Can the Market Multiply and Divide? Non-Proportional Thinking in Financial Markets

Nominal stock prices are arbitrary. Therefore, when evaluating how a piece of news should affect the price of a stock, rational investors should think in percentage rather than dollar terms. However, dollar price changes are ubiquitously reported and discussed. This may both cause and reflect a tendency of investors to think about the impact of news in dollar terms, leading to more extreme return responses to news for lower-priced stocks. We find a number of results consistent with such non-proportional thinking. First, lower-priced stocks have higher total volatility, idiosyncratic volatility, and market betas, after controlling flexibly for size. To identify a causal effect of price, we show that volatility increases sharply following pre-announced stock splits and drops following reverse stock splits. The returns of lower-priced stocks also respond more strongly to firm-specific news events, all else equal. The economic magnitudes are large: a doubling in a stock's nominal price is associated with a 20-30% decline in its volatility, beta, and return response to firm-specific news. These patterns are not exclusive to small, illiquid stocks; they hold even among the largest stocks. Non-proportional thinking can explain a variety of asset pricing anomalies such as long-run and short-run reversals, as well as the negative relation between past returns and volatility (i.e., the leverage effect). Our analysis also shows that the well-documented negative relation between risk (volatility or beta) and size is actually driven by nominal prices rather than fundamentals.