The Lepto-Variance of Stock Returns

TL;DR

This paper introduces the concept of lepto-variance to quantify the residual variance in stock returns that cannot be explained by regression trees of a given depth, providing insights into the structure of stock return data.

Contribution

It defines the k-bit lepto-variance as an upper bound on variance explained by regression trees, offering a new statistical measure for analyzing stock return structures.

Findings

Lepto-variance effectively quantifies unexplained variance in stock returns.

Regression tree depth limits the variance explanation, as captured by lepto-variance.

Application to IBM stock returns demonstrates the concept's practical utility.

Abstract

The Regression Tree (RT) sorts the samples using a specific feature and finds the split point that produces the maximum variance reduction from a node to its children. Our key observation is that the best factor to use (in terms of MSE drop) is always the target itself, as this most clearly separates the target. Thus using the target as the splitting factor provides an upper bound on MSE drop (or lower bound on the residual children MSE). Based on this observation, we define the k-bit lepto-variance ${\lambda}k^2$ of a target variable (or equivalently the lepto-variance at a specific depth k) as the variance that cannot be removed by any regression tree of a depth equal to k. As the upper bound performance for any feature, we believe ${\lambda}k^2$ to be an interesting statistical concept related to the underlying structure of the sample as it quantifies the resolving power of the RT for the sample. The max variance that may be explained using RTs of depth up to k is called the sample k-bit macro-variance. At any depth, total sample variance is thus decomposed into lepto-variance ${\lambda}^2$ and macro-variance ${\mu}^2$. We demonstrate the concept, by performing 1- and 2-bit RT based lepto-structure analysis for daily IBM stock returns.