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

When pricing financial assets, rational agents should think in terms of proportional price changes, i.e., returns. However, stock price movements are often reported in dollar rather than percentage units, which may cause investors to think that news should correspond to a dollar change in price rather than a percentage change in price. Non-proportional thinking in financial markets can lead to return underreaction for high-priced stocks and overreaction for low-priced stocks. Consistent with a simple model of non-proportional thinking, we find that total volatility, idiosyncratic volatility, and absolute market beta are significantly higher for stocks with low share prices, controlling for size. To identify a causal effect of price, we show that volatility increases sharply following stock splits and drops following reverse stock splits. The economic magnitudes are large: non-proportional thinking can explain a significant portion of the “leverage effect” puzzle, in which volatility is negatively related to past returns, as well as the volatility-size and beta-size relations in the data. We also show that non-proportional thinking biases reactions to news that is itself reported in nominal rather than scaled units. Investors react to nominal earnings per share surprises, after controlling for the earnings surprise scaled by share price. The reaction to the nominal earnings surprise reverses in the long run, consistent with correction of mispricing.