Parametric vs Non-Parametric VaR

Key Takeaways

• Parametric VaR fits returns to a distribution, usually the normal, and reads the loss straight off a formula using volatility and a z-score.
• Non-parametric VaR, most commonly historical simulation, ranks actual past returns and reads the loss off a chosen percentile, assuming no shape at all.
• Because the normal curve has thin tails, parametric VaR tends to understate losses when real returns are fat-tailed; historical VaR keeps whatever tail the data contained.
• Parametric VaR is fast and scales to large portfolios; historical VaR is more faithful to real distributions but depends entirely on the length and relevance of its sample window.

What It Is

Parametric VaR (also called the variance-covariance or analytical method) assumes returns follow a known probability distribution, typically the normal. Once you accept that shape, the loss at a confidence level is a closed-form expression: multiply portfolio volatility by the z-score for that confidence and by portfolio value.

Non-parametric VaR makes no assumption about the distribution. The dominant version is historical simulation: collect the actual periodic returns over a lookback window, sort them from worst to best, and take the loss at the percentile matching your confidence level. Nothing is fitted; the empirical distribution is the model.

The Intuition

A parametric model says, "tell me the volatility and I will draw you the whole curve." That is powerful, but the curve it draws is smooth and symmetric, and its tails shrink fast. Markets do not behave that way. Crashes cluster and outliers arrive more often than a bell curve predicts.

A non-parametric model says, "I will not guess the shape; here is exactly what happened." If the last year held a brutal down day, that day sits in the sample and shows up in the tail. The cost is the mirror image of the benefit: if the window contains no stress, the model has never seen one either.

How It Works

Parametric VaR at confidence c over one period:

• VaR = z * sigma * V, where z is the standard-normal quantile (1.645 for 95%, 2.326 for 99%), sigma is the return standard deviation, and V is portfolio value. A nonzero expected return can be subtracted, but over short horizons it is usually dropped.

Historical (non-parametric) VaR:

1. Gather n actual returns over the lookback window.
2. Sort them from worst to best.
3. The 95% VaR is the loss at the 5th percentile, found at observation (1 - c) * n. With 250 days at 99%, that is the 2.5th worst return, interpolated between the 2nd and 3rd worst.

The parametric route needs only two inputs, volatility and a z-score. The historical route needs the full return series and inherits its skew and kurtosis for free.

Worked Example

A $1,000,000 portfolio, one-day horizon, 99% confidence.

Parametric. Assume a zero mean daily return and a daily standard deviation of 1.5%. The 99% z-score is 2.326.

• VaR = 2.326 * 0.015 * 1,000,000 = $34,890.

The model says there is a 1% chance of losing more than about $34,900 tomorrow.

Historical. Take the last 250 trading days of actual returns and sort them. The 1% tail sits at the 2.5th worst observation. Suppose the three worst days returned -4.8%, -3.9%, and -3.6%. Interpolating between the 2nd and 3rd worst gives a tail cutoff near -3.75%.

• VaR = 0.0375 * 1,000,000 = $37,500.

Same portfolio, two numbers: $34,900 parametric versus $37,500 historical. The historical figure is larger because the actual return series had a fatter left tail than the normal curve allows. The normal assumption quietly trimmed the tail; the empirical data did not. That $2,600 gap is precisely the information the comparison is meant to expose.

Common Mistakes

1. Assuming normality without checking. Parametric VaR is only as good as its distribution. If returns are skewed or fat-tailed, the number understates risk. Test the assumption before trusting it.
2. Treating a short lookback as safe. Historical VaR can only report losses the window contains. A calm 250-day sample produces a low VaR right up until a regime it never saw.
3. Confusing VaR with a worst case. Neither method says anything about the size of losses beyond the cutoff. That is the job of expected shortfall, not VaR.
4. Comparing figures across confidence levels or horizons. A 95% one-day VaR and a 99% ten-day VaR are different animals; scaling one to the other requires care.
5. Ignoring volatility clustering. Equal-weighted historical VaR treats a 2008 day and a quiet 2017 day the same. Recent volatility often matters more.

Frequently Asked Questions

Q: What is the core difference in parametric vs non-parametric var? Parametric VaR assumes returns follow a distribution, usually the normal, and computes the loss from volatility and a z-score. Non-parametric VaR, such as historical simulation, makes no distributional assumption and reads the loss directly from ranked actual returns.

Q: When should I choose parametric vs non-parametric var? Use parametric VaR when returns are roughly normal and you need speed across a large book. Use non-parametric VaR when returns are skewed or fat-tailed and you have a long, representative sample of actual data.

Q: Why does parametric VaR often understate risk? The normal distribution has thin tails, so it assigns tiny probabilities to extreme moves that markets actually produce more often. This means the analytical formula tends to report a smaller loss than the true tail warrants.

Q: Is Monte Carlo VaR parametric or non-parametric? It sits between the two. Monte Carlo VaR simulates returns from a model you specify, so it depends on assumptions like parametric VaR, but it can use non-normal distributions and complex payoffs that a closed-form formula cannot handle.

Q: Does historical VaR need a distribution assumption? No. That is the defining feature of the non-parametric approach. It uses the empirical distribution of past returns directly, keeping whatever skew and kurtosis the sample contained rather than imposing a curve.

Sources

1. Investopedia. "Value at Risk (VaR)." https://www.investopedia.com/terms/v/var.asp
2. Investopedia. "Variance-Covariance Method." https://www.investopedia.com/terms/v/variance-covariance-method.asp
3. Investopedia. "Historical Simulation." https://www.investopedia.com/terms/h/historical-simulation.asp
4. Basel Committee on Banking Supervision. "Minimum Capital Requirements for Market Risk." https://www.bis.org/bcbs/publ/d457.htm

Disclaimer

This article is educational content only and is not financial advice. Nothing here is a recommendation to buy, sell, or hold any security. Consult a licensed advisor before making investment decisions.