VaR Backtesting Skaičiuotuvas

Backtesting is the empirical validation that a VaR model produces the calibration it claims. A bank claiming 99% VaR should, by definition, experience losses exceeding VaR approximately 1% of the time. Without backtesting, a model could systematically understate risk (producing too-low VaR, which reduces capital requirements) with no accountability mechanism. Backtesting creates a regulatory feedback loop: models that fail are penalized with capital add-ons, creating financial incentives for banks to maintain well-calibrated models. Backtesting also helps risk managers identify when models are becoming obsolete due to changed market dynamics.

Clean P&L (hypothetical P&L) measures the gain or loss on yesterday's portfolio positions repriced using today's market prices — isolating pure market risk exposure without the effect of new trades, fees, or portfolio changes. Dirty P&L (actual P&L) includes all P&L from all sources: trading revenues, new positions, fee income, and operational items. Basel FRTB requires backtesting against hypothetical clean P&L, because the VaR model estimates the risk of the existing portfolio, not the evolving portfolio. If VaR is compared to dirty P&L, good trading days can obscure risk model failures, and vice versa.

The minimum regulatory requirement is 250 trading days (≈1 year). However, this is statistically quite limited: at 99% VaR, only 2.5 expected exceptions occur per year. The standard error of the estimated exception rate with 2.5 observations is very high — it is impossible to distinguish between a model that produces 0.5%, 1%, 1.5%, or 2% true exception rates with only 250 observations. For robust validation, 3–5 years of data are preferred. Even then, the Kupiec test has low power at distinguishing small systematic errors. This is why multiple statistical tests and supplementary validation methods (stress testing, P&L attribution) are required alongside backtesting.

The Basel traffic light system (introduced in 1996 and updated in Basel III) classifies VaR model performance based on the number of backtesting exceptions in the most recent 250 trading days: Green Zone (0–4 exceptions) — model passes, no capital penalty; Yellow Zone (5–9 exceptions) — capital multiplier k increases from 3.0 to 3.4–4.0 depending on the number, and regulatory scrutiny increases; Red Zone (10+ exceptions) — model fails, k=4.0 or higher, and the bank may be required to switch to the standardized approach. The thresholds are set to balance Type I error (rejecting good models) and Type II error (accepting bad models) for a 250-day sample.

P&L attribution (PLA) requires banks to explain each day's actual P&L by decomposing it into contributions from the risk factors captured in the VaR model. Under Basel FRTB, the difference between the VaR model's theoretical P&L (using model risk factors) and the actual hypothetical P&L must be small and uncorrelated with actual P&L. If significant P&L unexplained by the model exists (model residuals are large), this suggests the VaR model is missing important risk factors and may be understating risk. PLA is performed at the trading desk level, while backtesting is also conducted at the desk level under FRTB.

Standard Kupiec backtesting only detects incorrect exceedance frequency. It cannot detect: (1) incorrect tail shape — if the model underestimates loss severity on exception days; (2) incorrect risk factor sensitivities — if the model correctly predicts the frequency but for the wrong reasons; (3) correlation misspecification — if diversification benefits are overstated. The Christoffersen test adds independence testing. Loss function-based tests (compare expected vs. realized P&L distribution) and regression-based tests provide additional diagnostic power. A complete model validation program uses multiple tests, stress testing, sensitivity analysis, and expert model review to identify different types of misspecification.

Excess exceptions trigger a structured model review process: (1) Investigate each exception: was the loss driven by a specific event, data error, or genuine model shortcoming? (2) Check whether volatility estimates are current — stale volatility assumptions are a common cause of VaR underestimation in rapidly changing markets. (3) Review correlation assumptions — check whether crisis correlations are being used when appropriate. (4) Test alternative distribution assumptions — replace normal with t-distribution or historical simulation. (5) Examine concentration risks not fully captured by the model. (6) If the model cannot be quickly remediated, increase capital charges or reduce position limits until the model is recalibrated.