# A Stock Market Model Based on CAPM and Market Size

## TL;DR

This paper develops a stochastic differential equation model linking market beta and risk to size, using real market data, and reproduces the linearity of the capital distribution curve.

## Contribution

It introduces a novel model incorporating price and total returns differences and applies it to actual market data, enhancing understanding of size-dependent risks.

## Key findings

- Model fits size decile data well
- Reproduces linear capital distribution curve
- Price and equity premium modifications show similar results

## Abstract

We introduce a new system of stochastic differential equations which models dependence of market beta and unsystematic risk upon size, measured by market capitalization. We fit our model using size deciles data from Kenneth French's data library. This model is somewhat similar to generalized volatility-stabilized models in (Pal, 2011; Pickova, 2013). The novelty of our work is twofold. First, we take into account the difference between price and total returns (in other words, between market size and wealth processes). Second, we work with actual market data. We study the long-term properties of this system of equations, and reproduce observed linearity of the capital distribution curve. Our model has two modifications: for price returns and for equity premium. Somewhat surprisingly, they exhibit the same fit, with very similar coefficients. In the Appendix, we analyze size-based real-world index funds.