How I might tweak my consumption utility simulator in the future

I wanted to look, in this post, at what might be missing in my lifetime consumption utility simulator. Specifically I wanted to anticipate how I might adjust it if I were to think more carefully about the presence of things like annuities, bequest motives, spend shocks such as long term care, the need to inter-temporally shift consumption in the future, support a desire for optionality on future consumption increases and so forth. None of that is "answered" here in this post. I just wanted to think about how I might deal with it at some point. The goal is not to educate a reader, by the way, it is rather to commit to "paper" what I think I'm trying to do for myself. I find that writing something down makes me think more carefully which I can then use in the future.

The Basic Consumption Utility Function

In my attempt to wing a consumption utility simulator earlier this year, I tailored the core value function, coded in R, to be like Case A in Yaari's 1965 paper. That was the case with what he called Fisher style utility (with constraints) without insurance (annuity) markets.  His Case A function -- without much explanation of terms except that Tbar is random life, V is an expected discounted utility of lifetime consumption, c is consumption, g is utility of consumption and alpha is a subjective discount -- looked like this:

Yaari case A (eq 13)

My amateur value function, the one I try to simulate, looks like this below which is more or less the same thing as above, given my non-economist efforts, except it's discrete and simulated...if I understood things right...which is a big if because it's hard for me to read calculus and impossible for me to read the related differential equations. Let's just say I made a good faith effort:

RH EDULC value function

This does a journeyman's job for what I am trying to do personally and works well enough for evaluating basic consumption utility. The terms and sim structure can be understood here.


Consumption Utility with Insurance

But I happened to recall yesterday that I had also thrown some annuity logic into the simulator as well which opens the way for me to take an amateur look at my version of his Case C which is Fisher style utility with insurance markets.  That case C happens to be "the well known one" where, in the absence of a bequest, it becomes clear that one would be inclined to hold all wealth in actuarial notes (annuities) in order to maximize consumption utility when insurance is available.

There is a vast literature on annuities, optimal time for annuitization, annuity paradoxes, mortality credits and so forth. I can't touch that but I can at least take a look to see if my simulator is up to the basic task it's supposed to do or whether it is out to embarrass me in public.  So, for a set of generic assumptions[1], let's annuitize wealth in increments to see what happens to the expected discounted utility of lifetime consumption (EDULC). It might look different for a different parameterization but this is what happens to lifetime consumption utility as I annuitize wealth->income in $5k increments of payout for my generic set-up:

Phew...at least this is not totally embarrassing because with insurance available it looks like I would annuitize all my wealth to maximize consumption utility. This outcome is based on a decreasing probability of and severity of what happens when or if wealth is depleted before the end of a random lifetime...due to the presence of lifetime income.  The outcome appears consistent with the theory expressed in case C. Left on the table, however, are questions of when exactly to annuitize if not immediately, how much to annuitize at any given time if not all of it, and how to allocate what is not annuitized.  Big questions, right? My current structure can deal with some of this (e.g., deferral of the annuity choice) but not all of it.  [Note again that the annuity is nominal and the spending inflates. That's why we can annuitize payouts over the top of spending at the beginning. ]


Consumption Utility with Insurance, Bequest Motive, and Other Stuff

I have not read the lit on the annuity paradox nor could I summarize it here. I just know that it exists. Basically: notwithstanding the previous chart, people are reluctant to annuitize.  I am myself reluctant to annuitize.  There are plenty of ways to try to explain this, from excessive loads in products to incomplete markets to loss of optionality on wealth or consumption to bequest motives etc etc.  The list probably approaches infinity but let's think about a few that interest me: bequest, spend shocks, separable big-ticket consumption goals, inter-temporal consumption shifts, and foregone optionality. Remember, the goal here is not to decode the annuity paradox, it is to contemplate how I might adjust a lifetime consumption utility simulator so that it does not over-reward immediate annuitization if I believe there are things more important than steady lifetime income forever...at least at the beginning, anyway.

1. Bequest

It's not clear to me how I'd  handle bequests inside the simulator. I have two versions of the value function, one with random draws on lifetime tenure and one that is weighted by a conditional survival probability over infinity. The programmatic handling of bequest would be shaped differently in each case.  Certainly it would be some function of net wealth (saving) at either time T (end of life) or probability weighted over an infinity of times t. It might also be adjusted for general time preferences and perhaps, as Yaari pointed out, also have higher utility at certain middle ages rather than those close in or those far out, giving it a hump shape.  In his case B (what he calls Marshall utility using a penalty/bonus function in the absence of insurance) he adds a term to the integral seen above in his eq 13:

Yarri case B (eq 21)

Where S is net wealth (saving), ϕ is the utility function on wealth and β is the subjective weighting.  Here is Yaari: "A consumer very likely values a given bequest not only according to its size but also according to the time at which it is made. To accommodate this possibility we introduce a subjective weighting of bequests, to be denoted β ... In most cases one would expect β to be a hump shaped function because the importance of bequests is foremost when the consumer dies in his middle years." 

I've been thinking about that last sentence for a year and it gets odder every time I read it. My best amateur guess on how I might handle that is via standard discounting but discounting turned inside out with respect to time.  By that awkward phrase I mean maybe something like this

where h is some preferred bequest planning horizon. That means we would discount heavily now, then low at horizon h, and then heavier thereafter. At least I'd get the hump.  I'm not really sure what Yaari intended or if this gets at it in any constructive way but this would be a starting point to play around with to see how it works if I were to try.

The problem though is really ϕ because I have no idea on its form.  For consumption utility I can use the CRRA power function because, though flawed, it is common and well known and easy to work with.  No idea what to use here on wealth, though; maybe CRRA, maybe not. This kind of problem may be why Professor Milevsky waved me off when I asked about bequest utility and suggested that I focus my energies elsewhere.

Fortunately I am saved not only by the professor's wave-off but also by Yaari's case D where insurance is introduced.  To quote: "The most important feature of equations (46) and (47) [differential equations] is the former does not involve S* and the latter does not involve c*. This means that when insurance is available the consumer can separate the consumption decision from the bequest decision, in the sense that the savings plan S affects the consumption plan c only globally, but not locally. We have seen that in case B that in the absence of insurance such a separation is not possible."

This saves me a lot of work and worry and it means, because I am anticipating the idea of insurance, I don't have to deal with bequest at times t or time T*, I can deal with it at time t=0.  In my simple version of the simulator, the initial endowment w, which is also net worth, is also the wealth at time zero = w(0). With a bequest involved I'd do something simple like this where "B" is the bequest (which is probably deeply discounted from the bequest horizon h and is managed either separately or as a virtual embedation in my portfolio):

w(0) = w - pv(B(h))

Problem solved for now.

2. Spend shocks

An often cited reason for avoiding full annuitization is that it kills flexibility in the face of large un- or partially- expected expenditures like health shocks or long term care.  How to handle this in a consumption utility sim is challenging. This type of consumption is not "positive" consumption, it is a buzz-kill (I get positive utility from consuming a nice bottle of burgundy but maybe less so from ventilator fees while in the care facility).  I can't model it as either planned or random consumption because it would receive positive utility which doesn't make sense to me.  It might be better conceived as a wealth shock, a sudden demand on and reduction of wealth. In this sense it is akin to a retiree having a second type black swan risk not available to those oh-so-lucky single-period no-consumption portfolio managers...as if a mere single-source black swans from the markets were not nearly enough for a hardy soul like me.

If I were to embed the shock in the simulator at times t I could do it either as a curveball to consumption but outside the utility function so that wealth get's ding-ed but consumption utility is passed over or I can model it as some kind of return or wealth effect. There would then be a global effect on lifetime utility via the implicit sequence risk that occurs and the corresponding diminution of portfolio longevity and extension of the wealth depletion time probabilities. But even then I'd still only be able to model it as a random "regular probability" process which would likely under-sell the full force of the chaos of spending black swans. Dirk Cotton did some good work on spending, bankruptcy, and chaos theory a while back.

An alternative is to punt and model it as we did above with bequests. This is perhaps a little easier when "budgeting" for something like long term care where there are some known estimates for likely lifetime spending on LTC (the estimate I recall is ~260k). It's harder, and maybe not even constructive or maybe too paranoid, to mentally conceive of all the other uncertain risks that might befall us.  So, let's keep it simple. Let's separate, at t=0, a budgeted amount for spend shocks plus a conservatising "other" that isn't so obsessively big that it forces us into a state of excessive self denial forever. It'd look something like this when we add it to the bequest version above:

w(0) = w - pv(B(h)) - E[pv(LTC)] - E[pv(o)]

where the expected value terms for LTC and "other" can be either simple plugs or a more complex stochastic present value analysis where we pick some metric like median or percentile p for the deduction. I vote for the former (plug).

3. Separable big-ticket consumption goals

I'll cut to the chase and say that I am going to do the same thing here that I did for bequest and spend shocks. This is standard (to me) balance sheet management.  The example I use for myself is that while I may have a holistically managed meta portfolio, I also have a separate liability (goal) on my BS for spending on independent schools and colleges for my kids. That means that that "spending" does not exist on my income statement because (a) it is not a normal recurring operating cost, and (b) it has already been "spent" because I have capitalized it on to the BS.  Then, any time I do personal spending analysis -- such as stochastic present values, monte carlo fail rate sims, life time probability of ruin calcs, or consumption utility -- the denominator, to the extent that I am working with a rate, is never w (net worth) it is w* which I call "monetizable-for-retirement-consumption net worth." That mans that the PV of college comes off of w. As does most (but not all) of my home value. As do (now) bequests and/or long term care budgets.  But the w(0) equation is now getting longer.  For fun, let's just call it like this:

w(0) = w - stuff

where "stuff" is a reasonable list of things to deduct off the endowment during financial analysis to either protect for known planning goals or to conservatise the plan in the face of uncertainty -- but without burning the plan to the ground entirely out of fear. None of this in necessarily best managed via annuities, by the way, which shine better when managing superannuation risk in isolation. I can now see where the full annuitization of wealth at t=0 would impinge on my ability to flexibly plan for either known things or fuzzy uncertainty, either now or later, as I best see fit.  Before we get to #4 below, which will change things, I might actually still want to fully annuitize wealth at t=0 but now it'd only be w* and not w. Enumeration of the stuff in order to determine w* might look like this:

Stuff =
  a. present value set-aside for bequests at time 0
  b. stochastic present value budget for expected long term care
  c. stochastic present value budgetary provision for vague uncertainty: "other" spend shocks
  d. separable big-ticket non-operating consumption or other planning goals
  e. annuity purchases at time 0 that are converted into lifetime income [4]
  f. other

4. Inter-temporal consumption shifts and foregone optionality

Other common thoughts about the idea of full annuitization are that it: (a) limits somewhat the ability to shift spending around inter-temporally and (b) there is option value, time value, in not annuitizing now.  I won't dwell on (a) because it is probably complex to model and I don't have a good mental picture of how to do it.  I just know that I do it all the time in real life via deferred maintenance or borrowing (and paying back) from a line of credit[2]. This, to give it a name, is consumption smoothing and it would be much harder for me to efficiently achieve in the face of full annuitization, especially if I lost my line of credit. I will leave this unmodeled for now.  While the inter-temporal shift idea is not the same as (b) I will let (b) subsume (a) for now due to some similarities that may or may not be self evident.

The option value of not annuitizing now [i.e., "(b)"] is a big story covered better elsewhere by others (see Milevsky's body of work) but let's assert here without evidence (a former boss once called it proof by intimidation) that there is some utility value to at least some un-annuitized wealth over a random lifetime such that we need to account for it in our simulator to keep the simulator from over-rewarding full annuitization at the beginning. This is a little open ended but even after we have buried some of our favorite considerations above into w* I think we still need some kind of "discounted utility of wealth" term that rewards the retention of at least some un-annuitized w* and, if there are no appropriate constraints, also penalizes negative wealth.  My guess is that in continuous form it would look something a lot like term above in Yaari's equation 21.  How to instantiate that in discrete form in a simulator written in R by an amateur is another matter altogether. My first guess/pass might be to go with the Yaari style approach[3], say like this:

RH adjusted EDULC value function[3]

Subject to:
w(0) = w* = (w - stuff)
w(t) > 0, c(t) is the plan
w(t) <= 0, c(t) = available income

where in my model, right or wrong, omega is a survival probability (should it be?), alpha is the same discount as above just hidden in a greek letter now, and g is as before, a CRRA function of consumption utility. So, the left side of the right side is the same as before, just re-written now to try to approximate Yaari's notation. What we've added is the right side of the right side. pi, here, I think is the same as omega which in my amateur version is a conditional survival probability but I am not certain and this could use some serious scrutiny at some point. My insecurities abound. epsilon is a subjective discount on wealth utility which I think could be the same as alpha but I am also not certain with this either; it might also have that inside-out discounting thing.  h is the utility of wealth function. As above, I have no idea what math form h is supposed to take so let's call that a "to-do" and a placeholder for later.

The other thing I'm thinking about here is that h might not really be a direct function of S(t) properly thought. Rather -- since I am marginally aware of an annuity boundary, below which I don't really want to go in order to not irretrievably lose future consumption utility and above which I have wealth I don't want to annuitize (yet) because of the extreme upside potential -- we are really talking about an option on wealth with the strike being the full annuitization we were trying to avoid (or some other appropriate policy boundary).  That means the h term might instead be h[m(S(t))] where m is a real derivative of an expected net wealth process. Ignore for the time being that I don't know economics at all and that I have no idea how to shape the math and/or equate or scale the marginal utilities of the two terms. I'm just playing around to see where I might start someday. I'd need some serious help to get to where I want to go. But given all that self-effacement, we still at least know how to try to do "m" from a past post. There, we evaluated, via simulation, a call option valuation on net-wealth like this

                                       C = ∑ { P(W_T) * PV(max[0, W_T - k]) }

where T was a tenure date and k was a strike in terms of a policy boundary which could easily be the annuity boundary.  I'm not sure exactly how I'd translate any of this to a working utility simulator but it's all just a concept for now anyway.  Also, my guess is that the nature of m, h, epsilon, and pi would all ensure, as in Yaari's comment, that the second term would be quite hump shaped across time since it seems intuitive that the importance (utility) of optionality of wealth will be bigger tomorrow than it is today but very much smaller at the "very end," net bequest.

So, what's the purpose for all this utility of optionality in the model? The purpose is whatever the upside buys. It could buy capacity for un-anticipated but needed inter-temporal shifts. It could buy risk capacity for raw unplanned uncertainty. It could liberate the capital previously dedicated at t(0) to the LTC budget and transform it into bequest at (h), it could ratchet up consumption to a new state not even possible at inception that gives higher lifetime utility than would have ever been available if wealth had been made fully illiquid via annuitization at t(0) i.e., a new optimal EDULC that was not possible without the option. In the end, I think something like this needs to be added to the sim.  Or at least one needs to be able to consider subjectively what is potentially forgone by locking up capital early even if that choice looks optimal in a more simplified analysis.


Note: this was supposed to be a 10 minute drive-by post.  Three days later...

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[1] $1M endowment, age 60, 50/50 allocation, a little bit of social security at 70, 4% constant spend rate.  The annuity is immediate and nominal. I priced the annuity at 80k per $5000 of payout which is not exact or fair but close-ish; it underprices a bit since a real price is closer to 85k or 87k or something.  80 was just a round number for illustration.  For each increment 5k in payout wealth at t(0) is reduced by 80k

[2] I have seen no small number of articles and advisor blogs that either caution about or rip on secured lines of credit. Certainly, intemperate use of credit for ill-thought-out purposes (or forbidden reg-D purposes) is not constructive but all of them forget or suppress discussion on the real utility value of lines of credit for well thought out consumption smoothing.

[3] this is effectively the same as Yaari's eq 45 in case D though I am still a little fuzzy on interpreting his math which does not surprise me.

[4] It may seem odd to deduct annuities when we are evaluating lifetime income (annuities) and consumption but in a slick 2007 paper by Babble & Merrill from Wharton, "Rational Decumulation" they point out that "we follow Richard and others in modeling the allocation of wealth in excess of that which is necessary to provide for the minimum acceptable level of consumption." i.e., the utility analysis the do is incremental wealth over an annuitized floor that immunizes the spend liability. I don't do that but I could, hence the footnote.