Value at Risk (VaR) Explained: Measuring a Portfolio's Loss Risk

Value at Risk, or VaR, has become the common language of market risk since the 1990s. A bank, a fund, or an individual trader can summarize in a single number the potential loss of a portfolio over a given horizon, at a given confidence level. This apparent simplicity explains its success, but also its limits, which must be understood before relying on it blindly.

Concretely, a 1-day 95% VaR of 50,000 euros means there is a 95% chance the portfolio's loss will not exceed 50,000 euros the next day, under normal market conditions. In other words, 5% of scenarios remain where the loss could be larger — and that is precisely where VaR becomes insufficient, as we will see further below.

What Exactly Is Value at Risk?

VaR answers a simple question: what is the maximum loss I can expect, with a given probability, over a given period? It therefore combines three inseparable parameters: a time horizon (1 day, 10 days, 1 month), a confidence level (95%, 99%), and the currency or unit of the portfolio in question.

It is essential to understand that VaR is not an absolute maximum loss. It is a statistical threshold: it defines the boundary beyond which the worst-case scenarios lie, without describing them. Two portfolios can have exactly the same 99% VaR while having radically different extreme-risk profiles beyond that threshold.

Banking regulators, notably under the Basel framework, have largely contributed to popularizing VaR as the reference for calculating capital requirements. A bank must therefore demonstrate it holds sufficient capital to absorb losses corresponding to its VaR over a given regulatory horizon, usually 10 days at 99% confidence.

The Three Methods for Calculating VaR

Historical Simulation

The most intuitive method consists of directly applying past market changes to the current portfolio. For example, one takes the last 500 days of returns, recalculates the portfolio's value for each of these historical scenarios, then reads off the loss corresponding to the desired percentile (the 5th percentile for a 95% VaR).

The advantage of this approach is that it makes no assumption about the shape of the return distribution — it naturally captures the fat tails and asymmetries observed in real markets. Its major drawback is that it assumes the past is representative of the future, which is particularly fragile during regime shifts.

The Parametric (Variance-Covariance) Method

This method assumes the portfolio's returns follow a normal distribution, characterized solely by its mean and standard deviation. VaR is then calculated analytically from the portfolio's volatility and a multiplicative factor corresponding to the chosen confidence level (1.65 for 95%, 2.33 for 99%).

Fast and easy to implement, this method becomes misleading as soon as returns deviate significantly from normality — which is almost always the case in practice, financial markets being known for tails fatter than the normal distribution predicts.

Monte Carlo Simulation

This third approach consists of generating a very large number of random scenarios for the underlying risk factors, using a statistical model chosen by the analyst (not necessarily a normal distribution), then recalculating the portfolio's value for each simulated scenario. VaR is then read directly from the simulated distribution of gains and losses.

This is the most flexible method, capable of handling non-linear instruments like options, but also the most computationally expensive and the most sensitive to the modeling assumptions used to generate the scenarios.

A Worked Example on a Sample Portfolio

Consider a 200,000-euro portfolio invested 60% in European equities and 40% in government bonds. Based on daily returns observed over the past two years, the portfolio's annualized volatility comes out to about 12%, or a daily volatility close to 0.75%.

Using the parametric method, the 1-day 95% VaR is obtained by multiplying the portfolio value by the daily volatility and by the factor 1.65: 200,000 x 0.0075 x 1.65, or about 2,475 euros. This means that, under normal conditions, the portfolio's loss should not exceed 2,475 euros over one day, 95% of the time.

Using historical simulation over the last 500 days, one might obtain a slightly different figure, say 2,800 euros, if the portfolio's past returns show a more pronounced negative skew than the normal distribution assumes. This gap illustrates well why the choice of method is never neutral.

The Limits of VaR: What It Doesn't Tell You

The most fundamental criticism leveled at VaR is that it completely ignores the magnitude of losses beyond the chosen threshold. A 99% VaR of 10,000 euros is identical whether the worst possible loss is 10,001 euros or 500,000 euros — the crucial information about the tail of the distribution is simply absent from the number.

It is precisely to fill this gap that Expected Shortfall, also called Conditional VaR or CVaR, was developed, calculating the average expected loss in scenarios where VaR is exceeded. CVaR thus offers a complementary, more conservative view of exposure to extreme events, and has gradually become the reference measure in the most recent regulatory frameworks.

VaR also suffers from other well-documented weaknesses: it is not always subadditive (the combined risk of two portfolios can, in certain mathematical cases, exceed the sum of their individual VaRs), it relies on historical data that may not anticipate structural market changes, and it can give a false sense of security if used as the sole risk management tool.

Practical Takeaways for an Individual Quant Trader

For an individual trader managing their own portfolio, VaR remains a valuable position-sizing tool, provided it is used judiciously. In particular, it allows setting a daily risk limit consistent with one's overall loss tolerance, then sizing each position so that its marginal contribution to the portfolio's VaR stays under that cap.

It also helps to track how the daily VaR estimate itself evolves over time on a simple chart, alongside the actual realized daily profit-and-loss. Repeated breaches of the VaR threshold within a short window, more frequent than the chosen confidence level would statistically predict, is an early warning that the risk model no longer fits current market conditions and needs recalibration.

It is recommended to calculate VaR using at least two different methods (historical and parametric, for example) in order to spot cases where the two methods diverge sharply — a sign that the portfolio's return distribution has atypical characteristics deserving particular attention.

Finally, VaR should never be used in isolation: it benefits from being systematically complemented by Expected Shortfall, by stress tests on extreme historical scenarios, and by regular monitoring of the maximum drawdown actually observed on the portfolio.

Another useful habit is to periodically review the historical window used for the calculation, especially after an episode of unusual volatility. A VaR calculated over an overly calm period will underestimate future risk if the market then enters a more turbulent regime, while a window that is too short right after a shock can conversely overstate risk for a long time once volatility subsides.

Conclusion

Despite its limits, Value at Risk remains an indispensable starting point for quantifying and communicating a portfolio's risk in a standardized way. Understanding what it measures — and above all what it does not measure — is the condition for using it wisely rather than as a false sense of security.

If you want to calculate the VaR and CVaR of your own portfolio, compare several methods, and visualize your stress scenarios, the quantitative analysis tools available at /outils on TrueVerdikt let you explore these concepts concretely and pedagogically, without ever constituting personalized investment advice.