Robust Structure Without Predictability: The “Compass Rose” Pattern of the Stock Market




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    • Crack is from the School of Business, Indiana University; and Ledoit is from the Anderson Graduate School of Management, University of California at Los Angeles. This research was conducted while both authors were at MIT. We thank Paul Asquith, Kenneth French, Roger Huang, and Andrew Lo for advice and encouragement. We thank the editor (René Stulz) and an anonymous referee for constructive criticism. We thank Darrin King for attracting our attention to this area of research. Any errors are ours.


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.