Riff on time averages and geometric means

I've done posts on this before but it was on my mind again. The analysis of single period finance usually relies on arithmetic returns but real people live in time so it looks different on a realized, multiplicative (geometric mean) basis. Even Markowitz (2016) made this point on his own methods.  

MPT has it's charms. Geometric mean maximization has its own (Kelly, Hakansson, Latane, Thorp, Michaud, Markowitz, not-Samuelson). I don't have intellectual access to the math and notation...yet...so I have to use the proletarian tool of simulation where I am allowed to play. Here is a simulated visualization of N(.07, .25) and it's many possible compound (and annualized) paths in all it's multiplicative glory, using the following formula:

 

We see here, I hope, that 

- over the long run (limited to t=100 here) the annualized geometric mean return of the many
  individual paths [only ~100 of 1000 shown] --> E[r(g)], the expected long term geometric return, 

- the mean geo return [black line], if not actually the real expectation, declines monotonically in the presence of volatility. This is, if I recall, called volatility drag but I also hear that is a crappy divisive term,

- as time passes, both the paths and the mean tend to approach the expected value. A common estimate for this expectation is  g = r - V/2 but I hear it is not quite that, 

- at t=100 which is my sorta reasonable proxy for "a long time" we see that while the paths start to converge towards the expectation there is still wide variance at t=100 that puts some paths underwater over that particular interval. I hear it can take a really really long time to converge even closer depending on how "closer" is defined,

- for shorter horizons -- I mean me? I have only 18 years to age 80 when I might start annuitize stuff, so a reasonable but short portfolio management horizon -- the uncertainty and distance around the expectation would be excruciating and the feeling of convergence would almost be nil. This is, I think, the core point of the critiques of geometric mean maximization (eg Samuelson), a psychology with which I might have some sympathy at this age,

- I've read estimates of how long it might take to converge to some reasonable proximity to the expectation but I can't remember. I want to say 1000 years or so is a possibility. I'll have to look it up,

- We assume things like IID and stability of our assumptions over 100 years which seems a little silly.

---

What we don't see above is any comparison to the timed geo mean of a different, lower volatility, strategy. In Michaud 2003 he makes this point of strategy selection favoring lower vol strategies (below) and the possibility of taking tax hits on highly concentrated stock holdings (not shown) over the long term. 

Here is 

N(.07, .25) [blue] vs 

N(.06, .12) [orange]:

I can see here in this figure that:

- the lower-vol-lower-return strategy "wins" the return war, in the multiplicative time average sense, over time,

- the "win" happens pretty quickly at the mean but we know nothing about the individual paths here,

- this is independent of spending so we can say little here to a real retiree yet,

- this strategy selection is akin to an allocation choice but that has not been made explicit here. The effect on an efficient frontier btw, if it were to be rendered in geometric terms, is that it (EF) might or might not bend "down" with a critical point on the interval [0,100%] allocation to risk. That might mean lower vol portfolios make sense over the long haul, which it often does. Certainly a levered portfolio might be a little fraught. Add spending to the equation and I am even more convinced of being a little circumspect,

- what is also not shown is that Michaud (03) uses this kind of thinking to rationalize de-risking from a concentrated stock position even if a tax hit were to be involved. Over the long run the vol will kill and it might pay to exit even with steep costs. That's another post

---

These mean paths over time can be estimated rather than simulated. Here is one imperfect formulation (from Michaud 03) for geometric mean return expectations that depends on time < infinity:

Applied to the distributions above (.07 red, , .06 purple) it looks like this. The variance between orange and red and blue and purple is because I only simulated 1000 iterations:

If one were to then create a matrix of realized geo returns (over, say, 1000 periods) using the estimator for different arithmetic returns and standard deviations, it'd look like this:

Would that we could all have high-return-no-vol but that is not in our cards I take it. It's more like set an expectation on your theoretical "return engine," using multiplicity and good intentions, and then hope it all plays out.  Doesn't really work like that but whatever. Also, while I love the math of geometric mean maximization, I am sympathetic to the harshness of what might happen if my horizon is short and I am not on the mean path. Or, rather, that path might actually be a kind of "mean path" heh. Note that we skip right over contributions and withdrawals in all of this. 

---------------------------------------------------------------------------------------

Markowitz, H., Blay, K. (2014) Risk Return Analysis, Volume 1. The Theory and Practice of rational Investing. 

Markowitz, H. (2016) Risk Return Analysis, Volume 2. The Theory and Practice of rational Investing.  McGraw Hill.

Michaud, Richard (2003, 2015) A Practical Framework for Portfolio Choice

Samuelson, P (The "Fallacy" of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling) 1971 

Hakansson, Kelly, Thorp, Latane, etc...