Wealth Depletion Time - an Hypothesis and a Self-Challenge

In several papers I've read lately I keep bumping up against the phrase "wealth depletion time" (WDT) or "wealth depletion age." My guess at the time of reading was that it was more or less the same as portfolio longevity or ruin risk but I gather that it is something quite a bit more subtle than that.  One of the subtleties is evidently that -- if one were to be enamored of utility analysis and formal mathematics, and I am not saying that I am -- capitalizing pension income onto a balance sheet and doing a PV analysis therein misses some of the nuances in evaluating the lifetime utility of wealth and consumption especially when consumption hits a "depletion inflection" and then reduces itself to the available pensionized income for some uncertain amount of time.  And this brings me to my hypothesis, which is:

My Hypothesis

I don't know much yet about WDT,  but my guess, before I start to dig into this, is that:

a) WDT is subtle but probably matters -- evidently way more often than is the norm in academic papers -- to formal analysis of retirement finance, lifecycle economics, and utility theories of consumption and wealth and that there are formal and serious ways to express and interpret it, and that it is probably worth me personally knowing at least a little bit more about it, but that...

b) to an average retiree the formal expressions and interpretations of WDT might not matter so much or, rather, can be interpreted on the street as "don't run out of money because leaning on a much lower level of income from social security kinda sucks...or if you do run out, try to make it (WDT...and this is counterintuitive) longer rather than shorter...maybe by pensionizing some assets."  These are mostly already known and it's possible that the subtleties of WDT won't or shouldn't filter down to a retail level...maybe.

Here in Figure 1 is a picture of my hypothesis. Tolerate the notation with compassion ; I'm an amateur.   p is pension or pension-like income (or funded consumption). C(t) is consumption in time t. wC(t) is direct consumption from the portfolio. U(c(t)) is the utility of consumption at time t. U(a(x)) is the utility of annuitizing remaining wealth at age x. w(t) is residual wealth at time t.  T(d) is the wealth depletion point. T(omega), if you didn't intuit it, is death.

I didn't notice until I copy/pasted that this looks like a cash register which, if you think about it for a minute, is pretty amusing...

Figure 1. My Wealth Depletion Time hypothesis

[post script edit - item 3: the rate of wealth depletion might be 

either slower or faster depending on circumstances at the time 

like maybe the absolute level of wealth available...tbd...]

My Self-Challenge

Usually when I don't understand something very well it is a sign that some self-learning is in my near future.  And so it is in this case.  I commit myself here and now to: 1) learning more on this WDT thing and 2) trying to see if I can disprove or substantiate hypothesis b.  Here is my reading pile which is already printed out, stapled, and sitting on my office floor about a foot from me:

• Lachance, M. (2012), Optimal onset and exhaustion of retirement savings in a life-cycle model, Journal of Pension Economics and Finance, Vol. 11(1), pp. 21-52.  

• Leung, S. F. (2002), The dynamic effects of social security on individual consumption, wealth and welfare, Journal of Public Economic Theory, Vol. 4(4), pg. 581-612.  

• Leung, S. F (2007), The existence, uniqueness and optimality of the terminal wealth depletion time in life-cycle models of saving under certain lifetime and borrowing constraint, Journal of Economic Theory, Vol. 134, pp. 470-493.   

• The Utility Value of Longevity Risk Pooling: Analytic Insights, and the Technical Appendix,  Milevsky and Huang 2018   

• Notes and Comments - Uncertain Lifetime, the theory of the consumer, and the lifecycle Hypothesis, Leung 1994    

• Uncertain Lifetime, Life Insurance, and the Theory of the Consumer. Yaari 1965 .  

• Approximate Solutions to Retirement Spending Problems and the Optimality of Ruin, Habibm, Huang, Milevsky 2017  

There are probably others.  I will understand almost none of this and will suffer greatly for my choices. But my goal is to come back from the wilderness and desolation of partial differential equations and utility theory with an ever so slightly better understanding of WDT so that I can answer "b" better than I can today. Even Job himself will pity me -- because I had a choice -- yet it needs to be done.  See you on the other side...

Some current views of WDT

This is from "Approximate Solutions..."

This is from "The Utility Value of Longevity Risk Pooling..." Page 8. Note the split utility and compare to the cash register above.

This reduces further to...

Which makes Milevsky's case that knowing T(d) is both subtle and interesting in the context of utility math as well as bolstering my interest in my own self-challenge.