How to Calculate Value at Risk (VaR): Methods and Formula

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Value at risk (VaR) is an estimate of the loss a portfolio should not exceed over a set period at a chosen confidence level. A one-day 95% VaR of $10,000 means that, under the model’s assumptions, losses on 95% of trading days should be $10,000 or less, and on about 5% of days they should be larger. You can calculate VaR in three standard ways: the historical method, the variance-covariance (parametric) method and Monte Carlo simulation. The parametric version has the simplest formula:

VaR = z × σ × V

Here z is the number of standard deviations for the confidence level, σ is the standard deviation of returns over the horizon, and V is the position or portfolio value. That version assumes an expected return of zero.

What Value at Risk Measures

VaR answers one kind of question: with 95% (or 99%) confidence, how much could I lose over the next day, week or month? “Our VaR is $1 million” is incomplete. “Our one-day 95% VaR is $1 million” is a usable risk figure.

The same figure can be written two ways. A one-day 5% VaR of $1 million is identical to a one-day 95% VaR of $1 million. Both mean an estimated 5% probability of losing more than $1 million in a single day, and a 95% probability of losing $1 million or less.

Higher confidence levels and longer horizons produce larger VaR figures, because they reach further into the loss tail and allow more time for prices to move. A high VaR can reflect a larger or more volatile portfolio or simply a stricter setting, so two VaR numbers can be compared only if they share the same horizon, confidence level and method.

The VaR Formula: Variance-Covariance Method

The variance-covariance method treats returns as normally distributed. Under that assumption, the worst 5% or 1% of outcomes sit at a fixed number of standard deviations below the mean. VaR uses one tail of the distribution, so it uses one-tailed z-values:

The full formula includes the expected return (μ) over the horizon:

VaR = (z × σ − μ) × V

For daily horizons, μ is usually small relative to σ and is often set to zero. That gives the simpler formula in the opening.

Example 1: Single position

Take a hypothetical $100,000 portfolio with a daily return standard deviation of 1.5%, and assume an expected daily return of zero.

• 95% one-day VaR = 1.645 × 1.5% × $100,000 = $2,467.50
• 99% one-day VaR = 2.326 × 1.5% × $100,000 = $3,489

On roughly one trading day in 20, the model expects a loss larger than $2,467.50. On roughly one day in 100, it expects a loss larger than $3,489.

Scaling VaR to longer horizons

A one-day VaR is often converted to a longer horizon with the square-root-of-time rule:

VaR(T days) = VaR(1 day) × √T

Using the 95% figure above, a 10-day VaR is $2,467.50 × √10 ≈ $7,803. The 99% 10-day VaR is about $11,033. The rule holds only if daily returns are independent and identically distributed and the positions stay unchanged over the period. Returns that trend or cluster in volatile stretches break those assumptions, and the scaled figure can then understate risk.

Example 2: Portfolio VaR with correlation

For a portfolio, you can’t add up the VaR of each holding unless the holdings are perfectly correlated. Portfolio volatility depends on how the assets move together. For two assets with weights w₁ and w₂, volatilities σ₁ and σ₂, and correlation ρ:

σp = √(w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρσ₁σ₂)

Portfolio VaR = z × σp × V

Consider a hypothetical $100,000 portfolio split as follows, with a correlation of 0.2 between the two assets:

• 60% stocks with a daily σ of 1.5%
• 40% bonds with a daily σ of 0.5%

The calculation runs in three steps:

1. Weighted volatilities: 0.6 × 1.5% = 0.9% and 0.4 × 0.5% = 0.2%.
2. Portfolio variance: 0.9² + 0.2² + 2 × 0.2 × 0.9 × 0.2 = 0.81 + 0.04 + 0.072 = 0.922 (in percent squared), so σp ≈ 0.960%.
3. 95% one-day VaR: 1.645 × 0.960% × $100,000 ≈ $1,580.

If you simply added the two stand-alone VaRs, you would get 1.645 × 1.1% × $100,000 ≈ $1,810. The $230 gap is the diversification effect of a correlation below 1. With many assets, the same idea is written in matrix form: σp = √(wᵀΣw), where w is the vector of weights and Σ is the covariance matrix.

Historical Simulation Method

The historical method makes no assumption about the shape of the distribution. It assumes the recent past is a fair sample of what could happen next. The steps are:

1. Choose a lookback window, such as the last 500 trading days.
2. Calculate daily returns for every position over that window.
3. Apply each historical day’s returns to today’s positions to produce a hypothetical profit or loss for each day. This step means a portfolio that changed composition is measured on what it holds now.
4. Sort the hypothetical results from worst to best.
5. Read the loss at the chosen percentile. For 95% VaR on 500 days, 5% of 500 is 25, so VaR sits at about the 25th to 26th worst outcome. Conventions differ on which observation to use or whether to interpolate.

In formula terms, historical VaR is the negative of the (1 − confidence) quantile of the simulated P&L distribution:

VaR = −Quantile(1 − c) of P&L

Example 3: Reading the percentile

Suppose the 26th worst simulated day for a hypothetical $100,000 portfolio is a loss of 2.1%. The 95% one-day historical VaR is then $2,100.

Because the method uses actual returns, it captures fat tails and skew that occurred in the window. The trade-off is dependence on that window: a calm two-year sample produces a low VaR even if a turbulent period lies just outside it, and a single extreme day can shift the result as it enters or leaves the window.

Valuation approaches: sensitivity, full revaluation and grid-based

Each scenario in historical or Monte Carlo VaR requires revaluing the portfolio. Sensitivity-based (delta or delta-gamma) approximation is fast but weakens for large moves and for options with strongly nonlinear payoffs. Full revaluation reprices every instrument under every scenario and is the most accurate and most computationally expensive approach. Grid-based valuation interpolates scenario values from prices computed in advance at a grid of risk-factor levels, sitting between the other two in speed and accuracy.

Monte Carlo Simulation Method

Monte Carlo VaR replaces the historical record with randomly generated scenarios drawn from a model you specify. The steps are:

1. Specify a model for each risk factor, for example a distribution with chosen volatilities and correlations. Fat-tailed or volatility-clustering models can be used instead of the normal distribution.
2. Generate many random scenarios, often thousands or tens of thousands of trials over the chosen horizon.
3. Revalue the portfolio in each scenario, using one of the valuation approaches above.
4. Sort the simulated outcomes and read the loss at the target percentile.

For instance, with 10,000 hypothetical one-month trials, 95% VaR is the loss at roughly the 500th worst trial. If that trial shows a 12% loss, the conclusion is: with 95% confidence, the model does not expect a monthly loss greater than 12%.

Monte Carlo handles options, nonlinear payoffs and complex portfolios well, but its output is only as reliable as the model and parameters behind it.

Comparing the Three Methods

Because the assumptions differ, the three methods can give noticeably different numbers for the same portfolio. That gap is useful information in its own right. When historical VaR sits well above parametric VaR, the actual return history has had fatter tails than a normal curve allows.

Checking a VaR Model: Backtesting

A VaR model can be tested by counting exceptions, the days when the actual loss exceeded the VaR estimate. A 99% one-day VaR should be exceeded about 1% of the time, or roughly 2 to 3 times in 250 trading days. Far more exceptions suggest the model understates risk. Far fewer suggest it is too conservative.

Bank regulators formalized this idea. Under the Basel Committee’s backtesting framework for 99% VaR over 250 days, banks fall into three zones:

• Green: 0 to 4 exceptions.
• Yellow: 5 to 9 exceptions.
• Red: 10 or more exceptions, which signals a model problem.

Banks in the yellow and red zones faced higher capital multipliers.

Limitations of Value at Risk

• It ignores the size of tail losses. Two portfolios can share the same 99% VaR while one loses slightly more than that on bad days and the other loses many times more.
• It can penalize diversification. VaR is not always subadditive, so the VaR of a combined portfolio can exceed the sum of its parts in some cases. This matters most for positions with skewed or option-like payoffs.
• Correlations change under stress. Correlations estimated in normal markets often rise sharply in a crisis, which weakens the diversification benefit that portfolio VaR relies on.

These weaknesses led to wider use of expected shortfall (also called conditional VaR). Expected shortfall measures the average loss in the cases beyond the VaR threshold. The Basel Committee’s revised market risk framework replaced 99% VaR with 97.5% expected shortfall for regulatory capital.

For individual investors, VaR is most useful as a consistent yardstick for comparing positions or tracking how a portfolio’s risk changes over time.

FAQs

The Bottom Line

VaR gives traders, risk managers and individual investors a single loss threshold, but only when the horizon, confidence level and method are stated together. Because it ignores how large tail losses get and relies on normal-market correlations, it works best alongside backtesting and expected shortfall. For individual investors, its main value is as a consistent yardstick for comparing positions and tracking how portfolio risk changes over time.