Noise, CAPM and the Size and Value Effects

Robert Arnott and Jason Hsu

Volume 6, Number 1, First Quarter 2008

We model a continuous time one factor economy where stock prices are noisy proxies of the informationally efficient stock values. The pricing error process is modeled as a mean-reverting process, which gives us a well-defined notion of over-pricing (positive pricing error) and under-pricing (negative pricing error) in the market. We show that in this economy, cap-weighting is a sub-optimal portfolio strategy. This is because, in a capweighting scheme, portfolio weights are driven by market prices; as such, more weights are allocated to over-valued stocks and less weight to under-valued stocks.

More importantly, we show that the CAPM would be rejected in this one factor economy with noise. Regressing portfolio returns against market clearing portfolio returns, non-capweighted portfolios exhibit significant alpha on average!

Additionally, a value tilted or size tilted portfolio is predicted to outperform(risk-adjusted). By construction, value and size are not risk factors in our one factor economy. However, in the cross-section, large cap stocks and high price-to-book stocks (growth stocks) tend to underperform. This is because higher capitalization stocks and higher price-to books stocks are indeed more likely to be stocks with positive pricing errors.

We note that prices are explicitly inefficient in our economy. However, the inefficiency does not lead to arbitrage opportunities. We carefully show conditions which prevent arbitrage in our informationally inefficient economy.

The paper contributes to the anomalies literature by showing that mean-reversion in stock returns and the Fama-French size and value effects are driven by the same market defect pricing noise! This suggests that models, such as disposition effect and information herding, which can generate stock price over-reaction and therefore mean-reversion in stock prices, can also explain the value and size puzzle.