2: subordinated debt and general provisions; absorbs losses in wind-down, not going-concern.

The buffer stack (Basel III). Above the 4.5% CET1 minimum sit buffers that must be met with CET1:

$$\text{CET1 requirement}=4.5\%+\underset{\text{capital conservation}}{\underbrace{2.5\%}}+\underset{\text{countercyclical, CCyB}}{\underbrace{0\text{–}2.5\%}}+\underset{\text{G-SIB}}{\underbrace{1\text{–}3.5\%}}.$$

The MDA (Maximum Distributable Amount) restricts dividends and buybacks once a bank dips into the conservation-buffer band - the concrete teeth of the stack.

Why RWA is a model. Every $r{w}_i$ is a regulatory assumption about loss in stress. The sensitivity is stark: re-weighting one $500 exposure from 100% to 20% cuts RWA by $400 and lifts the ratio by several points without changing a single underlying asset. That is the arbitrage surface (05).

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3. Computational Implementation - building the ratio end-to-end

Standard library. This constructs a bank’s RWA from a balance sheet, computes the three ratios, the buffer stack, and the sensitivity of the ratio to a risk-weight change.

Read the last block carefully. Re-labelling one $500 exposure - corporate loan → rated A → securitised AAA tranche - moves RWA from 1277.5 to 877.5 and the CET1 ratio from 9.39% to 13.68%, a 4.3-point regulatory improvement from a change in classification, not in risk. This is the single most important fact about capital regulation and the reason the output floor exists (§06).

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4. Failure Modes & First-Principles Breakdowns

1. Numerator quality is as important as size. A high ratio built from low-grade Tier 2 debt is not the same as one built from CET1 - the loss-absorption differs at the moment it matters. Always read the CET1 line, not the headline.
2. RWA is not comparable across banks. A standardised-approach RWA and an IRB RWA for the same exposure can differ by tens of percent. Cross-bank ratio comparisons without adjusting for approach are apples-to-oranges (a core motivation for the output floor).
3. The denominator can be engineered. Deleveraging weighted assets (securitisation, guarantees, model approval) raises the ratio without reducing economic risk - the arbitrage failure in miniature (§05).
4. Buffers get spent. In stress, banks run down the conservation/countercyclical buffers; the constraint that bites is then the MDA and eventually the 4.5% floor. Treating “the requirement” as a single 8% number understates how close to the edge a stressed bank is.

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5. References

• BCBS - Basel III: A Global Regulatory Framework (2010, d189). The capital-quality ladder, the $4.5/6/8\%$ minima, and the buffer stack, in the primary text. Read from the corpus PDF.
• BCBS - Basel III: Finalising Post-Crisis Reforms (2017, d424). The revised RWA definitions and the $72.5\%$ output floor that constrains how far the denominator may be modelled down. Read from the corpus PDF.
• Hull, John C. - Risk Management and Financial Institutions (5
• Hull, John C. - Options, Futures, and Other Derivatives (11th ed.)

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6. Connected Graph Bridges

• Back: 01 · From Zero · Index Hub
• Forward: 03 · Market Risk & FRTB · 04 · Credit & Operational Risk
• Related: Credit Risk & the Merton Model · 06 · Leverage & Output Floor