Value at Risk (VaR)

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What is Value at Risk (VaR)?

Value at Risk (VaR) is a statistical measure used to estimate the potential loss in value of a portfolio, position, or firm over a specified time horizon and at a given confidence level. It answers questions like: “How much could I lose over one day or one month with 95% confidence?” VaR is expressed either as a currency amount or as a percentage of portfolio value.

Key points
* VaR combines a time horizon (e.g., one day, one month) with a confidence level (e.g., 95%, 99%).
* A 95% one-month VaR of 2% means there is a 5% chance the portfolio will lose at least 2% in one month.
* VaR reports a threshold loss (the minimum loss at that confidence level), not the worst possible loss.

Why use VaR?

Benefits
* Simple single-number summary that is easy to communicate and compare across portfolios and asset classes.
* Useful for setting risk limits, capital allocation, and regulatory reporting.
* Widely implemented in risk-management systems and financial software.

Common VaR methodologies

There are three primary approaches, each with different assumptions and data requirements:

  1. Historical method
  2. Uses actual past returns to create a distribution of hypothetical future outcomes.
  3. Steps: collect historical returns (often daily), sort them from worst to best, and take the return at the chosen percentile (for 95% VaR take the 5th percentile). Multiply that return by current portfolio value to get the VaR.

  4. Strengths: nonparametric (no assumed return distribution). Weaknesses: assumes history is representative of the future and can underweight rare events not present in the sample.

  5. Variance–covariance (parametric) method

  6. Assumes portfolio returns follow a known distribution (commonly normal) and uses estimated mean and covariance (or standard deviation) to compute VaR.
  7. In practice the mean is often small relative to volatility, so VaR is approximated from volatility and a z-score corresponding to the confidence level.

  8. Strengths: fast and scalable for large portfolios. Weaknesses: sensitive to the normality assumption and to estimation error in volatilities and correlations; may understate tail risk.

  9. Monte Carlo simulation

  10. Generates many simulated return paths using a model of risk-factor behavior, then measures the loss percentile across simulated outcomes.
  11. Strengths: flexible and capable of modelling complex payoffs and non-normal behavior. Weaknesses: computationally intensive and model-dependent.

How to calculate VaR (historical method — practical steps)

  1. Choose a time horizon (e.g., daily, monthly) and confidence level (e.g., 95%, 99%).
  2. Gather a sample of historical returns for the portfolio or its risk factors (common choice: 252 trading days for one year).
  3. Compute period returns (percent changes) and apply those returns to current portfolio value to generate a distribution of hypothetical future values.
  4. Sort the results from worst to best and pick the loss at the chosen percentile (e.g., the 5th percentile for 95% VaR).
  5. Report VaR as the dollar (or percentage) loss corresponding to that percentile.

Limitations and criticisms

  • VaR gives the minimum loss at a confidence level but says nothing about the tail beyond that threshold — extreme losses can be much larger.
  • Results depend on methodology, sampling period, and model assumptions (e.g., normality). Using low-volatility historical periods or inappropriate models can understate risk.
  • VaR can produce a false sense of security if treated as a complete measure of risk, especially for portfolios with nonlinear exposures (derivatives) or fat-tailed returns.
  • Historical episodes (e.g., the 2008 crisis) exposed how VaR-based approaches can underestimate both the probability and magnitude of losses in stressed conditions.

Related measures

  • Standard deviation measures volatility (dispersion of returns) but does not provide loss thresholds tied to confidence levels. VaR focuses on loss quantiles rather than overall dispersion.
  • Marginal VaR estimates how much total portfolio VaR would change if a small additional position is added — a measure of incremental contribution to risk.
  • Incremental VaR measures the change in portfolio VaR from adding or removing a specific position (often computed for finite position changes rather than an infinitesimal one).

Practical considerations

  • Choose the time horizon and confidence level consistent with the decision or regulatory use (e.g., daily VaR for trading limits, monthly VaR for strategic capital planning).
  • Use complementary risk measures — expected shortfall (CVaR), stress tests, scenario analysis, and sensitivity metrics — to capture tail risk and model uncertainty.
  • Regularly recalibrate models and examine sensitivity to sample period, volatility regimes, and correlation shifts.

Bottom line

VaR is a useful, widely adopted metric for summarizing potential losses under normal-to-moderate conditions, enabling comparisons and risk limits. However, it should not be the sole measure of risk: its assumptions and focus on a single percentile make it blind to tail outcomes and model error. Combine VaR with stress testing, expected shortfall, and robust modelling practices to get a fuller picture of potential losses.