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Value at Risk (VaR)

Risk Metric Statistical Method Regulatory

Value at Risk (VaR) is a statistical measure that estimates the maximum expected loss of a portfolio over a specific time period at a given confidence level. It is one of the most widely used risk metrics in finance, adopted by banks, asset managers, and regulators worldwide.

VaR answers a straightforward question: "What is the most I could lose, under normal market conditions, over a given period?" For example, a one-day 95% VaR of $1 million means there is a 95% chance the portfolio will not lose more than $1 million in a single day. It compresses a complex distribution of possible outcomes into a single dollar figure, making it useful for quick communication of risk exposure.

Definition

VaR has three components: a time horizon (how far ahead you are looking), a confidence level (how certain you want to be), and a loss amount (the dollar or percentage figure). All three must be stated together for the number to be meaningful.

The Three Components of VaR

Time horizon: The period over which losses are measured. Common choices are one day (for trading desks) or ten days (for regulatory capital calculations).
Confidence level: The probability that actual losses will not exceed the VaR estimate. Typical values are 95% or 99%.
Loss amount: The resulting dollar or percentage figure. A 95% one-day VaR of $500,000 means there is a 5% chance of losing more than $500,000 in one day.

Changing any one of these components changes the VaR number. A 99% VaR will always be larger than a 95% VaR for the same portfolio and time horizon, because a higher confidence level demands a more conservative estimate. Similarly, a ten-day VaR will be larger than a one-day VaR because losses can accumulate over longer periods.

Calculation Methods

There are three main approaches to calculating VaR. Each makes different assumptions about how portfolio returns behave, and each has distinct strengths.

Historical Simulation

Historical simulation uses actual past returns to provide an estimate of potential risk based on historical parameters. The method collects a window of historical returns (often 250 to 500 trading days), sorts them from worst to best, and picks the loss at the desired percentile. For a 95% VaR using 500 days of data, the VaR is the 25th-worst daily return (the 5th percentile).

The advantage is simplicity: no assumptions about the shape of the return distribution are required. The disadvantage is that results depend entirely on what happened during the historical window. If the window does not include a crisis period, the model will understate risk.

Variance-Covariance (Parametric) Method

The variance-covariance method assumes that portfolio returns follow a normal distribution (a bell-shaped curve where most outcomes cluster near the average). Under this assumption, VaR can be calculated using just the portfolio's mean return and standard deviation (a measure of how spread out returns are).

Parametric VaR Formula

VaR = Portfolio Value × (Mean Return − Z-score × Standard Deviation)

The Z-score converts the confidence level into standard deviations. For 95% confidence, the Z-score is 1.65. For 99% confidence, it is 2.33. A portfolio worth $10 million with a daily standard deviation of 1% has a 95% one-day VaR of approximately $165,000.

This method is fast and easy to compute, but it relies on the assumption that returns are normally distributed. In practice, financial returns have "fat tails," meaning extreme losses occur more often than a normal distribution predicts.

Monte Carlo Simulation

Monte Carlo simulation generates thousands of hypothetical future return scenarios using random sampling. The method defines a statistical model for how each asset in the portfolio behaves, then simulates many possible paths. VaR is read from the resulting distribution of simulated portfolio outcomes.

Monte Carlo is the most flexible method because it can accommodate non-normal distributions, changing correlations, and complex instruments like options. The tradeoff is computational cost and the need to specify a model for how returns are generated, which introduces its own assumptions.

Interpreting VaR: The 95% Confidence Example

Consider a portfolio worth $1 million with a one-day 95% VaR of $15,000. This means that on 95 out of 100 trading days, the portfolio is expected to lose less than $15,000. On roughly 5 out of 100 trading days (about 12 to 13 days per year), losses could exceed $15,000.

Confidence Level	Meaning	Expected Exceedances per Year (~250 trading days)
90%	Losses exceed VaR about 10% of the time	~25 days
95%	Losses exceed VaR about 5% of the time	~12–13 days
99%	Losses exceed VaR about 1% of the time	~2–3 days

A critical point: VaR says nothing about how bad the losses will be on those exceedance days. The portfolio could lose $16,000 or $100,000, and both would count as VaR breaches. This is the single most important limitation of VaR as a risk measure.

Practical Example

A portfolio manager oversees a $50 million equity portfolio. Using 500 days of historical returns, the risk team calculates the following VaR estimates.

Metric	95% Confidence	99% Confidence
One-day VaR	$750,000 (1.5%)	$1,150,000 (2.3%)
Ten-day VaR	$2,370,000 (4.7%)	$3,640,000 (7.3%)

The ten-day VaR is roughly √10 (about 3.16) times the one-day VaR, a scaling rule often used under the assumption that daily returns are independent. This "square root of time" rule is a convenient approximation but can understate risk if losses tend to cluster together during stress periods.

Regulatory Context

VaR became a cornerstone of banking regulation through the Basel Accords, a set of international standards for bank capital requirements. The Basel Committee on Banking Supervision first endorsed VaR in its 1996 Market Risk Amendment (often called Basel I.5), allowing banks to use internal VaR models to determine how much capital they needed to hold against trading losses.

Under Basel II and Basel III, banks using the Internal Models Approach typically calculate a 99% ten-day VaR. The resulting capital charge is the higher of either the previous day's VaR or the average VaR over the past 60 trading days, multiplied by a scaling factor (usually 3, but it can increase if the model has too many exceedances during backtesting).

More recently, Basel III's Fundamental Review of the Trading Book (FRTB) introduced Expected Shortfall (ES) as a replacement for VaR in capital calculations. Expected Shortfall measures the average loss in the worst cases beyond the VaR threshold, addressing VaR's inability to capture tail risk. However, VaR remains widely used in day-to-day risk management and reporting outside of regulatory capital.

Known Limitations

Limitations to Keep in Mind

Does not measure tail risk. VaR tells you the threshold but nothing about what happens beyond it. A portfolio that occasionally loses 5% beyond VaR is very different from one that occasionally loses 50%, even if both have the same VaR number. Expected Shortfall (also called Conditional VaR) addresses this gap.
Assumes historical patterns continue. All three calculation methods rely on historical data or models calibrated to historical data. If the future departs significantly from the past (a regime change), VaR will understate risk. The 2008 financial crisis exposed this limitation when many banks' VaR models failed to predict the magnitude of losses.
Not subadditive. In some cases, the VaR of a combined portfolio can be larger than the sum of the individual VaRs, meaning diversification appears to increase risk. This mathematical property makes VaR inconsistent as a measure of portfolio risk in certain edge cases. Expected Shortfall does not have this problem.
Confidence level creates a false sense of precision. A 99% VaR sounds reassuring, but it still implies roughly 2 to 3 exceedances per year. Over a decade, that is 20 to 30 days when losses exceed the estimate, and the model says nothing about the severity of those losses.
Sensitive to parameter choices. The historical window length, confidence level, time horizon, and calculation method all affect the result. Two risk teams using different parameters on the same portfolio can produce substantially different VaR numbers.