In the last post "the challenge of retirement finance in a few charts" I tried to make a case that behind the fancy precise answers that advisors and academics devise for us there is really a big fat cloud of possibilities that we have to navigate by way of judgement and a continuous process of evaluation and adaptation. Actually I didn't make the case for that last part very well, it is implicit. Here is another example in this post, though.
In the last post we looked at portfolio longevity for different spend rates. Here I am -- for each "life" within each of 25000 iterations -- calculating or summing lifetime consumption utility for a random uniform spend rate between 1 and 12%. We will chart Life Utiles across the different spend rates.
The portfolio is still ~4%/12%. The spend is still spread out between 1 and 12%. Now, however, we add consumption utility and a start age since the weighting will be different at each age.
- Age: [63, 75]
- Life Utility: sum[1:100](U(w=[w,min(w)] )*tPx)
- U(w(t)) = ln(w(t))
- min(w) = 1000
- Longevity: Gompertz conditioned on age and M=88, b=9
- "Scale" is 10000 so a spend rate of 4 = 40000. This is to solve some math problems
- 25000 iterations
- Horizon is 100 years
- Note that the survival probability approaches zero after about age 100 or 37 y for age 63
- spend rate is ~4% ccr
Run it for age 63 and then age 75 and we get this in Figure 1.
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| Figure 1 |
The Charts: X is spend rate, Y is discounted life consumption utiles. grey are all of the outcomes of randomizing spending and returns over 100 years. Note that many or most but not all portfolios for higher spend rates will fail before 100 years...the lower spend rates can last forever more or less. The black plot is the mean of the utiles at each spend rounded to something like .001 or .0001 increments...I'd have to go back and look. What is clear in retrospect is that at, say, a 12% spend rate the distribution (vertical) of Utiles has a center weighted pretty low on the chart. It's hard to see here.
Side Comment: Ok, you ask, "what are the striations?" No idea. I'm embarrassed that I have no idea and I looked at this for a good long while this afternoon. Best guess is that there is some kind of discrete step I'm taking or some rounding thing in the code. But at 63 I realize I have lost my desire to chase the explanation too far. Basically it still works but I am mystified for now. Maybe tomorrow.
My Point, Such that it is: For the panel on the left, given log utility (fairly risk seeking), and looking at the black mean plot, the optimal spend given this setup only is maybe 4 1/2 or 5 percent. In reality there is a big fat fuzz cloud of scenarios, especially for higher spend rates where the ex-post utility evaluated all over the place. The optimal point is one of many possibilities within the cloud of simulated outcomes, something we'd never know in advance but probably know intuitively. Your optimized spend rate here is a statistical artifact of a fake world. Do you trust it? Do you know what log utility means? So in the end, my point is that we should probably be a little circumspect about point answers and recognize that there is a lot of fuzz in these models.
The reason that the 12% spend looks like it does is that for those portfolios where 12% worked for a long time they evaluate very very well; that'd be a fun spend, right?. For those paths at 12% that didn't last they died very very early and the convexity of the utility math is extremely punishing on that over the depletion time. The reason it is not even worse than it is is that the survival probability weighting zeros everything out after about 30 some years and moots some of the punishing effects over 100 years.
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