On the Fractal Geometry of the Balance Sheet and the Fractal Index of Insolvency Risk

TL;DR

This paper introduces a novel fractal-based measure of insolvency risk, the Firm Insolvency Risk Index (FIRI), which captures asset risk properties and allows for scale-invariant comparisons across firms.

Contribution

It develops a fractal geometry framework for analyzing balance sheets and proposes a new risk index that is symmetrical, proportional, and scale invariant.

Findings

The balance sheet exhibits fractal patterns.

The fractal index effectively measures asset risk dissimilarity.

FIRI serves as a risk thermometer bounded between 0 and 2.

Abstract

This paper reviews the economic and theoretical foundations of insolvency risk measurement and capital adequacy rules. The proposed new measure of insolvency risk is constructed by disentangling assets, debt and equity at the micro-prudential firm level. This new risk index is the Firm Insolvency Risk Index (FIRI) which is symmetrical, proportional and scale invariant. We demonstrate that the balance sheet can be shown to evolve with a fractal pattern. As such we construct a fractal index that can measure the risk of assets. This index can differentiate between the similarity and dissimilarity in asset risk, and it will also possess the properties of being self-similar and invariant to firm characteristics that make up its asset composition hence invariant to all types of risk derived from assets. Self-similarity and scale invariance across the cross section allows direct comparison of degrees of risk in assets. This is by comparing the risk dissimilarity of assets. Being naturally bounded to its highest upper bound, (0,2], the fractal index is able to serve like a risk thermometer. We assign geometric probabilities of insolvency P (equity is equal or less than 0 conditional on debt being greater than 0).