Capital Regulation Before the Global Financial Crisis: formula sheet

Key formulas

Risk-weighted assets

RWA = Σ (exposure × risk weight)

For off-balance-sheet items, first convert to a credit equivalent amount using the conversion factor, then apply the counterparty risk weight.

Minimum total capital ratio

(Tier 1 + Tier 2) ÷ RWA ≥ 8%

Minimum required capital = 8% × RWA.

Minimum Tier 1 ratio

Tier 1 ÷ RWA ≥ 4%

Tier 1 must be at least half of required total capital.

Tier 2 limit

Tier 2 (eligible) ≤ Tier 1

Excess Tier 2 does not count toward the 8%.

Credit equivalent for derivatives

Credit equivalent = current exposure + add-on (notional × add-on factor)

Current exposure is the replacement cost, floored at zero. The result is then multiplied by the counterparty risk weight.

Typical risk weights

0% / 20% / 50% / 100%

0%: cash, OECD government claims. 20%: OECD banks. 50%: residential mortgages. 100%: corporate and most other claims.

Basel I minimum capital

Total capital ÷ credit RWA ≥ 8%; Tier 1 ÷ RWA ≥ 4%

Credit risk only before 1996. RWA = Σ (exposure × risk weight).

Common Basel I risk weights

0% cash and OECD government; 20% OECD banks; 50% residential mortgages; 100% corporates

Weights depend on the borrower category, not on the credit quality of the individual borrower.

IMA market risk capital

Max(VaR(t−1), mc × average VaR over last 60 days) + SRC

VaR is 10-day, 99%. mc ≥ 3, up to 4 depending on backtesting. SRC is the specific risk charge if not covered by the model.

Market risk RWA equivalent

Market risk RWA = 12.5 × market risk capital charge

12.5 = 1 ÷ 8%. Lets market risk be combined with credit RWA in the 8% ratio.

Overall capital requirement with market risk

Capital ≥ 8% × credit RWA + market risk charge

Equivalent: 8% × (credit RWA + 12.5 × market risk charge).

Tier 3 limits

Tier 3 only for market risk; Tier 2 + Tier 3 capped; Tier 3 ≤ 250% of Tier 1 capital allocated to market risk

Original Tier 1 capital must be at least 28.57% of market risk capital; Tier 3 is short-term subordinated debt with a minimum original maturity of two years.

Minimum total capital ratio

Total regulatory capital ÷ (Credit RWA + Market risk RWA + Operational risk RWA) ≥ 8%

Basel II Pillar 1. Market and operational risk capital charges are converted to RWA by multiplying by 12.5.

Converting capital charge to RWA

RWA equivalent = Capital charge × 12.5

12.5 = 1 ÷ 8%. Use it for market and operational risk charges.

Credit RWA

Credit RWA = Σ (Exposure × Risk weight)

Under the standardized approach, risk weights depend on external ratings (for example 100% for unrated corporates, 20% for AA- to AA+ rated corporates or banks in the relevant category).

Three pillars

Pillar 1 = minimum capital; Pillar 2 = supervisory review (ICAAP/SREP); Pillar 3 = market discipline (disclosure)

Most questions test matching a scenario to the correct pillar.

Expected loss (IRB link)

EL = PD × LGD × EAD

Used in the IRB approach under Pillar 1.

Minimum capital

Capital = 8% × RWA

Applies to total credit risk RWA under all three approaches.

Standardized RWA

RWA = Σ (Exposure × Risk weight)

Corporate: AAA to AA- 20%; A+ to A- 50%; BBB+ to BB- 100%; below BB- 150%; unrated 100%.

Other standardized weights

Sovereign: AAA to AA- 0%, A+ to A- 20%, BBB+ to BBB- 50%, BB+ to B- 100%, below B- 150%, unrated 100%. Retail 75%. Residential mortgage 35%.

Banks have two options (based on sovereign rating or on the bank's own rating). Check which one the question uses.

Expected loss

EL = PD × LGD × EAD

Not covered by IRB capital K. It is handled through provisions.

IRB capital per unit of EAD (corporate)

K = [LGD × N((N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ)) − PD × LGD] × (1 + (M − 2.5) × b) ÷ (1 − 1.5 × b)

N is the standard normal CDF, N⁻¹ its inverse. The first bracket is the 99.9% worst-case loss minus expected loss.

Asset correlation

ρ = 0.12 × (1 − e^(−50×PD)) ÷ (1 − e^(−50)) + 0.24 × [1 − (1 − e^(−50×PD)) ÷ (1 − e^(−50))]

Ranges between 12% (high PD) and 24% (low PD) for corporates.

Maturity adjustment factor b

b = (0.11852 − 0.05478 × ln(PD))²

Used in the maturity term. M is floored at 1 year and capped at 5 years.

IRB RWA

RWA = K × 12.5 × EAD

The 12.5 is 1 ÷ 8%. Capital = K × EAD.

Foundation IRB supervisory inputs

LGD = 45% (senior unsecured), 75% (subordinated); M = 2.5 years

Bank supplies PD only. Advanced IRB: bank supplies PD, LGD, EAD and M.

Basic indicator approach

K(BIA) = [Σ(GI₁…ₙ × α)] ÷ n, with α = 15%

GI is annual positive gross income over the previous three years. n is the number of years in which gross income was positive. Years with zero or negative income are excluded from both numerator and denominator.

Standardized approach

K(TSA) = {Σ over years 1-3 of max[Σ(GI₁₋₈ × β₁₋₈), 0]} ÷ 3

Within a year, negative income in one business line can offset positive income in others. If the year's total is negative, it counts as zero, and the zero stays in the denominator of 3.

Beta factors by business line

Corporate finance 18%; Trading and sales 18%; Payment and settlement 18%; Commercial banking 15%; Agency services 15%; Retail banking 12%; Asset management 12%; Retail brokerage 12%

Memorize the three groups. 18% for corporate finance, trading and sales, and payment and settlement.

Gross income

Gross income = net interest income + net non-interest income

Gross of provisions and operating expenses. Excludes realized profits or losses from the sale of securities in the banking book, extraordinary or irregular items, and income derived from insurance.

AMA standard

99.9% confidence level, one-year holding period

Four required inputs: internal loss data, external data, scenario analysis, business environment and internal control factors.

Basel II minimum capital ratio

Total capital ÷ (Credit RWA + 12.5 × (Market risk capital + Operational risk capital)) ≥ 8%

Market and operational capital charges are converted to RWA by multiplying by 12.5 (the reciprocal of 8%).

Basel 2.5 market risk capital (internal models)

VaR + stressed VaR + IRC + CRM (for correlation trading), each with multipliers

Stressed VaR uses a 12-month period of significant stress. IRC is a 99.9%, one-year measure of default and migration risk.

Unweighted leverage ratio (Basel III fix)

Tier 1 capital ÷ Total exposure ≥ 3%

Non-risk-based backstop. Basel II had no such measure.

Procyclicality mechanism

Boom: lower PD → lower RWA → lower capital. Bust: higher PD → higher RWA → higher capital

Stated as direction of effect, not a fixed number. It is strongest with point-in-time ratings.

Quick revision

• Basel I focused on credit risk and set a minimum capital ratio against risk-weighted assets.
• Basel I used a small number of broad risk-weight categories, so risk sensitivity was low.
• The 1996 Market Risk Amendment added capital requirements for trading book market risk.
• Basel II rests on three pillars: minimum capital requirements, supervisory review and market discipline.
• Pillar 1 covers credit, market and operational risk capital.
• Pillar 2 is supervisory review: supervisors assess a bank's overall capital adequacy and risk management.
• Pillar 3 is market discipline through public disclosure.
• The standardized approach for credit risk relies on external credit ratings to set risk weights.
• The IRB approach uses bank estimates such as probability of default, loss given default, exposure at default and maturity.
• Operational risk approaches rise in sophistication: basic indicator, standardized, advanced measurement.
• Risk-sensitive capital rules can be procyclical, raising requirements in downturns.
• Weaknesses seen before the crisis include regulatory arbitrage, reliance on ratings and inadequate trading book capital.

Common mistakes

• Treating required capital as 8% of total assets rather than of RWA. Fix: Always compute RWA first. Capital ratio = capital ÷ RWA.
• Applying a 100% weight to every loan, including mortgages and OECD bank claims. Fix: Memorise the four buckets and the example assets in each.
• Saying Basel I covered market risk Fix: Basel I (1988) covered credit risk only. Market risk came in 1996 and was implemented by 1998.
• Using a 1-day or 95% VaR for IMA capital Fix: Regulatory IMA VaR is 10-day and 99%. Backtesting uses 1-day 99% VaR.
• Saying Pillar 2 sets the 8% minimum ratio. Fix: The 8% minimum is Pillar 1. Pillar 2 can add capital above it based on the bank's overall risk.
• Adding market and operational capital charges directly to credit RWA. Fix: Multiply those charges by 12.5 before adding to credit RWA.
• Forgetting the 12.5 multiplier, or using it the wrong way round. Fix: K is capital per unit of EAD. RWA = K × 12.5 × EAD. Capital = 8% × RWA gives back K × EAD.
• Applying the 45% LGD to every exposure under foundation IRB. Fix: 45% is for senior unsecured claims. Subordinated claims get 75%. Collateral can lower it under the supervisory rules.
• Dividing by 3 in the BIA when one year has negative gross income. Fix: In BIA, exclude non-positive years from both the sum and the count. In TSA, keep the floored zero in the divisor of 3.
• Treating the BIA 15% as a beta applied by business line. Fix: Alpha is a single 15% for the whole bank under BIA. Betas of 12%, 15% and 18% apply to business lines under TSA.

Exam tips

• Expect questions asking which asset gets which risk weight; know the four buckets cold.
• Questions often hide the trap in Tier 2 eligibility or an off-balance-sheet item; read the exposure list twice.
• For conceptual items, name the weakness precisely: no differentiation of credit quality within the 100% bucket, no recognition of diversification or hedging, and regulatory arbitrage.
• Link Basel I to what came next: the 1996 Market Risk Amendment added market risk capital, and Basel II introduced ratings-based and internal-model approaches.
• State ratios as a percentage of RWA, never of total assets.
• Expect questions that ask which feature belongs to Basel I and which to the 1996 amendment. Anchor on credit-only versus market risk.
• Memorize the IMA parameters: 10-day, 99%, at least one year of data, multiplier at least 3.
• For arbitrage questions, pick the answer that exploits a gap between regulatory and economic risk, such as securitizing high-quality loans.