This post is based on Chapter 10 of Statistical Consequences of Fat Tails by Nassim Nicholas Taleb. The chapter emphasizes practical diagnostics to test if data behaves as if it comes from thin-tailed (Gaussian-like) distributions, or if it exhibits fat-tailed characteristics — a major issue in finance.
Here, we apply similar tests on S&P 500 data, to explore if the behavior aligns more with thin-tailed or fat-tailed assumptions.
Key Diagnostics from the Chapter:
- Log-return behavior and survival functions
- Conditional expectations
- Zipf and Pareto-style plots
- Max drawdowns
- Record-breaking behavior
- Mean-stability and kurtosis aggregation
Below each chart is a brief explanation of its diagnostic value.
SnP500 Price at Close
This chart shows the raw time series of SnP500 closing prices.
Daily Log Returns
Log returns are calculated as log(P_t / P_{t-1}). This transforms multiplicative changes into additive and helps us analyze the distribution better.
Drawdown Charts
Maximum drawdown is defined as the maximum observed loss from a peak to a trough over a given time window. It captures tail risk that standard deviation misses.
We plot drawdowns for rolling windows of 252, 100, 30, and 5 trading days:
Zipf/Pareto Survival Plot
Here we plot the log-log survival function for absolute log returns:
Survival function: ( P(|X| > x) )
Comparing SnP500 with Gaussian simulations shows whether tails are heavier than the normal distribution.
Conditional Expectation
\[ \frac{\mathbb{E}[-X \mid -X > K]}{K} \] This is used to measure the expected size of losses conditional on exceeding a threshold ( K ). In thin-tailed distributions, this converges to a constant.
Fat tails show divergence, i.e., large negative values dominate the average.
Positive vs Negative Return Survival Functions
We analyze survival probabilities for both positive and negative returns, over aggregation windows from 1 to 15 days. It checks for asymmetry between gains and losses.
QQ Plot against Student-t Distribution
QQ plots compare empirical quantiles with theoretical quantiles of a reference distribution. Here, we use a Student-t with 3 degrees of freedom.
Deviation from the diagonal line suggests fat-tailed behavior.
Aggregated Kurtosis vs Time Window
Kurtosis measures the weight of tails. We plot how kurtosis behaves as we aggregate returns over longer intervals. In thin tails, kurtosis should decrease and stabilize.
Rolling Kurtosis
The 4th moment of returns computed in a rolling 250-day window. Helps detect time-localized fat-tail behavior.
Mean Stability
Running average of log returns. Thin-tailed distributions converge quickly to a stable mean. Slower convergence implies higher variance and/or fat tails.
Record Counts vs Expected (log n)
This test checks if the number of record-breaking highs and lows in returns align with expectations under a Gaussian process (which predicts ( \log(n) ) growth).