Value at risk, or VaR, is a quantitative measure of downside risk.
Value at risk, or VaR, is a quantitative measure of downside risk. It states a loss threshold that a portfolio is not expected to exceed over a specified time horizon at a chosen confidence level. In practice, a 95% one day VaR of $1 million means that, under the model and assumptions used to estimate it, the portfolio is expected to lose no more than $1 million on 95% of trading days. The remaining 5% of days are the tail beyond that threshold.
The concept has three required parts. The first is the portfolio whose risk is being measured. The second is the time horizon, such as one day or ten days. The third is the confidence level, such as 95% or 99%. VaR is therefore not a single universal number. It changes when the portfolio changes, when the holding period changes, and when the confidence level changes.
Mathematically, VaR is tied to a percentile of the portfolio return or profit and loss distribution. Let portfolio profit and loss over horizon h be denoted by PnL_h. The VaR at confidence level c is the loss threshold tied to the lower tail probability 1 minus c. Equivalently, if returns are used instead of profit and loss, VaR is linked to the corresponding lower quantile of the return distribution and then translated into currency units through portfolio value. This percentile view is the core mechanism. VaR does not attempt to describe every possible outcome. It summarizes one point on the left tail of the distribution: the cutoff exceeded only with probability 1 minus the confidence level.
Two common estimation approaches are historical simulation and the parametric approach.
Historical simulation estimates VaR directly from realized past portfolio moves. The usual procedure is to assemble a sample of historical daily returns or daily profit and loss observations for the current portfolio or for the portfolio revalued under past market moves. These observations are sorted from worst to best. The chosen tail percentile is then read from the ordered sample. For a 95% one day VaR, the threshold corresponds to the 5th percentile of the one day profit and loss distribution. This approach is simple and model light because it uses empirical outcomes rather than imposing a specific distributional shape. Its output depends strongly on the historical sample window and on whether the recent past is representative of current conditions.
The parametric approach estimates VaR from a distributional assumption, commonly using the mean and volatility of portfolio returns over the selected horizon. Under a normal return assumption, one computes a lower tail quantile from the normal distribution and converts that quantile into a loss amount for the portfolio. In plain terms, the method asks what loss corresponds to the selected tail probability if returns follow the assumed distribution. This approach is compact and easy to update, but its result depends on the quality of the distributional assumption and on the stability of estimated volatility and correlations.
A simple numeric example illustrates the calculation. Suppose a portfolio has a market value of $10,000,000. Assume its one day return volatility is estimated at 2%, and assume the daily mean return is 0 for simplicity. Under a normal approximation, the lower 5th percentile is about 1.645 standard deviations below the mean. The 95% one day VaR in return terms is therefore about 1.645 times 2%, which is 3.29%. Converting that return threshold into dollars gives 3.29% of $10,000,000, or $329,000. Under this model, the portfolio is not expected to lose more than $329,000 over one day with 95% confidence.
The same idea can be shown with historical simulation. Suppose the portfolio has 100 past one day profit and loss observations. After sorting them from worst to best, the 5th worst observation is negative $300,000. The estimated 95% one day VaR is then $300,000. The interpretation is the same. The threshold marks the loss level exceeded by the worst 5% of the observed one day outcomes in that sample.
VaR is useful because it compresses portfolio downside risk into a single number that can be compared across books, horizons, and confidence levels. It is often used as part of a broader risk framework rather than as a complete description of risk by itself. The Sonar Sciences fundamentals research page presents foundational quantitative topics and supports this kind of concept-first framing for research and risk analysis.
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