Thematischer Pillar-Hub · 20 Veröffentlichte Analysen
Parametric Var Expected Shortfall research topics and source records from QuantCalc Tools. Under Basel III MAR33, 97.5% Expected Shortfall replaces 99% Gaussian VaR to penalize tail kurtosis; this quantitative guide reconciles Cornish-Fisher quantiles, Student-t scaling, EWMA decay, and backtesting.
Under Basel III MAR33, 97.5% Expected Shortfall replaces 99% Gaussian VaR to penalize tail kurtosis; this quantitative guide reconciles Cornish-Fisher quantiles, Student-t scaling, EWMA decay, and backtesting.
Modified Cornish-Fisher Value-at-Risk expands normal quantiles via third- and fourth-order Hermite polynomials, lifting 99% portfolio risk metrics by 32% to 48% on negatively skewed strategies.
Unstandardized Student-t critical values inflate variance by nu/(nu-2); this quantitative guide derives unit-variance Student-t VaR and Expected Shortfall formulas across nu = 3 to 30.
Basel III MAR33.4 replaces uniform 10-day VaR scaling with a nested root-sum-of-squares Expected Shortfall cascade across 10, 20, 40, 60, and 120-day liquidity horizons.
RiskMetrics EWMA (lambda = 0.94) weights yesterday’s shock at 6.0% with an 11.2-day half-life but lacks mean reversion; GARCH(1,1) anchors multi-day VaR to long-run variance.
Kupiec’s Proportion of Failures (POF) test evaluates binomial VaR breach counts via a 1-df chi-squared log-likelihood ratio, governing Basel Green, Amber, and Red capital multipliers.
Christoffersen’s conditional coverage backtest decomposes VaR validation into Kupiec unconditional frequency plus a 1-df Markov transition test that catches consecutive breach streaks.
Scaling 1-day parametric VaR by sqrt(10) assumes zero serial autocorrelation; positive AR(1) persistence of +0.25 inflates true 10-day portfolio risk by 26.8% above naive root-time scaling.
Because parametric VaR is homogeneous of degree one in portfolio weights, Euler’s theorem guarantees that Component VaRs sum to 100% of diversified portfolio risk without order dependence.
Linear Delta-Normal VaR treats short straddles as zero-risk portfolios; second-order Delta-Gamma Taylor expansions capture negative convexity mean shift, variance uplift, and left-tail skewness.
Barone-Adesi Filtered Historical Simulation (FHS) marries GARCH conditional volatility responsiveness with non-parametric empirical fat tails and cross-asset rank dependence.
By the Pickands-Balkema-de Haan theorem, tail losses exceeding a high threshold converge to a Generalized Pareto Distribution, enabling closed-form 99.9% VaR and Expected Shortfall extrapolation.
Correlated Monte Carlo VaR engines crash with square-root domain errors whenever asynchronous closes or stress overrides create negative eigenvalues; Higham’s alternating projections restore PSD validity.
Because full modern risk-factor histories rarely exist back to 2007-2009, Basel III MAR33.5 scales reduced-factor crisis Expected Shortfall by the current full-to-reduced portfolio ratio.
Gneiting proved standalone Expected Shortfall lacks a strictly consistent scoring function, yet Acerbi and Szekely’s unconditional Z2 statistic directly backtests joint VaR-ES tail severity without nuisance parameters.
When asset count N approaches sample length T, sample covariance matrices invent spurious near-zero eigenvalues that understate minimum-variance portfolio VaR by 35% to 60% out of sample.
Diversification vanishes fastest when needed most: flipping stock-bond correlation from -0.35 to +0.55 and equity-FX correlation to +0.80 inflates balanced portfolio VaR by 52% overnight.
Close-to-close volatility treats an intraday +/-4% V-reversal ending unchanged as zero variance; Yang-Zhang OHLC estimators achieve up to 14x higher statistical efficiency while capturing overnight gaps.
Linear arithmetic VaR reports absurd 116% losses on unlevered 50%-volatility positions over 1-year horizons; exact lognormal transformation bounds losses below 100% while accounting for Ito drift.
Risk factors failing Basel MAR31.12 real-price observation thresholds (24 per year or 100 without 90-day gaps) are stripped from diversified Expected Shortfall and charged zero-correlation stress capital.
CAGR calculator →
Compound annual growth rate between two values, with a year-by-year projection.
FX forward rate calculator →
Forward exchange rate and forward points from spot and two interest rates (covered interest parity).
Value at Risk (VaR) calculator →
Parametric VaR and Expected Shortfall for a position, at 95% or 99% confidence.
Gaussian 99% VaR ignores loss severity beyond the 1% threshold and violates subadditivity, occasionally penalizing portfolio diversification. Calibrating Expected Shortfall at 97.5% preserves the exact normal-distribution capital baseline (2.3378 sigma vs 2.3263 sigma) while penalizing fat-tailed kurtosis and satisfying coherent risk axioms.
Multiplying 1-day VaR by sqrt(10) assumes independent, identically distributed zero-mean returns with constant variance. If daily returns exhibit positive AR(1) autocorrelation (rho = +0.20), true 10-day volatility exceeds sqrt(10) scaling by roughly 20%, whereas GARCH(1,1) mean reversion pulls post-spike multi-day volatility below sqrt(T).
Because Cornish-Fisher truncates an asymptotic Hermite polynomial expansion at skewness and excess kurtosis, extreme moment combinations violate the Maillard domain bound, causing the cubic z-score polynomial to fold backward so 99.5% VaR erroneously prints lower than 99.0% VaR.
Under Basel desk-level and bank-wide 99% VaR backtesting over 250 business days, 0 to 4 exceptions keep the model in the Green Zone (1.50x ES multiplier), 5 to 9 exceptions place it in the Amber Zone (1.70x to 1.92x multiplier), and 10 or more breaches trigger Red Zone model disqualification.
In the fourth-order Hermite term (Ke / 24) * (z^3 - 3z), the polynomial crosses zero at z = sqrt(3) = 1.73205, corresponding to 95.84% one-tailed confidence. At 95.0% confidence (z = 1.64485), z^3 - 3z equals -0.4865, reflecting thinner intermediate shoulders in leptokurtic distributions.
If portfolio return skewness S is negative (left-skewed crashes) and z_alpha is entered as a positive number (+2.32635 for 99% loss), the skewness term must be subtracted—that is, -(S/6)*(z^2 - 1)—so negative S produces a positive addition to the loss quantile.