Charta is on Google Play: practise on the go. Get the app
Charta

Give the reasons why return distributions have fat tails and outline what follows from them.

FRM® Part I · Valuation and Risk Models · Measuring and Monitoring Volatility · FRM1.VR.3.b

The idea

Fat tails mainly arise because the conditional distribution changes over time. If each day's return is roughly normal but the volatility itself moves between calm and turbulent levels, the unconditional distribution is a mixture of normals with different variances, and any such mixture with a common mean has excess kurtosis. Other causes are jumps from news and announcements, shifts between market regimes, and leverage and forced selling in stress. Financial data also keep producing extremes, even in large samples, rather than converging to a normal. The consequences for a risk manager are practical. A normal VaR calibrated to the standard deviation understates losses at 99% and beyond, and extreme events are far more frequent than normal tables suggest. Tail risk needs stress testing, heavier-tailed distributions, historical or simulated tails, or models of time-varying volatility. Scaling a lower-confidence quantile by normal z ratios is the classic error.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which explanation best accounts for fat tails in the unconditional distribution of daily returns?

  1. Returns are sums of many independent shocks, so the central limit theorem makes the tails heavy.Why not: Misapplies the central limit theorem, which pushes sums toward a normal distribution, not toward fat tails.
  2. Returns are measured as logarithms, and log returns are inherently fat-tailed.Why not: Log and simple returns are almost identical at daily horizons; taking logs does not create fat tails.
  3. Samples are small, and with enough observations the tails always converge to those of a normal.Why not: Fat tails persist in very large samples; they are a feature of the process, not sampling error.
  4. Volatility varies through time, so the unconditional distribution mixes calm days and turbulent days. - correct

Even if each day were normal given that day's volatility, changing volatility makes the unconditional distribution a mixture of normals with different variances, which has excess kurtosis. Jumps and regime shifts add to this.

A harder one

A risk manager estimates the 99% one-day VaR of a portfolio at USD 2.00 million from the empirical distribution, then approximates the 99.9% VaR by scaling with the ratio of normal quantiles (3.090 / 2.326). Which figure results and what is the likely bias?

  1. About USD 2.66 million, and the true 99.9% VaR is probably lower.Why not: Gets the arithmetic right but reverses the bias; fat tails make extreme quantiles larger than the normal ratio implies.
  2. About USD 2.66 million, and the true 99.9% VaR is probably higher. - correct
  3. About USD 1.51 million, and the true 99.9% VaR is probably higher.Why not: Inverts the ratio of quantiles (2.326 / 3.090), producing a 99.9% VaR below the 99% VaR.
  4. About USD 6.18 million, and the true 99.9% VaR is probably higher.Why not: Multiplies by the 99.9% z-value (3.090) instead of the ratio of the two z-values.

Normal scaling gives 2.00 x 3.090 / 2.326 = 2.66 million. Because tails are fatter than normal, the quantile grows faster than the normal ratio as confidence rises, so the figure is likely too low.

The hardest tier

Daily returns come from a mixture of two zero-mean normal distributions: with probability 0.8 the standard deviation is 1%, with probability 0.2 it is 3%. For a zero-mean normal with variance s2, the fourth moment is 3 x s2². What is the kurtosis (not excess kurtosis) of the unconditional distribution?

  1. 7.54 - correct
  2. 4.54Why not: Reports excess kurtosis (kurtosis minus 3) rather than kurtosis.
  3. 2.51Why not: Omits the factor 3 in the normal fourth moment when computing the mixture's fourth moment.
  4. 3.00Why not: Assumes a mixture of normals is itself normal, so kurtosis stays at 3.

Fourth moment = 3 x (0.8 x 1² + 0.2 x 9²) = 3 x 17 = 51. Variance = 2.6, so variance² = 6.76. Kurtosis = 51 / 6.76 = 7.54, well above 3, even though each regime is normal.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Measuring and Monitoring Volatility