Wealth Depletion Time Simulator - an update and summary

While I'm waiting for the software I built to simulate the below to be pulled off a crashed hard disc now in a geek squad clean room in Kentucky (have I posted on doing backups more frequently? I think I have) I thought I'd update my WDT notation and comments to more accurately reflect what I was trying to do. The past post I'm updating is here:

The Wealth Depletion Time EDULC Simulator is live!

Now, from that post but with some new edits...


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BACKGROUND

I described the basic idea for my WDT simulator here (Putting it all together: a wealth depletion time "utility simulator") and in a rudimentary form here (3D "utility surface" in a Wealth Depletion Time model: spend rates vs annuity) and also here so I won't completely repeat all that again.  That last link is probably the best one to help visualize the basic concept.

The basic idea 
The sim -- over x,000 iterations, each of which have a random lifetime -- sums discounted utiles of real consumption over each individual life during an iteration and then averages over all of the the iterations to calculate the "expected discounted utility of lifetime consumption" (EDULC).  There is a lot going on in there but the major things to note include things like:

(a) Lifetime is random -- the modeled lifetime for each iteration is random but random to probabilities inherent in the SOA IAM annuitant life table 2012 with G2 extension to 2018 reflecting changes in longevity expectations,

(b) The primary focus is on consumption -- This model and software do not make wealth or, in a strict sense, income the center of focus. The essential focus is on spending and its "utility."  Things like wealth, returns, volatility and income are necessary components, of course, since that is what (1) enables the voluntary spending strategy in a pre-wealth depletion state and (b) determines the length of the "involuntary" wealth depletion state, if any, prior to death,

(c) There is a consumption floor -- If there is one thing to pay attention to in the model it is that spending snaps to available income (e.g., SS, purchased annuities, and pensions) when wealth runs out. That snap is important and has a hard, blunt force effect on the expected value of discounted lifetime consumption utility,  and

(d) There is no "risk of ruin" as such -- while this model is based on a simulation framework, note that there is no "failure" or ruin probability per se since assumption (c) precludes a hard fail. Yaari, in his 1965 paper, comments that given social structures (let's say government programs, associations, and family not to mention behavioral and adaptive considerations that would anticipate and intervene before a ruin state would occur) it is more or less difficult for someone to actually fail in modern society. The homeless might disagree with this comment but this model will assume some de minimis (or higher) income floor. So the risk is less of "ruin" than it is of a hardscrabble lifestyle forced by some combination of consumption behavior, returns, volatility, pooling choices and longevity.

The heart of the math framework for the "expected utility" part of the sim is this:


where the following items describe both the formula above and also generally what I am trying to do in the sim as a whole:

ELEMENTS OF THE WDT SIM

1. E[V(c)] is the expected discounted utility of lifetime consumption and is the main output of the simulation and depends on the other items described below. V(c) is a random variable because lifetime (T) is a random variable.

2. c - is a custom consumption plan, c(t) is consumption in period t.  The following is attempted in the sim. For:

• Wealth(t) > 0: consumption c(t) is the "custom" plan (could be constant or rules based)
• Wealth(t) = 0: consumption snaps to available income = SS + pension + annuity [1]
• Wealth(t) < 0: borrowing not allowed so same as W=0

3. S - is the number of iterations, I typically do 10-20k. Maybe more. Depends on my boredom level.

4. k - is a subjective discount on utiles. Small in my case. Based on outside advice, say .005. [2]

5. g[c(t)] - is the CRRA utility function where g[c(t)] is in the following forms. I often use a gamma of 1 (log utility which has some interesting attributes) but maybe gamma is better set at 3 or 4 based on what I read about real life experiments in risk aversion. The constant "-1" (more or less common in the lit) was used because, as an amateur, I totally could not get my head around discounting negative utiles:

for gamma < > 1

for gamma = 1

         g[c(t)] = ln(c(t))

6. Wealth at time t "W(t)" (not shown in formula but it is in sim) is roughly like this:

W(t) = W(t-1) + r*W(t-1) - c(t) + income(t) - annuity.purchase(t).  

where c(t) is as described elsewhere. I'll have to check the order I did this when I get my hard drive back but I think this is how I did it. The annuity purchase is as described below. Income as described below. But "r" is something to pay attention to (next item).

7. Return modeling is either of the following:

• a normally and randomly distributed return based on arithmetic input r and a standard deviation sd.  The time averaging effects of the simulation will force a geometric return outcome.  There should be enough sims that reflect bad sequences of returns that we can see some sequence risk. Since the sequence risk can be overwhelmed by the number of sims there is, if I recall, a feature to force a more direct regime of good and bad returns where r is -x in the first half of expected lifetime and +x in the last half or vice versa. This might have been only in my deterministic spreadsheet sim; I have to wait for my hard drive. Easy enough to program, though.
• a fat tailed randomized gaussuan mix distribution based, roughly, on X% return1 sd1 plus Y% return2 sd2. This can force a fat tail that can be matched to what is expected in, say, a 60/40 or 100/0 portfolio based on historical data, or it can also go extreme into terra incognita if one wishes. [3]

8. Income: Income in the sim is only SS + pension + annuity (purchased from W which decrements at time of purchase). These start at different times but continue for random lifetime T*.  I have not modeled any employment or continuing pre-retirement income (or taxation for that matter). SS is inflation adjusted...as is the annuity in the current implementation of the software.  Pension income is, right now, a stub that is not implemented yet.

9. Annuity: the annuity that is calculated at time t (if we decide to purchase at t) is a hypothetical, mythical beast based on the concept of an an idealized real annuity priced using the purchase age, a conditional survival probability based on that purchase age and an SOA life table, and an annuity discount rate that is based on, well it's based on not much.  Right now I'm using around 3% but that should move in the future or maybe be more flexible in the model.  There is a baked in load too of, I think, 10%; I'll have to check when and if I get my hard drive back.  The math is more or less like the equation below where a(x) is the annuity price at age x, tPx is a vector of probabilities in future times "t" for start age x of survival at t, R is the annuity discount and L is the load. The cash flow is either an implied nominal 1 or an implied inflation adjusted 1 based on the c(t) at the time of purchase at t. I think I have this right though there is maybe a better way to notate it, say by adding a cf(t) to the numerator to make it more clear. Again, I have to check my software when I get it back. Totally ignore for now whether any of this is realistically purchasable in a real, complete, and non-defaulting market:


10. T* - is the random lifetime and is the main, and very important, reason we came to play this game. I used the SOA IAM table for a male if I recall. I modified it using the SOA adjustments for longevity changes over time.  The maleness is a proxy because I made the sim for me and the shape of the curve can be more important than the gender. At some point I will work on this a bit more.

11. Other.  Note that this is a sorta complex but very simplified model. I have no explicit modeling of  things like taxes or fees and so forth.  These, in particular, could be baked into the return assumptions, though.  I have this theory that the general shape of the model is way more important than sweating all the details. This is debatable but then again this model is for me and not a commercial or academic proposition so who knows? Another missing piece that should be obvious is the concept of a bequest utility. I am ignoring this for several reasons.  First, the bequest, in theory, is seperable at time zero and so is at least potentially ignorable. Second, I'm not sure how to do it.  Third, Prof. Milevsky, for both theoretical and common sense reasons, discouraged me once from spending too much time on it.  For my purposes I will assume that bequest is either zero or seperable at T0.  Convenient, eh?

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[1] I have some unrealistic simplifying assumptions for when wealth is less than the current period's consumption but slightly above zero.  This may change but I don't think it matters much at this point when we are looking at the broad effects of consumption on utility.

[2] This was suggested by Gordon Irlam.  We did not have a comprehensive dialogue on this so we can re-examine this at some point.

[3] I have had criticism before on overly simple return modelling.  That is why I added the non-normal option with its fat tails.  But I don't really think this matters much and the effects, I believe, will be dwarfed quite a bit by other considerations like spending and vol and overall level of returns. I have seen this corroborated by academic researchers way smarter than me.  This is why I don't sweat things like autocorrelation and mean reversion overmuch like others do (plus I have no one to please but myself).  Also, note that I am not really modelling multiple asset classes and their various correlations. I have only one point of connection in the sim with a portfolio return and std dev (whether normal or fat).  However, this means that if I am generally aware of a plausible efficient frontier (stable or otherwise), I can model the portfolio stats (allocations) just fine along the EF though it is a bit more manual than one might want.  It also means that I have a free hand to model points off of the efficient frontier if I know how adding alt-risk assets really works in real life in terms of portfolio level effects over some unknown horizon with some unknown correlation.  I can also, btw and for the same reason, "burn down" efficiency by modeling under the EF. In the end there are no portfolio combinations that are not model-able in the 2D EF space with a simple proxy. You just have to know what you are doing and also, fwiw, know something about portfolio covar math, linear vs compound returns, the inputs to MVO, and the effects on long term outcomes of time and multiple periods and consumption over a horizon.  It's not really rocket science but does take a little thought and effort. I'm not sure I have it down cold yet but I also think the criticism I've received is sometimes well informed and sometimes not so much. The latter can go (politely) hang. The former can write to me in great depth and the correspondence will be more than welcome.