Temporal Aggregation of Random Walk Processes and Implications for Asset Prices

This paper examines the impact of time averaging and interval sampling data assuming that the data generating process for a given series follows a random walk with uncorrelated increments. We provide expressions for the corresponding variances, and covariances, for both the levels and differences of the aggregated series, demonstrating how the degree of temporal aggregation impacts these particular properties. Moreover, we analytically derive any differences that arise between the aggregated series and its disaggregated counterpart, and show that they can be decomposed into a distortionary and small sample effect. We also provide exact expressions for the variance and sharpe ratios, and correlation coefficients for any level of aggregation. We discuss our results in the context of asset prices, which have utilized these extensively.