HI

@RajivBoolell Not stupid, thought provoking! It is true that the randomization is random discrete uniform variable; e.g., if there is a history of 21 daily returns, then the probability of selecting a specific historical day is 1/21. But this reflects our want to conduct a genuinely

**random sample** (see

https://en.wikipedia.org/wiki/Simple_random_sample ... "In statistics, a simple random sample is a subset of individuals (a sample) chosen from a larger set (a population). Each individual is chosen randomly and entirely by chance, such that each individual has the same probability of being chosen at any stage during the sampling process, and each subset of k individuals has the same probability of being chosen for the sample as any other subset of k individuals")

By randomizing a uniform discrete variable as the basis

*for selection* we are ensuring that our sample (as the size increases) is

*representative* of the population, which in this case, each

*day does have an equal weight *in the actual history (although history is a sample of the unknown population such that bootstrapping is re-sampling: taking several samples of a single sample). If the "true" (unseen) population returns are normal, then we can expect the historical sample (of daily returns) to be approximately or roughly normal, and a random re-sampling (bootstrapping) selects the actual days based on uniform random discrete, but that's just "being random," such that we should expect the re-sample to also be roughly normal (size depending).

So i don't think the uniform random variable, as a basis for selection, is imposing any expectation on the underlying stock price distribution. I think it's ensuring a representative re-sample. Although i am not highly confident that I've I nailed the statistical property, frankly, because it raises an interesting question: if we alter to a non-uniform random variable, which properties do we distort ...

I would add that this simple HS is just one approach. Another is filtered historical simulation: however, the bootstrap step in FHS also re-samples by giving each standardized, day an equal probability of being selected (ie, random uniform discrete). The issues that typically are related to cross-correlation (that correlation among stock returns on a given day, which bootstrapping does maintain because it selects the day's vector of returns) and correlation over time (aka, auto-correlation, which simple HS does not maintain because it conducts an i.i.d. random sample of the historical days). I hope that's helpful .. not stupid, thought provoking to me!

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