Ch3. FRM Financial Risk Manager — Operational Risk & Quantitative Analysis

Operational Risk: Definition and Scope

Operational risk is defined by Basel as “the risk of loss resulting from inadequate or failed internal processes, people, systems, or external events.” It explicitly excludes strategic and reputational risk (though these may result from operational failures).

The Seven Basel Operational Risk Event Categories

Category	Example
Internal Fraud	Employee embezzlement, unauthorized trading
External Fraud	ATM skimming, phishing attacks
Employment Practices	Discrimination lawsuits, workplace injuries
Clients, Products & Business Practices	Mis-selling, market manipulation
Damage to Physical Assets	Natural disasters, terrorism
Business Disruption & System Failures	IT outages, power failures
Execution, Delivery & Process Management	Settlement errors, data entry mistakes

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Measuring Operational Risk

Basel III/IV Approaches

Basic Indicator Approach (BIA): Capital = 15% × Average Gross Income (3 years)

Standardized Approach (SA): Similar to BIA but broken down by business line, each with its own alpha multiplier.

Advanced Measurement Approach (AMA): Internal loss data + external data + scenario analysis + business environment factors. Allowed pre-Basel IV; being phased out.

Standardized Measurement Approach (SMA) — Basel IV:

• SMA replaces AMA for international banks
• Capital = Business Indicator Component (BIC) × Internal Loss Multiplier (ILM)
• Penalizes banks with high historical losses

Loss Distribution Approach (LDA)

The standard quantitative model under AMA:

1. Fit a frequency distribution to loss counts (Poisson is typical)
2. Fit a severity distribution to loss sizes (lognormal, Weibull, GPD)
3. Convolve the two using Monte Carlo to get the aggregate annual loss distribution
4. Read off the 99.9% quantile as the operational VaR (OpVaR)

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Quantitative Analysis: The FRM Math Foundation

FRM Part I requires strong statistical and econometric skills. This section covers the highest-yield topics.

Probability & Statistics

Descriptive statistics:

• Mean, variance, standard deviation, skewness (asymmetry), kurtosis (tail fatness)
• Excess kurtosis > 0 (leptokurtic / fat tails) is the norm for financial returns

Key distributions:

Distribution	Use Case
Normal	Benchmark; VaR parametric method
Lognormal	Asset prices (always positive)
Student-t	Fat tails; better for financial returns
Poisson	Operational loss frequency
Generalized Pareto (GPD)	Extreme Value Theory (EVT) tail losses

Regression Analysis

OLS (Ordinary Least Squares): y = α + β₁x₁ + … + βₙxₙ + ε

Key FRM applications:

• Beta estimation: Regress stock returns on market returns → β = Cov(R_i, R_m)/Var(R_m)
• Factor models: Fama-French three-factor, Carhart four-factor
• Multicollinearity, heteroskedasticity, autocorrelation: Understand how each violates OLS assumptions and remedies

Time Series Analysis

Autocorrelation: Past returns predict future returns — violates the efficient market hypothesis (weak form). Measured by ACF/PACF plots.

ARCH/GARCH models: Capture volatility clustering (calm periods followed by turbulent periods):

• GARCH(1,1): σ²ₜ = ω + α·ε²ₜ₋₁ + β·σ²ₜ₋₁
• Parameters: α (reaction to shocks) + β (persistence) < 1 for stationarity

EWMA (Exponentially Weighted Moving Average): Special case of GARCH(1,1) with ω = 0:

• σ²ₜ = (1−λ)·r²ₜ₋₁ + λ·σ²ₜ₋₁ (λ ≈ 0.94 for daily data — RiskMetrics standard)

Extreme Value Theory (EVT)

EVT addresses the FRM’s central problem: standard distributions underestimate tail risk.

Peaks Over Threshold (POT) method:

1. Set a high threshold u
2. Fit a Generalized Pareto Distribution to exceedances (losses above u)
3. Extrapolate to get VaR and ES at extreme quantiles (99.9%+)

This is the theoretically sound approach for operational risk capital and stress VaR.

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Correlation, Copulas, and Tail Dependence

Standard linear correlation (Pearson’s ρ) is insufficient for capturing how assets co-move during crises.

Copulas separate the marginal distributions from their dependence structure:

• Gaussian copula: Zero tail dependence — underestimated CDO losses in 2008
• t-copula: Positive tail dependence — more realistic for stress scenarios
• Archimedean copulas: Clayton (lower tail), Gumbel (upper tail)

The 2008 financial crisis exposed the dangers of assuming Gaussian copulas for correlated mortgage defaults.

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Key Formulas to Memorize

Formula	Use
σ²ₜ = ω + α·ε²ₜ₋₁ + β·σ²ₜ₋₁	GARCH(1,1) conditional variance
EWMA: λ ≈ 0.94	RiskMetrics daily volatility
EVT GPD: ξ, β parameters	Tail loss modeling
OLS β = Cov(x,y)/Var(x)	Regression/Beta estimation
Skewness < 0	Left tail / negative skew (common in equity returns)

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Summary

Area	Key Exam Points
OpRisk categories	7 Basel event types; internal vs. external
Basel IV SMA	Replaces AMA; uses BIC × ILM
LDA	Poisson frequency × severity → convolve → 99.9% OpVaR
GARCH(1,1)	Volatility clustering; α + β < 1
EVT	GPD for tail modeling; superior to normal for extremes
Copulas	Capture non-linear correlation; t-copula for tail dependence

Chapter 4 provides a comprehensive review of FRM high-yield formulas and a 20-question mock exam to test your readiness.