Range of Practices and Issues in Economic Capital Frameworks: formula sheet

Key formulas

Economic capital (unexpected loss basis)

Economic capital = Loss at confidence level α − Expected loss

Over a stated horizon, usually one year. Some banks instead use the full loss quantile. State which definition is used.

Unexpected loss

UL = Loss quantile (VaR at α) − EL

Economic capital covers UL, not EL.

Capital adequacy comparison

Available capital ≥ Economic capital required

Shortfall means the bank is under-capitalised on its own risk view.

RAROC

RAROC = (Revenue − Costs − Expected loss) ÷ Economic capital

Compared with a hurdle rate based on the cost of equity.

Economic capital (VaR basis)

EC = VaR(α) − Expected Loss

Unexpected loss at confidence level α over the horizon. Some banks hold the full VaR instead, so read the question.

Expected shortfall

ES(α) = E[Loss | Loss ≥ VaR(α)]

Average tail loss. ES(α) is always at least VaR(α).

Link of confidence level to rating

Confidence level α = 1 − target one-year default probability

Example: target default probability 0.03% gives α = 99.97%.

Normal VaR

VaR(α) = μ + z(α) × σ

For normal losses over the horizon. z(99%) ≈ 2.33, z(99.9%) ≈ 3.09, z(95%) ≈ 1.645.

Normal ES

ES(α) = μ + σ × φ(z) ÷ (1 − α)

φ is the standard normal density. At 99%, ES ≈ μ + 2.665σ.

Square-root-of-time scaling

VaR(T days) ≈ VaR(1 day) × √T

Valid only for independent, identically distributed returns with zero mean. Weak for credit and illiquid positions.

Economic capital (credit, typical form)

EC = Loss at confidence level α − Expected Loss

Capital covers unexpected loss only. Expected loss is covered by provisions and pricing.

Expected loss

EL = PD × LGD × EAD

Per exposure, over the horizon.

Horizon scaling of VaR (square-root-of-time)

VaR(T days) ≈ VaR(1 day) × √T

Valid only under i.i.d. returns and constant positions. It is a rough approximation.

Operational loss distribution approach

Annual loss = sum of N individual losses, N ~ frequency distribution, each loss ~ severity distribution

Combined by Monte Carlo. Capital is read at a high percentile of the aggregate loss.

Aggregation with correlation (two risks)

EC_total = √(EC₁² + EC₂² + 2ρ × EC₁ × EC₂)

A variance-covariance shortcut. Total is below the simple sum if ρ < 1.

Simple summation

EC_total = EC_1 + EC_2 + ... + EC_n

Equivalent to assuming correlation of 1 between all risks. Gives no diversification benefit. It is the upper bound for any correlations up to 1.

Variance-covariance aggregation

EC_total = √(Σi Σj ρij × EC_i × EC_j)

ρii = 1. For two risks: √(EC_1² + EC_2² + 2ρ × EC_1 × EC_2).

Diversification benefit

DB = Σ EC_i − EC_total

Can also be quoted as DB ÷ Σ EC_i. Zero when correlations are all 1.

Zero correlation case

EC_total = √(Σ EC_i²)

Square root of the sum of squares. Gives the largest benefit among non-negative correlations. Negative correlations would give an even lower aggregated figure than the zero-correlation case.

Copula aggregation (Sklar idea)

Joint distribution = Copula(marginal_1, ..., marginal_n)

Marginals describe each risk. The copula describes dependence. Student-t copula has tail dependence; Gaussian copula does not.

Expected exceedances over a horizon

Expected number = (1 − confidence level) × number of independent periods

At 99.9% over one-year periods, you expect one exceedance per 1,000 years. This is why direct backtesting of economic capital fails.

Economic capital (unexpected loss view)

Economic capital = loss at chosen confidence level − expected loss

Capital covers unexpected loss. Expected loss is covered by pricing and provisions. Check how the question defines it.

Confidence level and target rating

Confidence level = 1 − target annual default probability

A target of 0.03% annual default probability for the bank implies a 99.97% confidence level.

RAROC

RAROC = (Revenue − Costs − Expected loss + Return on economic capital) ÷ Economic capital

Numerator is risk-adjusted net income. Check whether the question credits a return on the capital itself. Some use a simpler version without it.

Economic capital (unexpected loss)

Economic capital = Loss at chosen confidence level − Expected loss

Loss at the percentile minus the mean loss. A higher confidence level means more capital.

Value-creation test

Create value if RAROC > Hurdle rate (cost of equity)

Compare RAROC with the shareholders' required return. Equal means break-even.

Economic value added (risk-based)

EVA = Risk-adjusted net income − (Hurdle rate × Economic capital)

A positive EVA means return above the cost of capital, in money terms.

Diversification benefit

Benefit = Sum of stand-alone capital − Diversified total capital

Always zero or positive for a sound measure. It is zero only when risks are perfectly correlated.

Capital allocation by contribution

Component capital of unit i = Share of total capital attributed to i, with all components summing to total capital

Component (Euler-type) allocation adds up to the total. Stand-alone capital does not.

Quick revision

• Economic capital is the internal estimate of capital needed to cover unexpected losses at a chosen confidence level and horizon.
• Regulatory capital follows prescribed rules; economic capital reflects the bank's own risk view.
• Confidence level is often linked to the bank's target credit rating: a higher target rating implies a higher confidence level.
• A one-year horizon is the common choice for economic capital.
• Expected losses are covered by pricing and provisions; economic capital targets unexpected loss.
• Expected shortfall looks at the tail beyond VaR and is more informative about severe losses.
• Credit and operational risk have fat tails, so tail estimates carry large uncertainty.
• Adding stand-alone capital figures assumes perfect dependence and ignores diversification.
• Diversification benefits depend on correlation or dependence assumptions, which are unstable in stress.
• Limited data on rare events is a main source of model risk in operational and credit risk.
• Validation covers inputs, model logic, outputs and back-testing where possible.
• Governance requires senior management understanding, clear ownership and use in decisions, not just calculation.

Common mistakes

• Treating economic capital as the same as regulatory capital. Fix: Remember regulatory is rule-based and minimum; economic is internally modelled and bank-specific.
• Forgetting to subtract expected loss. Fix: Unless told otherwise, economic capital covers unexpected loss: quantile minus EL.
• Treating a higher target rating as needing a lower confidence level Fix: Better rating means lower default probability, so a higher α and more capital.
• Saying VaR shows how large losses are beyond the cut-off Fix: VaR is only a threshold. ES measures the average size of losses beyond it.
• Treating economic capital as total expected plus unexpected loss. Fix: Remember capital covers unexpected loss: percentile loss minus expected loss, unless the question defines it otherwise.
• Assuming the same horizon and confidence level apply to all risk types. Fix: Check each measure's horizon and confidence level, and adjust to a common basis before aggregating.
• Treating simple summation as the average case Fix: Remember it assumes correlation of 1. It is a conservative upper bound with zero diversification benefit.
• Adding capital figures and then taking the square root without squaring first Fix: Square each EC, add the cross terms with correlation, then take the root.
• Saying economic capital can be backtested like a 99% daily VaR. Fix: Count the expected exceedances. At 99.9% over one year there is almost none to observe, so test components and use other tools.
• Treating stress tests as giving a probability of loss. Fix: Stress tests show impact under a chosen scenario. They complement statistical models but carry no confidence level.

Exam tips

• Expect comparison questions: who sets it, what basis, what purpose. Learn the three-way contrast cold.
• Watch the confidence level and horizon. They change the number.
• Check whether the question deducts expected loss before computing capital.
• Questions on range of practices stress that banks differ on risk coverage, confidence levels and diversification, so figures are not directly comparable.
• Expect case-style questions that give a rating and ask for the confidence level. Convert rating to default probability first.
• Know that ES is coherent and VaR is not, and that VaR ignores tail size. These appear as conceptual options.
• Check whether the question defines capital as VaR or VaR minus expected loss.
• Watch for horizon traps: one year is the usual economic capital horizon, and √T scaling needs stated assumptions.