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Abstract
There are many applications in which the long-term statistical properties of short-term financial time series are required. Requirements for such statistics arise, for example, when the authors look at value-at-risk estimates using the historical simulation method. In this article, the authors derive a mathematical formula that estimates the under-reporting of volatility by using overlapping data. They also estimate the under-reporting of the 1% (or, in general, x%) worst return statistics by using simulation techniques. Regarding overlapping variance, in this article they examine the method that obtains an unbiased estimate by averaging variances based on non-overlapping intervals. They found that, as expected, the longer the overlapping period, the more severely they underestimate risk. The larger the number of samples, the less important this issue becomes. Even though practitioners often use overlapping data, there is very little in the literature on how severely this underestimates the risk.