Skip to content
DP
Value at Risk: Methodsintermediate

Fat Tails and Skewness

Real financial returns are not normal. They show leptokurtosis, a kurtosis above the normal value of 3, meaning fat tails where extreme moves happen far more often than a bell curve predicts. Returns are also often negatively skewed, with crashes larger than rallies. Because parametric VaR assumes normality, it systematically understates the probability and size of large losses. This is the central reason historical-simulation and fat-tailed models exist, and why a normal-based VaR can look reassuring right up to a tail event.

Why it matters

A normal distribution treats a five-standard-deviation move as essentially impossible, yet markets deliver them every few years. Fat tails mean the rare disaster is not as rare as the bell curve claims, and negative skew means the disasters are losses, not windfalls. A VaR built on normality therefore draws its tail line too close to the centre, and the comforting number it reports is exactly the one that fails when it matters most. Kurtosis measures how heavy those tails are; skew measures which side they lean to.

Formulas

Excess kurtosis (fat tails)
Kurt(r)=E ⁣[(r−μ)4]σ4,excess=Kurt−3\mathrm{Kurt}(r) = \frac{E\!\left[(r - \mu)^4\right]}{\sigma^4}, \quad \text{excess} = \mathrm{Kurt} - 3
The normal has kurtosis 3. Excess kurtosis above 0 signals fat tails: more probability mass in extremes than the normal allows.
Skewness (asymmetry)
Skew(r)=E ⁣[(r−μ)3]σ3\mathrm{Skew}(r) = \frac{E\!\left[(r - \mu)^3\right]}{\sigma^3}
Negative skew means a longer left (loss) tail: large losses outweigh large gains, which normal-based VaR ignores.

Worked examples

Scenario

A normal model puts the probability of a daily move worse than −4σ-4\sigma at about 0.003 percent, roughly one day in 31,500. Equity indices have historically delivered such moves several times a decade. What does this say about normal-based VaR?

Solution

The empirical frequency of −4σ-4\sigma days is orders of magnitude higher than the normal 0.003 percent, direct evidence of fat tails (excess kurtosis). A 99% parametric VaR, calibrated to the bell curve, therefore understates both how often and how badly the tail is breached. Historical simulation or a fat-tailed (e.g. Student-tt) model would assign realistic, higher weight to these extremes.

Scenario

On Black Monday, 19 October 1987, the Dow Jones Industrial Average fell 22.6 percent in a single session. With a pre-crash daily standard deviation near 0.8 percent, this was roughly a 20-sigma move. How likely is such a day under the normal assumption, and what is the lesson for VaR?

Solution

Under a normal distribution a 20-sigma daily move has a probability so small it should not occur even once in many times the age of the universe, yet markets produced one in 1987. The crash is the textbook proof that return tails are vastly heavier than the bell curve allows, so a parametric VaR built on normality assigns essentially zero probability to the very event that wipes out a desk. The practical response is to lean on fat-tailed models, historical simulation, and stress testing rather than trust a Gaussian tail.

Common mistakes

  • ✗Financial returns are approximately normal. Empirically they are leptokurtic and often negatively skewed, so extreme losses occur much more frequently than a normal model implies.
  • ✗Fat tails only matter for exotic assets. Plain equity, FX, and bond returns all show excess kurtosis; fat tails are the rule, not the exception.
  • ✗Skewness and kurtosis measure the same thing. Skewness measures asymmetry (which tail is longer); kurtosis measures tail heaviness; a distribution can have one without the other.
  • ✗A high VaR already accounts for fat tails. If VaR is computed under normality, raising the confidence level does not repair the understated tail; the model itself is wrong in the extremes.

Revision bullets

  • •Returns are leptokurtic: kurtosis above the normal value of 3
  • •Fat tails mean extreme moves are more frequent than normal predicts
  • •Returns are often negatively skewed (crashes beat rallies)
  • •Parametric VaR under normality understates large losses
  • •Motivates historical-simulation and fat-tailed models

Quick check

Leptokurtosis (excess kurtosis) in returns means that, compared with a normal distribution,

Negative skewness in equity returns implies that

Connected topics

Parametric VaRHistorical SimVaR LimitsExpected Shortfall

Sources

  1. Jorion (2007), Ch. 5
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007. ISBN 978-0-07-146495-6.
    Documents fat tails and skewness in returns and their effect on normal-based VaR.
  2. Hull (2018), Ch. 12
    Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018. ISBN 978-1-119-44811-2.
    Discusses the failure of the normal assumption and the prevalence of heavy-tailed return distributions.
  3. Carver / CFA Institute (2012)
    CFA Institute. "Fact File: S&P 500 Sigma Events." Enterprising Investor, 2012.
    Documents the 19 October 1987 crash as an approximately 20-sigma daily move, far outside any Gaussian tail.
How to cite this page
Dr. Phil's Quant Lab. (2026). Fat Tails and Skewness. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-fat-tails-skew
← Back to the atlasSee in the network →