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 is that the sim -- over x,000 iterations that each have random lifetime -- sums discounted utiles of real consumption over each individual life and then averages over the iterations to calculate the expected discounted utility of lifetime consumption (EDULC). The major things to note here are that:
(a) the modeled lifetime for each iteration is random but random to probabilities inherent in the SOA IAM annuitant life table, and
(b) spending snaps to available income (e.g., SS, purchased annuities, and pensions) when wealth runs out.
That snap is important and has a blunt force effect on utility which is why I am doing this. This post, by the way, is mostly me announcing to myself that I am done. I will also take a quick stab here at running a comparison between a constant spend and my RH40 spend rule (like RMD) to see what happens. Nothing rigorous yet. This is just an initial shake out.
MATH - A
The math framework for the sim proper is this:
where
E[V(c)] is the expected discounted utility of lifetime consumption
c - is a consumption plan, c(t) is consumption in period t
S - is the number of iterations
k - is a subjective discount on utiles
g(c) - is the CRRA utility function but using a constant (-1) in the numerator
T* - is the random lifetime
(income - in the sim; that and the snap can't be seen in the above formula)
MATH - B
The lifetime consumption utility math shows up in the ret-lit fairly often but most of the time I see it in papers by Milevsky. However, let's in this post tie the concept we are talking to the work by Yaari 1965. Here is an excerpt from the Yaari paper where he first frames the issue before getting into the weeds. The sim above is, to the best of my amateur ability if I have everything right, an expression of this concept:
TRIAL RUN IDEA
See the first link above at the top for the sim framework. I'll do a "for-fun" trial run of two things with no graphs or charts this time. For each of the below I'll calculate the expected discounted utility of lifetime consumption, or EDULC:
A - constant 4% spend
B - RH40 adaptive spend rule, which I'll explain in a minute
No judgments or calculations related to bequests or bequest utility is done here. That is for later...
ASSUMPTIONS
The core assumptions are not really supposed to be realistic, just placeholders for my initial test of the software. Assumptions include but are not limited to:
age - 60
lifetime - random per SOA IAM 2012 with G2 extension
consumption - 4% constant inflated else a spend rule I call RH40
endowment - 1M
return - .08 (normally distributed but fwiw I do have a live feature to mix returns into a fat left tail)
std dev - .11 (this return and sd are vaguely similar to a 20th century 60/40 portfolio...but not quite)
iterations - 20k x 5 = 100k
CRRA gamma: 1
SS - ~16k at age 70
no purchased annuities in this post though that has a pretty big impact
inflation - .03
subjective utility discount - .005
I am probably missing things here....
THE RH40 SPEND RULE
RH40 is a spending rule that adjusts by age. It is very similar to the RMD approach but is more conservative. The rule was based on an adaptation of a paper by Evan Inglis that described his "divide by 20" rule. The math for RH40 is this:
spend rate against wealth in points = age /(40-age/3)
for the interval 60 to 90 it looks like this:
The basic idea is that since we face an uncertain lifetime we should probably be conservative early to reserve for superannuation risk and loosen up later. It looks like self denial but it might also beat running out of money. The WDT sim will tell us... You can search the blog for RH40 for additional background.
THE RESULTS OF THE BAKE OFF
A. EDULC of a constant spend: 254.33
B. EDULC of RH40: 261.00
RH40 wins the utile game but I knew it would. The constant spend will cause wealth to fail under more scenarios which forces the utility snap shock that will dominate the math. The RH40 conservative approach mitigates the likelihood of those failures so that even though we spend less upfront the overall utility over a random lifetime has a better chance of being higher when we check over a ton simulated random lifetimes*. A more interesting test would be to compare rules like RH40, RMD, ARVA, Guyton, etc. Also, at some point I have to figure out the weighted utility of bequest because we are kind of leaving that issue to rot here.
* postscript 5/29/18 - I forgot to mention that since the rule responds to ambient wealth it will generally perform better than a constant spend but introduces lifestyle vol. I have not measured this but it would be important to check how bad it can get in bad scenarios. But then again this was intended to be a software shakeout not science.
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