Calculating Value At Risk

Calculating Value At Risk

When To Use

• Computing portfolio VaR at specified confidence levels (typically 95% or 99%) and holding periods (1-day, 10-day)
• Comparing risk estimates across parametric (variance-covariance), historical simulation, and Monte Carlo approaches
• Backtesting an existing VaR model against realized P&L to assess model adequacy
• Supporting regulatory capital calculations under Basel framework internal models approach [VERIFY: Basel III/IV applicability for jurisdiction]
• Producing risk reports for portfolio managers, risk committees, or regulators

Inputs To Gather

• Portfolio composition: Asset classes, positions, notional values, currency denominations
• Market data: Historical returns series (minimum 1 year for historical simulation; 3-5 years preferred), closing prices, FX rates, yield curves
• Parameters: Confidence level (e.g., 99% for regulatory, 95% for internal), holding period, lookback window length
• Correlation/covariance data: Correlation matrix or raw return series for parametric method; decay factor if using exponentially weighted moving average (EWMA)
• Distribution assumptions: Normal, Student-t, or empirical for parametric; number of simulations and random seed for Monte Carlo
• Benchmark P&L: Realized daily P&L series for backtesting (minimum 250 trading days)

Workflow

1. Data Preparation

• Collect and align time series to consistent trading calendar (handle holidays, missing data)
• Compute log returns or arithmetic returns (state choice and rationale)
• Check for stale prices, outliers, and corporate actions; flag gaps with [VERIFY]
• Convert multi-currency positions to base currency using consistent FX rates

2. Methodology Selection

• Parametric (Variance-Covariance): Best for linear portfolios (equities, FX). Compute portfolio variance as w'Σw. VaR = z_α × σ_p × √(holding period). Note: assumes normal returns — underestimates tail risk for fat-tailed distributions
• Historical Simulation: Rank historical portfolio P&L; VaR = the (1-α) percentile loss. No distributional assumption required. Sensitive to lookback window choice — shorter windows react faster to regime changes
• Monte Carlo: Fit stochastic model to risk factors, simulate N paths (10,000+ recommended), revalue portfolio on each path, extract percentile. Required for portfolios with options or path-dependent instruments. Specify random number generator and variance reduction technique (antithetic variates, importance sampling)

3. VaR Calculation

• Run selected method(s); compute VaR at each requested confidence level and holding period
• For 10-day VaR from 1-day: use square-root-of-time scaling only if returns are i.i.d. [VERIFY: whether autocorrelation or volatility clustering invalidates scaling rule]
• Calculate Component VaR and Marginal VaR to attribute risk to individual positions or asset classes
• If running multiple methods, present side-by-side comparison table

4. Backtesting

• Compare realized P&L breaches against VaR predictions over the test window
• Kupiec POF test: Binomial test on number of exceptions vs. expected count — report p-value
• Christoffersen test: Check for independence of exceptions (clustered breaches indicate model failure)
• Classify model in Basel traffic-light zones: Green (0-4 exceptions at 99%/250 days), Yellow (5-9), Red (10+) [VERIFY: current regulatory thresholds for applicable regime]
• Document exception dates and magnitudes; investigate any breach exceeding 2× VaR

5. Stress & Sensitivity Analysis

• Compute Conditional VaR (Expected Shortfall / CVaR) as the average loss beyond VaR threshold
• Run sensitivity on key parameters: vary lookback window (250, 500, 750 days), confidence level, and decay factor
• Report how VaR changes under stressed correlation assumptions (e.g., correlations → 1 in crisis)

Output

Produce a VaR Calculation Worksheet containing:

• Summary table: VaR figures by method, confidence level, and holding period
• Component VaR breakdown: Top 10 risk contributors by position or asset class
• Backtest results: Exception count, Kupiec p-value, Christoffersen p-value, traffic-light classification
• CVaR / Expected Shortfall: For each method and confidence level
• Sensitivity table: VaR under alternative parameter choices
• Assumptions log: Every distributional, data, and modeling assumption stated explicitly
• Methodology narrative: Plain-language explanation of approach suitable for risk committee review

Quality Checks

• VaR at 99% must exceed VaR at 95% for the same method and holding period — flag if violated
• Component VaR values should sum approximately to diversified portfolio VaR (exact for parametric; approximate for simulation methods)
• Monte Carlo VaR should converge as simulation count increases — run at 10K and 50K and confirm difference < 5%
• Backtest exception rate should fall within 2 standard deviations of expected rate — otherwise flag model inadequacy
• Confirm holding-period scaling is appropriate: if significant autocorrelation exists in returns, do not apply square-root-of-time rule without adjustment
• Cross-check: historical simulation VaR should be broadly consistent with parametric VaR for linear portfolios with near-normal returns; large divergence signals fat tails or non-linearity
• All market data sources and observation dates must be cited; any interpolated or proxy data marked [VERIFY]