# Projection pursuit based generalized betas accounting for higher order co-moment effects in financial market analysis

## TL;DR

This paper introduces a novel method for calculating generalized betas that incorporate higher order moments, enhancing financial market analysis by capturing skewness and kurtosis effects while remaining interpretable.

## Contribution

It proposes the co-moment analysis projection index (CAPI) within the projection pursuit framework to compute higher-order moment-aware betas, including a robust version using trimmed means.

## Key findings

- Simulation confirms the statistical properties of the projections.
- The method effectively captures higher order moment effects in financial data.
- A small example illustrates practical application and interpretability.

## Abstract

Betas are possibly the most frequently applied tool to analyze how securities relate to the market. While in very widespread use, betas only express dynamics derived from second moment statistics. Financial returns data often deviate from normal assumptions in the sense that they have significant third and fourth order moments and contain outliers. This paper targets to introduce a way to calculate generalized betas that also account for higher order moment effects, while maintaining the conceptual simplicity and interpretability of betas. Thereunto, the co-moment analysis projection index (CAPI) is introduced. When applied as a projection index in the projection pursuit (PP) framework, generalized betas are obtained as the directions optimizing the CAPI objective. A version of CAPI based on trimmed means is introduced as well, which is more stable in the presence of outliers. Simulation results underpin the statistical properties of all projections and a small, yet highly illustrative example is presented.