Plotting daily stock returns against themselves with one day's lag reveals a striking pattern. Evenly spaced lines radiate from the origin; the thickest lines point in the major directions of the compass. This 'compass rose' pattern appears in every stock. It is caused by discreteness. However, counterexamples demonstrate that the existence of exchange-imposed tick sizes (e.g., eighths) is neither necessary nor sufficient for the compass rose. The compass rose cannot be used to make abnormal profits: it is structure without predictability. Among other consequences, the compass rose may bias estimation of ARCH models and tests for chaos.

Plotting daily stock returns against themselves with one day's lag reveals a striking pattern. Evenly spaced lines radiate from the origin; the thickest lines point in the major directions of the compass. This “compass rose” pattern appears in every stock. It is caused by discreteness. However, counter‐examples demonstrate that the existence of exchange‐imposed tick sizes (e.g. eighths) is neither necessary nor sufficient for the compass rose. The compass rose cannot be used to make abnormal profits: it is structure without predictability. Among other consequences, the compass rose may bias estimation of ARCH models, and tests for chaos.