We compare performance of US stocks based on their size (market capitalization). We regress alpha and beta over sizeand other factors for individual stocks in Standard Poor 500, and for randomly generated portfolios. In addition, we compare exchange-traded funds (ETFs) consisting of large-, mid- and small-cap stocks , including international ETFs. Conclusions: Size and market exposure (beta) are inversely related (strong evidence for ETFs and portfolios, weaker evidence for individual stocks) .
 A rank-based model of competing Brownian particles, introduced in (Banner, Fernholz, Karatzas, 2005), captures the dependence of stock dynamics on size: Small stocks have higher growth rate but higher volatility than large stocks, on average. However, in this model, geometric returns of stocks have Gaussian tails. We modify this model using Capital Asset Pricing Model: The market exposure depends on the rank, and the benchmark moves as a general L\'evy process. We show that this model fits real market data. We discuss stability and long-term convergence .
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We compare performance of US stocks based on their size (market capitalization). We regress alpha and beta over sizeand other factors for individual stocks in Standard
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Poor 500, and for randomly generated portfolios. In addition, we compare exchange-traded funds (ETFs) consisting of large-, mid- and small-cap stocks , including international ETFs. Conclusions: Size and market exposure (beta) are inversely related (strong evidence for ETFs and portfolios, weaker evidence for individual stocks)
A rank-based model of competing Brownian particles, introduced in (Banner, Fernholz, Karatzas, 2005), captures the dependence of stock dynamics on size: Small stocks have higher growth rate but higher volatility than large stocks, on average. However, in this model, geometric returns of stocks have Gaussian tails. We modify this model using Capital Asset Pricing Model: The market exposure depends on the rank, and the benchmark moves as a general L\'evy process. We show that this model fits real market data. We discuss stability and long-term convergence