Value at Risk (VaR): Formula, Worked Example, and Its Blind Spot

Direct Answer

Value at Risk (VaR) estimates the dollar loss a portfolio is unlikely to exceed over a given time horizon, at a chosen confidence level, under normal market conditions. A $250,000 portfolio with 2.2% daily volatility has a 1-day 95% VaR of about $9,075 — meaning roughly a 5% chance of losing more than that in a single day. VaR says nothing about how much worse that remaining 5% could get; that question belongs to Conditional VaR, not VaR itself.

The practical objective is not to memorize a formula. It is to understand what the number is actually claiming, compute it correctly for a real portfolio, and recognize the specific blind spot that has caused VaR to be misused — sometimes catastrophically — by desks that treated it as a hard ceiling rather than a probabilistic threshold.

Key Takeaways

• VaR = Portfolio Value × Z-score × σ × √(time horizon), where σ is the portfolio's return standard deviation for the base period.
• A 95% VaR uses a Z-score of 1.65; a 99% VaR uses 2.33 — higher confidence produces a larger threshold, not a safer ceiling.
• Scaling a 1-day VaR to a longer horizon multiplies by the square root of the number of days, not by the number of days itself.
• Parametric VaR assumes normally distributed returns, an assumption that breaks down for fat-tailed assets like crypto.
• Historical simulation and Monte Carlo simulation are the two common alternatives, each with different strengths and different failure modes.
• VaR describes a threshold, not a worst case — it says nothing about the size of losses beyond that threshold.

What Value at Risk Measures

VaR compresses a portfolio's downside risk into a single dollar figure tied to a probability and a time horizon.

Value at Risk answers a specific, narrow question: over the next N days, how much could this portfolio lose, such that losses larger than that amount happen only X% of the time? The confidence level (typically 95% or 99%) and the time horizon (commonly 1 day or 10 days) have to travel with the number itself — "$9,075" means nothing on its own; "a 1-day, 95% confidence VaR of $9,075" is a complete statement.

VaR became a standard risk metric in the 1990s after J.P. Morgan popularized the variance-covariance approach through its RiskMetrics methodology, and was later written into international bank capital regulation. That pedigree gives VaR an authority its actual scope doesn't fully support: it answers a threshold question well, but it was never designed to describe tail severity.

Practical checklist

• Always state VaR with both its confidence level and its time horizon attached, never as a bare dollar figure.
• Treat VaR as an estimate conditioned on the model and inputs used, not an objective property of the portfolio.
• Recalculate VaR whenever position sizes, volatility, or correlations change materially.
• Pair VaR with a tail-severity measure such as Conditional VaR before treating any risk limit as complete.

Common mistake: quoting a VaR figure without its confidence level and horizon, which strips the number of the context that makes it meaningful and invites it to be misread as an unconditional loss estimate.

The Parametric (Variance-Covariance) Method

Parametric VaR multiplies portfolio value by a confidence-level constant, the portfolio's volatility, and a time-scaling factor.

The parametric method, also called the variance-covariance method, is the fastest of the three common VaR approaches to compute because it reduces the whole problem to a single formula built on the assumption that portfolio returns are normally distributed:

VaR = Portfolio Value × Z-score × σ × √(time horizon)

Each term does a specific job. Portfolio Value is the current dollar value of the position or account. Z-score is the standard normal critical value for the chosen confidence level — 1.65 for 95%, 2.33 for 99% — the point beyond which that share of a normal distribution's probability mass sits. σ (sigma) is the portfolio's return standard deviation for the base period, almost always a daily figure estimated from a trailing window of historical returns. √(time horizon) is the square-root-of-time factor that converts a 1-day volatility estimate into an estimate for a longer holding period.

The square-root-of-time rule works because variance — not standard deviation — scales linearly with time under the assumption that daily returns are independent and identically distributed (no autocorrelation, no trending, no regime change). Since standard deviation is the square root of variance, scaling a 1-day VaR to a 10-day VaR means multiplying by √10, not by 10 — a convenience, not a law of markets, since the independence assumption fails during trending or accelerating volatility regimes, exactly when accurate scaling matters most.

Portfolio-level parametric VaR for more than one position also requires a covariance matrix, not a simple sum of each position's individual VaR. Summing individual VaRs implicitly assumes every position is perfectly correlated, which overstates diversified portfolio risk; a proper multi-asset calculation blends each position's volatility with the correlations between them.

Practical checklist

• Use 1.65 for a 95% confidence Z-score and 2.33 for a 99% confidence Z-score; do not interpolate casually between them.
• Estimate σ from a consistent, clearly stated lookback window, and update it as new return data arrives.
• Scale time horizons with the square root of the number of days, never a linear multiple.
• For multi-position portfolios, incorporate a correlation or covariance matrix rather than summing individual VaRs.

Common mistake: scaling a 1-day VaR to a 10-day VaR by multiplying by 10 instead of the square root of 10, which overstates the multi-day figure by roughly a factor of three.

Worked Example: A $250,000 Portfolio

Apply the parametric formula to a concrete portfolio, then scale the result to a longer horizon.

Inputs

• Portfolio value: $250,000
• Daily volatility (σ): 2.2%, or 0.022
• Confidence level: 95% (Z-score: 1.65)
• Time horizon: 1 day

Calculation

VaR = Portfolio Value × Z-score × σ × √(time horizon)
VaR = $250,000 × 1.65 × 0.022 × √1
VaR = $412,500 × 0.022
VaR ≈ $9,075

That result means there is roughly a 5% chance this portfolio loses more than about $9,075 in a single day under normal market conditions, given a 2.2% daily volatility estimate and the assumption that returns are approximately normally distributed. Read the other direction: on roughly 19 out of 20 trading days, the loss (if any) is expected to stay within about $9,075 — a threshold, not a maximum.

Scaling to a 10-day horizon

A risk desk that wants the equivalent VaR over a 10-day holding period does not recalculate from scratch — it applies the square-root-of-time factor to the 1-day figure already computed:

VaR (10-day) = VaR (1-day) × √10
VaR (10-day) = $9,075 × 3.1623
VaR (10-day) ≈ $28,698

The 10-day figure is not ten times the 1-day figure — $90,750 — but only about 3.16 times larger, because risk compounds with the square root of elapsed time under the model's independence assumption, not linearly. Multiplying by the number of days directly instead is the single most common arithmetic error in applying VaR, and it overstates multi-day risk by a wide margin. The accuracy of the scaled figure also depends on daily volatility and return independence holding steady across all ten days, an assumption that erodes the further out the horizon stretches.

Confidence level changes the threshold, not the risk

Holding the portfolio and volatility inputs constant, raising confidence to 99% changes only the Z-score: $250,000 × 2.33 × 0.022 ≈ $12,815. That figure is larger than the $9,075 95% figure because it describes a narrower, more extreme tail — the worst 1% of days rather than the worst 5% — not because the portfolio became riskier. Both numbers share the same blind spot: neither says how large the loss could be once that tail is actually entered.

Practical checklist

• Compute the 1-day VaR first, then scale to longer horizons using the square root of the number of days.
• Never multiply a 1-day VaR by the raw number of days to estimate a multi-day figure.
• State the volatility estimate's source and lookback window alongside any VaR figure reported.

Common mistake: reading a larger VaR figure at 99% confidence as proof the portfolio grew riskier, when the same inputs simply describe a narrower, more extreme slice of the same underlying distribution.

Historical Simulation VaR

Rank actual past returns and read the VaR off the appropriate percentile, with no distributional assumption.

Historical simulation sidesteps the normal-distribution assumption entirely. It takes the portfolio's actual historical returns over a lookback window — commonly one to several years of daily data — applies today's position sizes to each historical day's return, and sorts the resulting hypothetical profit-and-loss figures from worst to best. The VaR at a given confidence level is simply the loss at the corresponding percentile: with 500 trading days of history, a 95% VaR corresponds to roughly the 25th-worst day (5% of 500), and a 99% VaR to roughly the 5th-worst day.

The appeal is that it captures whatever fat tails, skewness, and asymmetry actually showed up in the data, without forcing a bell curve that may not fit — three separate 8% single-day drops in the lookback window show up directly, where a parametric model calibrated to average volatility might never predict a move that size. The limitation is equally direct: it can only reflect regimes actually present in the chosen window. A calm window understates the risk of a future crisis; a window dominated by a past crisis can overstate ordinary-day risk going forward. Newer assets — many crypto tokens included — may simply not have enough trading history to populate a meaningful sample.

Practical checklist

• Choose a lookback window long enough to include at least one stress period, when available data allows it.
• State the exact lookback window length whenever a historical VaR figure is reported.
• Recognize that a shorter or unusually calm lookback window will understate tail risk.
• Check that enough historical data exists for a young asset before relying on this method alone.

Common mistake: choosing a lookback window that happens to exclude the asset's worst historical drawdown, producing a historical VaR that looks reassuring purely because of when the sample starts and ends.

Monte Carlo Simulation VaR

Generate thousands of simulated return paths from a modeled process and read VaR off the simulated outcome distribution.

Monte Carlo simulation builds a statistical model of how the portfolio's returns behave — a normal distribution, a fitted distribution that better captures fat tails, a bootstrap resampling of historical returns, or a more sophisticated stochastic process — and uses it to generate a large number of random simulated future paths, often tens of thousands. For each path, the portfolio's hypothetical value at the horizon is calculated; sorting the results from worst to best and reading off the appropriate percentile produces the VaR estimate, on simulated rather than actual past data.

The advantage is flexibility: it can incorporate correlations between many assets and model nonlinear instruments such as options, where a simple linear formula breaks down — making it the preferred method for portfolios containing derivatives. The tradeoff is that it is only as good as the model feeding it; choosing the wrong underlying distribution reintroduces the same problem parametric VaR has, just hidden behind a more elaborate process that can look more rigorous than it is, and it is the most computationally intensive of the three methods.

Practical checklist

• Document which distribution or stochastic process underlies the simulation, since this choice drives the result.
• Run enough simulated paths that the tail percentile estimate is stable, not sensitive to the random seed.
• Prefer Monte Carlo when the portfolio holds options or other instruments with nonlinear payoffs.
• Validate simulated output against historical outcomes where data exists, rather than trusting the model blindly.

Common mistake: treating a Monte Carlo VaR figure as more trustworthy than a parametric one purely because the process involved thousands of simulations, when a poorly chosen underlying distribution can produce the same understated tail risk either way.

Choosing a Method: Why Parametric Breaks Down for Crypto

The three methods trade off speed, data requirements, and how faithfully they capture fat tails.

Parametric VaR is fast, requires only a volatility estimate, and is easy to explain — which is exactly why it became the industry default in the 1990s for diversified portfolios whose returns approximate a normal distribution reasonably well. That approximation breaks down for crypto assets, where single-day moves of 10% or more are common rather than exceptional; a normal distribution assigns vanishingly small probability to moves that size, so a parametric model calibrated on typical daily volatility will systematically understate both the frequency and magnitude of the worst days. Using parametric VaR on a fat-tailed asset without adjustment is one of the most consequential mistakes in applying this framework to crypto.

Historical simulation is generally the better-suited choice for crypto because it makes no distributional assumption at all — it uses whatever fat tails, sudden gaps, and asymmetric moves actually happened, constrained only by how much relevant price history exists. Monte Carlo simulation earns its computational cost when the portfolio includes derivatives or nonlinear payoffs, or when correlations across many assets need to be modeled explicitly; for a straightforward spot portfolio, that added complexity often isn't worth it relative to historical simulation.

Practical checklist

• Default to historical simulation for fat-tailed spot portfolios, crypto included, when sufficient price history exists.
• Reserve Monte Carlo simulation for portfolios with derivatives, options, or complex multi-asset correlations.
• Only use parametric VaR on crypto with an explicit adjustment for fat tails, or clear disclosure of the assumption's limits.
• Match the method to the data available, not to whichever method is fastest to compute.

Common mistake: applying an equity-style parametric VaR model to a crypto portfolio without adjustment, carrying over an assumption built for calmer, more normally distributed return series into an asset class that regularly violates it.

The Critical Limitation: What VaR Doesn't Tell You

VaR describes a threshold you are likely to stay within, not the severity of the losses that fall outside it.

This is the single most important thing to understand about Value at Risk, and it is the reason VaR has been criticized so sharply after major market events: a 95% VaR tells you the loss level you are likely to stay within on about 19 out of 20 days. It says nothing about what happens on that 20th day. The loss on the worst 5% of days could be barely larger than the VaR threshold, or several multiples of it — VaR, by construction, only measures where the tail begins, not how far it extends.

This blind spot is not a minor technicality — it is precisely the failure mode that has blindsided institutions relying heavily on VaR during past market crises: portfolios that looked well-contained by their daily VaR figures still experienced catastrophic losses because the tail was far fatter than the model implied, especially once correlations spiked during the crisis itself. A stable, unremarkable VaR number day after day can lull a manager into treating the metric as a complete risk picture, right up until a tail event the metric was never designed to size occurs.

This is exactly the gap Conditional VaR — also called Expected Shortfall — exists to close. Rather than identifying only the threshold, it estimates the average loss across the scenarios that fall beyond it, answering "given that we're in the worst 5% (or 1%) of outcomes, how bad does it tend to be on average?" That question mattered enough that the Basel Committee's post-2008 Fundamental Review of the Trading Book shifted regulatory capital calculations from VaR toward Expected Shortfall for this reason. A full treatment of how to calculate it belongs in its own discussion, but no VaR-based framework should be treated as complete without knowing this question is left unanswered by VaR alone.

Practical checklist

• Never describe a VaR figure as the "worst case" or a hard loss ceiling in any risk documentation.
• Pair every VaR figure with an explicit note about what it does not measure: tail severity beyond the threshold.
• Treat a stable VaR reading as reassurance about frequency, not about the size of a rare event should one occur.
• Consider Conditional VaR / Expected Shortfall as the next metric to add once a basic VaR process is in place.

Common mistake: reporting VaR as if it caps the portfolio's downside, when it only marks the boundary of the most common outcomes and says nothing about how severe the excluded tail could be.

Misconceptions Versus Reality

Common Mistakes, Risks, and Limitations

Most VaR errors come from misapplying the model's assumptions, not from arithmetic mistakes in the formula itself: using parametric VaR on fat-tailed assets like crypto without adjustment, and treating any VaR figure as a hard risk cap rather than the probabilistic statement it actually is, are the two most consequential.

• VaR does not measure or bound the severity of losses beyond its stated threshold — that is Conditional VaR's job, not VaR's.
• Historical simulation is bounded by whatever regimes appeared in the lookback window and cannot reflect a scenario that never occurred within it.
• Monte Carlo simulation's accuracy depends entirely on the chosen underlying model, which can reproduce parametric VaR's blind spots while appearing more sophisticated.
• The square-root-of-time scaling rule assumes constant volatility and no autocorrelation, both of which weaken over longer or more volatile horizons.
• Correlation inputs estimated from calm markets often understate how closely positions move together during broad stress, exactly when it matters most.
• None of the three VaR methods account for liquidity risk — the possibility a position cannot be exited at the assumed price during a stressed market.

Practical Implementation Checklist

1. State the portfolio value, confidence level, and time horizon before computing any VaR figure.
2. Choose the calculation method — parametric, historical simulation, or Monte Carlo — based on the asset's return distribution and available data, not convenience.
3. For parametric VaR, confirm the volatility input is current and its lookback window is documented.
4. For historical simulation, confirm the lookback window includes a representative range of market conditions where possible.
5. For multi-position portfolios, use a covariance-based calculation rather than summing individual position VaRs.
6. Scale any horizon change using the square root of the ratio of days, never a linear multiple.
7. Report the VaR figure alongside its confidence level and horizon, every time, without exception.
8. Add a Conditional VaR / Expected Shortfall estimate wherever tail severity, not just tail frequency, needs to be understood.
9. Re-run the calculation whenever position sizes, volatility, or correlation estimates change materially.
10. Document the calculation method and assumptions used so the figure can be checked later against what was actually known at the time.

Tool Opportunity

A dedicated Swoopr tool should calculate parametric, historical, and Monte Carlo VaR from a portfolio's real open positions and price history.

Recommended inputs: open position list with current market values, historical daily returns for each asset, a chosen confidence level and time horizon, a correlation or covariance estimate across positions, and a selectable calculation method.

Expected outputs: a 1-day and scaled multi-day VaR figure at both 95% and 99% confidence, a side-by-side comparison across the three calculation methods, an accompanying Conditional VaR / Expected Shortfall estimate, and a plain-language statement of what the figure does and does not represent.

Validation requirements: flag insufficient historical data for historical simulation, disclose the distributional assumption used for parametric or Monte Carlo calculations, warn when an asset's realized volatility diverges sharply from the volatility input used, and never present the VaR output as a maximum possible loss.

Frequently Asked Questions

Sources and Methodology

This guide's formula, terminology, and confidence-level conventions follow standard, widely cited risk-management references. Key sources include:

• RiskMetrics Technical Document (J.P. Morgan/Reuters, 4th edition, 1996): the methodology paper that established the variance-covariance approach now known as parametric VaR, including the standard use of 1.65 and 2.33 as the 95% and 99% Z-scores.
• Basel Committee on Banking Supervision, Minimum Capital Requirements for Market Risk (Fundamental Review of the Trading Book), Bank for International Settlements: the post-2008 regulatory framework describing the shift from VaR to Expected Shortfall for bank market-risk capital, motivated in part by VaR's blindness to tail severity.
• Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk: a standard academic and practitioner reference covering the parametric, historical simulation, and Monte Carlo approaches to VaR in detail.

The worked numeric examples on this page are illustrative calculations using the stated hypothetical inputs, not live figures for any real account or asset.

Conclusion

Value at Risk estimates the loss a portfolio is unlikely to exceed at a given confidence level over a given horizon — nothing more, and nothing less.

Compute it correctly, scale it correctly, and choose the method that fits the asset's actual return behavior — historical simulation over parametric for fat-tailed assets like crypto, absent a good reason otherwise. Then pair the figure with a clear-eyed understanding of what it deliberately leaves unanswered: how bad the excluded tail could be. That question is what portfolio volatility and drawdown metrics help contextualize, and what Conditional VaR exists to answer directly.

Related Reading

• Portfolio performance metrics hub — the parent guide this page belongs to.
• Portfolio volatility — the standard-deviation input that feeds directly into the parametric VaR formula above.
• Maximum drawdown duration — a complementary way to see how long a portfolio can stay underwater after a loss, beyond what a single VaR figure captures.
• Portfolio risk: correlation, concentration, drawdowns, and portfolio heat — the broader framework VaR fits into alongside heat and concentration checks.