Since simulated terminal wealth distributions are, well, distributed, we should be able to play the same game to come to some conclusions about retirement strategies...I think...this may be the exact point where I start to go wrong...
Here is the density function and empirical cumulative distribution of terminal wealth for different allocations (100 to 0% risk asset in 10% increments vs. safe asset; 50/50 is red; 100% stocks is the dispersed distribution of course) for a given spend rate (4%) and a given set of simulator assumptions [1].
We should be able to split the distribution along a threshold to evaluate good distribution vs bad. Since we usually measure fail rates as wealth going to (or below, in sim world) zero let's assume terminal wealth at or below zero is bad and positive wealth is good (we'll gloss over discussion of fail magnitude in terms of duration or whether lifestyle degradation is the beginning of the fail process for now) . Let's also assume negative wealth actually means something as opposed to being a simulated fiction or evidence of poor modeling or programming skills which it may in fact be. We'll roll with this in the hopes that it can lead to some insight along the way. I once tried to figure out what negative wealth might actually mean and all I could come up with is that it might be some type of opportunity cost or maybe what one might need to beg or borrow from family or neighbors or something. I don't think, other than in sim mind-games like this one, that there is much use or justification for this kind of nonsense. But as long as we're here...
Here are the fail metrics for different allocations and spend rates from the sims I ran:
Pick your conclusion. Mine is spend less than 4% and keep your allocation conservative, say 30-50% invested in the risk asset [2]. If your spend rate happens to be unsustainably high, higher allocations may be the only way out, coming at the price of more intense and earlier retirement fails. Better to cut spending. Or earn more. Or retire later.
Here is the same data showing the certainty equivalent of terminal wealth (terminal wealth that is positive, that is, over a 3x initial spend hurdle so that the math works) using a CRRA utility function. This is not part of the analysis, I just happened to have this in these sim runs. I'm not sure what to make of this yet but you can take a shot at it if you wish.
Then, taking the terminal wealth distributions at face value, rather than fighting the impulse to reject the idea of negative wealth, this is what the omega ratio might look like when using a threshold level of zero. I can imagine other thresholds with different results (as in investment return analysis, a high hurdle will favor stocks while a low one will disfavor stocks and their variance). Note that one of the many things I don't know here is whether the wealth distribution needs to be normalized in some way. Let's go with no for now since I don't have any statisticians or quants to lean on.
Hmmm. If we are to believe any of this then my conclusion is still more or less the same as what I had before I did this post: spend less and allocate relatively conservatively if the planning horizon is long and allocate more assertively if you can't control spending (because it might be the only lotto ticket out of the fail zone, that's why the fail "magnitude" goes way up for higher allocations to risk).
But if true, which might actually require some effort to believe, I'd add a couple other things: 1) for high spend rates, asset allocation -- in and of itself and in a broad zone between, say, 30 to 100% risk -- doesn't look like it contributes nearly as much as spending control, and 2) for low spend rates, where the endowment does not get consumed as fast, it looks like there might be an optimum around a 40% allocation to risk in this overly simplified 2-asset 25-period sim world. That happens to confirm a previous bias of mine so I thought it was interesting. It also matches the Markowitz MVO portfolio using the last 3 years of data for a couple of ETFs as well as the conclusion I came to about a "safe allocation" that came from doing a backward induction exercise late last year. While I'm not going to run to a trademark office with my "retirement omega" because, not least, I think there are too many flaws and unknowns in my thinking, I might, since my personal spend rate is relatively conservative, add this type of analysis to my own planning to see if it adds any value.
notes ------------------------------------------
[1] $1M age 65, term age fixed at 90, tax effects, fee .006, historical returns, returns suppressed 1st 10 years, no SS or annuity, no spend shocks or random variance, etc.
[2] I was asked once why I only do two assets (risk vs safe) asset vs more complex allocation schemes. The answer is laziness and complexity. 5 assets allocated in 1% increments have ~4.6 million different combinations. Even moving in 10% increments its 1001 if I have my combinatorics right. Too much trouble for now.
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