I can think of no reason, other than when we consider what is already embedded in common sense, for why this post would be of practical use. How's that for sandbagging, huh? But I had a thought and the thought was:
"if I had a 'slider' that could slip spending between "constant" and "percent-of-portfolio" in a continuous fashion, what would happen in a consumption utility model, especially one that knows that wealth depletes and spending snaps to income when that depletion event happens?"If you are an avid reader of this blog, and I have my eye on all three of you, you may have noticed that I do not trade much in spend rules (except for my own RH40 rule and that was half tongue-in-cheek). Spend rules seem to be a big thing in retirement lit. I don't trade in them because rules tend to be a wee bit ad-hoc, arbitrary, and not always mathematically necessary, not to mention blind to optimality along some dimension or other. This is a point that Patrick Collins made in Monitoring and Managing a Retirement Income Portfolio. But rules, when they are thrown out there into the ret-fin universe, generally try to do one thing if they are honest. They adapt to the circumstances that arise after retirement-initiation vs., say, a Bengen 4% rule or academic papers that use constant inflation adjusted spending for convenience. Let's also ignore a strict application of macro econ and consumption smoothing for now even thought that is more or less what we are doing in this post. In my 5-process paper I made a case (that, embarrassingly, I can't remember) that this smoothing can sometimes be self-contradicting when put into practice.
The Slider Concept
It's crystal clear (to me, anyway) that a plan that adapts has some advantages, or at least should have some advantages, to one that doesn't adapt. In this post I'll use a little naive amateur hack to create a "slider" and pretend that this is a proxy for all rules that lie on a continuum between constant-spend and percent of portfolio. Of course, anyone with even rudimentary tutoring in ret-fin will immediately object that this is false. They would be right. That continuum, as I have pointed out before, is a fake straw-man. While there can in fact be a continuum, spending can also be irrational (i.e., step functions and various discontinuities, random, bizarrely ad-hoc, etc.) or even willfully destructive (e.g., markets are low, spend is high...and vice versa. This is what I did to myself in '08 and '09). They can also have more rational forms. Michael Edesess did a recent piece (can't find or remember the title) where he created a flex spend rule with a floor and for which he evaluated it (vs annuities) in an econ context that is probably better than anything posted here. I'll look for it. I am also ignoring what is actually often observed in real life (say a down-trend with a late uptick, i.e., a "smile" ala Blanchett (2016) or Banerjee (2014)).
Slider concept:
Constant spend <------------------------------------------> Percent of portfolio spend
/\lifestyle
Other, less rational or less "continuous" or even "destructive" rules:
Ignored
The basic idea is to allow spending (lifestyle), as compared to a constant spend, to flex but to also have some memory of what was consumed in a prior period as well as what is going in in the current period. It's like an auto-regressive model for inflation, sorta. Full 100% memory/habit would be the constant spend. "No memory" would be percent-of-portfolio spend. To do this kind of thing I adapted one of the Diamond and Mirrlees (2000) equations to work for what I am doing and ignored whether I can really do this in the first place. We're all about practical application here. The framework looks like this:
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| Eq 1. Lifestyle habit model |
Where S(t) is the current year's lifestyle which is a function of the prior period [S(t-1)], an awareness of what lifestyle should be if consumption C(t-1) happened to be a percent-of-portfolio, and a coefficient of habit "a." This effort is also modeled in nominal terms in the sim, btw, so i is inflation which, in this case, I keep deterministic for now.
To illustrate what is going on here we'll use a simple excel model to show three things:
1) the constant spend in nominal terms (grey dashed)
2) the percent of portfolio spend (blue)
3) the habit-model spend/lifestyle (grey)
and we'll use two illustrations (figure 1). First, the top of figure 1 is the illustration where "a" is closer to zero and we have "high" memory. This will be very close in practice to a constant-spend which can be seen by comparing the two grey lines. For the numbers to make sense, assume we have a $1M endowment, a 40k spend, a return, I think, of 4%, and a std dev of 12%. These are not supposed to be realistic yet, just an illustration. Second, the lower part of figure 1 is where "a" is closer to infinity and we have something that is very close to a "percent of portfolio" spend model. These, of course, will last forever but have drawbacks in spending vol and consumption utility, as we will see later:
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| Figure 1. Habit model illustration |
Starting to Simulate
Ok, now to extend our illustration, assume that we are going to simulate this by throwing random returns (and, later, random inflation) at the net wealth process that is going on here. And for a point of reference, lets recall that the net wealth process can be roughly described as a recursive beast that looks like this, though my notation probably sucks a bit:
W' = W*r' - S*i' + I - A(t,x)
Eq. 2 - Net wealth process in simulation
W is prior period wealth
S is the lifestyle consumption from Eq 1
r' is random returns (let's leave them at normal distrb for now)
i' is random inflation but we are forcing this to deterministic for now
I is income from both exogenous (say SS or pensions) and endogenous sources (say annuity alloc)
A(t,x) is the price of an annuity purchased in period t by old guy aged x
Milevsky uses this:
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| Eq 3. more formal description of recursive net wealth process |
If we were to animate Eq 2 in simulation and randomize returns and we also did it 10,000 times we'd get a whole bunch of scenarios, 10,000 of them, in fact. Here is an illustration using a rudimentary Excel model (not tuned for wealth depletion, but) which will make the point. I copied 5 scenarios to show what happens to the habit line under different sim regimes for an intermediate habit coefficient (arbitrary) of .2. Recall that the blue line would be a percent of portfolio and the grey dashed would be the constant spend...except that that would crash to income when wealth depletes:
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| Figure 2. Habit Scenarios |
The Utility Model and its Assumptions
I am not going to regurgitate the model again. The whole idea is in this page/post. The basic idea is that we are calculating, additively over a simulated lifetime, the expected discounted utility of lifetime consumption. We assume here that it is ok to do this and that this idea has no detractors, which is likely false. The concept is simple:
1. Simulate a net wealth process like the one in Eq 2
2. When wealth depletes, force spending to whatever income exists
3. Calculate a CRRA utility on real consumption in each period t
4. Weight U(c(t)) with a conditional survival and a subjective discount
5. Add the U(c(t))s
6. Average the E[V(c)]= Sum[U(c)] over the sim iterations
7. Start looking for max's of E[V(c)]
The permutations of the input parameters is legion. My goal is not to exhaustively explore them and create a unified field theory. I was just curious about what would happen to the value function results in terms of
- shapes
- behaviors
for different spend rates and for different values of the habit coefficient for just one illustrative contextual parameterization.
Some basic assumptions used in this post:
Return <-- .05Model output
SD <-- .12
Endowment <-- $1M
Spend <-- varies. used .03 to .10
Coeff of risk aversion (Gamma) <-- 2
Iterations <-- 10,000 per scenario so ~1.3M iterations overall
Social Security <-- 20k at 70 in age 70 dollars
Habit coefficient <-- varies but slid on a geometric scale from zero to 1000 in 10x incr.
Annuities and pensions not modeled
inflation <-- 3% deterministic (for now)
Age 61, so 9 years to SS
Subjective discount <-- .005
When we take the assumptions above and jam them into the model, for which I am not totally sure the programming is 100% airtight, we get this:
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| Figure 3. EDULC for different spend rates and different coefficients of habit/adaptability |
where the colored lines are the different spend rates, the x axis is the geometric scale for the habit coefficient (zero is constant spend, 1000 is a proxy for percent of portfolio though infinity would be better). and the Y axis is the value function described above.
This is a little hard to see but it's a surface. So, let's do a surface. The following is hard to take in because it'd be better if you could rotate it. Fly to Fort Lauderdale and we can do that together. I'm sure a smart kid could create a java or python object to do it but I'm not the kid.
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| Figure 4. Same as fig 3 except rendered as a surface |
Just to round it out, lets look at what the actual simulated (real) spend paths look like in the simulation. First, Figure 5 is a sample of 50 iterations over 35 periods for: (a) a middle of the road spend, and (b) middle of the road habit coefficient. It is roughly the top optimal spot in figure 4 (yellow spot at the top).
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| Figure 5. Simulated spend paths for something in the middle of figure 4 |
Then, let's take a high spend rate (say 9%) and moderate adaptability (coeff =1) -- this would be the middle of the right side of figure 4, so still relatively "optimal" when compared to the last example. We get this:
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| Figure 6. Simulated spend paths for something on the right side of figure 4 |
Discussion
Since we are doing nothing to tease out other permutations of the infinite number of parameter choices, there is little we can really, truly say here. But given just what is in front of us?
- low spend rates can be sub optimal, high can be wicked
- high spend rates and low adaptability is a pretty bad strategy
- low spend rates don't really need to be adaptable at all
- high spend rates can have relatively high lifetime utility if the spend strategy adapts
- adaptable strategies can have very volatile consumption which if even "optimal" would be hard to swallow in real life. Figure 6, for example, evaluates to reasonable lifetime utility relative to the others in this illustration but I seriously doubt I'd go down that path in this world.
- 100% adaptability, surprisingly to me, doesn't work all that well except at the very high-risk spend rates. You'd think it would show better.
- There appears to be a very complex and convoluted "sweet spot" at about 5% spend and middle of the road spend adaptability for this run only.
- The model, and this is not shown, is quite sensitive to the presence of income, as it should be.
- I still think all of this is common sense. Spend low and hang onto maybe suboptimal steady consumption. Spend high and you probably have to tune into the level of your portfolio at some point. Go too far into perseverating on the level of wealth every year and you might just be a bit too far out onto the spectrum for your own good. That'd be me, by the way.
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Banerjee, S. (2014), How does Household Expenditure Change with Age for Older Americans, ERBI.org
Blanchett, D. (2016), Estimating the True Cost of Retirement, Morningstar.
Diamond, Peter and Mirrlees, James (2000), “Adjusting One’s Standard of Living: Two-Period Models,” in Incentives, Organization, and Public Economies, Papers in Honor of Sir James Mirrlees.
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