One of the big complaints I mutter to myself when I read finance theory and portfolio choice is that it seems like it focuses more on institutional or theoretical abstract retail investors that have very long or infinite time horizons that can be treated as "one period" and where the investors often have no spending requirements or constraints[1]. This looks nothing like a retiree-investor who has:
a) A spending constraint. Yes we spend. That spending has a material impact on the process. And it is a process. We, the retirees, are a what I want to call a net-wealth-engine-with-volatility. Unlike a mutual fund that has no spending constraint, or a venture or private equity fund that has no similar constraint, and unlike an endowment that has spending but is more or less an infinite-life entity we can spend an endowment to zero and then below. That is a problem and the lit does not respect it as much as it should (or usually treats it separately from the portfolio choice topic) except when we talk about things like the 4% rule which is important but flawed (and generally separate) or maybe Monte Carlo simulation that is also important but flawed (and sometimes separate) or maybe spending "rules" that are important but typically treated independently of portfolio choice as well.
b) A multi-period process vs single period. the spending thing immediately puts us retirees into a "process." The spending process is what, in my opinion, "discretizes" the investment horizon into multiple periods and makes it something that is now more subject than not to: 1) the vagaries of geometric returns and the influence that volatility has on geometric returns (think sequence risk), and 2) net compound returns of zero or below zero are a real possibility at time horizons that matter. (think fail-rate estimates and simulation; Markowitz, 2016 p 150, for his part generally assumes in his arguments that [net wealth in terms of] R is > -1)
c) Possibly very short horizons. Similar to point b above, a bunch of the retirement lit I see treats retirement investing like institutional investing with not only no spending but single time frames that are effectively infinite or very long. Yes, as an early retiree I could have considered myself facing up to 50 years of planning when I started. Yes, I still play and should play a long game because superannuation is a risk until it isn't. Yes, Markowitz (2016 p 164) states "Levy concludes that for 'For T=30 year horizon or more [MEL [my comment: read that as kelly math for now]] dominates other strategies by Almost FSD...' " [the implication, I gather, after a number of derivations, proofs, and discussion of utility is that Markowitz and Mean Variance vibrates in sympathy with max geo growth and Kelly up to a point and that long horizons are where Markowitz and Kelly are in synch as well as where the two are relatively tractable and important analytically]. BUT, interesting things happen for horizons less than 30 years (this supposedly was Samuelson's point in addition to the utility argument; but I don't know him or Merton or others which is why my stuff will always be an amateur endeavor). For example here is what I was thinking about: 1) geo growth in annualized return terms that are net of spending can look pretty bad between 5 and 15 years before time and compound returns start to redeem the net-wealth-volatility process a bit, 2) we may be both aged and planning; planning at 75 or 80 is different than at 59 or 40, 3) our planning horizon may be deliberately short due to health concerns or because we are targeting annuitization (at say 75) rather than targeting some terminal age (95) for planning purposes. Max net wealth at 75 might give me more choices I need and otherwise might not have if I am not explicitly planning for it. I look at a too-short 15 year horizon below just to do it and for the reasons just mentioned. Me? I'm about 15 years from a age gateway for optimal (all or nothing) annuitization so this could make sense. Here is a teaser related to the point in c)1. This figure is annualized compound geometric return (m=.09,sd=.20) over time but here the second line is net of spending (.04) using the ERN math I've used before. Short time frames (in EV terms here) are are particularly hurt by spending in early years. Some of the individual "sim" paths would be even worse.
Figure1. Geo returns over time; lower: with .04 spending
But I didn't want to look at geo returns as such. I wanted to look at geometric return frontiers (see Michaud 2003) and add spending even if the theory says I shouldn't (more accurately it seems silent). With some trepidation at embarrassing myself, here goes. I started with the proposition that if Michaud can do a geo frontier (ok, he's a math PhD but let's keep playing) so can I. In this post, even though there are other ways of estimating geometric returns for long and short horizons I'll use g = a - V/2 (a being the linear return) because it is easy. For a portfolio with two components (mean and sd of .04/.04 and .10/.25, these are arbitrary for illustration only) the traditional arithmetic frontier (the usual portfolio math) with a superimposed geo frontier -- where the linear returns re-cast as geo for different levels of vol that are equivalent to different allocation choices -- might look like this:
Figure 2. Geo frontier
[eq1]
Figure 3. Geo frontier confirmed with simulation
Let's call it confirmatory. Behold, a geo-frontier confirmed. But now that we are using eq1 for confirmation, why can't we evaluate the frontier with respect to spending as well? You may say no but I'm an amateur hack so I say yes. Let's plug .04 in there for w and see how it goes. This is how it goes. The return values in the red lines are annualized compound returns net of spending if I have done this right:
Figure 4. Geo frontier with spending - two ways, 40 period sim
But this is where I ran into my first design dilemma and why I revisited my code in a past post. Because in eq1 the right side of the equality (left version of the equation, when p = 0) is a net wealth process -- as is the second term of the Kolmogorov equation in a past post by the way -- that means net wealth can go below zero in future periods just like in a MC simulator which this resembles. That means that if I try to calc the EV of annualized compound returns inside the iterating loops (let's call it "method 1") then I am doing exponential math on negative numbers at some even years and that doesn't work very well and I don't have the math chops to work around it yet. On the other hand if I take the FVs implied by eq1 at each year/step and calculate the median and then annualize it ("method 2"; and here we are, I think, ever so slightly sympatico with Kelly math because he maximizes future fortune at the median, right? Markowitz, in some cases, if I understand him, is evidently on board here too. See end quotes) then I appear to be accepting the concept of negative wealth (neg wealth is in the distribution from which we calculate the median) which has it's problems. I don't mind it because I view negative wealth as either an opportunity cost thing or a magnitude thing so I lean towards that. The same problem occurs in MC simulation by the way, to which this is related. In the chart above, line "C" is method 1 and I replace the irrational numbers (or is it undefined? I can't remember) when they arise with zero and then calc a mean (EV). In that case, higher vol portfolios will, in my opinion, overstate the geometric return frontier. C' is method 2 and the one I will lean on going forward despite the concerns.
Now, remember figure 1? Short horizons might hurt more than long horizons when spending and volatility are in play? Lets take C' in figure 4 and re run the code with 15 periods instead of 30 and see what happens to the geo-spend frontier.
Figure 5. 15 period sim, .25 sd
Yes, the shorter time frame hurts across the board. The other interesting thing here to note is that while for these assumptions the geo frontier before spending has diminishing returns, it does not yet quite inflect (calculus critical point) down. The geo-spending does, however, and that probably has implications in portfolio choice when we are faced with making decisions about high vol portfolio components when factoring in consumption.
Now let's run everything at 15 periods again and amp up the vol component a little bit more. Here, instead of an "equity" component with 10% return and 25 % vol, let's make vol 30% and then lets take green line D above (15 periods 25% vol) -- and in this case below that "D" will now be called C' (grey) just to confuse you -- and compare it to 15 periods with a max 30% vol and that'll be called "C" (red).
Figure 6. 15 period sim, .30 sd
Short time frames hurt and vol hurts. Also the inflection (critical) point looks like it is moving left and now puts us in the meat of the "mid allocation" range. I once did a forward simulation using the dynamic asset allocation that came from backward induction. Playing around with the assumptions led me to the conclusion that allocation in the presence of spending and vol favored something between 40/60 to 70/30 or 80/20. This chart gives me the same feel. But because I am playing and not researching I don't think I can really make any hard conclusions here.
How about a variation in spend rate this time? Here is figure 7 which is the same as figure 6 but now with an added geo-spend frontier (same as C) except with a .03 spend rate.
Figure 7. 15 periods, .30 sd, 3% spend vs 4%
So, a nice move up and the inflection point looks like it moves right a bit which I guess means that slightly higher allocations to risk can make sense with a lower risk spend.
Conclusions?
I don't think I would make too many serious conclusions from playing around like this. It is of a piece, though, with other things I have read and seen: spend less if you can, be wary of very low and very high allocations to risk -- especially low though high can be a problem too, geometric returns can affect portfolio choice while a spending factor on top of that makes it even harder, and volatility is not always your friend. The newest take away is maybe that short time frames can hurt attempts to hit max wealth at that horizon more than I thought but I'd have to take this another step maybe to be sure.
"Markowitz (1959) does not recommend that in investor choose the MEL portfolio [growth optimal or Kelly criterion]. Rather, it recommends that the investor not select a portfolio that is higher on the efficient frontier, since such a portfolio has greater short-run volatility but less long-run return--in the Kelly sense--than the MEL portfolio." --Markowitz 2016
"The MEL criterion was proposed by Kelly (1956) and embraced by Latane (1957,1959). The Markowtiz (1959) Chapter 6 position was that the cautious investor should not choose a mean-variance efficient portfolio with a higher arithmetic mean...than that of the efficient mean-variance combination that approximately maximized expected log" [where Kelly and utility meet if I recall] -- Markowitz 2016
Some additional readings:
- On the Relationship between Arithmetic and Geometric Returns, Dimitry Mindlin, CDI Advisors 2011
- Richard Michaud "A Practical Framework For Portfolio Choice" Journal of Investment Management 2003.
- Diversification, Rebalancing, And The Geometric Mean Frontier, William J. Bernstein and David Wilkinson 1997.
- Quant Nugget 2: Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management, A. Meucci 2010.
- “Geometric Mean Maximization: An Overlooked Portfolio Approach?” Journal of Investing, Winter (2010), 134-147. Javier Estrada.
- Harry Markowitz, Risk-Return Analysis Vol 2, 2016 pages 141+
- Corey Hoffstein, Growth Optimal Portfolios 2017
- Ethier, S. (2004). The Kelly system maximizes median fortune. Journal of Applied Probability 41, 1230–1236.
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[1] the area of retirement finance that addresses this kind of thing seems to have exploded over the last 5-10 years.
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