Rudimentary spreadsheet sim vs. imperfect estimator for N-period geometric mean and sd

...or a re-title might be "why did I feel like I had to simulate other than to see how it works?"

I have a few spreadsheets and R-scripts that model the behavior of geometric returns over finite time horizons.  I did this mostly to educate myself and because I think mid-tier time horizons like 15 or 20 years can be of practical interest to retirees where that kind of horizon might be less interesting to a younger accumulator or an institution.  I know from what I've read (Michaud, Mindlin, etc) that there are estimators out there that can do the same thing as the sim just fine, but I just have not used them much.  Today I was curious how big a difference (not measured exactly, just visually compared) there would be between a simple spreadsheet sim that uses random normally distributed returns as part of the return generation process and the estimator offered by Michaud in A Practical Framework for Portfolio Choice [2003]. 



Sim.  The sim uses an arithmetic return of .07 and a std deviation of .20 and chains together 20 periods and calculates the cumulative geometric mean return along the way.  The sim in this case is run 60000 times (or more accurately we use the CLT and sample 20 times with 3000 iterations each...this is an old version of excel).  The image of the output was posted before and looked like this:

Estimator.  The estimator (6a below), which has other cousins out there, was taken from the link above. To quote directly: 

"A number of formulas are available for describing the N-period mean and variance of the geometric mean in terms of the single-period mean and variance of return.22 Such formulas do not typically depend on the characteristics of a particular return distribution and range from simple and less accurate to more complex and more accurate.23 The simplest, but pedagogically most useful formulas, given in terms of the portfolio single- period mean, m, and variance of return, s² , are: 

Formulas (6a) and (6b) provide a useful road map for understanding the multiperiod consequences of single-period efficient investment decisions. Note that (6a) explicitly shows the horizon dependent character of expected geometric mean return." 

So when I set the rudimentary sim up against the "simplest, but pedagogically most useful" formula we get the following comparison:

So not too bad.  I'll probably lean on the formula more often now for simple analytical work.