How is VaR actually calculated?

Three methods dominate practice. They answer the same question with different assumptions, and — this is the useful part — they disagree in informative ways. When two methods diverge on the same portfolio, the divergence is telling you something about the shape of the distribution.

Method 1 — Historical simulation

Take the actual returns of the last N days. Sort them. Read off the percentile. That's the method from lesson one.

What it assumes: the future will resemble this particular slice of the past.

Strengths: no distributional assumption at all. Whatever fat tails, skew and crash days lived in the window are automatically included. It is the easiest to explain to a non-specialist.

Weaknesses: it can only produce losses it has already seen. A 500-day window that happens to exclude every crisis will report a comfortable number right up until the crisis arrives. And every day, the oldest observation silently drops out of the window — so a big loss can vanish from your risk estimate on a calendar technicality rather than an economic one.

Method 2 — Parametric (variance-covariance)

Assume returns follow a normal distribution. Then you only need one number — the standard deviation — and the formula is arithmetic:

VaR = z × σ × Portfolio value

For a $10 million portfolio with a daily standard deviation of 1.2%:

• 95%: 1.645 × 1.2% × $10,000,000 = $197,400
• 99%: 2.326 × 1.2% × $10,000,000 = $279,120

Strengths: fast, transparent, scales to thousands of positions with a covariance matrix, easy to decompose by desk or asset.

Weaknesses: the normal assumption. Real return distributions have far more extreme days than a bell curve allows, so parametric VaR systematically understates deep-tail risk. It also handles options poorly, because option payoffs are curved and this method treats everything as linear.

Method 3 — Monte Carlo simulation

Specify a model for how the risk factors move — volatilities, correlations, and a chosen distribution — then generate ten or a hundred thousand random future scenarios, revalue the portfolio in each, and take the percentile of the simulated loss distribution.

Strengths: handles non-linear instruments (options, structured products, path-dependent payoffs) that break the parametric method. You can feed it fat-tailed distributions instead of a normal one.

Weaknesses: computationally heavy, and — the important one — it is only as good as the model you specified. Monte Carlo produces enormous quantities of output that all inherit whatever assumption you typed in at the start. Precision is not accuracy.

Reading the disagreement

Run all three on one portfolio and you might get:

• Parametric 95%: $197,000
• Historical 95%: $240,000
• Monte Carlo 95% (fat-tailed): $255,000

The parametric figure is the lowest. That's not a bug — it's the normal assumption quietly trimming the tail. The gap between parametric and historical is itself a measurement: it tells you how far the real data departs from the bell curve. A wide gap is a flag that the parametric number should be treated as a floor, not an estimate.

Professionals rarely pick one method. They run several and pay attention to when the answers separate.

In the data

The parametric method's sigma is easy to pick up in the wrong units. Charting tools publish a "standard deviation" of the price itself; here is one for the S&P 500 fund, over the fifty sessions to the end of 2025:

That figure is in dollars. The price wandered that far around its fifty-day average, on a close of $681.92 that day, and the wandering includes the trend as well as the noise. The formula wants the standard deviation of daily returns, in per cent, which is the 1.032% in the table below. Pass it the price figure and it produces a number with no meaning.

Try it now

1. The window is 2 January 2024 to 31 December 2025 for the S&P 500 fund: 501 daily returns built from adjusted closes, not a standard deviation of prices. Five hundred returns are not a hand calculation, so the two inputs are below, computed on 28 September 2026 for the index fund and for Apple. Multiply the S&P 500 fund's standard deviation by 1.645 and note that parametric 95% VaR in percent.

2. Compare it to the historical 95% figure, the 25th worst day in the table. Which is larger, and by how much?
3. Repeat both for a single volatile name rather than an index, the Apple row. Does the gap between the two methods widen or narrow, in points and as a share of the historical figure? Describe what you see; don't conclude which method is "right."