On some futility in thinking about consumption smoothing rules

I spent the better part of an afternoon trying to excelify some math on consumption smoothing, succeeded, and then gave up after the fact for reasons below.

Let's say, as a convenient-for-me strawman, that there are four broad categories of spending in Ret-fin models:

• Constant inflation-adjusted spend
• Percent of portfolio
• Honorable attempts to be somewhere in the middle of the last two for reasons, and
• Irrational or non-mathematically necessary rules or heuristics that might/not accidentally work

On preservation of capital over long horizons

I know I've done the kind of charts in this post before, but whatever. David C pointed out to me I've been kinda re-hashing my past stuff lately but sometimes that's necessary to pound it into one's own head. Here  (Figure 1) I was running a "portfolio longevity" calc for a .04/.12 consumption portfolio (different spend rates) 4 million times (uh, there is a reason for that big nbr) to see how many portfolios "tip over" into portfolios that last to infinity (or in this case 100 years which is a convenient proxy for forever but not really). 

Spend ranges for different decumulation strategies and risk aversion params

This is very subjective and depends on both my code correctness and my assumptions/parameters. I am skeptical of everything here but I ran it to see what it looks like anyway. 

The goal here was to run a sim to calc the expected discounted utility of lifetime consumption for these main variables:

- different risk aversion coefficients between 1 and 3 [1], and

- two main reductive spending strategies (constant and percent of portfolio)

The assumptions are in digest form:

Real absolute spend after 50 years of %Portfolio spending

As before, I hope I didn't botch the code but that is standard sandbagging for me. This is the real absolute spend scaled to an initial 1M portfolio (.04/.12) at year 50. 50 could represent a very very long retirement or a long dated trust or an endowment. The past posts using a weighted utility calc miss some of this because it is survival weighted and at 50 years the conditional survival prob is approaching zero (.0000001101895 using recent params). So basically at 50 most retirees wouldn't care though when I retired at 50, 50 years was at least within the realm of possibility.  

On starting to phase out personal records

Over the last six decades I have certainly had my fair share of competition with other people but the main competition was always the mirror ... in other words with myself. It certainly wasn't with my pretty grey face. Self-drive is a powerful engine to which to harness what one does in the world and I have worked pretty hard at it in various fields of human endeavor since 1958.  But I am starting to see some fraying among the threads I have woven for myself and I am starting to second guess the idea of personal records or personal bests. At least in the gym. 

What happens to consumption utility if I make spend a % of portfolio [amended]

I usually trade in "constant spend" because it is easy to work with. A % of portfolio isn't all that much harder and lasts forever, right? Sorta. The trade off is consumption volatility and the possibility of really low consumption in absolute terms late in the game. In this post I am simulating (50k times) randomized spend rates against a .04/.12 portfolio over 100 years. Lifetime is a survival probability laid over the 100 years, an interval which is arbitrary and used here as a proxy for forever. The utility score is sum of "consumption utility results over all 100 years weighted by the life survival probability in each year." The lifetime parameters are somewhere in a generic middle between average health and annuitant. Risk aversion coefficient is arbitrarily set to 3 in Figure 1.  

[note: fixed an error post-publ]

Portfolio Longevity Strawman with a Trend-Following-Like Inflection

 

Summary stats might mislead given the defective distribution so I just dropped the image. The finite P mode bumps out a bit but not much but the whole thing does shift right a little bit. This is why I fold some trend-following into my Portfolio. In theory -- tho no guarantees -- the payoff structure can hedge portfolio vol a bit and goose portfolio longevity as a result. Here I am merely asserting a vol reduction rather than simulating something more complex. Note that more pink portfolios will tip over to infinite (using 100 as a proxy here) but are not shown due to the cropping.

Fuzzy # 3 - optimal spend rates with CRRA Utility > 1 as a cloud of solutions

 This is the third post on fuzzy clouds, the first two being:

• The Challenge of Retirement Finance in a Few Charts, and
• Another Fuzzy Cloud Chart Now with Consumption Utility

Here I am modifying the code a bit by

1. Adding CRRA utility for risk aversion coeff !- 1 where 1 is log utility, and 
2. rounding spend rates -- uniformly random -- to 2 digits from 1
3. Using 100k iterations in the sim vs 25k before (changes little if anything)

When I do this, I still get the look of optimality at some spend rate except now it is more of a cloud within a cloud [1].

Another Fuzzy Cloud Chart Now with Consumption Utility

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.

The Challenge of Retirement Finance in a Few Charts

I won't repeat Sharpe's over-used quote about the "retirement problem" but let's at least stipulate that retirement finance is a bit of a challenge. The intersection of uncertain returns, uncertain longevity and uncertain consumption in the absence of lifetime income creates a difficulty in how much to save, how much to spend and how to allocate resources.  The self-serving illusion articulated by much of financial services or in academic studies is that there is "a" solution -- i.e., your "number" or some other optimal calculation distilled down to a point -- where in reality there is just fuzz: fuzz now and then even more fuzz as time unfolds.  The point here is to highlight some of the idea of "fuzz" rather than solutions or points. 

Small differences in assumptions

I got confused the other day, when playing around with an Rscript for portfolio longevity. I've always used discrete returns (1+r(t)) in simple sims because  1) it's easy, 2) I don't know continuous time(CT), and 3) I am working inside the sim with very discrete annual steps. In the script I got from Professor Milevsky he uses e^(vt), a continuous form of return, but told me later that either approach is fine as long as the rates are matched to the compounding intervals.

So, why either? What difference does it make?  

Real Yield Curve April 6 2022

 fwiw...