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Use of the Geometric Mean Return Formula. The formula for the geometric mean return is used specifically for investments that are compounded. By contrast, a simple interest account would use the arithmetic average which is summing the rates and dividing by the number of periods.
The log return for the year is the sum of the daily log returns. The log return for the year for the S&P is 12.8% (the data are in percent already). We can use the bootstrap to get an idea of the variability of that figure. There are 251 daily returns in the year. One bootstrap sample is 251 randomly sampled daily returns.