Value at Risk (VaR)
EN: Value at Risk / VaR PT: Valor em Risco
La métrica estándar de Wall Street para quantify potencial de pérdida — "95% VaR de $1M" significa que hay 95% probability de no perder más de $1M en el período. Widely used pero critizada post-2008 por underestimar tail risks. Complement con stress testing y expected shortfall.
Qué es Value at Risk
El Value at Risk (VaR, en portugués Valor em Risco) es la métrica más estándar de Wall Street para quantify potential loss de un portfolio durante un specific time horizon con given probability. Definición formal: "VaR de X% significa que hay X% probability de que la pérdida no exceda el VaR amount". Ejemplo: "95% 1-day VaR de $1M" means que hay 95% probability de que portfolio no perderá más de $1M en 1 día. Equivalentemente: 5% probability that loss exceeds $1M. Standard convention: VaR reported como positive number representing potential loss. Tres parámetros críticos: (1) Confidence level: típicamente 95% o 99% (sometimes 99.9% for regulatory). Higher confidence = higher VaR estimate. (2) Time horizon: 1 day (daily VaR), 10 days (regulatory standard), 1 month, 1 year. Longer horizon = higher VaR. (3) Scenario: historical simulation, Monte Carlo, o parametric method. Different methods can produce different VaRs. VaR emerged en los 1990s como standard entre banks following J.P. Morgan's RiskMetrics (1994). Became foundation de Basel II (2004) banking regulations. Basel III (2013) expanded beyond VaR due to criticism. VaR está en virtually every professional risk management system globally — banks, hedge funds, asset managers, corporate treasuries all compute y report VaR daily. Simple example: $10M equity portfolio, daily volatility 1.5%, normal distribution assumed. 95% VaR = $10M × 1.645 × 1.5% = $246,750. Meaning: 95% probability daily loss ≤ $246,750. 5% probability loss exceeds this. Calculation simple, intuitive, communicable to non-quants.
Métodos de Cálculo
Existen 3 métodos principales para calcular VaR. (1) Parametric (Variance-Covariance): asumes returns normally distributed. Formula: VaR = Portfolio Value × Z-score × Volatility × sqrt(Time Horizon). Z-score para 95% = 1.645; 99% = 2.326. Ejemplo: $1M portfolio, 2% daily volatility, 95% confidence, 1 day → VaR = $1M × 1.645 × 0.02 × 1 = $32,900. Pros: simple, fast, intuitive. Cons: assumes normal distribution (false — markets have fat tails), doesn't capture non-linearity (options), underestimates tail events. (2) Historical Simulation: usar historical returns directly. Ranking past daily returns, 95% VaR = 5th percentile worst return × portfolio value. Ejemplo: 1,000 historical daily returns, 50 worst-loss days, 50th worst = -2.5% → 95% VaR = 2.5% × portfolio. Pros: no distribution assumptions, captures actual historical behavior. Cons: limited to observed history (future may be different), requires long data history, equally weighs all historical periods. (3) Monte Carlo Simulation: generate thousands de potential future scenarios based on statistical model. Rank results, extract percentile. Pros: flexible, handles complex portfolios (options, derivatives), captures non-linearity. Cons: computationally expensive, depends on model accuracy, requires correlation estimates between assets. Comparison example: $10M equity portfolio. Parametric VaR: $246K. Historical VaR: $310K (captures 2008-2020 tail events). Monte Carlo VaR: $295K (similar to historical if model well-calibrated). Banks frequently compute all three y report max o weighted average. Discrepancy between methods reveals model risk. Regulatory VaR: Basel requires 10-day 99% VaR for banks (= approximately 1-day VaR × 3.16). Capital requirement = 3 × VaR × multiplier based on model accuracy (backtesting). Poor backtesting = higher multipliers y more capital required.
Limitaciones del VaR
El VaR tiene limitaciones serias que llevaron a critiques post-2008. (1) Tail risk underestimation: VaR communicates nothing about magnitude of losses BEYOND the threshold. 95% VaR $1M tells you about 95% de días. Nothing about worst 5% — losses could be $1.01M or $50M. 2008 crisis revealed extreme tails frequently occur. Markets cannot distinguish between "normal" y "catastrophic" loss days using VaR alone. (2) Non-normal distributions: parametric VaR assumes normal returns (thin tails). Real returns have fat tails (more frequent extreme events than normal predicts). Black Swan events (9/11, 2008, COVID 2020) invalidate parametric assumptions. (3) Non-stationary: volatility changes over time (volatility clustering). Historical VaR during low-vol periods underestimates future high-vol periods. GARCH y other volatility models attempt to address but imperfectly. (4) Correlation assumptions: portfolio VaR depends on asset correlations. Historical correlations can shift dramatically during stress (correlations tend to 1 during crashes — "diversification fails when you need it"). (5) Option non-linearity: parametric VaR poor for options portfolios with convexity y gamma. Can underestimate tail risk badly. Monte Carlo better but complex. (6) Model risk: different models produce different VaRs. Choice of method creates incentives — banks may choose methods giving lowest VaR (lower capital). Basel regulations attempt to address via backtesting. (7) Psychological false security: "95% VaR $1M" feels safe but 5% × 252 days = ~13 loss days per year. During those days, losses can be catastrophic. Nassim Taleb critique: extensive criticism of VaR in "The Black Swan" (2007). VaR underestimates tail risk by design. "Turkey problem" — 1000 days of Thanksgiving turkey's VaR suggests minimal risk until slaughter day, when VaR completely fails to predict. Replacement/complement metrics: Expected Shortfall (ES) / Conditional VaR (CVaR) — average loss during VaR breach. Captures tail severity. Basel 2019 transitioned from VaR to ES as standard. Stress testing: simulate specific scenarios (2008 repeat, 9/11 repeat, GFC 2.0, stagflation). Scenario analysis: reverse engineer — what market moves would cause $X loss? Are those moves plausible? Complementary to VaR.
Expected Shortfall y Modern Risk Metrics
El Expected Shortfall (ES), también llamado Conditional VaR (CVaR), es la successor metric post-2008 que addresses VaR limitations. Definición: ES α = E[Loss | Loss > VaR α]. Es el promedio de pérdidas que exceden el VaR threshold. Ejemplo: 95% VaR = $1M. 5% worst days have losses: $1.1M, $1.5M, $2M, $3M, $10M (extreme). Average of these = ES. Captures tail SEVERITY, not just probability. Basel III transition: desde 2019, banks required to use 97.5% ES instead of 99% VaR for regulatory capital. ES captures tail risk better mientras being similarly computable. Stress testing: complement to VaR focused on specific historical o hypothetical scenarios. Historical stress tests: replay 2008 crisis, COVID crash, 9/11, various severe periods against current portfolio. Hypothetical stress tests: construct scenarios — "Fed tightens 400bps rapidly," "major geopolitical disruption," "black swan type event." Portfolio loss under each scenario. Central banks (Fed CCAR) require banks to stress test quarterly. Reverse stress testing: start with unacceptable outcome (e.g., "portfolio loses 30% capital"). Work backward to identify what market moves would cause this. Helps identify blind spots. Crisis VaR: VaR calculated using only crisis-period historical data. Generally much higher than normal-period VaR. Prudent capital allocation uses crisis VaR for stress scenarios. Maximum Drawdown vs VaR: MDD is retrospective (historical worst). VaR is forward-looking (probabilistic estimate of worst). Both provide complementary information. Professional risk management uses both. Liquidity VaR: adjusts for liquidation cost in stress. Positions in illiquid assets may take days to exit, during which losses continue. Liquidity VaR captures this. Critical for hedge funds with illiquid alternative investments. Options VaR: sophisticated methods account for gamma (non-linear risk), vega (volatility sensitivity), theta (time decay). Greeks-based approach better than parametric for options portfolios.
Operativa Práctica para Traders
La aplicación práctica de VaR para traders retail/intermediate. Portfolio-level VaR: calculate daily VaR of your portfolio. Rule of thumb: VaR should be < 1-2% of portfolio value for conservative traders, 2-5% for aggressive. If VaR exceeds comfort level, reduce exposure. Trading software con VaR: Interactive Brokers, TradeStation, NinjaTrader, ThinkorSwim provide built-in VaR calculations. Educational brokerages increasingly include basic VaR reports. Simplified VaR for personal use: Estimate portfolio volatility (use 20-day standard deviation of daily returns). Daily 95% VaR ≈ Portfolio × 1.645 × Daily Vol. Example: $100K portfolio, 1.5% daily vol → 95% VaR = $100K × 1.645 × 0.015 = $2,468. Probability daily loss exceeds $2,468 = 5%. Probability loss exceeds $2,468 × 2 = $4,936 (approximate 99% VaR) = 1%. VaR for options positions: complexity higher. Simple approach: estimate likely worst-case (delta-adjusted), compare with premium paid. For spreads, max loss is defined — this is absolute VaR ceiling. Portfolio risk budget: allocate VaR across positions. Example: $100K portfolio, 2% VaR budget = $2K total. Allocate: Position A $1K VaR, Position B $500 VaR, Position C $500 VaR. Total VaR ≤ $2K. Correlation in VaR: portfolio VaR ≠ sum of individual VaRs unless correlated perfectly. Uncorrelated positions provide diversification benefit — portfolio VaR < sum. During crises, correlations rise, benefit shrinks. Stress scenario: assume correlation = 1 during crash. Compare with normal-correlation VaR to understand worst-case. Cuando usar VaR: VaR is useful for: (a) comparing portfolio risk across strategies, (b) setting position limits, (c) communicating risk to stakeholders, (d) complying with regulatory requirements. VaR is insufficient for: (a) tail risk assessment (use ES), (b) liquidity risk (use liquidity-adjusted VaR), (c) black swan scenarios (use stress testing), (d) options with complex payoffs (use scenario analysis). Combined approach provides robust risk framework.
VaR Methods y Modern Alternatives
Each method has tradeoffs; professional risk management uses multiple.
| Métrica | Strength | Weakness | Best Use |
|---|---|---|---|
| Parametric VaR | Fast, intuitive | Assumes normal distribution | Simple stock/bond portfolios |
| Historical VaR | No distribution assumption | Limited to history | Mean-reverting strategies |
| Monte Carlo VaR | Flexible, handles complex | Computationally expensive | Options/derivatives |
| Expected Shortfall | Captures tail severity | More complex calculation | Post-2008 standard |
| Stress Testing | Scenario-based, intuitive | Limited scenarios tested | Black swan preparation |