In honor of Gauss B-day

Since it's his 241st birthday I thought I'd do a little Gaussian mix experiment I've been meaning to try out.  The basic mix concept is described here: Deconstructing one non-normal SPY distribution into two normals -- just for fun -- and then putting it back together again. The purpose was to get a smooth random distribution that has a fat, non-normal tail vaguely akin to market data.

Now I'm going to harness it to a net wealth process starting with $1M and a constant inflated 40k spend rate and see what happens.  The formal notation for the process, if I have it right, looks like this:

which is from Habib, Faisal and Huaxiong, Huang and Milevsky, Moshe A., Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin (March 31, 2017). Available at SSRN: https://ssrn.com/abstract=2944125 or http://dx.doi.org/10.2139/ssrn.2944125.  In the case below alpha will be 100% and pi is zero.

That's a bit much for me so we'll describe a net wealth process with a mean terminal wealth "W" expectation like this since we are going to simulate rather than do calculus:

E(W) = 1/S * sum[1:S](p(w(t),s(t),k(t))) 

where p is a simulated process of wealth, returns and spending iterated S times and where p can be described by this

for (t in 1:30) { w(t) = w(t-1)*(1+k) - s(t) }

given that w(0) = $1M, s(1) = 40k inflated at 3% thereafter, S is 10,000 and k is either 

a) normally (gaussian) distributed return with mean 11.4%, std dev of .198 [1], or 

b) a gaussian mix fit to the Stern data[1] to pick up a non-normal fat left tail 

The question is: "how much impact does adding the non normal fat tail affect the simulated net wealth process given the rather arbitrary and relatively non-scientific mix-fit attempt by an amateur?" The answer appears to be: "not that much" since this below is what the terminal wealth distributions for (a) and (b) look like where (b) is in red. We're using the R density function. 

There are in fact some differences in the summary stats done on the two distributions but (1) they are more artifacts of the simulation than anything else and (2) I don't find them (the differences) all that compelling after stepping back and looking at the picture.  Maybe I mis-fit the tail.  

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[1] data was annual data from A Damodaran at the Stern school, 1928-2016 with 100% allocated to large cap.