Describe the difficulties met in implementing an economic capital framework when defining and computing risk measures, aggregating risk, validating models, modeling dependency in credit risk, evaluating counterparty credit risk, and assessing interest rate risk in the banking book.

FRM® Part II · Operational Risk and Resilience · Range of practices and issues in economic capital frameworks · FRM2.ORR.19.a

The idea

Economic capital is the capital a bank's own models say it needs to absorb unexpected losses at a chosen confidence level over a horizon. The horizon is usually one year, the confidence level 99.9 percent or higher, and the level is often tied to a target rating. The difficulties sit in six places. The first is defining the measure, because VaR or expected shortfall, confidence level, horizon and risk coverage differ across banks. The second is aggregation. Adding stand-alone figures assumes perfect correlation and overstates capital, while correlation matrices and copulas rest on thin data and give very different diversification benefits. The third is validation: a one-in-a-thousand-year standard cannot be backtested, so banks rely on benchmarking, stress tests and review. The fourth is credit dependency, where asset correlation and copula choice drive the tail but are hard to calibrate. The fifth is counterparty credit risk, where exposure is stochastic and wrong-way risk links it to default. The sixth is banking-book interest rate risk, where behavioural assumptions for non-maturity deposits and prepayments dominate. For the exam, tie each difficulty to its cause: scarce data and unobservable dependence.

Formulas

Aggregate economic capital with a correlation matrix

EC = √(Σ Σ ρij × ECi × ECj)

ρij: the correlation between risk types i and j, with ρii equal to 1
ECi, ECj: the stand-alone economic capital of risk types i and j

The double sum counts each cross term twice. Thin data make the correlations uncertain, and different choices give very different diversification benefits.

Worst-case default rate in a one-factor Gaussian copula

WCDR = N((N⁻¹(PD) + √ρ × N⁻¹(0.999)) ÷ √(1 − ρ))

N: the cumulative standard normal distribution, with N⁻¹ its inverse
PD: the average default probability
ρ: the asset correlation

A large homogeneous portfolio at 99.9 percent. The tail default rate rises steeply with ρ, which is hard to calibrate.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why is validating an economic capital model by backtesting much harder than backtesting a 10-day 99% trading VaR model?

At a 99.9% one-year standard a breach is expected about once in a thousand years, so history cannot test the tail. - correct
Supervisors do not allow economic capital to be compared with realised losses, which makes any comparison impossible in practice.Why not: Confuses supervisory use with statistical testability: nothing bars the comparison, the problem is too few observations.
Economic capital covers only credit risk, for which no loss data exist, whereas trading VaR has daily data.Why not: Economic capital spans several risk types and credit loss data do exist; the issue is how rarely tail events occur.
Trading VaR is a coherent risk measure while economic capital is not, so only VaR can be tested.Why not: Reverses the facts: VaR is not coherent, and coherence has no bearing on whether a model can be backtested.

With a 99.9% one-year standard the expected number of exceedances in any feasible history is far below one, so a backtest has almost no power. Banks therefore combine benchmarking, stress testing, sensitivity analysis and qualitative review.

A harder one

A bank has stand-alone economic capital of USD 600 million for credit risk, USD 300 million for market risk and USD 200 million for operational risk. It aggregates with a correlation matrix: credit-market 0.5, credit-operational 0.2 and market-operational 0.2. Aggregate economic capital is closest to:

USD 1,100 millionWhy not: Simple summation: assumes perfect correlation, so it gives no diversification benefit.
USD 861 million - correct
USD 700 millionWhy not: Assumes zero correlation: square root of the sum of squared stand-alone amounts.
USD 819 millionWhy not: Includes only the credit-market cross term, omitting the credit-operational and market-operational terms and the factor of two.

Aggregate = sqrt(sum_i sum_j rho_ij x EC_i x EC_j). Squares: 360,000 + 90,000 + 40,000 = 490,000. Cross terms: 2 x (0.5 x 600 x 300 + 0.2 x 600 x 200 + 0.2 x 300 x 200) = 2 x (90,000 + 24,000 + 12,000) = 252,000. Total 742,000, square root = 861. Diversification benefit is 1,100 - 861 = 239.

The hardest tier

A bank buys credit protection from a dealer that is heavily exposed to the same sector as the reference entity. When computing counterparty credit risk economic capital, the bank assumes the dealer's default is independent of the value of the protection. What is the likely effect?

Economic capital is overstated, because assuming independence is the most conservative possible case.Why not: Treats independence as the worst case; positive dependence between exposure and default is worse and sits in the tail.
Economic capital is unaffected, because expected exposure is unchanged and only the price of protection responds to dependence.Why not: Conditional exposure at default differs from expected exposure; capital is set in the tail where the dependence acts.
Understated: the protection is worth most when the dealer is most likely to default, which needs joint modelling. - correct
Economic capital is overstated, because netting and collateral already remove any link between exposure and default.Why not: Netting and collateral reduce exposure but do not remove the wrong-way dependence, and margin lags during stress.

This is wrong-way risk: exposure rises as the counterparty's credit quality falls. Assuming independence understates the tail loss. Capturing it needs simulation of market risk factors jointly with counterparty default, which is one reason counterparty credit risk is hard to include in economic capital.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

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