What Value at Risk Measures
Value at Risk (VaR) is a statistical measure of the potential loss on a portfolio or position over a specified time horizon at a given confidence level. A formal statement of a VaR estimate might be: "The one-day 99% VaR of this portfolio is $1.2 million."
That statement means: with 99% probability, the portfolio will not lose more than $1.2 million over the next trading day. Equivalently, there is a 1% probability that the loss will exceed $1.2 million over the next trading day.
Three numbers define every VaR estimate: the portfolio or position, the time horizon, and the confidence level. Changing any of them changes the VaR figure.
Time horizon is typically one day (for trading book positions, where Basel requires daily VaR) or ten days (for regulatory capital calculations, since Basel translates one-day VaR to ten-day VaR by multiplying by the square root of 10, which assumes returns are independent and identically distributed). Longer time horizons produce larger VaR estimates.
Confidence level determines how far into the tail you are measuring. A 99% confidence level includes losses that occur more than 1% of the time. A 95% confidence level is less conservative, since it includes the loss level exceeded 5% of the time. Moving from 95% to 99% confidence increases the VaR figure. The relationship between confidence levels depends on the shape of the loss distribution.
The Three Main VaR Estimation Methods
The FRM Part I curriculum covers three approaches to estimating VaR, each with distinct assumptions, strengths, and limitations.
Historical Simulation
Historical simulation constructs the VaR estimate from actual observed returns over a historical period. The method ranks historical returns from worst to best and reads off the loss at the relevant percentile.
For a one-day 99% VaR using 500 days of historical returns: rank the 500 daily returns from worst to best. The 1% tail corresponds to the five worst observations (500 × 0.01 = 5). The VaR estimate is approximately the loss at the fifth-worst observation.
The advantages of historical simulation are that it does not require a distributional assumption and it automatically captures the fat tails, skewness, and correlations that were present in the historical period.
The limitations are equally important for the exam. Historical simulation is backward-looking. If the historical window used for calibration does not include a stress event similar to the one you are trying to measure, the VaR estimate will understate tail risk. The method also gives equal weight to all observations in the window. A loss that occurred 499 days ago and a loss that occurred yesterday influence the estimate equally. EWMA-based extensions of historical simulation give more weight to recent observations, but this introduces its own choices and complications.
Parametric (Variance-Covariance) VaR
The parametric approach assumes that returns follow a normal distribution and uses the estimated mean and standard deviation of returns to compute VaR analytically.
For a portfolio with daily return mean μ and daily return standard deviation σ, the one-day 99% VaR is:
VaR = (−μ + 2.33 × σ) × Portfolio Value
The 2.33 is the z-score corresponding to the 99th percentile of the standard normal distribution (the 1% left-tail cutoff). For a 95% confidence level, the z-score is 1.645.
If the mean daily return is assumed to be zero (a common simplification for short horizons), the formula reduces to VaR = 2.33 × σ × Portfolio Value.
The advantages are computational simplicity and easy aggregation across positions (portfolio variance can be computed from individual positions and their correlations using matrix algebra). The limitation is the normality assumption. Financial returns are empirically non-normal: they have fatter tails and more pronounced skewness than the normal distribution. Using a normal distribution understates VaR in the tails, which is precisely where VaR is meant to measure.
For options positions, parametric VaR requires additional care because option payoffs are nonlinear. The delta-normal approach applies the delta (the linear sensitivity of the option to the underlying) to approximate the option's exposure, but this approximation breaks down for large moves or highly nonlinear positions.
Monte Carlo Simulation
Monte Carlo simulation generates a large number of hypothetical return scenarios by drawing randomly from a specified distribution (or set of distributions with specified correlations) and computes the portfolio loss for each scenario. The VaR is then read from the distribution of simulated losses at the relevant confidence level.
Monte Carlo simulation is the most flexible of the three methods. It can accommodate nonlinear positions (options), complex instruments (path-dependent derivatives), and non-normal distributional assumptions. It is also computationally intensive, which was historically a constraint but is less so for modern risk systems.
The limitation is model dependence. The simulation is only as good as the distributional assumptions driving it. A Monte Carlo simulation calibrated to normal distributions will produce similar answers to parametric VaR and will share the same understatement of tail risk.
What VaR Does Not Tell You
The FRM curriculum places significant emphasis on VaR's limitations, and the exam tests these directly.
VaR says nothing about the distribution of losses beyond the confidence level. A one-day 99% VaR of $1.2 million tells you that losses will exceed $1.2 million with 1% probability. It does not tell you what those losses might be when they do exceed $1.2 million. They could be $1.3 million or $50 million. This is where Expected Shortfall (ES, also called Conditional VaR or CVaR) becomes important: ES measures the average loss in the tail beyond the VaR threshold.
VaR is not sub-additive in general. A coherent risk measure must satisfy sub-additivity: the risk of the combined portfolio should be no greater than the sum of the individual risks (diversification should reduce risk, not increase it). VaR does not always satisfy this property, which means that in some cases the VaR of a combined portfolio can exceed the sum of the VaRs of its components. Expected Shortfall is sub-additive, which is one reason regulators have moved toward ES for market risk capital requirements under Basel IV (the Fundamental Review of the Trading Book).
VaR depends heavily on the estimation method. The same portfolio measured by historical simulation, parametric VaR, and Monte Carlo simulation will produce different numbers, sometimes substantially different. The choice of method is a model risk question, and Part II's market risk domain covers this in depth.
Backtesting VaR
Backtesting is the process of comparing VaR estimates to realised losses to assess whether the model is well-calibrated. For a 99% one-day VaR, you expect losses to exceed the VaR estimate approximately 1% of trading days — roughly 2.5 days per year. An exception is a day when the actual loss exceeds the VaR estimate.
If exceptions occur much more than 1% of the time, the VaR model is underestimating tail risk. If they occur much less often, the model may be too conservative. Basel's traffic light approach classifies VaR models based on their exception count over a 250-day backtesting period:
- 0–4 exceptions: Green zone (model appears well-calibrated)
- 5–9 exceptions: Yellow zone (increased regulatory scrutiny)
- 10+ exceptions: Red zone (model is likely flawed; capital multiplier applied)
The exam tests the interpretation of backtesting results and the limitations of backtesting itself. Backtesting over a calm market period will show few exceptions even for a poorly calibrated model, since losses in calm periods rarely reach the tail threshold.
Preparing for VaR Questions on the Exam
VaR questions appear across both Part I (primarily in Valuation and Risk Models) and Part II (primarily in Market Risk Measurement and Management). The exam tests calculation, interpretation, and critical evaluation of VaR in roughly equal measure.
For calculation questions, you need to be comfortable computing parametric VaR from a stated mean and standard deviation, scaling VaR across time horizons using the square root of time, and reading VaR from a stated historical return distribution.
For interpretation questions, the exam frequently presents a VaR figure and asks what it does and does not imply, or it presents a backtesting result and asks what the exception count signals about the model.
Part I Valuation and Risk Models practice questions cover VaR extensively. Working through the VaR subtopics systematically and reviewing the explanations for incorrect answers will clarify exactly which aspects of the concept the exam expects you to apply rather than simply recognise.