Capital Asset Pricing Model with Size Factor and Normalizing by Volatility Index

Abstract

The Capital Asset Pricing Model (CAPM) relates a well-diversified stock portfolio to a benchmark portfolio. We insert size effect in the CAPM, capturing the observation that small stocks have higher risk and return than large stocks, on average. Our goal is to make the resulting linear regressions have independent identically distributed Gaussian residuals. In some cases, we find that including the Volatility Index as a multiplicative factor by these residuals makes them closer to that ideal. In this article, we combine these ideas to create a new discrete-time model, which includes volatility, relative size, and the CAPM. We fit this model using real-world data, prove the long-term stability, and connect this research to Stochastic Portfolio Theory.

18 pages, 2 tables, 4 figures. Keywords: Capital Asset Pricing Model, stochastic volatility, ergodic Markov process, stationary distribution, size effect, autoregression, capital distribution curve