Merton and a special case of optimal consumption

I think I've done this before but what the heck? If a retirement blog can't drool and repeat itself every once in a while, then is it really a retirement blog?[1] The occasion here is some thoughts coming out of reading Merton's 1970 paper on "Optimum Consumption and Portfolio Rules in a Continuous Time Model."  Either a hat tip or an accusatory finger pointed at David Cantor for this outrage  

Ok, I will confess that I can't really "read" Merton. It reminds me a little of when I try to read French. Of course if there is a word in English in the text I'll think "hey, I recognize that" or if there are cognates (say, restaurant) or maybe some phrases - "jeune fille" or "tout le monde" then I'm ok. But unless it is juvenile lit or a simple short story...I'm usually lost, though I can get the general drift.  In this case I was buzzing along and more or less getting it -- I mean I actually recognized the differential equations for a net wealth process -- but then I got really lost.  On the other hand, late in the paper, when he opened his assumptions out to a new set of restrictions, something simple popped out.  When the constraints include, but are not limited to: continuous time, dynamic price expectations outside of  Geometric Brownian Motion, HARA utility including Bernoulli Log utility, consumption and asset demands are linear in wealth, some case definitions I didn't follow, and filtration on current price, etc then the optimal consumption, on that page anyway, collapses to 

C* = W/(T-t)            [1]

where C* is optimal consumption, W is wealth at time t (I presume), T is the conditional (I presume) longevity expectation and t is the current period. Unless I am mistaken about this equation, my reaction, as it is in seeing an English word in a French text was "hey, I recognize that!" It's basically an RMD calc, often touted by some as "close enough" to optimal solutions. It also reminded me that Gordon Irlam suggests this construct, acknowledging Merton if I recall, as a sorta-near-optimal rough rule of thumb. It also reminded me of some work by Milevsky where in certain cases the issue of longevity dominates the consumption calc...wish I could put my finger on the papers.  Anyway, [1] is something easy to work with and I'm pretty sure I've worked with it before.  Certainly it is easier than this, which appears earlier in Merton...

The Work Zone

Even though I know I've played almost this exact game before, we'll run through it again. And, while we are doing this I'll point out I am not playing a recursive simulation game here. I am playing constant-wealth-dollar game. I explain it like this. There are two ways to roll with these kind of things:

1. Recursive

wealth in time 1 W1 - age 60 - is a function of W0 and random returns and consumption

wealth in time 2 W2 - age 61 - is a function of W1 and random returns and consumption

2. Constant Wealth

if I had $1 and retired at age 60 what would I spend?

if I had $1 and retired at age 61 what would I spend?

and so forth...

I am working in zone 2 so there is no confusion.

Assumptions

To set this up I'll assume a few things

- age 60 to age 80 is my interval of interest; could have gone longer but didn't

- spend is per dollar of wealth. Note that spend can in some models be a function of wealth. For example, in a life consumption Utility sim if there is high wealth, the drop to ambient income in the case of wealth depletion can be pretty far and the convexity of the Utility math punishes that so that higher wealth can, in that universe only, mean more circumspect spend rates especially at early ages. But that's another story. 

- longevity model is Gompertz with mode 90 and dispersion 8.5. This puts it somewhere between average expectations ala Social Security and annuitant expectations ala SOA IAM. The use of Gompertz is conditioned on age so that in estimating T at age 70 is conditional on reaching 70 and so forth. 

- the three models are: 1) the simple special case Merton using eq [1] plus the stated Gompertz for estimating T , 2) a rude composite of my reinforcement-trained-algo spend estimate using consumption utility framework and risk aversion coeff of 2. and 3) for fun I'll throw in my RH40 age based spending rule of thumb.

The Models

1. Merton as in [1]. T is estimated using Gompertz as mentioned. W is $1. Irlam condenses Merton to W/e but it is the same thing. RMD calcs can use the RMD tables of the IRS but that is limiting, especially for early retirements or idiosyncratic longevity expectations, and this is the same principle. In this case I am using the 90th and 95th percentile for conditional longevity. 70th would have higher spend rates, 99th lower. 90-95th is arbitrary and tendentious in a way.

2. My Machine Algo. I won't regurgitate here but I tried to train a program to teach itself what to spend using a value function for lifetime consumption utility and gompertz mortality with some parameters I can't recall and a risk aversion of 2 which is arbitrary but useful. The spend rate line is both a composite of several machine attempts as well taking that composite and fitting a line using a 2nd order poly trend. This is all illustrative anyway.   My cover is here. The Value function, if you need to know, is as follows where only the left side to the right of the summation is used and the terms of which can be decoded by way of the link. 

3. RH40. This was a bit of a lark. I took a formula for age based spending created by Evan Ingliss and converted it to something that would bend with age. I wrote it up here. It's here just because I wanted to see if it lined up which I expect it to do. Note that RH40 only works ok over the intverval of, say 60-90 or 95. The formula is, for reference 

withdrawal = Age / (40 - Age/3) + n  [ / 100]

where n is a fudge for risk tolerance based on absolutely nothing I can back up. In this case n is set to 0.

The Goal

At this stage of my blogging life I have lost sight of some of my goals or at least sometimes I do. Maybe here we can just say: "I wanted to check out this Merton special case optimum that is both simple to understand and simple to deploy...against two of my ego-invested personal projects to see what a genius I either am or might have been." Heh. Kidding. Or maybe the question at a minumum is: "are they all three at least playing, very roughly, the same game?" 

The Output

I am not entirely sure I set up the background info and assumptions well enough to jump to the output. But in this case I will do exactly that. If there is either confusion and/or interest then email me. 

Discussion

You know, it's not exact but in terms of the goals I set I can at least say everyone is playing the same game here...roughly speaking. 

Longevity expectations would dominate here. A slight shift younger would push these Merton lines up quite a bit

The RH40 rule for all it's dumb brute force doesn't seem that bad

The Machine, probably because by design it is adaptive, seems to know something about the early years that the others don't. I have not explored that nor can I really explain it well. 

Sub 3% spend rates at 60 seem absurd based on other work I have done but then sometimes I am personally trying to go sub-3 so idk. Ed Thorp called 2% consumption a perpetuity and he is probably right but then he didn't live in 2021 US or 1989 Japan so who knows. 

No one line here at 60 or 65 is at 4% or the new 4.5% that Bengen just came out with.  But then there are some other issues to deal with.  First, no one should skate on the ice near the open water. Second, many models ignore at least a bit the superannuation risk mitigated by annuities. Third, few modelers think in Talebian terms of robustness where survival is more important than optimality or at least it has first priority. Robustness in simple terms can be articulated as redundancy. Redundant capital is the same thing as very low spend rates at least initially. More risk can be taken in spending as longevity risk comes in and the redundancy is cashed in. Even my machine figured that out though I have a better graphic for that than the one I used here.  

W/(T-t) as a type of dynamic RMD calc is probably useful for pedagogic reasons as well as something that is relatively easy to use and explain.

Spend rates in adaptive real life are not lines or optima, they are zones within which the right answer is fuzzy and readdressed each year. Most real older retirees already know this without knowing any math. This is why they sometimes laugh when they hear what I do for a hobby.

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[1] I get savaged on Twitter for playing the old man game, but I amuse myself...