Noise, Capm and the Size and Value Effects

Abstract

We model a continuous time one factor economy where stock prices are noisy proxies of the informationally efficient values. The pricing error process is modeled as a mean-reverting process; this gives us a well defined notion of over-pricing and under-pricing in the market, and stocks fluctuates between being over-priced and under-priced. We show that in this economy, cap-weighting is a sub-optimal portfolio strategy. A non-cap-weighted portfolio outperforms a cap-weighted portfolio with the same factor exposure. This is because, in a cap-weighting scheme, portfolio weights are driven by market prices; as such, more weights are allocated to overvalued stocks and less weight to undervalued stocks. Following the same intuition, we show that in the cross-section, large cap stocks tend to underperform small cap stocks and high price-to-book stocks (growth stocks) tend to underperform low price-to-book stocks (value stocks). In this economy, if we proxy the hidden risk factor with the market clearing portfolio and regress stock returns against market returns (a CAPM regression), a non-cap-weighted portfolio exhibits significant alpha! However, when we regress the same non-cap-weighted portfolio against market, size and value, the Fama-French style regression exhibits factor loadings on value and size with insignificant alpha. Clearly, value and size are not risk factors in our economy! Value and size shows up in our regressions because the market portfolio is a poorly constructed proxy for the hidden risk factor. 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 effects 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.