That settled that. But I was interested in more and worked through Michaud, Mindlin , Meucci and some others like Bernstein on geometric frontiers . I did some spreadsheets. I wrote some simulation programs isolating geometric returns. I rewrote the programs to consider, like a good retiree, spending. After that I just tried to pay attention to what I was doing and what others were writing.
The hindsight portion of this post,then, is that I wish I had known earlier about some of the characteristics of geometric returns in a multi-period (every time I say multi-period, think retirement or retirees) context. It has been useful so far and I guess I'm glad I tried to pay attention. The very basic, thinnest summary of what I learned can perhaps be described like this:
1. In the long run, as Markowitz says, one gets the geometric return.
2. What "one gets" is always going to be less than the arithmetic return.
3. The geometric return, for reasons you can read about in the links below, is affected by volatility and can be roughly estimated as: ar - V/2 where ar is the arithmetic return and V is variance or standard deviation squared. In all of this we are totally ignoring the dicey issue of how returns (and vol) are projected from one horizon (say monthly if that is all you have) to another (say annual). That is another story altogether.
4. That estimation above is just that, an estimation. There are also other, different estimation formulas (see Mindlin). The estimation, notably, is also typically true-ish only at time = ∞. More interesting things, though, happen to geometric returns in the time before infinity but that is another point below.
5. When comparing two strategies with one expected arithmetic mean return higher than another, it is not necessarily the case that at infinity the higher strategy will still be higher at infinity when calculated as a multi-period geometric return. That makes geometric return analysis important in things like strategy comparison, evaluation of tax effects in strategy switching, and other analysis (see Michaud). Given point 4 it is also important to look at what is happening between t=0 and t=∞. Since retirees can sometimes plan over relatively short horizons compared to institutions or young accumulators, the point (time/year) where strategy A might "cross over" strategy B is good to know. Strategy A might have higher arithmetic mean at t=0 and lower geometric mean at t= infinity and the crossover might be in 5 years, or 20, or never.
6. When spending is factored into the analysis in point 5 it can get even trickier. In a past post (I'll find it at some point) I tried to show that, as in point 5, a strategy might have higher expected arithmetic mean return at t=0 but in geometric return terms starting with t=1 through t=∞ Strategy A (the higher one I guess) might be higher than strategy B early, lower for a while, and then higher again. On the other hand that analysis depends on how one accounts in the statistics for the portfolios that go below zero. That means this is a sketchy point but one that I'll assert anyway. Prove me wrong and you get a guest post.
7. Using geometric returns to re-cast an efficient frontier developed from arithmetic (estimated forthcoming mean) returns (see Bernstein) can be a useful exercise. For each point in mean-variance space for a given standard deviation, if the arithmetic portfolio return is recast as geometric, and if the volatility is high enough as you go right, the efficient frontier can inflect down or have a critical point in calculus terms. That is useful to know. Markowitz makes a similar point in a different way in Vol2 of his book (2016) when he says you should only go so far up the EF (max expected log) and no further. He makes a nod to Kelly and Latane on this. I think these are both saying the same thing but I have not done the work to prove it to myself. This is another guest post if you've got the goods.
8. With respect to all the points above but in particular #6 it is important to tune into words like mean and expected. When you run a simulation it is easier to see. The mean or expected value of the many paths in a simulation is a nice tidy curve down and to the right. Individual paths of the simulation, however, just like a single person (retiree) in real life, can have a wild ride and in terms of #6 that ride can drive itself straight through zero in a smallish number of finite years while the mean can stay positive forever. Hence the complexity and anxiety in retirement planning.
In hindsight I wish I had known all of this stuff sooner. This is useful knowledge capital for someone self managing a portfolio-with-spending over time. If not self managed, this kind of knowledge can sometimes at least save a person from the excesses of advisor wizardry and sleight of hand or, if you have a good CFA, it allows you to speak the same language.
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[1] sources for info on geometric/compound returns and geometric frontiers:
A Practical Framework for Portfolio Choice, Richard Michaud, 2015
On the Relationship between Arithmetic and Geometric Returns, Dimitry Mindlin, 2012
Linear vs. Compounded Returns – Common Pitfalls in Portfolio Management, Attilio Meucci, 2010
Risk Return Analysis, Volume 1, Markowitz and Blay 2014, chapter 3.
Diversification, Rebalancing, And The Geometric Mean Frontier, William J. Bernstein and David Wilkinson 1997.
Geometric Mean, Wikipedia
[2] How exactly does one get out of a two year MBA without knowing this stuff. Maybe that's what the quantitative finance programs and MS and PhDs are for (speaking of which, is quantitative finance a tautology?).
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