Value at Risk (VaR)

Value at Risk (VaR) is a risk metric that summarizes the potential loss of a portfolio into a single number. Specifically, VaR answers the question: "What is the maximum loss I should expect over a given time period, with a given level of confidence?"

For example, a 1-day 95% VaR of $500,000 means: "On 95% of trading days, the portfolio will lose less than $500,000. On the remaining 5% of days (roughly 12-13 days per year), losses could exceed $500,000."

VaR was developed at JPMorgan in the early 1990s as part of their RiskMetrics system and quickly became the industry standard for risk measurement. It is now mandated by banking regulators under the Basel III framework for determining minimum capital requirements.

The appeal of VaR is its simplicity: it reduces the complex, multi-dimensional risk profile of a portfolio to a single dollar figure that executives, regulators, and risk committees can understand. However, this simplicity is also its weakness — VaR tells you how bad things could get on a "normal bad day" but says nothing about how bad things could get on a truly catastrophic day.

There are three primary approaches to computing VaR:

1. Parametric (Variance-Covariance) VaR:

Assumes returns are normally distributed. Under this assumption, VaR is simply:

VaR = μ - z_α × σ × √T

where z_α is the normal distribution quantile (2.33 for 99%, 1.65 for 95%). This method is fast and easy but fails when returns have fat tails (which they often do).

2. Historical Simulation VaR:

Uses actual historical returns to build the loss distribution. Sort the last N days of returns, and the VaR is simply the loss at the appropriate percentile. For 1-day 99% VaR with 1,000 days of data, VaR is the 10th-worst day's loss. This method makes no distributional assumptions but requires sufficient historical data.

3. Monte Carlo VaR:

Simulates thousands of random scenarios for how the portfolio might evolve, using a stochastic model. The VaR is the loss at the appropriate percentile of the simulated distribution. This is the most flexible method and can handle complex portfolios with options and other nonlinear instruments, but it is computationally expensive.

Let's compute the VaR for a simple equity portfolio using the parametric method:

Given:

• Portfolio value: $10,000,000
• Expected daily return (μ): 0.04% (approximately 10% annualized / 252 days)
• Daily standard deviation (σ): 1.2% (approximately 19% annualized)
• Confidence level: 99% (z = 2.33)

Calculation:

VaR = Portfolio × (z × σ - μ)

VaR = $10,000,000 × (2.33 × 0.012 - 0.0004)

VaR = $10,000,000 × 0.02756 = $275,600

Interpretation: On 99% of days, the portfolio will not lose more than $275,600. On the remaining 1% of days (~2.5 days per year), losses could exceed this amount — and VaR gives no indication of how much more.

Scaling: To convert a 1-day VaR to a 10-day VaR (commonly required by regulators), multiply by √10: $275,600 × √10 = $871,500. This assumes daily returns are independent — a simplification that breaks down during market crises when returns become serially correlated.

VaR has well-known limitations that every risk professional must understand:

• Tail risk blindness: VaR tells you the loss at a specific percentile but says nothing about what happens beyond that percentile. A 99% VaR of $1M means the worst 1% of days have losses exceeding $1M — but those losses could be $1.1M or $100M. VaR cannot distinguish between the two.
• Not subadditive: Parametric VaR is subadditive (the VaR of a combined portfolio is less than or equal to the sum of individual VaRs), but historical and Monte Carlo VaR are not necessarily so. This means diversification doesn't always reduce VaR, which is counterintuitive.
• Model risk: Parametric VaR assumes normality, but real returns have fat tails. Historical VaR assumes the future will resemble the past. Both assumptions can fail dramatically during crises.
• False precision: VaR gives a precise-looking number ($275,600) that can create false confidence. In reality, the estimate has significant uncertainty.

Expected Shortfall (ES), also called Conditional VaR (CVaR), addresses the tail risk limitation by computing the average loss conditional on exceeding VaR. If the 99% VaR is $1M and the 99% ES is $2.5M, it means the average loss on those worst 1% of days is $2.5M. Basel III has begun requiring Expected Shortfall alongside VaR for regulatory capital.