FRM Part II cheat sheet

Parametric VaR: VaR = (z_alpha x sigma - mu) x V. Relative VaR omits mu. Both assume normal i.i.d. returns and therefore understate fat-tail and option risk.

Expected shortfall is the average loss conditional on exceeding VaR; ES is coherent and subadditive, whereas VaR can violate subadditivity across positions.

Basel FRTB replaced 99%/10-day VaR with 97.5% ES, roughly comparable in magnitude but sensitive to tail severity rather than a single quantile.

EWMA: sigma^2_t = lambda x sigma^2_(t-1) + (1 - lambda) x r^2_(t-1). RiskMetrics uses lambda = 0.94 daily and 0.97 monthly; higher lambda gives smoother, slower-reacting volatility.

GARCH(1,1): sigma^2_t = omega + alpha x r^2_(t-1) + beta x sigma^2_(t-1). Persistence is alpha + beta and long-run variance is omega / (1 - alpha - beta), requiring alpha + beta < 1.

Fixed income: dP/P = -D_mod x dy + 0.5 x convexity x (dy)^2; DV01 is the price change per 1bp. Positive convexity means duration overstates the loss for a yield decline and understates it for a yield rise.

Square-root-of-time scaling (VaR_T = VaR_1 x sqrt(T)) is valid only for i.i.d. returns; positive autocorrelation makes short-horizon VaR understate longer-horizon risk.

Backtesting uses 250 days at 99% VaR with 2.5 expected exceptions; Basel green is 0-4 exceptions, yellow 5-9 (multiplier 3.00 to 3.99), red 10 or more.

Portfolio variance is w' x Sigma x w; the diversification benefit equals the sum of standalone VaRs minus portfolio VaR and is driven by correlations.

Expected loss = PD x LGD x EAD. Unexpected loss is the standard deviation of loss around EL and drives economic capital, while EL is priced into spreads.

Merton model treats equity as a call option on firm assets with strike equal to the face value of debt; default occurs when asset value at maturity is below debt face value.

Distance to default = [ln(V/F) + (mu - 0.5 x sigma_V^2) x T] / (sigma_V x sqrt(T)); the EDF is N(-DD).

Vasicek-style worst-case default rate: WCDR = N[(N^-1(PD) + sqrt(rho) x N^-1(confidence)) / sqrt(1 - rho)]; IRB capital uses EAD x LGD x (WCDR - PD) x maturity adjustment.

In Basel IRB, asset correlation rho declines with firm size for SMEs and is floored at 12% for large regulated financial firms, reflecting higher systemic linkage.

Reduced-form models use hazard rates: cumulative PD over t is 1 - exp(-lambda x t), and for small PD the CDS spread approximates lambda x (1 - R).

Structural models use balance-sheet asset value and are forward-looking but assume lognormal asset value and simple debt structure; reduced-form models calibrate to market spreads and price credit derivatives more tractably.

CreditMetrics builds a portfolio value distribution from rating migration probabilities plus forward zero curves; CreditRisk+ is default-only, actuarial, with Poisson default frequency and no migration.

Single-name and sector concentration raise portfolio unexpected loss; the IRB granularity adjustment explicitly charges for undiversified idiosyncratic risk.

Basel event-type taxonomy: internal fraud; external fraud; employment practices and workplace safety; clients, products and business practices; damage to physical assets; business disruption and system failures; execution, delivery and process management.

Standardised Measurement Approach: BIC = 15% of the three-year average of positive annual gross income; ILM = ln[e - 1 + (average annual operational losses / BIC)]; capital = BIC x ILM.

Loss distribution approach multiplies a frequency distribution (typically Poisson) by a severity distribution (typically lognormal or extreme value) and convolves or simulates to obtain the aggregate loss distribution.

Aggregate operational VaR is dominated by severity tails: small changes in the tail index or the severity threshold shift capital estimates materially.

RCSA identifies risks, assesses inherent likelihood and impact, maps controls, and records residual risk; scenario analysis calibrates low-frequency, high-severity events that historical data miss.

Business continuity: the business impact analysis identifies critical processes; RTO is the maximum tolerable downtime and RPO the maximum tolerable data loss, both validated through testing.

Third-party risk covers vendor concentration, subcontracting chains and exit strategy; outsourcing an activity does not transfer regulatory accountability.

Model risk management requires a model inventory, tiering by materiality, independent validation, ongoing monitoring and documented limitations; cyber and conduct risks are leading current operational loss drivers.

LCR = high-quality liquid assets / net cash outflows over 30 days, required to be at least 100%; HQLA is split into Level 1 (no haircut), Level 2A (15% haircut) and Level 2B (25-50% haircuts) with cap constraints.

NSFR = available stable funding / required stable funding, required to be at least 100% over a one-year horizon; ASF weights rise with funding maturity and stability, RSF weights rise with asset illiquidity and maturity.

Deposit runoff assumptions distinguish stable retail, less-stable retail and wholesale funding; unsecured wholesale funding from financial institutions is assumed to run off almost entirely.

Liquidity-adjusted VaR = VaR + 0.5 x bid-ask spread x position for exogenous spread cost; an endogenous add-on grows with position size relative to average daily volume.

Roll's spread estimator infers the effective spread from negative serial covariance in transaction price changes: spread = 2 x sqrt(-cov).

Funding and market liquidity interact: margin calls on leveraged positions force fire sales, converting funding stress into market price impact and further margin calls.

Liquidity stress testing combines contractual cash-flow gap analysis with behavioural assumptions for wholesale funding rollover and drawdown of committed credit lines.

A contingency funding plan specifies the liquidity buffer, early-warning indicators, funding diversification targets and a documented escalation sequence.

Sharpe = (Rp - Rf) / sigma_p; Treynor = (Rp - Rf) / beta_p; Jensen's alpha = Rp - [Rf + beta_p x (Rm - Rf)]; M-squared = Sharpe x sigma_m + Rf.

Information ratio = active return / tracking error; leverage that scales return and active risk equally does not improve it.

The Sortino ratio replaces total volatility with downside deviation in the denominator and is preferable when returns are skewed.

Marginal VaR_i = beta_i x VaR_p in relative form; component VaR_i = w_i x marginal VaR_i; components sum to total portfolio VaR.

Risk budgeting allocates total risk using component VaR or risk contributions; risk parity equalises risk contributions across assets rather than equalising capital weights.

Hedge fund database biases: survivorship (failed funds are dropped), backfill (successful funds report history retroactively), and stale pricing that smooths returns and understates volatility.

Illiquid or infrequently priced assets create serial correlation, which biases Sharpe ratios upward and market betas downward; unsmoothing techniques correct for it.

Performance attribution decomposes excess return into allocation and selection effects, and the result depends critically on the chosen benchmark and universe.

USD LIBOR panels ceased for new contracts on 30 June 2023; SOFR is a secured overnight rate derived from US Treasury repo transactions, and term SOFR is a forward-looking derivative of overnight SOFR.

Climate risk splits into physical risk (acute event and chronic trend) and transition risk (policy, technology and preference shifts); NGFS scenarios stress both channels jointly.

Climate stress tests typically run to 2050 and are not comparable with one-year regulatory capital horizons; outcomes are highly sensitive to assumed policy timing and carbon price paths.

DeFi replaces central intermediation with smart contracts: principal risks are contract bugs, oracle manipulation, governance attacks and cascading collateral liquidations.

Stablecoins differ as fiat-backed, crypto-collateralised or algorithmic; depeg risk transmits through collateral haircuts and redemption runs into leveraged crypto positions.

Machine learning model risk includes overfitting, non-stationarity, data leakage and limited interpretability; controls require out-of-sample and adversarial validation plus continuous monitoring.

CBDC design trade-offs involve privacy versus AML compliance, bank disintermediation versus payment efficiency, and retail versus wholesale scope.

Private credit expansion raises concerns about mark-to-model valuation opacity, liquidity mismatch in open-end vehicles, and borrower concentration among sponsor-owned issuers.