What is the Hurst exponent, and how does it distinguish between mean-reverting, random, and trending time series?

The Hurst exponent (H) quantifies the degree of long-range dependence (or anti-persistence) in a time series. It reveals whether price movements tend to continue in the same direction (trending), reverse direction (mean-reverting), or move randomly.\n\nInterpretation:\n\n| Hurst Value | Behavior | Description | Trading Implication |\n|---|---|---|---|\n| H = 0.5 | Random walk | No memory; past returns don't predict future | No edge; passive investing |\n| H > 0.5 | Persistent / Trending | Up moves followed by up, down by down | Momentum strategies work |\n| H < 0.5 | Anti-persistent / Mean-reverting | Up moves followed by down, and vice versa | Mean-reversion strategies work |\n\nEstimation Method (R/S Analysis):\n\nThe Rescaled Range (R/S) method works as follows:\n\n1. Divide the time series into sub-periods of length n\n2. For each sub-period, compute the range R (max cumulative deviation minus min) and standard deviation S\n3. Average R/S across all sub-periods\n4. Repeat for different values of n\n5. The Hurst exponent is the slope of log(R/S) vs log(n)\n\nE(R/S) ~ c x n^H\n\nWorked Example:\n\nAnalyst Rowan at Ferndale Quant examines daily returns of two assets over 1,000 trading days:\n\n| Sub-period length (n) | log(n) | log(R/S) Asset A | log(R/S) Asset B |\n|---|---|---|---|\n| 10 | 1.00 | 0.65 | 0.40 |\n| 25 | 1.40 | 1.01 | 0.53 |\n| 50 | 1.70 | 1.27 | 0.61 |\n| 100 | 2.00 | 1.52 | 0.68 |\n| 250 | 2.40 | 1.87 | 0.78 |\n\nSlope for Asset A: (1.87 - 0.65) / (2.40 - 1.00) = 1.22 / 1.40 = H = 0.87 (strongly trending)\nSlope for Asset B: (0.78 - 0.40) / (2.40 - 1.00) = 0.38 / 1.40 = H = 0.27 (strongly mean-reverting)\n\nTrading Strategy Mapping:\n\nFor Asset A (H = 0.87): Rowan deploys a trend-following strategy:\n- Enter long when 20-day SMA crosses above 50-day SMA\n- Hold until the trend reverses\n- Expected Sharpe ratio improvement vs. buy-and-hold: +0.3\n\nFor Asset B (H = 0.27): Rowan deploys a mean-reversion strategy:\n- Buy when price drops 2 standard deviations below 20-day mean\n- Sell when it reverts to the mean\n- Expected Sharpe ratio improvement: +0.4\n\nLimitations:\n\n1. Non-stationarity: the Hurst exponent can change over time. A market may be trending in one regime and mean-reverting in another.\n2. Sample size sensitivity: reliable estimation requires long time series (1,000+ observations)\n3. Short-range vs long-range: R/S analysis can conflate short-range autocorrelation with true long-range dependence. DFA (Detrended Fluctuation Analysis) is a more robust alternative.\n4. Not a crystal ball: even a high Hurst exponent doesn't guarantee future trending behavior\n\nLearn more quantitative techniques in our FRM study resources.