Data atlas / fourteen fits
Applications
Our financial system is full of fat tails. These are the market series that make the case most clearly, from daily Dow returns to the Swiss franc.
This page presents the best known financial time series fitted by the skew lognormal cascade distribution. The fat tails are clear under this microscope. A fat tail distribution needs at least 4,000 data points, or about 15 years of daily data. I would like to use the distribution on other real world data as well. If you have a series to try, let me know.
The table is sorted by kurtosis, with eta roughly following the same order. Kurtosis scales up with total duration and down with sampling interval. The mean is usually very small next to the width of the daily returns. Long term growth is almost negligible when compared with the daily fluctuation.
Sorted by kurtosis
Fat tails in familiar markets.
| Kurtosis | Asset class | Series | Chart |
|---|---|---|---|
| Above 20 | Stock index | Dow daily log return, 1928 to 2008 | |
| 10 to 20 | Stock index | Dow weekly log return, 1928 to 2008 | |
| 10 to 20 | Commodity | WTI oil spot price, 1986 to 2008 | |
| 10 to 20 | Commodity | Gold against the US dollar, 1972 to 2009 | |
| 10 to 20 | Stock index | Nasdaq Index, 1971 to 2009 | |
| 10 to 20 | Stock index | Dow monthly log return, 1928 to 2008 | |
| Near 10 | US stocks | Blue chip stock collection, 1980 to 2010 | |
| 5 to 10 | Stock index | Russell 2000, 1988 to 2008 | |
| 5 to 10 | Stock index | Nikkei 225, 1984 to 2009 | |
| 5 to 10 | Commodity | XAU Index, 1983 to 2008 | |
| 2 to 5 | Bond | 10 year Treasury, 1962 to 2008 | |
| 2 to 5 | Volatility | VIX, 1990 to 2008 | |
| 2 to 5 | Currency | Japanese yen against the US dollar, 1975 to 2008 | |
| 2 to 5 | Currency | Swiss franc against the US dollar, 1975 to 2008 |
Source shelf
Data and further reading
- Yahoo FinanceMarket data and historical quotes used for several series.
- Nasdaq CompositeOfficial index information for the Nasdaq Composite.
- Nikkei marketsJapanese market data and index information from Nikkei.
- FTSE Russell US indicesIndex information for the Russell 2000 and related benchmarks.
- Federal Reserve Economic DataEconomic time series from the Federal Reserve Bank of St. Louis.
- US TreasuryDaily Treasury yield curve and interest rate data.
- EIA WTI spot pricesWest Texas Intermediate spot price history.
- LBMA precious metal pricesLondon benchmark prices and market data.
- Cboe VIXOfficial volatility index information from Cboe.
- Cboe GVZThe Cboe dashboard for the gold volatility index.
- World Gold Council dataGold price data and market research.
- Lihn 2008 on SSRNThe 2008 working paper on the scale invariant Brownian motion model.
- Lihn 2009 on SSRNThe working paper on higher order lognormal cascade distributions.
- Clark 1973Subordinated stochastic process model for speculative prices.
- Fernholz 2002Stochastic portfolio theory and the diversity index.
- Heavy tailed distributionsA reference overview of distributions with heavy tails.
- Lognormal distributionDefinition and properties of the lognormal distribution.
- Stable distributionBackground on stable distributions and heavy tails.
- Stochastic portfolio theoryA reference overview of stochastic portfolio theory.