What Is Value at Risk?
Value at Risk answers a deceptively simple question: what is the maximum loss a portfolio could suffer over a given time horizon at a specified confidence level? A statement like 'the one-day 99% VaR is $4.2 million' means there is a 1% probability that the portfolio will lose more than $4.2 million on any given day, under normal market conditions.
VaR became the industry standard for market risk measurement in the 1990s and was formally incorporated into banking regulation through the Basel Accords. For FRM candidates, VaR appears across multiple readings in both Part I and Part II, making it one of the most heavily tested topics on the exam.
Parametric (Variance-Covariance) VaR
The parametric approach assumes portfolio returns follow a normal distribution. Under this assumption, VaR is simply a function of the portfolio's mean return, standard deviation, and the z-score corresponding to the chosen confidence level.
For a 99% confidence level, the z-score is 2.326. If a portfolio has a daily standard deviation of $1.8 million and a mean daily return near zero, the one-day parametric VaR equals 2.326 times $1.8 million, or approximately $4.19 million.
For multi-asset portfolios, the approach requires a covariance matrix. The portfolio variance is computed using the standard matrix formula, and VaR follows directly from the portfolio standard deviation.
Advantages: computationally simple, fast to calculate, easy to decompose into component and incremental VaR contributions by asset. Limitations: assumes normality (underestimates tail risk from fat tails), assumes linear payoffs (fails for portfolios with significant option positions), and relies on the accuracy of the covariance matrix, which can become unstable during crises.
Historical Simulation VaR
Historical simulation makes no assumption about the distribution of returns. Instead, it takes the actual historical returns for each risk factor over a look-back window (typically 250 to 500 trading days), applies those returns to the current portfolio, and sorts the resulting profit-and-loss outcomes from worst to best.
For a 99% VaR using 500 days, you identify the fifth-worst loss (the 1st percentile of the sorted distribution). That value is the VaR estimate.
Consider a simplified example with 250 historical days. You reprice the current portfolio under each day's market moves, generating 250 hypothetical P&L values. Sorting from worst to best, the 2.5th observation (interpolated between the 2nd and 3rd worst) gives you the 99% VaR.
Advantages: captures fat tails and non-normal features present in the actual historical data, naturally handles non-linear instruments like options, and requires no distributional assumptions. Limitations: entirely dependent on the look-back window (if the window misses a crisis, VaR will be understated), treats all historical days as equally likely, and cannot generate scenarios worse than those observed in the data.
Monte Carlo Simulation VaR
Monte Carlo simulation generates thousands of hypothetical future scenarios by sampling from a specified statistical model of risk factor behavior. Each scenario produces a portfolio P&L, and VaR is extracted from the resulting distribution in the same way as historical simulation.
The approach involves three steps: define a stochastic process for each risk factor (often geometric Brownian motion with correlations), generate a large number of simulated paths (10,000 or more), and fully reprice the portfolio under each scenario.
Advantages: the most flexible method — can accommodate any distributional assumption (fat tails, skewness, regime switching), handles complex instruments including exotic derivatives, and can generate scenarios more extreme than any historical precedent. Limitations: computationally intensive (full repricing for thousands of scenarios), results depend on the chosen model (model risk), and the approach is a black box that can be difficult to explain to stakeholders.
Expected Shortfall: Beyond VaR
VaR tells you the threshold loss at a given confidence level, but says nothing about how bad things could get beyond that threshold. A portfolio might have a 99% VaR of $5 million, but the average loss in the worst 1% of scenarios could be $5.2 million or $15 million — VaR cannot distinguish between these two very different risk profiles.
Expected Shortfall (ES), also called Conditional VaR (CVaR), addresses this limitation by calculating the average loss in the tail beyond the VaR threshold. The 99% ES is the expected loss given that the loss exceeds the 99% VaR.
ES has superior mathematical properties: it is a coherent risk measure (satisfying subadditivity, meaning the risk of a combined portfolio is never greater than the sum of individual risks), whereas VaR can violate subadditivity for non-normal distributions. The Basel III framework adopted ES at the 97.5% confidence level as the primary market risk measure, replacing VaR.
Backtesting VaR Models
Backtesting compares actual portfolio losses against the VaR forecast to assess model accuracy. If the model predicts a 99% VaR, you expect approximately 2.5 VaR breaches (exceptions) per year out of 250 trading days.
The Basel framework uses a traffic-light system based on the number of exceptions observed over a 250-day window. The green zone (0 to 4 exceptions) indicates acceptable model performance. The yellow zone (5 to 9 exceptions) triggers increased scrutiny and a higher regulatory capital multiplier. The red zone (10 or more exceptions) indicates a seriously flawed model requiring immediate remediation.
The Kupiec proportion-of-failures test formalizes this comparison using a likelihood ratio statistic. The Christoffersen test extends this by also checking whether exceptions are independently distributed — clustering of exceptions (multiple breaches in consecutive days) suggests the model fails to capture volatility dynamics.
Basel Regulatory Requirements
Under the Basel III Fundamental Review of the Trading Book (FRTB), banks must calculate the internal models approach (IMA) market risk charge using Expected Shortfall at the 97.5% confidence level, computed using stressed calibration windows. The ES must be calculated for each risk class (interest rate, equity, foreign exchange, commodity, credit spread) and aggregated using a prescribed correlation structure.
The standardized approach (SA) serves as both a fallback and a floor. Banks that fail backtesting or P&L attribution tests on specific desks must revert those desks to the SA, which uses prescribed risk weights and correlations.
For the FRM exam, understand the hierarchy: ES replaces VaR for regulatory capital, but VaR remains widely used for internal risk management and limit-setting. Know why the transition occurred (VaR's lack of coherence and insensitivity to tail shape) and the practical implications.
Exam Strategy
VaR questions on the FRM exam span calculation, interpretation, and critique. Be prepared to calculate parametric VaR for a two-asset portfolio, extract historical simulation VaR from a sorted dataset, interpret backtesting results, and explain why ES is preferred over VaR for capital adequacy.
The most common pitfall is applying the parametric method to a portfolio with significant option positions — always note when linearity assumptions break down. Another frequent error is using the wrong scaling factor when converting between time horizons: VaR scales by the square root of time only under specific assumptions (i.i.d. returns), which may not hold in practice.
Deepen your preparation with our FRM practice questions on market risk, or discuss challenging VaR scenarios in the community Q&A.