I won't claim I can read or deeply understand the continuous math, the utility functions, or the optimization setup (which, if you really think about it, more or less takes the whole paper off the table for me) but: a) I get a gold star for trying, and b) in taking a look-see I did find at least some additional surface confirmation of my intuition on both my custom spend calc and the idea of stochastic present value of spending. I tried to tie these two together in several recent posts. Mid-paper he gets to the nub and lays out his core contention that there are four consumption related "cases" to be examined. Think of a consumption-utility matrix in four quadrants: with/without a wealth constraint[1] and with/without insurance (e.g., annuities). The Yaari paper appears to be quite well known for case C (with-constraint-with-insurance) where the logical conclusion was that in the absence of bequest and in the presence of annuities the retiree would hold all assets in actuarial notes (annuities). That is no doubt a mathematically tight conclusion given the set-up but I think it is off in practice. But that is a distraction and not why we are here. It is the with-constraint-without-insurance scenario (his "case A") that I am primarily interested in. This is the same proposition mentioned by Milevsky in a past post that he called the "discounted lifetime utility of wealth...without annuities" which, if I recall correctly, he attributes to Yaari. I am interested because these two continuous time expressions are close kin to my hacked out spend calc attempts.
The Reader's Digest version of Case A, found in his equation 13 in the middle discussion about case A, is this below (I'll try, if I can, to define terms later), the goal of which he describes as "Find an admissible plan c* such that V[c*] >= V[c] for all admissible plans c" (i.e., it's an optimization gig. Blogger.com, by the way, hates the "bar notation" but it should be there over the Vs):
Let's walk through this formulation and relate it to my naive, amateur attempts at hacking out spending (consumption) math. First, though, recall the deterministic version (terms later) of my custom spend liability (CSL) as this:
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| (1) deterministic version of a RH spend calc |
And then below in eq2 is the animated version with simulation where the result is a stochastic present value (spv) spending distribution. npv(c) is random because the terms (especially D [the discount], N [a random lifetime process in years; this is sorta similar to tPx or Ω(t) above] and parts of Ct [the spend plan, similar to "c" above and Yarri's c(t)]) are random; more on terms later. EV is the expected value but I think that extracting the Pth percentile is as or more interesting. There are other ways to do this sim, by the way. This is just the way I recently did it and the equation was my amateur attempt at describing it. Note that I had an earlier version of the simulation below where the discount was neither stochastic nor chained in a series which means it might be a closer (discrete non-utility) proxy to Yaari's eq13:
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| (2) dynamic spend liab. calc (stochastic PV) |
1) Yaari's T[bar] . Right off the bat (and this is the whole point of Yaari's paper I gather) we need to point out that T is an expectation about the duration of life and that many papers prior to Yaari (and way way too many after) treat T as a constant where here it is turned into a random variable that "assumes values in the interval [0,T-bar]". He then makes the additional point that "In the first place, the Fisher utility function V depends directly on T and thus if T is a random variable, so is V." So this brings up another point: before we even get to something like Mindlin's discussion (a past post of mine) of a stochastic present value for spending (ignoring utility functions and calculus for a minute) because discount rates might be uncertain we have to entertain stochastic longevity first. That variability in lifespan, by it's very nature, makes a spending evaluation for a balance sheet a stochastic problem before we ever get to mess around with discount rates. This was my main point in my various spending calc efforts, or at least it was in the dynamic first version of my simulated spending (not shown)...as well as maybe 3/4ths of all my past blog posts.
2) V[bar](c). This is Yaari's "utility of the consumption plan c" or rather, I gather, an expected value of utility, from a distribution, from a stochastic process. This is also Milevsky's "discounted lifetime utility of wealth...without annuities." Unpack the "utility" part of this and squint one's eyes hard enough to obscure essential details and processes and one might be considered as having a distribution of consumption plan NPVs which was the goal I had in my second formula. I'll assert with lukewarm confidence that, adjusted for my lack utility stuff and the continuous math that I struggle with, we, they and I, are playing a similar game. Not the same, just close enough to make this post.
3) Ω(t). This is, in his words, "the probability that the consumer will be alive at time t." I'll call that the same idea as tPx in my deterministic calc and the same idea as the simulated process for t=1 to N in my simulation where N is a random pull from a mortality PDF. It is a conditional survival probability. I can't prove these are all precisely the same. I can imagine it though.
4) α(t). This may, in his words, "be interpreted as a subjective discount function." He has more complex economist stuff than I'm able to read on this but I'll take it, as an amateur, as the idea that a consumer can not only "subjectively choose" the proper and useful rate in high or low terms but also, if she were to have a quirky sense about time asymmetries in discounting, maybe she can adjust the discount unevenly over time, too. This is why I threw in the hyperbolic term in my deterministic formula.[2] That embellishment didn't make it into the simulated version but it could have. Either way my take away is that the discount rate is a malleable thing based on each individual consumer and it depends on his or her situation and sensibilities. Plus it might evolve. I can buy all that.
5) g[c(t)]. "c is any arbitrary consumption plan which the consumer might contemplate...c(t) describes the rate of expenditure on consumption...which would take place at time t if the plan c were to be adopted." g[] is the "utility associated with the rate of consumption at every moment in time. My point exactly in my Ct terms in my spend calcs! I just did not frame it as a utility function in continuous time. The key point I might make here is that it is, in fact, a plan, it is arbitrary, it is contemplated, and it is evaluated. It is not fixed by the universe or a "formulaic given" or a constant it is a well considered human construction made by and for real people. Plus it might evolve...
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This is interesting stuff but maybe interesting like comparing the facial features of second cousins. In any case in the end it is mostly beside the point. The point, in Yaari 1965, is to pick "the c" not merely "a c" or, rather, to maximize the utility function V[c] given admissible c-s (see note 1). And it it here that he starts to leave me behind by stepping into the rarefied air of math and methods where I cannot follow. If it were to be a climbing metaphor he is a seasoned technical no-O2 free-climber in the death zone of K2 while I am, well, I'm not even up to the base camp yet. I'm still in my pajamas on my couch watching some random Finnish language police procedural on Netflix while ignoring the dirty dishes (can be done tomorrow). But then again I think there is some utility (pun intended) to my cousin-versions of the spend calc because, while the real Yaari trick is to optimize this kind of thing with seriousness and rigor for serious and rigorous people, the trick before the trick for un-serious people like me is to be able to calculate or measure "a" spend plan or "any arbitrary consumption plan" at all in the context of stochastic longevity (but maybe without the advanced continuous form math that most of us can't touch). Then, even more importantly, the trick before the trick before the trick is to have a plan in the first place and I'd wager that no small number of retirees don't really. A constant spend plan doesn't count by the way although that is "a" plan I suppose.
What, then, can I say here? I'll wing a few ideas:
- The Yaari and Milevsky versions are the same which does not surprise me because the latter draws on the former. Plus Milevsky is a pretty bright guy anyway.
- My spend calcs are ok, if not pretty good, at describing "a" plan in the context of stochastic longevity (as was my MC simulator...and my ruin estimation tool...and my stochastic adaptations to Perfect Withdrawal Rates...there were others). They also appear to be quite sympathetic to all of the main issues described in the more formal theory. They have nothing to say at all however about "the" plan, the optimal one. In addition, I think I have probably under-appreciated the nature of some of the constraints described by Yaari that were always there but I was not paying attention to especially with respect to the condition that wealth S(T) at T is >=0 (in the Yaari world without annuities and without the possibility of negative wealth, anyway). On the other hand, since I would never do these kind of calculations while not being connected to a household balance sheet and since I would be doing the analysis in present time, not continuous, any consumption plan that was not feasible today (>= to wealth available) would not be admissible to begin with and so the Yaari constraints would probably hold in a way. My point is that I have too often lately been thinking about the spend process in isolation which is silly.
- I was surprised that there was no discussion of variable spending in a randomness or spend-shock sense (maybe hiding behind the notation? I might not have recognized it). In Yaari case C where annuitization becomes a foregone conclusion I'm not sure yet how he gets there. Income (the annuity) is never exactly the same as spending (the consumption plan). That's why we have things like bankruptcy courts and FICO scores. If I annuitize all assets and the cash flow from that annuitization perfectly matches a consumption plan, that's pretty cool[3] for a while but when the spend process later starts to wobble and destabilize I will no longer have an optimal plan. I'll go back and take a closer look at the paper when I have the stomach for calculus again.
- It would be interesting to see an optimal consumption path some day but I also think there would be too many modeling risks to be 100% confident in the result (see prev point). Also, even if we could gin one up at a given point today my guess is that it would be terribly unstable tomorrow. I'm sure there are formal ways to deal with this in the various economic/finance sciences that deal with decision making under uncertainty but that reminds me that I have a fear of academics that may have stupendous minds but little skin in my game.
- I can't prove it here but I'm thinking that a constant spend plan is not the optimal one. In the Case A set-up it would more than likely violate the constraints. It is a plan to be vetted, though, along with all the others. Whether it would be admissible or feasible is unknown. But maybe that's the point.
- As a non-student, non-practitioner, and non-teacher it was not all that easy for me to get my hands on the Yaari paper so I'm glad I've now had a chance to read it. It is helpful to see an iconic paper that an awful lot of others point back to over and over and it is useful to have some theoretical context, however sub-optimally understood, when I flail away at my amateur hacks.
Related and partly redundant post: My Excellent Spending Adventure
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[1] the absence of constraints (for case A he says: "we shall call a consumption plan c admissible if it is bounded, measurable, non-negative and, finally, if it satisfies the wealth constraint S(T) >= 0 with probability one." Emphasis in the original) opens things up to both bequests and negative wealth, the latter of which I mentioned twice as an odd duck idea in several prior posts.
[2] Yaari teases my ignorance here by hinting at the applicability of variable time preferences e.g., "In other words, the future is being discounted more heavily because of uncertainty about survival." and quotes Fischer on this ("Uncertainty of human life increases the rate of preference of present over future income for many people, although for those with loved dependents it may decrease impatience") and shows where it is more or less implicit in the integral...but...he then goes on to baffle an amateur with one of those econo-ninja-mind-tricks that abruptly negate the recently and temporarily held idea. This left me looking like one of those slack-jawed doofus kids that gets neither the joke nor the big words.
[3] Cool but also a pretty big "if." Before we even talk about randomly varying spending or perhaps chaotically spinning-out-of-control spend shocks I think that we'd have to admit, given the truth of most real world spend plans, that it will be a long time before any insurance company will have a price-able, customizable, adaptable cash flow "product" at a loaded cost that would make sense...or that would perfectly match anything. Ever.
[2] Yaari teases my ignorance here by hinting at the applicability of variable time preferences e.g., "In other words, the future is being discounted more heavily because of uncertainty about survival." and quotes Fischer on this ("Uncertainty of human life increases the rate of preference of present over future income for many people, although for those with loved dependents it may decrease impatience") and shows where it is more or less implicit in the integral...but...he then goes on to baffle an amateur with one of those econo-ninja-mind-tricks that abruptly negate the recently and temporarily held idea. This left me looking like one of those slack-jawed doofus kids that gets neither the joke nor the big words.
[3] Cool but also a pretty big "if." Before we even talk about randomly varying spending or perhaps chaotically spinning-out-of-control spend shocks I think that we'd have to admit, given the truth of most real world spend plans, that it will be a long time before any insurance company will have a price-able, customizable, adaptable cash flow "product" at a loaded cost that would make sense...or that would perfectly match anything. Ever.
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