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You really don't know much about math . . . or investment/retirement simulators.

That's very true.

If you are hoping for analytic predictive capability it doesn't matter how well you match the historical distribution, you won't be able to predict the future. But that's not the point. If you are able to approximate the historical distributions of the past, then you should at least be able to gain general understanding of how a portfolio might perform under conditions consistent with history.
Now this "general understanding" - what is that? It's great if you get an idea of how the portfolio has performed historically; but then what? Are not assuming it will do the same in future (which is the point of Monte Carlo trials - to predict several future paths)? To do so, you do have to predict, say, next year's returns (along with possible dispersion), do you not? And then for the year after that, and so on.

The specific numerical function approximation you choose will have insignificant impact on the final answer provided your approximation is kept to low delta error. For example, if you are trying to compute the area under an arbitrary curve, you can get arbitrarily close to the exact answer with rectangular or trapezoidal summations - or with spline fits. You just need to pay attention to the details.

Oh, is that what they call "integral"?

Back to your original assumption that this year's (or take any other period) returns are somehow a function of something - returns and dispersions from recent past for various asset classes as you assume - do you have any actual data to back it up?

The little math I know says that, despite the large dispersion (and corresponding low confidence), a simple average of past yearly returns is still the best (though not a good) predictor of the next year's returns. Let me know how more complexity helps.
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