I've been through this paper -- Uncertain Lifetime, Life Insurance, and the Theory of the Consumer, Menahem E. Yaari 1965 -- maybe 10 times now. I've done this because it was a seminal paper and and it frames the lifecycle dynamics of the retirement problem well in quantitative terms and it is a good read with some sly comments here and there. Most decent finance papers that go beyond simple finance and into econ and actuarial topics either do or should reference it. Here are some stray thoughts on my 10th (or so) read.
< with some corrections 2/8 >
1. Sandbag #1
Yeah, OK, I still can't read calculus and differential equations well. I mean, I tried to brush up a few years ago and I have read a giant stack of finance papers over a decade so it is kinda "familiar" and I can see into some of what is going on but it's a little like my French Skills. I can read juvenile literature well enough but were I to try to read science or philosophy it would it be more or less impenetrable to me. I can get by on my general "recognition" but wish I had a better command of the tools used.
2. Forwards = Backwards
He made a point I missed the last time reading through section VIII "Case C."
"The purpose of the foregoing paragraph has been the construction of an argument which would permit one to replace compounding forwards by discounting backwards in the global constraint."
Italics in the original. This is probably best narrowly construed to what is going on in Case C -- where the end bequest is zero and the constraint is that the consumers estate can never be non-negative -- But it does remind me of another later paper by someone else. In Robinson and Tahani (2007), they demonstrated, using Ito math, how the forward-backwards equivalency thing works. In my language I might say something like: a forward simulation (or dispersion process) is more or less the same as a backwards discounting stochastic present value exercise under some constraining assumptions. i.e., I guess I am saying feasibility (a balance sheet calc done at t=0) is sorta the same as sustainability (think Monte Carlo and fail rates kinda sorta). I am sure there are circumstances where if we let things get messy that feasibility=sustainability isn't exactly everywhere and always true but it's true enough that I can hold it generally in mind for second and not wig out.
Here is a tiny lift from that R&T 2007 paper (appendix) that explains what I am talking about. It looks like this https://rivershedge.blogspot.com/2020/09/on-alliance-between-fail-rates-and.html
"It may be well to mention here that a marginal utility condition for this maximization [Case D only, eq 45] is given by(48) α(t)g'[c(t)] = β(t)ϕ'[S(t)]for all t where the constraint c(t) >= 0 is ineffective. In other words it is optimal to equate the marginal utility of consumption to the marginal utility of bequests at every moment."
Like point one above I am going to do a sandbag #2 here. I am neither an economist nor a mathematician. As an amateur auto-didact, as is true for all auto-didacts, and despite some gnarly efforts at self ed over the years, I have some big fat holes in my understanding of marginal utility and differential calc.
I can't really manipulate or explicate 48 very well like the cool kids might -- let's say roughly: alpha and beta are weightings or discounts[2], g and phi are (often) power utility functions, ' is the derivative I hope, c is consumption at time t, and S is the estate remaining at t i.e. we are working in general with marginal weighted utility -- so...I'll tell a story of what eq 48 means to me by way of a different path.
I once attempted to deduce optimal consumption at my age for certain portfolio and horizon constraints by using backward induction and stochastic dynamic programming i.e., a Bellman type process. I thought I'd been the first to do that for spending but I'm not so sure anymore. I actually think Gordon Irlam did it first once. At least I've never actually seen it done by anyone on paper either then or since then so it's at least rare. [link below of my shot]
In that bellman-equation type process, rather than confront the combinatoric morass of moving forward in time, instead we do this: first, one works backwards from some one period horizon (say t=30) and optimizes there and, then 2) one steps backwards one time increment (say t=29) and then optimizes that given that one already knows what is optimal at t=30. So it becomes a relatively simple two period model rather than the 101^30th combinations or whatever it is over "30 years and different portfolio choices." Might have better luck with forward in 2023 since processing is cheap. This was more for fun.
The thing I noticed, though, was that -- since I was using power utility value functions -- that if I just focused on spending, unattached to any other principles, then obviously the value function on consumption must go towards 100% in any or all periods: spend it all and get all that utility and "screw the next period." [3] Portfolio choice would be a little irrelevant in that case, too. This was broken thinking and broken code. Therefore I figured out I needed to reserve in time t [we have only t and t+1 where t+1 might be t=30 = T], some piece of the pie for next period (or in the case of t=30 a real bequest to the time past 30) so the whole thing would work over multiple periods. In other words the code only knows t+1 or t and t+1 but is also aware in an abstract sense that there is, in fact a future that will have needs, too, if not entirely clear in t and t+1.
This reserve was either a naïve error or a little bit of an "aha!" or both. Let's go with "aha!" I realized that I had visualized my irl "bequest" as me being an old man handing my keys and money to my kids on a deathbed many years in the future. NOW, I realized that the bequest is really (or also) to "future me" next year in order to keep the game going until the end. I became my own legatee in time and the bequest is to either future me or my estate in any given instant (tho I don't really work in continuous time) which means future me and present me are in direct competition for resources to consume. That means I had to balance in my code consumption and bequest in every period [more in the link]. I mean, I never tested back then whether the marginal utilities of each term were equal as in eq48 but the principle of managing the tradeoff, on the margin, between the utility of consumption and my now re-defined bequest in any given period (or instant) makes intuitive sense now. So when I see eq48 now after my aha! it makes a lot more sense than it used to ...and his conclusion, which that quote above is in the paper, "pops" a little more than it used to [4][5][6]. It's like "getting" a hard phrase in French and not having to look it up for a change.
Notes -------------------------
[1] Think annuity where he says actuarial notes are purchased and life insurance for those that are sold. Professor Milevsky corrected me on this, though, and said in the continuous time context the annuity would more properly be considered an "instantaneous tontine."
[2] not 100% sure on this but typically -- a convention I've followed before -- it be a function of time preference, a conditional survival weighting for the age at "t" and in the case of Beta likely to also have some weighting I don't understand but reflects varying preference for bequest at different ages.
[3] as well... if I were to evaluate only the utility of wealth it would be optimal to spend zero which is dumb in a retirement context.
[4] keeping in mind this is for Case D with insurance available and no wealth constraint. In case B without insurance "these two marginal utilities were not necessarily equal." But that is still a little over my head until read number 11. Since we actually do live in a world with annuities even if our allocation may be zero%, Case D keeps my imagination warm and is more consistent with how I hacked out my code.
[5] I never really sorted out how to weight the bequest in anything other than a hackish way. Certainly there was no rigor, more trial and error. If I recall it just devolved to having optimal spend rates that looked familiar to other methods and then passing along the rest. I'd have to look at the code, which I won't today, but the beta discount was a flat .5 or something; no "humps." t=30 was tricky. Probably should have looked at marginal utility now that I see it after this post.
[6] The concept of present-me and future-me being in direct competition for money and future-me being a legatee of me in any instant makes sense of course for anyone doing ret-fin for more that a little bit. But exactly zero of the women I dated for 10 years understood any of that. Future me (or future kids or estate) didn't exist. Their consumption utility for my present resources was or would have been basically infinite if I'd had no backbone. Spend it all now...on them ;-)
References --------------------
Robinson & Tahani. Sustainable Retirement Income for the Socialite, the Gardener and the Uninsured 2007 York University
Yaari, M. Uncertain Lifetime. Life Insurance, and the Theory of the Consumer 1965 (jstor)
Me, My first kinda botched attempt at backward inducting spending via SDP Feb 2020
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