Fat tails describe the tendency of financial price changes to produce extreme moves far more often than a Gaussian, or normal, distribution predicts. Mandelbrot's 1963 paper, "The Variation of Certain Speculative Prices," analyzed daily and monthly cotton price changes from New York and the United States. In samples ranging from 200-odd to 1,000-odd observations across 1900-1905, 1944-58, and 1880-1940, he found an estimated exponent of 1.7 matching a stable Paretian distribution. That exponent implies price changes have infinite variance, undermining standard deviation as a measure of risk.
What the Study Found
Mandelbrot (1963) found that daily and monthly cotton price changes matched a stable Paretian distribution with an estimated exponent of 1.7, given delta=0 and beta=0. Near 1900, the ratio of T-day to 1-day scale coefficients for daily cotton price changes equaled 18 for both tails. The critical exponent a0, where the doubly logarithmic graph begins to overshoot its asymptote, fell in the neighborhood of 1.5. Each 15-year subsample of monthly cotton prices from 1816 to 1940 contained 200-odd months, with tails appearing straight and uniformly spaced except during 1880-96. In a refined model, the conditional distribution of L(t,1) given L(t-1,1) is asymptotically Paretian with an exponent of 2a+1.
Methodology
Mandelbrot (1963) tested unsmoothed daily and monthly cotton closing prices from New York and the United States. Samples ranged from 1,000-odd daily or monthly price changes to 15-year subsamples of 200-odd months each. Time periods examined were 1900-1905 and 1944-58 for daily changes, and 1880-1940 plus 1816-1940 for monthly changes. No explicit statistical controls were applied; unadjusted price-change series were compared directly against the theoretical stable Paretian curve.
Key Statistics
| Metric | Finding | Context |
|---|---|---|
| Stable Paretian exponent (a) | 1.7 (delta=0, beta=0) | Full sample: daily NY cotton changes 1900-1905, daily US cotton index 1944-58, monthly NY cotton changes 1880-1940 |
| Scale ratio C'(T)/C'(1) and C''(T)/C''(1) | 18 | Test period: daily cotton price changes, New York, near 1900 |
| Critical exponent a0 (asymptote overshoot threshold) | in the neighborhood of 1.5 | Full sample: numerical evaluation of symmetric stable Paretian densities |
| 15-year subsample size | 200-odd months | Test period: 1816-1940 stationarity test, monthly cotton price averages |
| Stability equation (Lévy characteristic function) | log f = i·delta·z − gamma·|z|^a[1+i·beta·(z/|z|)·tan(a·pi/2)] | Defines the stable Paretian family fitted to cotton price changes |
Why This Matters
The stable Paretian model implies infinite variance for price changes, meaning population moments beyond the first fail to converge. Least-squares forecasting and standard deviation-based risk measures both assume finite second moments, so neither retains its formal justification. Portfolio and risk models built on Gaussian assumptions may therefore understate the frequency of extreme market moves. Later research on power-law behavior in financial markets built directly on this framework.