How to use the VaR calculator
Parametric (normal) gives a quick estimate from volatility alone:
- Portfolio value: the current market value of the positions.
- Annual volatility: the annualized standard deviation of portfolio returns, from your own return history, a risk system, or the portfolio volatility calculator.
- Confidence level and horizon: 95% over 1 day is common for internal risk reporting; 99% over 10 days is the classic regulatory setting.
- Expected annual return is optional. Leave it empty to assume a zero mean, which is standard for short horizons.
Historical uses your own data. Paste daily portfolio returns in percent (a column from a spreadsheet works), then enter the portfolio value and confidence level. The example contains 250 days, about one trading year, of a fictional diversified stock portfolio.
How VaR and CVaR are calculated
The parametric tab assumes returns over the horizon are normally distributed:
VaR = V × (zc σh − μh)
CVaR = V × (σh φ(zc) ÷ (1 − c) − μh)V is the portfolio value, σ the annual volatility, μ the expected annual return, h the horizon in trading days and c the confidence level. zc is the standard normal quantile (2.326 at 99%) and φ the standard normal density. The quantile is computed with Acklam's rational approximation, accurate to about one part in a billion.
The historical tab makes no assumption about the distribution:
Expected shortfall = −V × average of all ri ≤ q1−c(r)q1−c is the (1 − c) percentile of the daily returns ri, interpolated linearly between neighboring observations (the method of Excel's PERCENTILE.INC). The result is a 1-day figure because the data are daily.
A worked example
Parametric. A $1,000,000 portfolio has 18% annual volatility. Over 10 trading days the volatility is 3.59%, and the 99% quantile sits 2.326 standard deviations below the mean. The 10-day 99% VaR is $83,416 (8.34%). On the 1% of periods that are worse, the average loss, the CVaR, is $95,566.
Historical. The 250 daily returns in the example have an annualized volatility of 15.42%. At 95% confidence, the 1-day historical VaR is $15,230 (1.52%): 13 days in the sample lost more. Those worst days averaged 2.40%, an expected shortfall of $23,992.
VaR and CVaR answer different questions
VaR tells you where the tail begins. CVaR tells you what the tail contains. Two portfolios with the same VaR can have very different CVaR if one of them occasionally suffers much larger losses, for example because it sells options or holds illiquid credit.
Under the normal model the gap is fixed: at 99% CVaR is 1.15× VaR, at 95% it is 1.25×. Real data have deeper tails. In the historical example the 95% expected shortfall is 2.40% against a VaR of 1.52%, a much wider gap than a normal distribution would give.
CVaR also has a property VaR lacks: subadditivity. The CVaR of a combined portfolio never exceeds the sum of the parts’ CVaRs, so diversification is always rewarded. VaR can violate this for portfolios with lumpy, skewed payoffs, which is why it is not a coherent risk measure in the sense defined by Artzner, Delbaen, Eber and Heath in 1999.
Historical against normal: the fat-tail problem
The example data come from a portfolio with ordinary volatility but occasional sharp drops. Their excess kurtosis is 4.66 and their skewness -0.834; a normal distribution has zero for both. That shows up as soon as you look past the 95% level:
| Confidence | Historical VaR | Normal VaR on the same data | Historical ES | Normal ES |
|---|---|---|---|---|
| 95% | 1.52% | 1.54% | 2.40% | 1.95% |
| 97.5% | 1.79% | 1.85% | 3.07% | 2.22% |
| 99% | 3.11% | 2.21% | 4.21% | 2.53% |
At 95% the two methods almost agree. At 99% the historical VaR is 1.41× the normal figure. This pattern is typical of financial returns: the normal model is roughly right in the body of the distribution and too optimistic in the far tail, exactly where VaR is supposed to warn you.
Historical VaR has its own weakness at high confidence. A 99% VaR from 250 days rests on the two or three worst observations, so a single day entering or leaving the window can move it sharply. Longer samples help but bring in stale market conditions; many risk teams weight recent days more heavily or combine a long sample with a volatility adjustment.
When square-root-of-time scaling fails
The parametric tab scales volatility by √h, so 4 times the horizon doubles the VaR. That rule relies on returns being independent and identically distributed with constant volatility. It breaks down when:
- Volatility clusters. After a calm period, √h understates the risk of a turbulent month; after a crash, it can overstate it once markets settle.
- Returns are autocorrelated. Trending markets, smoothed valuations or illiquid assets produce multi-day losses larger than √h suggests.
- Tails change with the horizon. Fat tails in daily data become less extreme over longer horizons, so scaling a daily historical quantile by √h distorts the long-horizon figure.
- The portfolio changes. A 10-day VaR assumes positions stay fixed for 10 days; stop-losses, rebalancing and margin calls make that untrue.
- The mean matters. Over a year, expected return is no longer negligible. With an 8% expected return, the 1-year 99% VaR is $338,743 instead of $418,743.
From VaR to expected shortfall: the Basel change
Bank capital rules for market risk relied for two decades on a 99%, 10-day VaR. The 2008 crisis exposed its blind spot: VaR ignores how bad losses are beyond the threshold, and models calibrated to calm years understated risk just before markets broke.
In January 2016 the Basel Committee published the Fundamental Review of the Trading Book (FRTB), later revised in 2019. Among many changes, it replaces 99% VaR with expected shortfall at 97.5% for internal models, calibrated to a period of significant market stress, with liquidity horizons that vary by risk factor instead of a uniform 10 days. Implementation dates have been pushed back in several jurisdictions.
The 97.5% level was not chosen at random. For a normal distribution, 97.5% expected shortfall is almost the same number as 99% VaR: here $83,826 against $83,416. For fat-tailed return distributions, such as a Student-t, 97.5% expected shortfall comes out above 99% VaR, and the heavier the tail, the larger the gap. The new measure therefore charges more for exactly the risk the old one ignored: the size of losses beyond the threshold.
What this calculator does not include
- Volatility forecasting. The parametric tab takes volatility as given; it does not model clustering (GARCH) or use implied volatility.
- Position-level risk. It works on total portfolio returns. To see which holdings drive the risk, use VaR attribution.
- Non-linear positions. Options and other convex payoffs need full revaluation or Monte Carlo simulation, not a normal approximation.
- Scenarios outside the data. Historical VaR cannot see a crash larger than the worst day in the sample. A stress test fills that gap.
- Arithmetic returns. The normal model uses simple returns, which can in principle fall below −100% at very long horizons or very high volatility.