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Value at Risk (VaR) and Conditional VaR

Difficulty advanced

Overview

VaR measures the maximum expected loss over a given time horizon at a specified confidence level. It's the standard risk metric used by financial institutions.

Definition

VaR CVaR (avg of tail) mean returns → freq 5% tail "95% of outcomes are better than VaR"
VaR cuts off the bad tail at the α percentile · CVaR is the average loss conditional on being in that tail
VaR(α, t) = Maximum loss not exceeded with probability α over time t

Example: VaR(95%, 1-day) = $10,000
Means: 95% confidence that loss won't exceed $10,000 in one day
Or: 5% chance of losing more than $10,000

where: α confidence level (e.g. 0.95 or 0.99) · t time horizon (typically 1 day, 10 days, or 1 year). does: the threshold loss that won't be exceeded with probability α. Does not describe what happens when the threshold is breached — that's CVaR's job.

Methods

1. Parametric (Variance-Covariance)

VaR = V_p × (μ - z_α × σ) × √t

where:
V_p = Portfolio value
z_α = Z-score for confidence level (1.645 for 95%, 2.326 for 99%)
σ = Portfolio volatility
t = Time horizon

does: assumes returns are normally distributed — VaR is then just a quantile of that normal. Fast to compute, but underestimates tail risk because actual returns have fat tails (use Cornish-Fisher or historical simulation if accuracy in the tail matters).

2. Historical Simulation

VaR = α-percentile of historical portfolio returns × Portfolio Value

does: non-parametric — just reads VaR off the empirical return distribution. Captures observed tail shape (skew, kurtosis) but is limited to events that actually happened in your sample. Add a stress overlay for events outside the sample.

Portfolio VaR

VaR_p = √(w' × Σ × w) × z_α × V_p × √t

where:
w = Portfolio weights
Σ = Covariance matrix

does: the parametric portfolio VaR — used for regulatory reporting, internal risk limits, and capital allocation across desks; the √(w'Σw) factor is the portfolio volatility, so correlations enter directly and diversification benefits show up automatically as lower VaR.

Conditional VaR (Expected Shortfall)

CVaR(α) = E[Loss | Loss > VaR(α)]

Average loss beyond VaR — captures tail risk

where: α confidence level matching the VaR threshold · E[· | ·] conditional expectation · Loss portfolio loss random variable does: the expected loss conditional on a VaR breach — used in place of VaR for tail-sensitive limits, regulatory FRTB capital under the new market-risk framework, and any optimization where you want a sub-additive (coherent) risk measure.

Limitations of VaR

Limitation Issue Solution
Not sub-additive Portfolio VaR can exceed sum of parts Use CVaR instead
Ignores tail shape Doesn't capture tail severity CVaR / Expected Shortfall
Normality assumption Returns have fat tails Historical or Monte Carlo
Static view Assumes fixed portfolio Dynamic/stress testing
Confidence choice 95% vs 99% changes everything Use multiple levels

Practical Guidelines

  1. Use Multiple Methods — Compare parametric, historical, and MC
  2. Report CVaR Too — VaR alone understates tail risk
  3. Backtest Regularly — Validate VaR predictions
  4. Stress Test — VaR doesn't capture extreme events
  5. Short Horizon — 1-day and 10-day are standard
  6. Component VaR — Understand which positions contribute most
  7. Marginal VaR — Impact of adding a new position

Key Formulas Reference

Parametric VaR: V × z × σ × √t
Historical VaR: Percentile of historical returns
CVaR: E[Loss | Loss > VaR]
Portfolio VaR: √(w'Σw) × z × V × √t
Marginal VaR: ∂VaR/∂w_i = z × V × (Σw)_i / √(w'Σw)

where: V portfolio value · z confidence quantile · σ portfolio volatility · t horizon · w weights vector · Σ covariance matrix · (Σw)_i ith element of Σw does: compact reference for the canonical VaR variants and the marginal sensitivity to each weight — used in code, in risk reports, and in allocation decisions where Marginal VaR identifies which positions to trim first to reduce total VaR per unit notional.

Next Steps