Risk-Return Relationship in High Frequency Data: Multiscale Analysis and Long Memory

Abstract

This study investigates the relationship between the return on a stock index and its volatility using high frequency data. Two well-known hypotheses are reexamined: the leverage effect and the volatility feedback effect hypotheses. In an analysis of the five-minute data from the S&P500 index, two distinct characteristics of high frequency data were found. First, it was noted that the sign of the relationship between the smallest wavelet scale components for return and volatility differs from those between other scale components. Second, it was found that long memory exists in the daily realized volatility. The study further demonstrates how these findings affect the risk and return relationship.

In the regression of changes in volatility on returns, it was found that the leverage effect does not appear in intraday data, in contrast to the results for daily data. It is believed that the difference can be attributed to the different relationships between scale components. By applying wavelet multiresolution analysis, it becomes clear that the leverage effect holds true between return and volatility components with scales equal to or larger than twenty minutes. However, these relationships are obscured in a five-minute data analysis because the smallest scale component accounts for a dominant portion of the variation of return.

In testing the volatility feedback hypothesis, a modified model was used to incorporate apparent long memory in the daily realized volatility. This makes both sides of the test model balanced in integration order. No evidence of a volatility feedback effect was found under these stipulations.

The results of this study reinforce the horizon dependency of the relationships. Hence, investors should assume different risk-return relationships for each horizon of interest. Additionally, the results show that the introduction of the long memory property to the proposed model is critical in the testing of risk-return relationships.