Risk Capital Attribution and Risk-Adjusted Performance Measurement: formula sheet

Key formulas

Economic capital (credit-style)

Economic capital = Loss quantile at confidence level α − Expected loss (EL)

Equals unexpected loss at that confidence level. Some sources measure capital from the quantile alone, so check the definition given.

Expected loss

EL = PD × LGD × EAD

Covered by pricing and provisions, so it is not what capital is held for.

Capital adequacy test

Available capital ≥ Required economic capital

Required capital rises with the confidence level and the horizon.

Diversification benefit

Diversification benefit = Σ stand-alone capital − Total (diversified) capital

Always non-negative for coherent measures; it is zero only if risks are perfectly dependent in the tail.

Capital ratio

Regulatory capital ratio = Eligible regulatory capital ÷ Risk-weighted assets

Regulatory measure; not the same as the economic capital calculation.

Diversification benefit

DB = Σ K_i(stand-alone) − K(firm)

Positive when units are not perfectly correlated. For a volatility-based measure, DB is zero only when every pair of units has a correlation of 1 (perfect positive correlation).

Pro rata scaling of stand-alone capital

K_i(allocated) = K_i(stand-alone) ÷ Σ K_j(stand-alone) × K(firm)

Spreads the diversification benefit in proportion to stand-alone size. It ignores each unit's correlation with the rest of the firm.

Incremental capital

K_i(incremental) = K(firm) − K(firm without unit i)

Sum across units is generally not equal to K(firm). Use it for add or remove decisions.

Euler allocation

K_i = w_i × ∂K/∂w_i, and Σ K_i = K

Holds when the risk measure is homogeneous of degree one and differentiable. The full-allocation property is the reason it is preferred.

Euler contribution with volatility

K_i = k × Cov(R_i, R_P) ÷ σ_P = k × ρ_iP × σ_i

Here capital K = k × σ_P (k is a multiplier) and R_i is the unit's loss or return contribution. For two units, Cov(R_A, R_P) = σ_A² + Cov(A, B). The Euler share is below stand-alone capital (k × σ_i) when ρ_iP < 1. Here ρ_iP is the unit's correlation with the whole portfolio, not with another unit.

Portfolio volatility, two units

σ_P = √(σ_A² + σ_B² + 2ρσ_Aσ_B)

Needed to find the firm capital and the Euler shares.

RAROC

RAROC = (Revenue − Costs − Expected Loss) ÷ Economic Capital

Some versions add return on capital and subtract tax. Use the items the question lists.

RORAC

RORAC = Net Income ÷ Risk Capital

Risk adjustment is in the denominator only.

Hurdle test

RAROC > hurdle rate → creates value

Hurdle is usually the cost of equity.

Economic value added (risk-adjusted)

EVA = Risk-adjusted income − hurdle rate × Economic Capital

Positive EVA is the same signal as RAROC above hurdle.

Sharpe ratio

Sharpe = (Rp − Rf) ÷ σp

Total risk. Best for a whole portfolio.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Systematic risk only.

Jensen's alpha

α = Rp − [Rf + β(Rm − Rf)]

Return above the CAPM-required return.

Information ratio

IR = (Rp − Rb) ÷ Tracking error

Active return per unit of active risk.

Risk-adjusted profit

Risk-adjusted profit = Revenue − Costs − Expected loss (+ return on allocated capital, if the bank credits it)

Expected loss is deducted from profit. Capital covers unexpected loss. Follow the question's definition of profit.

RAROC

RAROC = Risk-adjusted profit ÷ Economic capital

Compare with the hurdle rate. Some versions add the return on capital to profit. Use the data given.

EVA

EVA = Risk-adjusted profit − (Cost of capital × Economic capital)

Positive EVA means value created. Uses the same profit and capital as RAROC.

EVA from RAROC

EVA = (RAROC − Hurdle rate) × Economic capital

Valid when RAROC and the hurdle rate use the same profit and capital definitions.

Value-creation rule

RAROC > Hurdle rate ⇔ EVA > 0

The two tests give the same verdict for a given unit and capital base.

RAROC

RAROC = (Revenue − Costs − Expected Loss + Return on Capital) ÷ Economic Capital

Definitions vary slightly. Always use the numerator the question gives you.

Excess return over hurdle

RAROC − Hurdle rate > 0 means value is created

The hurdle is usually the cost of equity. Compare like with like.

Diversification benefit

Benefit = Σ Stand-alone capital − Firm capital

Allocation rules decide who receives this benefit.

Euler (component) allocation

Σ Component capitals = Firm capital

Components add up to total. Each depends on the correlations with the rest of the portfolio.

Incremental capital

Incremental = Firm capital with unit − Firm capital without unit

Incremental amounts usually do not sum to firm capital.

Quick revision

• Economic capital covers unexpected loss at a chosen confidence level.
• Expected loss is handled by pricing and provisions, not by capital.
• Regulatory capital follows Basel rules, while economic capital is the firm's own estimate.
• Stand-alone capital ignores diversification, so unit figures sum to more than total capital.
• Diversification benefit is the gap between summed stand-alone capital and diversified firm capital.
• Marginal or incremental capital measures the change in firm capital when a position is added.
• Component contributions are built to add up to total firm capital.
• RAROC = risk-adjusted return ÷ economic capital.
• Risk-adjusted return is revenue minus costs and expected loss, plus any return on the capital held.
• Compare RAROC with the hurdle rate; above it creates value.
• EVA = net operating profit after tax − capital charge, where the charge is cost of capital × capital.
• Positive EVA means the unit earns more than its cost of capital.

Common mistakes

• Saying economic capital is the same as regulatory capital. Fix: Economic capital is internal and risk-sensitive; regulatory capital is a supervisory minimum. They can differ by portfolio.
• Including expected loss in capital by default. Fix: Expected loss is covered by pricing and provisions. Capital covers unexpected loss, so subtract EL when the question says so.
• Treating stand-alone capital as the allocated capital. Fix: Stand-alone amounts sum to more than firm capital. Scale them or use Euler if the question wants a full allocation.
• Assuming incremental capital amounts add up to firm capital. Fix: Incremental amounts generally do not sum to the total. Say so, and use Euler when full allocation is required.
• Forgetting to subtract expected loss in RAROC. Fix: EL is a cost in the numerator. Capital covers unexpected loss.
• Treating RORAC and RAROC as identical. Fix: RAROC adjusts the numerator for EL and uses economic capital. RORAC uses unadjusted income over risk capital.
• Deducting economic capital instead of the capital charge when computing EVA. Fix: Subtract only cost of capital × capital. Capital itself is not an expense.
• Ignoring expected loss in risk-adjusted profit. Fix: Always deduct expected loss first. Capital covers unexpected loss, not expected loss.
• Assuming allocated capital from different methods is comparable. Fix: Stand-alone, incremental and component amounts differ. Compare units only under one method.
• Saying diversification benefit is stable. Fix: Correlations rise in stress, so the benefit can shrink exactly when losses occur.

Exam tips

• Read the definition in the stem. Some questions measure capital as the quantile, others as quantile minus expected loss.
• In comparison questions, key words are internal model and risk-sensitive for economic capital, and minimum and rule-based for regulatory capital.
• Expect questions linking confidence level to target credit rating and to the amount of capital.
• For allocation questions, remember diversification: sums of stand-alone figures overstate total capital.
• Name the purpose precisely: pricing, limits, performance measurement and incentives.
• Always check whether the question asks for a full allocation. If it does, the answer is Euler or a scaled method, not raw incremental or raw stand-alone.
• The diversification benefit, stand-alone sum minus firm capital, is a quantity a question may ask for directly. Compute it as a check even when it is not requested.
• Many questions test interpretation: which method suits closing a unit (incremental), setting limits (stand-alone) or performance measurement (Euler).