With/without replacements

Assume that we have the historical data, such as price and return, for a stock. Obviously, we could estimate their mean, standard deviation, and other related statistics. What are their expected annual mean and risk next year? The simplest, maybe naïve way is to use the historical mean and standard deviation. A better way is to construct the distribution of annual return and risk. This means that we have to find a way to use historical data more effectively to predict the future. In such cases, we could apply the bootstrapping methodology. For example, for one stock, we have its last 20-year monthly returns, that is, 240 observations.

To estimate next year's 12 monthly returns, we need to construct a return distribution. First, we choose 12 returns randomly from the historical return set without replacements and estimate their mean and standard deviations. We repeat this procedure 5,000 times. The final output will be our return-standard distribution....