Introduction to Value-at-Risk (VaR): Different Methodologies, Assumptions, and Limitations

What is Value-at-Risk?

Value-at-Risk (VaR) is a statistical measure that quantifies the potential maximum loss an investment portfolio could experience at a certain confidence level over a specified time horizon. It provides a single risk number that helps traders, risk managers, and financial institutions assess how much they can potentially lose under normal market conditions.

In simpler terms, VaR answers the following question:

"What is the worst-case loss I can expect on my portfolio with _% confidence over the next N days?"

Understanding the Mechanics of Value-at-Risk (VaR)

Value-at-Risk is defined by some critical parameters, each of which must be specified to interpret the risk figure meaningfully:

• Confidence Level (95%, 99%): VaR requires the probability threshold to determine potential losses. A 99% confidence level implies that the model estimates losses under normal market conditions where the losses will not exceed the VaR amount 99% of the time. It also implies a 1% chance of losses exceeding VaR, often referred to as tail risk—this tail risk falls outside the scope of traditional VaR and must be addressed using Expected Shortfall (ES) or stress testing.

  
  

• Time Horizon (1 day, 10 days): VaR requires the period over which the potential losses are to be measured.

  • Institutional traders typically use 1-day VaR for daily risk monitoring,

  • While regulators may require financial institutions to compute 10-day VaR for capital adequacy calculations.

  VaR is not additive across time—you cannot simply multiply a 1-day VaR by 10 to estimate a 10-day VaR. Instead, the square-root-of-time rule is used:

  

  
  

• Lookback Period (Historical Data Window): The lookback period defines the length of historical data used to estimate the volatility and loss distribution. This parameter is especially important in Historical Simulation VaR and Parametric VaR, where past returns are central to estimating future risk.

  
  

• Currency/Monetary Units: VaR is typically expressed in monetary/dollar terms, such as USD, EUR, or GBP, making it a practical measure and highly intuitive for stakeholders, internal risk reporting, and capital adequacy requirements. for example, a VaR of $10 million at 99% confidence over 1 day clearly communicates the maximum expected daily loss in dollar terms.

Methodologies for Calculating Value-at-Risk (VaR)

Value-at-Risk (VaR) can be implemented through different methodologies, each offering distinct theoretical assumptions, computational requirements, and modeling precision suited to different types of portfolios. The choice of method depends on the type of portfolio, the availability of data, and the desired level of accuracy.

• Parametric (Variance-Covariance) VaR: Also known as the analytical method or delta-normal method, this approach is based on the assumption that portfolio returns are normally distributed and that risk factors have a linear impact on portfolio value. Under these assumptions, Value-at-Risk can be calculated using a closed-form expression that incorporates the portfolio’s expected return, standard deviation, and a z-score corresponding to the selected confidence level.

• Historical Simulation VaR: Also known as a non-parametric method, this approach uses actual historical returns to estimate future risk, avoiding any assumption about the distribution of returns. It applies past changes in market variables to the current portfolio and derives the distribution of hypothetical portfolio returns. VaR is then determined by identifying the appropriate percentile based on the chosen confidence level.

• Monte Carlo Simulation VaR: This method is the most flexible and sophisticated VaR methodology. It involves generating a large number of potential future scenarios by simulating the behavior of risk factors based on user-defined statistical distributions and correlation structures. Each scenario is applied to the portfolio to estimate potential changes in value, capturing a wide range of outcomes. This approach is particularly well-suited for portfolios with complex or non-linear instruments, such as derivatives, and can model non-normal return distributions, fat tails, and volatility clustering.

In practice, risk managers often combine multiple approaches or use hybrid models to balance computational feasibility with accuracy and robustness.

In the next article of this series, we will explore each methodology in depth, with step-by-step calculation examples, use-case scenarios, and illustrations to help you understand and implement VaR across various portfolio types.

Interpretation of Value-at-Risk (VaR) Number

Scenario: A risk manager calculates a 1-day Value-at-Risk (VaR) of $10 million at a 99% confidence level for a trading portfolio.

What Does This Mean?

Under normal market conditions, there is a 99% chance that losses will be less than or equal to $10 million, with a 1% chance that the portfolio could lose more than $10 million in a single day.

Note: The VaR figure does not predict the actual loss, nor does it describe losses beyond the 99% threshold. It’s also important to understand that VaR is forward-looking, based on the past behavior of market variables. This highlights its dual nature as both a predictive and statistical tool—useful but inherently uncertain.

---

Limitations of Value-at-Risk (VaR)

While Value-at-Risk (VaR) remains one of the most widely used risk metrics in financial institutions—particularly within banks, asset managers, and regulators—it has significant limitations. These limitations become especially critical during market stress (stress events or crisis), for complex portfolios, or when risk managers rely on VaR without complementing it with other risk measures.

• Dependence on Historical Data and Lookback Periods:

  
  

  • VaR calculations rely heavily on historical market data to estimate future risk. This introduces a key limitation: if the historical dataset does not include periods of extreme volatility or crisis, the VaR measure will likely underestimate potential losses.

    
    

  • The lookback period used (the number of past days or years considered) has a major impact on the accuracy of VaR. A short lookback period may reflect recent calm markets and fail to capture rare but impactful tail events, while an excessively long period may dilute the relevance of recent market conditions. for example, a VaR model using data from 2017 to 2018 (a period of low volatility and steady returns) would completely miss the shock of COVID-19 in early 2020. Risk estimates from such a model would provide a false sense of security, showing artificially low risk just before a major market collapse.

  
  

  Solution: Stressed Value-at-Risk (SVaR) is specifically designed to address the limitation of relying solely on benign historical periods in VaR calculations. It uses historical data from a period of significant financial stress (the 2008 Global Financial Crisis) to model potential losses under extreme but plausible conditions—ensuring that the risk measure captures tail events and periods of market dislocation that standard VaR might ignore.

• VaR Does Not Capture Tail Risk (Fails in Extreme Events): VaR only provides the threshold loss at a given confidence level (95% or 99%), but it does not tell us the potential size of losses beyond this point. for example, a 1-day 99% VaR of $10 million implies that there is a 1% chance of losing more than $10 million. But how much more? $15 million? $100 million?, VaR does not quantify how much worse the losses could be.

  
  

  • In the 2008 Global Financial Crisis, many institutions relied on VaR for risk management and capital allocation. However, most VaR models missed the magnitude of the losses, they captured the “usual bad days” but not the rare catastrophic ones.

    
    

  • VaR at 99% confidence does not account for the 1% worst-case scenarios, which is where catastrophic losses occur.

  
  

  Solution: Expected Shortfall (ES, also known as Conditional VaR) estimates the average loss beyond the VaR threshold (it measures the risk of extreme losses in the tail, especially critical in risk-averse or regulated environments).

• VaR Assumes Normal Distribution (Underestimates Fat-Tailed Risks): Many implementations of VaR (especially the Parametric method) assume that asset returns are normally distributed, which means it fails to capture fat tails (extreme price movements). However, real financial markets often exhibit fat tails (extreme losses occur more often than predicted) and skewness (returns are not symmetrically distributed). for example, stock market crashes exhibit leptokurtosis (fat tails), which means extreme losses occur more frequently than normal distribution predicts. VaR does not adjust for this and that makes risk estimates unreliable during financial crises.

  
  

  The 2000 Dot-Com Crash, the 2008 Financial Crisis both saw extreme losses far beyond what normal distribution-based VaR predicted. Under a normal distribution, such losses were supposed to occur once in a million years, yet they happened within a single decade—exposing the flaws in the assumption.

  
  

  Models that incorporate fat-tailed distributions (t-distribution or EVT) or use non-parametric methods like historical simulation or Monte Carlo can mitigate this, but Expected Shortfall (ES) remains a better alternative, especially when used with fat-tail modeling.

• VaR Is Not Subadditive: A risk measure should satisfy subadditivity, which means that the risk of a diversified portfolio should not exceed the sum of the individual risks. VaR violates this in some scenarios: When two uncorrelated portfolios are combined, the total VaR can be higher than the sum of the individual VaRs (non-linear portfolios). This also occurs because VaR does not always reward diversification, especially when returns are non-normal or non-linear.

  
  

  Portfolio A and Portfolio B may each have a VaR of $10 million. When combined, the VaR of the aggregate portfolio could be $22 million, instead of being less than or equal to $20 million—which is counterintuitive and misleading, violating subadditivity—making VaR unreliable for portfolio risk aggregation and capital allocation for institutions that manage diversified portfolios.

  
  

  Solution: Expected Shortfall (ES) satisfies subadditivity, meaning it accounts for diversification benefits and provides a more accurate measure of total risk, making it a more coherent risk measure for portfolio risk aggregation.

• VaR Is Highly Model-Dependent (Sensitive to Methodology): VaR is not a single model but a framework that can be implemented using different methodologies. Different VaR calculation methods (Historical, Parametric, Monte Carlo) introduce different assumptions and produce different results, making VaR highly dependent on the chosen model. Different banks report different VaR values for the same portfolio due to different modeling assumptions.

  
  

  • A trading desk at Bank A may use Parametric VaR, assuming normality, which may understate risk in non-normal markets.

    
    

  • Bank B may use Historical VaR, reflecting recent market history, and fails to predict future crises that haven’t happened before.

    
    

  • Bank C may implement Monte Carlo VaR, generating thousands of random return paths—more flexible but computationally expensive.

  
  

  Despite managing similar portfolios, their reported VaR numbers may differ, making risk non-comparable across institutions.

  
  

  Solution: Expected Shortfall (ES) still depends on methodology (a model-dependent measure), but it consistently captures extreme risks better than VaR, regardless of the method used.

• When determining portfolio risk, a reliable risk measure is expected to incorporate correlations between each pair of assets. However, as the number of assets increases and the portfolio becomes more diverse across asset classes, sectors, and geographies, it becomes extremely challenging to estimate and maintain accurate correlations. Correlation matrices become unstable and noisy, especially when using limited historical data, leading to inaccurate aggregation of portfolio risk and ineffective diversification.

  
  

  In large institutional portfolios with hundreds or thousands of positions, even small errors in estimating pairwise correlations can distort the overall risk profile, causing underestimation or overestimation of VaR and, consequently, under-allocation of capital against high-risk exposures.

While Value-at-Risk provides a simple and interpretable measure of risk, its limitations—particularly in terms of tail risk, distributional assumptions, aggregation inconsistencies, and data/model sensitivity—highlight the need to supplement VaR with more robust measures. Expected Shortfall (ES), stress testing, scenario analysis, and model risk awareness are all essential tools to ensure more resilient and accurate risk management frameworks.