Value-at-Risk

Value-at-risk (VaR) is a statistical measure estimating the maximum loss a portfolio could experience over a defined period (e.g., one day) at a specified confidence level (e.g., 95% or 99%). It answers the question: “What is the worst loss I can expect with X% confidence over Y days?” Despite its limitations, VaR is the dominant risk metric in finance.

Understanding VaR with an example

A portfolio manager calculates a 1-day VaR of $2 million at 99% confidence. This means:

• There is a 1% chance (roughly 2.5 trading days per year) that the portfolio loses more than $2 million in a single day.
• There is a 99% chance that the portfolio loses less than $2 million in a single day.

This is useful: the manager knows that on a typical bad day, the loss is at most $2M. On really bad days (the 1% tail), the loss could be much worse — $5M, $10M, or more. But most days are not really bad, and VaR captures the threshold for the typical bad day.

Three methods for calculating VaR

Parametric VaR (Variance-Covariance): Assume returns follow a normal distribution with a mean and standard deviation. Calculate the percentile of the distribution corresponding to the confidence level. A portfolio with daily returns averaging 0.05% and standard deviation of 1% has a 1-day 95% VaR of roughly 1.65 standard deviations below the mean = 1.65%. For a $100M portfolio, this is $1.65M.

Advantage: Fast, simple. Disadvantage: Assumes normal distribution; underestimates risk if returns have fat tails.

Historical VaR: Use historical returns directly. For 1-day 99% VaR, take the worst 1% of past daily returns and use that as the VaR. If the worst 1% of days over the past 5 years (1,000 trading days) had losses of $2.5M or worse, the 1-day 99% VaR is $2.5M.

Advantage: No distributional assumptions; reflects actual market behaviour. Disadvantage: Assumes the past is representative of the future; misses new types of shocks; sparse data for extreme tail (1% of 1,000 days is only 10 days).

Monte Carlo VaR: Simulate thousands or millions of possible future scenarios using models for price movements, correlations, and volatility. For each simulation, calculate portfolio loss. Use the worst 1% (for 99% confidence) as VaR.

Advantage: Flexible; can model complex instruments and non-linear relationships. Disadvantage: Computationally intensive; depends on model assumptions; garbage in, garbage out.

VaR in practice: where it succeeds and fails

VaR works well for:

• Typical market conditions. When markets are calm and correlations are stable, VaR estimates are reasonable.
• Comparison. VaR allows different portfolios to be compared on a common metric.
• Regulatory communication. Banks report VaR to regulators; it is a standard language.
• Risk limits. A trading desk might have a limit of “$10M daily VaR,” which operationalizes risk management.

VaR fails:

• Tail events. VaR does not measure the size of the 1% tail loss. A portfolio with 1-day 99% VaR of $2M could lose $2M or $20M in that 1% tail; VaR does not distinguish.
• Crisis periods. VaR models fitted to calm periods underestimate risk when volatility spikes or correlations jump.
• Stress scenarios. VaR assumes the future is like the past. A new type of shock (a new kind of black swan) is not captured.
• Fat tails. Normal distribution assumption means VaR underestimates the probability of extreme losses.

The limitations are well-known

The 2008 financial crisis showed that VaR was systematically too low. Banks held what they thought were safe portfolios with calculated VaRs of 2-3%, but actual losses in September-October 2008 exceeded VaR by 5-10x. This prompted regulators to supplement VaR with expected-shortfall and stress-testing.

Beyond VaR: Expected-Shortfall

Recognizing VaR’s shortcomings, regulators now require banks to also calculate expected-shortfall (ES), also called conditional VaR or CVaR. ES measures the average loss in the tail — the average loss on the worst 1% of days. This directly addresses VaR’s main weakness: not measuring the size of tail losses.

See also