Since this WDT thing appears to be a popular topic, I'll record some of what I run into as I am on the run. For example, today and yesterday I read through Notes and Comments - Uncertain Lifetime, the Theory of the Consumer, and the Lifecycle Hypothesis, Leung 1994. Milevsky calls this paper the first to discover the idea of WDT. Those who know me will successfully guess that both of those days I was reading the paper at a bar. Forget the unconstructive personal dynamics of this choice, it at least makes for interesting conversations sometimes. The last conversation happened to be with someone who, while painfully attractive, is not only too young for me, her mother is probably too young for me as well. But this is why reading financial economics at a bar can kinda be fun sometimes.
So, with respect to my reporting today, let's be clear. This post is not an explication. This is really just me reporting on what I'm running into. It is also a placeholder for future posts where I will have a stronger point of view and hopefully a better understanding. My mental state in apprehending this stuff at this stage of the game can be summarized like this:
- Do I understand it or should I or can I?
- Is it true?
- If I were to read synoptically, is there coherence and continuity among the major threads?
- Will this add to my personal understanding and goals?
- Should stuff like this "pass through" to others by way of my blog?
- Is there any chance I can assimilate this into an RiversHedge point of view?
I can't answer any of that yet. But let's at least summarize some of my major take-aways at this point after two days at Fort Lauderdale restaurants hanging out with some pretty obscure stuff. Note that I will add these take-aways to my "hypothesis pile" from the previous post to be used for later judgment on whether it is useful or important or worthy of additional reportage. Based on my amateur reading of Leung 1994, which was more readable and interesting than I expected, my take-aways were (at least so far):
- There must always be a wealth depletion time less than (T[bar] = the end of life) given Leung's focus on Yaari's "Case A" (no annuity, no bequest). There will always be a segment of uncertain time where wealth is depleted and consumption = pensionized income,
- My conclusion from point 1 is that any WDT that is longer than or equal to the lifetime looks like it means that there is probably a suboptimal consumption plan in place but that might just be a function of the paper's constraints and Leung's focus on Yaari's case A...TBD,
- The ratio of wealth to income at the beginning of retirement has an outsized impact on WDT,
- WDT can be sooner than most people think, have a material impact, and has social implications too,
- There is an implied point here, not completely drawn directly from Leung but appropriated mostly from from Milevsky, that the demarcation of retirement into two pieces has a [somewhere between subtle and profound] impact on lifetime utility of consumption math.
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Here are some miscellaneous excerpts[1] from the paper with comments:
- Using optimal control theory with a state-variable inequality constraint, I prove that Yaari's model cannot have an interior solution that lasts until the maximum lifetime. Contrary to the basic life cycle hypothesis, savings must [emphasis added] be depleted at some time before the maximum lifetime and consumption will be equal to income thereafter.
- Proposition 2:
comments: See the last bullet below for definition of (P). Proposition 2 here doesn't seem to be as hard as it looks I think. He is just basically saying the same thing that is in bullet 1: that there must be a WDT after which income = consumption and that consumption utility can have infinitely negative utility (ran out of W) if there is income at end of life or there is no infinitely negative consumption utility for any of the "piecewise" continuous consumption utility functions. I will admit to being a little confused about the "or" though. It seems counter-intuitive unless he is either (a) saying that there is always explicit or implicit income at the end of life or (b) there is something going on in calculus in the last "instant" that I don't understand. For (a) I think he supports this idea a page later by saying
"In Proposition 2, the condition m(T[bar]) > 0 [income at end of life] is needed if g'(0) = ∞ [i.e., the infinite disutility (slope) of no income]. This is not a stringent condition. A positive post-retirement income stream (especially in the form of annuities) lasting through T may come from individual pension funds, family support, or government programs. For example, social security benefits in the U.S. are positive and do not decline with age. Many countries have some kind of social security program or old-age allowance."Yaari makes a vaguely similar same point (not directly about income as such but about the wealth constraint being more than zero so no borrowing to fund consumption. This is an indirect comment on the availability of income at end of life time) in his 1965 paper when he says
"Now a violation of the wealth constraint S(T) >= 0 is clearly a physical possibility, but some people think that the institutional framework makes it virtually impossible for a man in our society to die with a negative net worth."
As regards (b) maybe Leung's footnote 6 might address this; I'm not sure. A quant would have to read this for me and tell me a little more.
- positive savings throughout the life horizon is impossible. Savings must be exhausted at some time before the maximum lifetime. During this period of zero savings, consumption at each time instant is equal to whatever non-interest income the consumer has.
- These subtle results have been overlooked in all previous studies of uncertain lifetime...[they] have either ignored or mistreated the borrowing constraint in solving the optimal control problem.
- Equation (16) shows that it is the ratio of wealth to income at age 65, S(65)/m, that determines t*
- There are many notable results. The most important one is that more than half of the values of t* are less than 75, which suggests that many consumers will completely exhaust their wealth within ten years after their retirement
- Savings will be depleted earlier the higher the discount rate, the lower the degree of risk aversion, or the lower the wealth/income ratio.
- In 1975, a typical elderly household in the U.S. had an average wealth/income ratio of about 5
- these new characterizations can also be used to explain the puzzle of widespread under-saving reported in many national surveys
- Let c*(t) denote the optimal solution of (P)...Let ϕ be the set of piecewise continuous functions on [0, T] with values on [0,∞ ). The consumer solves the optimization problem (P):
This last equation should look familiar. It would be roughly what I tried to draw in the dotted line in my figure 1 in the last post in the link above (though, since he is working in piece-wise integration chunks, I suppose we could talk about optimization of my "blue" line as an alternative). The equation is also the same formulation as other lifetime consumption utilities in Yaari and Milevsky (and probably elsewhere by others) and is also the theory foundation for my amateur stab at modeling stochastic present values of spending over a lifetime.[2] Omega is a conditional survival probability or hazard rate (not sure of the technical name), a(t) is a discount rate and g(c(t)) is the consumption utility function. T[bar] is end of life. The similarity is not accidental since Lueng is platform-ing off of Yaari's paper.
End Note
I am going to Key West tomorrow, land of rum bars, with my pile of remaining research papers on WDT so we will see what comes out of that unholy combination...
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[1] This paper was actually paid for and this post is not commercial so I am comfortable with my references.
[2] This equation, by the way and in my opinion, is the formulation of the retiree "problem" that is almost always missed by advisors with their hyper-focus on portfolios-only, and returns over "single periods"...a focus that often precludes robust conversations on variable spending, multi-period processes, and random lifetimes.
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