Utility of Sim Wealth in an Amateur Experiment

I am obviously not an economist or a working finance professional and it would be embarrassing for me to claim that I knew what I am doing with utility functions because I decidedly don't but I thought I'd take a shot to see what something might look like for some sim data I ran recently with whatever little I do know (I'm running with scissors, mom).  The sims were for a $1M portfolio for a 65 year old with at least two spend rates, 3 and 4% over 25 years tested against 11 allocation choices.   In a recent post on "retirement Omega" I looked at the same data and asset allocations and looked at what might be optimal with respect to the allocations by using an omega-ratio approach to analyzing the distribution of final wealth. The answer there was to spend less and if spending less, allocate conservatively...somewhere in the middle between stocks and bonds...more or less...kind of....  Fuzzy, but that's the way it is.


Here, I wanted to extend whatever I fuzz I did in the last post.  One of the big problems with that post was that the simulator, because I did not program it otherwise, allows negative wealth as an artifact of the sim math by taking wealth that is going to zero or negative and then compounding it with negative returns and more spending.  Frankly it is insanity to consider this kind of thing but here in this post I have decided to embrace it.  I think that the shape of the wealth distribution in a sim should not be truncated at zero.  Why walk around with a mis-shapen distribution? I think the strategy that the sim is exploring should have a fully formed end-distribution of wealth, interpretation of negative-wealth be damned.  The negative wealth, though, in addition to being interpretationally complex also is a challenge when it comes to the math of utility evaluation.  An awful lot of finance U functions use some type of log function (of wealth, consumption, returns, etc) or some type of power function where negative values can get to be a problem.  Some day some smart reader will call me and educate me on this stuff because I think I'm a little too far over my skis on this one.  For now, I found some solace in a random web page that said for negative wealth I can use a U function that delivers zero when wealth is zero or below.  Ignore for the moment that running out of money can have an infinite dis-utility, the point here is to evaluate strategies that, in sim world anyway, go negative and where we have to evaluate the full, weird distribution to decide what to do.  Decisions coming out of that may be really, really, flawed but that is another post on another day. Maybe there is another utility function out there...there always is, right.

So here is what I did or tried to do...  I assumed that someone somewhere might consider maximizing log-wealth to be a proper utility maximizing endeavor.  So for my sims I concluded with [almost] no basis whatsoever that the expected value of ln(Wt) was the way to go...maximize E(log(Wt)).  Finance quants should clip whatever wings I think I have now.  The problem, though, was with the negative wealth, or more properly, the wealth less than $1.00 (or, I guess, some other threshold but lets use $1 for now; this is just a game).   So in place of the last post where I calculated the probability weighted ratio of up-wealth vs down-wealth with respect to a threshold (omega), here I decided to evaluate the expected value of Log(Wt) where Wt is any terminal sim wealth across the 10000 sims I ran. This was modified (to deal with "< 0") to assume that the Utility of any Wt that was at or below $1 would be...zero.   Not perfect, but whatever ("whatever" is pseudo-economist jargon.  I use it a lot but then again I have children in middle school so I might just hear it a lot...).  When I do this and calculate E(log(Wt)) for the sim in question I get something like this:

Is there anything of interest here, ignoring the various caveats above, like economic illiteracy? Well, I think it is all more or less the same as the last post.  Certainly there are optima here that we can consider which is always nice if one believes the analysis.  If so, the conclusion I might make, putting back on a more normal retiree hat, is that: 1) spending less is a good thing or at least better than spending more, 2) within a broad range (ignoring allocations of less than 30% to risk) for a given spend rate, asset allocation has relatively small effects, 3) going all in on equity is not necessarily a utility maximizing endeavor [a recent post on multi-period geometric returns made the same point], and 4) if one were to have the mis-fortune to be somehow committed to a spend-too-much path, then the lotto of high equity allocations might be the only way out...but watch out for those early fail dates if you do. Then again, we already knew this.