Random Thoughts on Portfolio Choice and its Discontents

This has never been a teaching blog but rather a reportage-of-my-learning blog. Two different things. That means a lot of my stuff comes out like any work-product of autodidacts: spotty, holes, not 100% coherent across topics, un-tutored in others, etc. On the other hand that allows me to roll with whatever - which is what I'll do here.  I don't really have a tight theme or thesis just some thoughts from some recent spreadsheeting last week. 

The Idea

The underlying idea is one I periodically run up a flagpole: 98% (I made that number up) of financial pros in advisory or academia will perseverate on teeny tiny nuances in portfolio design around elusive factors or adding analytical approaches involving things like Cholesky decompositions or autoregressive yield models (to which billions of dollars flow in compensation) but then miss other truths...not that I have found any truths myself of course. 

The main thought here, then, is to chase down some thoughts about the (minor) deficiencies I see in MPT and in an over-focus on portfolio engineering and asset management approaches...and then offer some context of why I think that way. As a retiree, of course, not as practitioner or academic of which, of course, I am neither. 

The Portfolios

As I start here, I am not sure exactly where it is going to go but let's dive in. First the portfolios. I have been challenged on precisely how I do this but whatever. I figure Markowitz earned a Nobel so it's good enough for me. Plus I am working in broad strokes not precision engineering. Also note that all data is illustrative and probably does not resemble a reasonable reality. We're just playing here.

• Asset 1 - N(.02, .04) Normally distributed expressly for my convenience
• Asset 2 - N(.06, .18)  " 
• Asset 3 - N(.07, .25)  "

• Portfolio A - combinations of 1&2 - in 10% steps 0->100%
• Portfolio B - combinations of 1&3 - in 10% steps 0->100%

For the combinations I will continue to use this approach because that's what I do. Tell me if I'm wrong. 

 
Eq 1&2 - MPT mean variance

I guess the biggest criticism might be the implication of a continuous rebalance in how I apply this in simulation but I'm ok with that for the broad strokes. Rendered in MV space, like this:

Figure 1 - Efficient Frontiers

where a 60/40 portfolio for B looks like it dominates A along at least the return dimension and B dominates A entirely at the end of the EF if return is the only consideration (which it is not). It is probably banal by now to depart the MPT path at this point due to the various discontents with MPT but depart we shall.  

My main discontent is that this doesn't get me very far as a retiree and even nudging out the frontier through rigorous design theory might not even do the trick. Who knows? MPT has zero to say about spending or the time[1] effects of the risk taken. It also does not consider life income like SS, pensions or annuities. Also, horizon matters in that the young have long ones and I have a short one. Also, in past work by G. Irlam, it is clear from other methodologies that the scale of assets involved might influence the optimizing solution.  So, imperfect[2]. 

First: Time  

I know I've done this before but if we overlay or transform the EF by way of estimating the geometric return[3], the one we realize, then it'd look like this (dotted lines): 

Figure 2 - Geometric Efficient Frontier

where the 60/40 portfolios are at parity under this stylized parameterization and A dominates B at higher allocations. A has no critical point at < 100% risk but B does at ~80% about which Markowitz reminded us in 2016 that in 1959 he didn't say choose the critical point (basically the Hakansson or Kelly portfolio) but just not go above it because it leads to lower return and higher volatility in the short run. 

Second: Time Again

The geometric return estimated above is: a) just an imperfect estimate, b) calculated at an infinite horizon, and c) an expected value so there is variance around the geo mean estimate even at very long horizons. The problem for someone like me is that my horizon is short. I'm 62 but let's say 65. At 85 I won't be dead with perfect certainty (according to actuarial the tables) and if I'm still alive my mental faculties might be in decline and the fact that I made it to 85 means my survival probabilities have counterintuitively gone up a bit. Therefore I might want to drop some or all of what I have left into life income if I haven't already done so. That means my investment horizon is 20 years. 

20 years is pretty short and there are a ton of ways to deal with that: bond ladders, TIPS ladders, option hedging, floor and upside via annuities, etc.  Most of those are outside of factor engineering and Cholesky decompositions so you can see the source of some more discontent. For now, let's look at the same geometric frontier idea but now at a 20 year horizon. For this I'll use some N-period estimators[4] I lifted from R Michaud though I think Markowitz had this in his 2016 book but I haven't checked. 

Here I am departing mean-var space and just charting the N-period estimate for the expected value of the geometric mean return for each asset allocation step along each frontier for Portfolio A and B in 10% increments.  Also, just to confuse you I converted return to real return after 2% (deterministic) inflation is considered because I need real returns later. Hopefully this isn't too much apples to oranges at this point.  

Figure 3 - N-period real geometric return

B dominates A up to about a 70% allocation and the 60/40 is trivially different in my opinion. But this is an expected value so there is some variance around it and at 20 years it is higher than it is at infinity though less than in the first several years. So let's look at that. Using 6b in note 4, and the 60/40 comparison, Portfolio A's geo mean estimate has a .074 standard dev. and Portfolio B has .087 if I got the math right. The difference between A and B is uncertain and the actual path taken in real life might be quite a bit below what is expected.  Discontent. 

Third: Considering Spending [via Merton or MC sim]

The elephant in the room for retirees is spending.  A lot of advisors help retirees with this problem, of course, but fewer than you think and the compensation arrangements often don't help. 35 year old portfolio engineers and asset managers, though, barely even know about spending. Academics do but usually they are economists. Some integrate portfolio and spending, the most notable that I know of is Merton with the very precise Merton Optimum where allocation and spending are, properly, a joint choice. 

My main amateur issues with Merton are 

• I can barely understand the math
• it demands normally distributed returns where I sometimes fatten the tails
• a true risk free asset which always seemed a little mythical to me
• no provision for annuities or SS or pensions. Yaari '65 showed the importance 
• no problems with borrowing
• optimal solutions, imo, go stale the minute they are solved

One way around some of this is through blunt-force adaptive planning via simulation. This (sim) has the advantage of being able to shape spending, customize taxation, have non-normal return assumptions, feather in life income and otherwise customize the hell out of things. The downside is that it seems rather arbitrary, prone to modeling error and modeler bias, blind to scope of issues outside the trial and error of a narrow-band model, slow turnaround time in planning, rarely considers random lifetime, 8/10 times (made that up) focuses only on an imperfect "fail rate" number that is easily misinterpreted and blind to the magnitude of the fail.   

But MC sim does provide some insight, especially when it comes to spending. At this point it'd be entirely logical that I'd trot out my MC simulator, set it to 20 years, and do a sh*t ton of trial and error to tease out some conclusions. Makes sense, right? But I broke my MC sim with some stray code a while back and I can't figure it out. It'd be easier to re-write it now which I won't.  I won't because I use my FRET tool (flexible ruin estimation tool - heh, that's a joke since I fret about ruin too much) instead. 

FRET has the advantage of being fairly simple, robust, and rigorous. Portfolio longevity is unconstrained here and asset exhaustion possibilities running to infinite horizons are considered along with the full "term structure" (Milevsky's phrase) of longevity. Since my 20 year horizon includes the non-zero probability of death within the interval I don't feel too bad about using it in this post though it might be a little bit of an apples->oranges thing. idk.  The basic simulation structure I use is this: 

Eq 2 - Lifetime probability of ruin

where tPx is a survival probability conditioned on attaining age 65 (x) and g(w,t) is a portfolio longevity function/distribution. Think of it as portfolio longevity in years weighted by a survival probability which it is. This construct happens to satisfy the Kolmogorov PDE for the Lifetime probability of ruin (LPR) like this:  

Eq 3 - Kolmogorov PDE

which is a diffusion equation related to the heat equation. I only know this because I had coffee with Prof. Milevsky once and he told me. Plus I put my results up against a finite differences solution to the PDE someone wrote: the same...

When I run FRET for the portfolio A and B assumptions, adjusted for real returns, tuned to modal life of 88 and dispersion of 8.5 (if I recall that is in-between the SS and SOA IAM life tables) and a 4% spend  -- note that for the remainder of this post I use 4% just to make my life easy rather than considering other spend plans...this is a flaw, btw; there is exactly zero discussion of "safe withdrawal rates" in a conventional sense in this post -- I get the following result in LPR terms

Figure 4 - Lifetime Probability of Ruin for different portfolios and allocations

Lower is better here so A dominates B entirely above about a 30% allocation to risk (again, only for these parameters). 

Fourth: Inspecting the 20 year diffusion distribution for tidbits of insight

One complaint I have about MC sims and their cousins is that the language is usually mostly in terms of ruin or fail rates. This is unreasonable and a slightly unfair criticism because better advisors and most academics will look a little further than just fail but here are my main "other" complaints about MC simulation which might be redundant with some of the complaints above: 

1. A single stat like "fail" is reductive and easily misinterpreted
2. Usually the magnitude of the fail in time or other measures is skipped over
3. In real life people do not typically "mathematically fail" or fall into a "gamblers ruin." In real life, spending is reduced, jobs are sought, family steps in, institutions of association and government help out, etc. etc. 
4. There are no objective thresholds for evaluating fail. 0% fail is unrealistic and would be infinitely expensive anyway[5]. Also, the difference between a 5 and 10% fail is meaningless to trivial. 
5. The role of MC sim as a simple recurring-dashboard-adaptive-red-flag metric is underplayed
6. The internal shape of the distributions evolving over time are lost
7. MC rarely considers economic consumption utility
8. MC doesn't think too deeply about the wealth depletion interval or the Life Cycle Model
9. MC doesn't think much about the optionality of wealth held in assets vs structured "floors"

Some of this can be remedied but much of it falls perilously close to over-analysis. But we are over-thinkers here so let's proceed anyway.  Going forth, I'll propose the same portfolios A&B as above but now rendered in: a) real terms with a 2% deterministic inflation, and b) "units" where a 4% initial spend unit is = 1 so initial wealth "units" at T=0 are 25. Again, I am not examining safe withdrawal rates here! Also, to the extent that I discuss an annuity boundary later I am referring to the cost to buy, say, 1 unit in (real) consumption at some age x conditioned on achieving age x, with the then operative survival probabilities, in an attempt to immunize the unit consumption we had at time zero but now at time=20. This is a moving target, of course, but at T=20 (age 85 for me) in this post I used an annuity pricing model to infer that the boundary would be 6 at year 20 and then slapped on a margin of error of 1.5 for a target goal at 20y = 9. In case we need that number. This should be examined because I'm not sure I have this entirely right. My three readers can disabuse me later. 

First, just for the sake of a visual, lets show what portfolio A looks like as a diffusion process over time.   

 

Figure 5. Net Wealth Diffusion, Portfolio A, t=1:20

• Initial Wealth units = 25
• Initial spend = 1
• At T=20, I guess that the cost to immunize a $1 real spend is 9 but might be wrong. 
• The black bars in the final distribution are:   

          - the initial wealth of 25, for reference

          - the annuity boundary

          - the zero wealth fail level

As I mentioned above we have to be tolerant here of the concept of unconstrained negative wealth. My guess is that this is something achieved via unconstrained borrowing but then I have no interest rate modeled so this is relatively naive but maybe gets me to my point. Again, my three readers can disabuse me of my naivete later. 

4.1 Inverted Option Type Idea

This might be a little fuzzy because this post ended up being different than I thought it would when I began. At the beginning I had what I thought was a slick approach that turned out to be nothing more than a type of Monte Carlo Sim. The idea was to evaluate two portfolio strategies by way of inverting an option i.e., evaluate them using a simulated (not Black-Scholes) real option but evaluate below the strike rather than above (in the call version). This Real Option concept leaned on the paper I've used before:  The Intuition Behind Option Valuation: A Teaching Note, where the basic idea is that the real option can be evaluated via this method: 

R = ∑ { P(MT) * PV(max[0, MT - k]) }

Eq 4. Real Option in simulation 

where k is the strike. I've done this before and have been able to price exchange traded options within a penny or two so it seems like a viable brute force method.  

When I imagine this idea inverted -- to evaluate or integrate the P(MT) from -inf to k -- basically I have a probability of failing to meet a goal at some future time. That sounds like, and probably is, a "fail rate" like we see in MC sims except that k is not necessarily a "zero" level among several other things.  But then again, here are some other thoughts on why maybe I am not strictly in Monte Carlo land: 

• The strike or boundary is not necessarily zero or a "fail" state except in the sense of not meeting the goal. 
• The strike would be variable if we need with horizon: age, interest rates, inflation, policy choice, etc all would affect the boundary/strike
• The strike is a "goal" or liability like a pension liability but here would be an "annuity boundary" that would defease a spending level at a future horizon age (say 20) out to infinity i.e., a place to pull a ripcord.
• I am imagining a shorter horizon like the 20 years between age 65 and 85 (85 is when I might annuitize) but 20 years is not radically different than the 30 years we might see in a vanilla MC sim.
• I think we are making a more full use of the full distribution of the net wealth diffusion at the horizon but I don't know other people's MC sims very well. I don't just have the number of fails here as a percentage but also the full distribution of relative frequencies and wealth values of states at T=20 (figure 5). No doubt better sims do this as well all the time but at least I could price over the strike and get a value a real option on wealth, which I have tried before to validate optimal ages to annuitize (I was trying to replicate a Milevsky paper; not sure how well it turned out). That means I could also price the fail as well, a type of magnitude if you will...but only if we believe in unconstrained negative wealth which I kinda don't. TBD
• The basic concept, though, is: "what is the probability of not meeting the policy-boundary goal?" But again, I don't really think we are too far away from (or exactly the same as?) basic MC simulation. 

So, in this section, basically what I am doing is integrating the front yellow distribution in Figure 5 from (because this is turned around) from right to the boundary (otherwise -inf to 9) where the sum[P(x)] from +/- inf = 1. If I do this for both strategies for the different allocations (I skipped the 0% risk allocation) I get this:  

Figure 6. Probability of missing boundary at T=20

Where lower is better and A dominates B from about 40% allocation to risk and has a min around 80% risk. Then, if I try to do the inverted option thing -- and this is intellectually the weakest thing I do here -- where we invert the sim to be k-MT rather than vice versa we get a kind of "pricing" of the probability of the option expiring out of the money...whatever that means. I just call it a magnitude but I haven't seen anyone else do it so I have no idea. Anyway, in that construct we'd see this: 

Figure 7. The scale of missing the Boundary (??) at T=20

where lower is better and A dominates B almost all the way and with a min near 40% risk.  I won't even try to interpret this because I got my head turned around like Linda Blair a couple times on this. In the end I am not sure if this is meaningful or that it could not be just a MC sim with magnitude in years rather than "price." Idk. 

4.2 Pulling the Rip-Cord on an Annuity Boundary While Monitoring the Diffusion

Each year from 1 to 20 we know what it would cost to defease the 1 unit spend. This is a moving target that changes with age, time, and interest rates among other things. I did my best to estimate it age T=20 where I came up with 9 which was really 6 x a margin of error. Even if I'm wrong I'll let that stand as illustrative rather than real. I'll illustrate this in a second.

First, the evaluative framework here will be a weighted life consumption utility at T=20. I'm not sure I can do it the way I want here; I'd have to ask an economist. I'll pretend. The idea is that: a) the wealth units achieved at T=20 can and will buy a life income, b) legacy is ignored or embedded in the consumption somehow, c) the income is evaluated for each future year via CRRA utility[7], d) the CRRA is weighted by a survival probability in each future year t conditional on reaching the age x (85) at the horizon = 20, e) real terms so no financial discount, f) lazy so no subjective time preference, g) that whole stream is summed, h) the sum (EDULC - expected discounted utility of life consumption) is weighted by relation to the wealth-unit outcome which is hard to explain and may not be economically justifiable anyway. 

The utility framework is like this:

Figure 8 Utility Framework

So, I am going to do this now in two ways:

a) I will, at T=20, use wealth units to buy income but assume that when -- at any time before the horizon that I hit the boundary -- presciently and courageously -- I pull the ripcord and use remaining wealth to defease the future spend of at least 1 unit.  So I will integrate the probability of the wealth states below the boundary (at T=20) and use that as a weight for the EDULC of 1 unit spend, then weight the EDULCs of incomes purchased above the barrier with the probability of being able to purchase that income with the wealth units achieved. Reference figure 5 & 9. To the best of my naïve amateur ability I think this is the notation but idk; you tell me, this stuff gives me nightmares. g here is the utility function and I kinked the U(c) above W(0) because it was too much work otherwise and it probably doesn't matter.   See note 6

Eq 5

b) I will, at all times 1:20, let wealth run to "whatever" under or over the boundary, including below zero but I'll assume I can buy no income below zero. I will also assume that there is some basic consumption level = .25 that won't be breached. So I will integrate the probability of the wealth states below zero and use that to weight the EDULC of 1/4 unit, then weight the EDULCs of incomes purchased above "W-->1/4 unit" with the probability of being able to purchase that income with the wealth units achieved. Reference figure 5 & 9. To the best of my naïve amateur ability I think this is the notation but idk; again, you tell me my 3 readers. g here is the utility function and I kinked the U(c) above W(0) because it was too much work otherwise and it probably doesn't matter.  See note 6

 

Eq 6

Ok, so I confused even myself at this point and was thinking about it in the shower this morning (a cry for help, right?) and this (figure 9) is the best way I can describe it. Otherwise DM me and I will try to explain.  But basically under "a" if we hit the boundary we buy income of $1, if it is above the boundary at the horizon we buy more. "b" is unconstrained except at zero. I think I am repeating myself. 

Figure 9. Converting Horizon Wealth Outcomes to income
and evaluating the consumption utility. To the right of
the vertical line we use Figure 8 as framework

Holy crap that was a lot of work and confusing. When I do all that stuff I just said above I get the following almost entirely -- except for a couple key points in the comments at the end -- unremarkable output like this:  

Figure 10. Watching the Boundary

If you've ever heard me use the metaphor of "riding a bike in first gear," this is what I'm talking about: a ton of heat and motion without a lot of forward progress.  I'll fold comments on this section above into the final thoughts rather than being redundant both here and there. 

Comments 

1. MPT doesn't really seem to be enough for me for evaluating portfolio choice when one decumulates over time, especially over a tight horizon, and considers issues like life income[8]. Me? I'll leave the finicky portfolio factor engineering and optimization to the single-period non-spending accumulators. But they are missing the bigger story though I think.
   
   
2. The concept of triangulation (sorry David) is, to me, king because different frameworks give different answers and sometimes they conflict. One has to think this through from multiple perspectives. Single solutions touted as optimal might be so but they only are "so" from one point of view, one model, and only in one particular instant of time.
   
   
3. Except in a couple situations the lower vol. strategy (given only this parameterization) generally wins and is prudent when spending and spending to a specific short horizon for reasons here and outside the scope of this post. MPT doesn't give this insight directly though Markowitz clearly addressed at least some of this in 2016. 
   
   
4. I can't tell if there is an analogue in pension fund management or consulting to the weirder stuff I did here but there might be. Idk, I don't work in that world but in a re-run in life maybe I should have. Might have been easier if I'd started my ed at 18 rather than 60. 
   
   
5. Figure 10 - Higher risk wins here if we manage and monitor the boundary (upper lines) and successfully pull the rip cord on income strategy
   
   
6. Figure 10 - The portfolio engineering implicit in the lower two lines appears quite meaningful but is very much dominated by the decision to manage to the boundary. This is something that no asset manager has ever told me...except maybe NewFound Research. 
   
   
7. Figure 10 - with the rip-cord strategy (upper) we see two different interesting inversions: 1) strategy B now tends to dominate A for a change in this post, and 2) the ability to carry higher risk to the horizon (in asset allocation terms) goes up quite a bit. I'd like to see if that holds if I were to be more rigorous and add more variation in parameters. I bet it might. 
   
   
8. With the clear dominance of a "managing and monitoring" strategy over fine tuning the difference between, say, a 50/50 allocation and a 60/40 -- or even adding something like trend following or options to either -- it seems clear that a premium would be paid for actually "managing and monitoring" and then easing into life income of some sort if it makes sense. This seems odd because almost every advisor I've ever talked to over decades -- ok, my current advisors are pretty sharp but that is a diff story -- has perseverated on asset allocation and alternative asset classes to enhance the portfolio engineering aspects (I mean they get paid for allocation choice via AUM, right?) at the same time telling me that annuities and annuitization suck. My financial advisor ex gf was stuck on the latter but in her defense she used products in the space that are now, later after we broke up, known to offer a viable strategy that is similar to the life income choice. But that is another post. 

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[1] Ok, that's not exactly true. Markowitz covers this in detail in his works. 

[2] As are other methods of getting to the allocation and spend choice (Merton, SDP, formulas, etc) but that is not for this post. 

[3] I'm using: g = r - V/2 here

[4] from "A framework for portfolio choice" R Michaud 2003, 2015 

[5] When I was first introduced to "fail" I wondered: "well, why wouldn't we plan for zero?" I didn't know then that that is basically not a reasonable thing to ask. 

[6] for the sake of not blogging this piece all week I decided to only evaluate consumption post horizon. Technically I should do it over the whole interval age 65+. In that event, the more robust plan should allow higher consumption in the early years consistent with the lifecycle theory of Modigliani and as illustrated my M E LaChance with a downward sloping consumption until the floor and wealth depletion is hit. I haven't done that and so I guess I implicitly assume that consumption washes before the horizon across all scenarios...which it wouldn't. 

[7] Fwiw, I used a risk aversion coefficient of 2. This is arbitrary but is is roughly mine. It is above risk taking but not too risk averse. I have used this in the past and it got stuck in my analysis.

[8] never even got to Soc Sec.