The Informational Content in Lepto-Variance and Its Relation to Higher Moments

1. Introduction and Main Findings*

• Lepto-regression constructs a regression tree of a target on itself; k-bit lepto-variance is residual variance after k steps, and lepto-ratio divides it by total sample variability.Macro-variance at depth k is the maximum variance explainable by any feature combination.Source: §1 · pp. 1–2
• 58% of historical US stock-return variability remains unexplained by a 1-bit regression tree, while 1-bit macro-variance explains 42%.At 2 bits, lepto-variance leaves 26.3% of total variance unexplained.Source: §1 · p. 2
• The study uses n = 100 normal samples to relate 1-bit lepto-variance and lepto-ratio to sample variance, skew, and excess kurtosis.A large-sample simulation examines the normal lepto-ratio’s convergence.Source: §1 · p. 2
• In n = 100 N(0, 1) samples, sample variance explains 16 bp of 36 bp total lepto-variance variability.Average lepto-variance is 0.35, with standard deviation 0.06 and values from 0.174 to 0.694.Source: §1 · p. 2
• For small normal samples, lepto-ratio is maximally correlated with excess kurtosis (0.78), averages 35%, and is orthogonal to sample skew.Lepto-ratio also correlates with lepto-variance; large-sample normal simulation shows it converging to 36.3%.Source: §1 · p. 2

2. Motivation

• Regression R-squared measures the share of target variability explained by included features, while residual variance depends on which financial factors are chosen.Residual variance is categorized as idiosyncratic risk, whose asset-pricing implications remain incompletely understood.Source: §2 · pp. 3–4
• Regression trees recursively split the sample on features to minimize residual target variance, assigning each terminal node its subset mean.Their predictions minimize residual MSE when new instances come from roughly the training sample’s distribution.Source: §2 · p. 3
• Regression trees are interpretable, non-parametric, and able to capture nonlinear relationships, but can overfit; pruning and ensembles address related modeling concerns.Random forests and gradient-boosted trees combine trees to improve predictive performance and robustness.Source: §2 · pp. 3–4
• Risk management depends on quantifying volatility and correlations across investable assets and understanding the sources of that volatility.A simple model-free method could assist investors with risk assessment and portfolio decisions.Source: §2 · pp. 4–5
• Financial researchers seek simpler, more intuitive model-free volatility methods because VIX calculations are described as neither simple nor intuitive.Investors use volatility to assess risk and inform portfolio construction and asset allocation.Source: §2 · p. 4

3. The Historical Lepto-Variance of US Stock Returns

• Sorted target splits minimize MSE and make lepto-regression an upper bound on tree-explainable variance; lepto-variance is the residual structure after self-regression.Total variance is decomposed into macro-variance, the explainable component, and lepto-variance, the residual component.Source: §3 · pp. 5–6
• The optimal 1-bit split divided market returns 30/70 at −0.264%, with child means near −1% and 0.5%.The smaller 30% subset was centered near −1%, and the larger 70% subset near 0.5%.Source: §3 · p. 6
• 58% of historical US stock-return variability remained unexplained by any 1-bit regression tree, while 1-bit macro-variance accounted for about 42%.The 1-bit lepto-variance was 0.678 of total variance 1.167; macro-variance was 0.489.Source: §3 · p. 6
• HML explained 0.0355 of total MSE, or 7.26% of 1-bit macro-variance, while SMB explained 0.025, or 5.11%.The comparison describes HML as slightly dominating SMB in explained variance.Source: §3 · p. 7
• At deeper tree depths, greedy splits can be locally optimal yet suboptimal for deeper variation capture in rare degenerate cases, particularly with small samples.For large samples and depths below roughly 3–4 splits, the distinction between average and maximum depth is described as unlikely to matter.Source: §3 · p. 7
• In the left subtree, 42% of its variability remained lepto after the second split; its 3.45% extreme-left leaf had residual MSE 1.968.The adjacent 26.5% subset had residual MSE 0.167; 0.373 of the left subtree's total MSE of 0.877 remained unresolved by the 2-bit tree.Source: §3 · p. 7
• In the right subtree, 47% of variance remained lepto; its 8.6% extreme-right leaf had residual MSE 1.393.The largest, 61.5% central subset had residual MSE 0.124.Source: §3 · pp. 7–8
• At two bits, lepto-variance was 26.3% of total variance in the historical sample.The reported 2-bit lepto-variance was 0.307 against total variance 1.167.Source: §3 · p. 8

4. The Lepto-Variance of the Uniform Distribution

• The continuous-uniform 25% lepto-ratio is related to the 6 dB-per-bit signal-to-noise result for uniform quantization.Source: §4 · p. 8
• Unlike quantization, lepto-regression determines each tree leaf’s support region from the sample rather than fixing it in advance.Quantization seeks efficient encoding for any message, whereas lepto-regression studies a specific sample.Source: §4 · pp. 8–9
• For the two-member set {0, 1}, the only split separates the values and leaves zero residual lepto-variance.That split produces a variance drop of 1/4.Source: §4 · p. 9
• For the equiprobable four-member set {0, 1, 2, 3}, the balanced split leaves residual variance 0.25 and yields a 20% lepto-ratio.The split produces a variance drop of 1 from total variance 1.25.Source: §4 · p. 9
• As the discrete-uniform sample size increases, the lepto-variance ratio converges to one-quarter of total variance, matching the continuous-uniform limit.The continuous-uniform 1-bit lepto-ratio is 25%.Source: §4 · pp. 8–9

5. The Lepto-Variance of Small Normal Samples

• Small N(0, 1) samples are used because large normal samples converge toward zero sample skewness and excess kurtosis, leaving little variability for correlation analysis.Source: §5 · p. 10
• The 1-bit lepto-variance is around 0.35, with standard deviation 0.06 and values from 0.174 to 0.694.Source: Table 2 · p. 10, §5 · p. 11
• Across 10,000 simulated n = 100 N(0, 1) samples, the 1-bit lepto-ratio averages 35.345%, with standard deviation 3.33% and a 23%–49% range.Source: Table 2 · p. 10, §5 · p. 11

6. Analysis of Lepto-Ratio

• Lepto-ratio is maximally correlated with excess kurtosis (0.78) and the 4th cumulant, is correlated with lepto-variance, and is orthogonal to sample variance and skew.Source: §6 · p. 11
• Lepto-ratio rises by an estimated 5.25% per unit of excess kurtosis, and the linear regression has R-squared 61.2%.The fitted intercept is the mean lepto-ratio value of 35.34.Source: §6 · p. 11, Figure 1 · p. 11

7. Regression Tree Analysis of Lepto-Ratio on Excess-Kurtosis

• The analysis fits trees predicting lepto-ratio from variance, cumulants, skew, excess kurtosis, and lepto-variance.Source: §7 · p. 12
• The 2-bit tree explains 6.5 bp of 11 bp total variability, leaving 4.5 bp residual MSE; excess kurtosis remains the strongest predictor at the second split.Source: Figure 2 · p. 12, §7 · p. 12
• The 1-bit tree explains 4.6 bp of 11 bp total variability, with residual MSE of 6.4 bp; excess kurtosis has the highest explanatory power.Squared error is reported in basis points, where 1 bp = 1% × 1%.Source: Figure 2 · p. 12, §7 · p. 12

• Lepto-variance correlates 0.83 with sample variance and 0.425 with excess kurtosis (and the 4th cumulant).Source: §8 · p. 13
• For small N(0, 1) samples, the least-square slope of lepto-variance on sample variance is 0.351.Source: §8 · p. 13
• The reported regression line is lepto = 0.0025 + 0.351var.Source: §8 · p. 13, Figure 3 · p. 13

• Sample variance has the highest explanatory power at both splits of the 2-bit tree, with balanced splits at 0.85, 1, and 1.15 and residual MSE of 14 bp.Source: Figure 5 · p. 14
• Across 10 simulations of n = 100,000 N(0, 1) samples, mean lR12 is 36.3% (std = 0.1%), ranging from 36.2% to 36.5%.Source: Table 3 · p. 15
• At n = 10,000,000, lepto-variance is always 36.3% of total sample variance.Source: Table 4 · p. 15

• For smaller N(0, 1) samples, lepto-variance is highly correlated with sample variance, whereas lepto-ratio is highly correlated with excess kurtosis.Source: §10 · p. 15
• Both lepto-variance and lepto-ratio are orthogonal to sample skew, and lepto-ratio is strongly correlated with lepto-variance but orthogonal to sample variance.Source: §10 · p. 15
• Large-sample simulation places the normal lepto-ratio convergence value at 36.3%; whether the orthogonality holds for skewed distributions remains unresolved.Source: §10 · p. 15