Value at Risk (VaR)
ES: Value at Risk (VaR) PT: Valor em Risco
The industry standard metric for quantifying potential loss. A one-day 95% VaR of one million means there is a 95% probability of not losing more than that figure. Widely used, and widely criticised after 2008 for underestimating tail risk.
What Value at Risk Is
Value at risk is the most widespread metric for quantifying a portfolio’s potential loss over a specific time horizon at a given probability.
Its formal definition is that an X% VaR indicates that probability of the loss not exceeding the calculated amount. A one-day 95% VaR of one million means there is a 95% probability of not losing more than a million in a session or, put the other way, a 5% probability of losing more.
It depends on three parameters: the confidence level, normally 95% or 99% and sometimes 99.9% for regulatory use, where more confidence implies a larger figure; the time horizon, one day, ten days — the regulatory standard — or a month, where a longer horizon also implies more risk; and the calculation method, which can be historical simulation, Monte Carlo or parametric, and which can give different results.
Its history begins in the 1990s, when it became the banking industry standard, and it went on to underpin Basel II regulation in 2004. Basel III, in 2013, extended the framework beyond this metric precisely because of the accumulated criticism. Today it is present in practically every professional risk management system in the world.
A simple example: a 10 million portfolio with 1.5% daily volatility, assuming a normal distribution, has a 95% VaR of 10,000,000 × 1.645 × 0.015 = 246,750. Its great virtue is exactly that: the calculation is simple, intuitive and communicable to non-specialists.
The Three Calculation Methods
There are three main methods, with different virtues and defects.
The parametric method, also called variance-covariance, assumes returns follow a normal distribution. Its formula is portfolio value × Z score × volatility × √horizon, with a Z score of 1.645 for 95% and 2.326 for 99%. A one million portfolio with 2% daily volatility gives 1,000,000 × 1.645 × 0.02 = 32,900. It is fast and intuitive, but it assumes normality — which is false, because markets have fat tails — it does not capture the non-linearity of options, and it underestimates extreme events.
Historical simulation uses past returns directly: they are sorted and the corresponding percentile marks the figure. With a thousand historical sessions, the 5th percentile gives the 95% figure. It requires no distributional assumption and captures real behaviour, but it is limited to what has already been observed, needs a long history, and weights every period equally.
Monte Carlo simulation generates thousands of future scenarios from a statistical model and extracts the percentile. It is flexible, handles complex portfolios with options and derivatives, and captures non-linearity, but it is computationally expensive and depends entirely on the quality of the model and the estimated correlations.
Comparing them on the same 10 million portfolio, the parametric might give 246,000, the historical 310,000 — because it picks up the extreme episodes of 2008 and 2020 — and Monte Carlo 295,000. Banks often calculate all three, and a large discrepancy between them is itself a signal of model risk.
Limitations
The metric has seven serious limitations that prompted its revision after 2008.
The gravest is that it says nothing about the magnitude of losses that exceed the threshold. A 95% VaR of one million tells you about 95% of sessions; about the remaining 5% it says nothing, and that loss could be 1.01 million or 50.
The second is the parametric method’s assumption of a normal distribution, when real returns have fat tails and extreme events are far more frequent than it predicts. The third is non-stationarity: volatility changes over time and clusters, so a measurement taken in a calm period underestimates the turbulent period that follows.
The fourth is the correlation assumptions: portfolio-level calculation depends on them, and during crises they converge toward one, so the diversification benefit evaporates exactly when it is needed. The fifth is the non-linearity of options, which the parametric method captures poorly and which can lead to seriously underestimating tail risk.
The sixth is model risk: different methods give different figures, and choosing one over another has consequences for required regulatory capital, which creates perverse incentives. And the seventh is the false sense of security: a 5% daily probability equates to around thirteen sessions a year in which the threshold is breached, and during those sessions the loss can be catastrophic.
Nassim Taleb has been its most persistent critic with the turkey problem: for a thousand days, the turkey’s risk metric suggests its life is safe, until Thanksgiving arrives and it fails completely.
Expected Shortfall and Modern Metrics
Expected shortfall — also called conditional value at risk — is the metric that succeeded VaR after 2008 and corrects its central defect. It is defined as the average loss across the scenarios that exceed the threshold.
An example: if the 95% threshold is one million and the sessions that breach it record losses of 1.1, 1.5, 2, 3 and 10 million, expected shortfall is the average of those figures. It therefore captures the severity of the tail, not just its probability. Since 2019, banking regulation has required expected shortfall at 97.5% instead of VaR at 99% for exactly this reason.
Stress tests are the indispensable complement. Historical tests replay specific crises against the current portfolio. Hypothetical tests construct bespoke scenarios, such as a rapid and sharp rate rise or a geopolitical disruption. And reverse tests start from an unacceptable outcome — losing 30% of capital — and work backwards to identify what moves would cause it, which helps detect blind spots.
There are also specialised variants. Stressed VaR is calculated using only data from turbulent periods and produces far higher figures. Liquidity-adjusted VaR incorporates the cost of unwinding positions in a stressed market, which is critical for portfolios with illiquid assets, where exiting can take days during which the loss keeps growing. And for options portfolios there are greek-based methods that decompose risk by delta, gamma, vega and theta, and which work far better than the parametric approach.
It is worth distinguishing it from maximum drawdown: that looks backwards and describes the worst that has already happened, while value at risk looks forward and estimates probabilities. They are complementary, and professional management uses both.
Practical Application
For the retail or intermediate trader, the metric applies on five fronts.
The first is calculating your own portfolio’s VaR. As a general rule, it should sit below 1% or 2% of total value for a conservative profile, and between 2% and 5% for an aggressive one. If it exceeds that, reduce exposure.
The simplified calculation is accessible: estimate daily volatility using the standard deviation of the last twenty days and apply portfolio × 1.645 × daily volatility. A 100,000 portfolio with 1.5% daily volatility gives 100,000 × 1.645 × 0.015 = 2,468, that is, a 5% probability of losing more than that figure in a session. Most professional platforms calculate it automatically.
The second is risk budgeting: distributing total VaR across positions. With a 100,000 portfolio and a 2% budget, you have 2,000 to allocate across all of them.
The third is bearing in mind the effect of correlation: portfolio VaR is not the sum of individual VaRs unless everything is perfectly correlated. Uncorrelated positions provide a diversification benefit that shrinks during crises, so it is worth also calculating the scenario with correlation equal to one.
The fourth is understanding what it is for: comparing risk between strategies, setting position limits, communicating risk and meeting regulatory requirements. And the fifth, understanding what it is not for: assessing tail risk, which requires expected shortfall; measuring liquidity risk, which requires the adjusted variant; anticipating extreme events, which requires stress testing; and valuing complex options structures, which requires scenario analysis.
Risk Metrics: What Each One Contributes
Every method has its trade-offs; professional management combines several.
| Metric | Strength | Weakness | Best use |
|---|---|---|---|
| Fast and intuitive | Assumes a normal distribution | Simple equity and bond portfolios | |
| No distributional assumption | Limited to what has been observed | Mean-reverting strategies | |
| Flexible, handles complex portfolios | Expensive to compute | Options and derivatives | |
| Captures the severity of the tail | More complex to calculate | The current regulatory standard | |
| Scenario based, very intuitive | Only covers the scenarios tested | Preparing for extreme events |
Frequently Asked Questions
What is the difference from expected shortfall?
An example: if the 95% threshold is one million and expected shortfall is 2.5 million, it means that in 95% of sessions the loss stays below a million, but in the remaining 5% the average loss is 2.5 million.
Expected shortfall therefore captures the severity of the tail, which is exactly what value at risk ignores by construction. For that reason banking regulation moved to requiring it as the standard from 2019.
Which calculation method is best?
The parametric method works well for simple portfolios of liquid shares and bonds: it is fast and intuitive. The historical method is preferable when you have a long history and expect a similar market regime, because it captures real behaviour. And Monte Carlo is the right one for complex portfolios with options or derivatives, however expensive it is to compute.
Professional managers calculate all three and compare them: a large discrepancy between them is a model-risk signal worth investigating. For the retail investor, parametric or historical is enough.
Why did it fail in 2008?
Tail risk was underestimated, because the parametric method assumed normality and markets have fat tails. The correlation assumption failed: during the collapse, diverse assets fell together and the diversification benefit evaporated. A liquidity crisis appeared that made selling at fair value impossible, something the standard calculation ignores entirely. There was systemic model risk, because banks used similar models and all underestimated risk the same way. The available history was insufficient, since few had lived through an episode of that magnitude. And the complexity of securitised derivatives introduced a non-linearity the simple calculation did not capture.
These lessons drove the subsequent reforms and the widespread adoption of expected shortfall alongside stress testing.
Can I calculate it myself?
With a 50,000 portfolio and 1% daily volatility: 50,000 × 1.645 × 0.01 = 822, that is, a 5% probability of losing more than that figure in a session.
For portfolios with options the calculation gets much harder, and it is worth leaning on your broker’s tools, which usually include built-in estimates.
The limitation is always the same: the simple calculation assumes normality, which is false in fat-tailed markets. Complement it with a simple stress test, asking what would happen to the current portfolio if an episode like 2008 repeated.
Does it work for options?
There are three better approaches. Delta-adjusted VaR approximates the linear sensitivity and already improves the result considerably. Full revaluation reprices every option under stressed scenarios and calculates the loss: the most accurate and the most expensive. And the greek-based approach decomposes risk into delta, gamma, vega and theta, offering the most nuanced view.
For the retail trader, the practical route is simpler: calculate each position’s maximum loss and add them for the portfolio’s worst case. In defined-risk spreads that figure is explicit; in uncovered positions you need scenario analysis, not a simple statistical estimate.