“Theorem: If Harry repeatedly invests in a portfolio whose E log(1+R) is greater than that of Paul [i.e., the growth-max proposition], then -- with probability 1.0 — there will come a time (T(0)) when Harry’s wealth exceeds Paul’s and remains so forever thereafter.” Harry Markowitz in a 2016 book poking fun at Paul Samuelson on their past argument about growth optimal investing criteria.
"Mean log of wealth then bores those of us with tastes for risk not real near to one odd (thin!) point on the line of all tastes for risk -- and this holds for each N with N as big as you like...For N as large as one likes, your growth rate can well (and at times must) turn out to be less than mine -- and turn out so much less that my tastes for risk will force me to shun your mode of play. To make N large will not (say it again, not) make me change my mind so as to tempt me to your mode of play. QED" Paul Samuelson (1979)
Intro
There seems to have been an uptick in the last few years of interest in the growth optimality consideration for portfolios. Ergodicity Economics feels like the new kid on the block, but this topic has a pretty long history, and the general interest bubbling up I find useful because it is an interesting and worthy topic. I won’t recapitulate all the math or notation in this post since it is tedious blogging [7] and one can read it comprehensibly, for the most part, and quite usefully, in the following bulleted references. Or check out the recommended reading list at the end and in particular pay attention to the references inside the various papers:
Seven Random Thoughts on Time Averages and Geometric Mean Maximization
This may be redundant with the critique section below but what the heck.
Thought 1 - Ensemble averages use a large number of parallel scenarios over long intervals to estimate a central tendency while a time average uses one scenario over a large (maybe but not necessarily infinite) number of steps to approach some final expected state.
Thought 2 - The realized geometric mean [2] of a stable, continuously rebalanced(?) portfolio, a time average process, approaches its median expectation at long horizons and is related to the median of terminal wealth. The ensemble average of terminal wealth diverges from the median due to extreme outcomes and therefore becomes less useful in multi period models with large dispersion. This is one reason why growth optimization (ie maximize median terminal wealth) makes a lot of sense to people: why not pick the portfolio with the highest estimated median terminal wealth outcome?
Thought 3 - Maximizing the geometric mean return, and hence median terminal wealth, is a mathematical optimizing framework for portfolios over very long horizons but may be problematic over short horizons. Life is lived neither in an ensemble average, nor the median time-averaged terminus, nor in some infinite future, it is lived on the one and only one relatively short path given to us, and that path can be a rough short ride in the moment given a goal that is less than, say, 30 or 40 years out. See critiques.
Thought 4 - A growth optimal portfolio can’t technically “ruin,” but it can fall very close to zero during lifetime, or after, and so it is perhaps a better fit for academics, very-long-horizon sequence gamblers, non-spenders, and institutions.
Thought 5 – Retirees spend – a choice made jointly with the portfolio risk composition – which means a retirement portfolio can go to and through zero, all else equal, with no adaptation along the way. A growth optimizing framework does not consider spending and therefore a retiree could crash hard following that path. As cool as I think the discussion of growth optimality is re time dynamics, and I do, it is just not a serious good-faith participant in the chat with retirees unless we are talking about an upside-no-touch-for-more-than-x-years portfolio with a floor of income to protect lifestyle.
Thought 6 – A multi-period growth optimizing portfolio choice may exist on the single period MPT efficient frontier but that does not always mean it is always qualitatively best. This Kelly-point may be one choice among many although to the right/up does not necessarily make sense. Markowitz (’59) “does not recommend that an investor choose the [growth optimal] portfolio, [just not one higher on the EF]…since such a portfolio has greater short run volatility but less long run return…”
Thought 7 – While any single geometric return time average path will converge towards the expectation for geometric mean at infinity (and median terminal wealth), even at long horizons, say 500 years, there is a fair amount of dispersion in returns even then. Certainly, the wealth paths diverge immensely.
Acknowledging some Critiques of Growth Optimal in the Literature
While I have not set up the math framework here in this piece, the properties of the geometric mean max (GMM, max growth criterion) have been of keen interest in either the special (Kelly and much of the financial lit on growth optimality) or general sense (EE???) for a long time. On the other hand, there are evidently reasons to be wary, some of which I touched on above. In a superficial canvass of the literature below as well as some of my memory of work I’ve done on this before, this list is what I came up with for now for what people worry about when attempting to select growth optimal portfolios. I reserve the right to adjust this later:
1. Time. a) The median of the geometric mean at intermediate real horizons is not the same as the median of the GM at infinity, a common mistake. Plus, more importantly b) the volatility and path dependency mean that it could take a ton of time for a time-average to approach what one might think it should approach. – See: Rubinstein, Rujeerapaiboon, Markowitz 2016, Michaud.
2. Epistemology, by which I mean: the GMM cannot be truly known or calculated due to the unknowability of the inputs and the instability and evolution of the environment enveloping the process over time, i.e., all the stuff required to get you to the place you thought you were going to get to but maybe never can. – See: Michaud
3. Countervailing optimality frameworks. One can solipsistically perseverate on growth optimality as an ultimate dominating desideratum but there are other ways of seeing reality, with: a) the rationality principles embedded in expected economic utility being one and, say, b) return probability frameworks being at least one other. Behavioral stuff might be a third if different from "a." All of any longer list would have their very own critiques, of course, but they are there. See note [1] – See: Rujeerapaiboon, Samuelson, Doctor, Hoffstein, Michaud.
4. Lack of diversification and extreme volatility especially when it’s terribly inconvenient (e.g., when spending and/or raising children are in play alongside the portfolio). This is tied at the hip to #1 but gets separate attention. LCM and consumption smoothing probably need to come into the chat right here but some growth optimalists seem to reject such things out of hand. Most of the growth optimal literature does little to acknowledge life cycle theory or consumption so #4 is either a very strong critique or a blogger-rube’s non-sequitur. TBD. Even without spending the extreme low wealth outcomes in the near-term could fire up behavioral issues an optimalist would never likely consider. – See: Estrada, Doctor, Michaud, Selden.
Observations on at Least One Critique
I am not going to dilate on everything related to geo mean critiques here. I am just going to take a look at #1, the time critique (see note 3), and maybe a little bit on #4. This objection (#1) seems to come up the most in the material I read on growth optimization. Here is a paragraph from Rujeerapaiboon that makes the point for me. I think this might’ve been where Hoffstein got his comments on Rubinstein.
Or b) one can tap into the same intuition via simulation. That’s pretty easy to do. Here, for example is a simulated geo mean process over a long horizon of 500 years, N[.04,.12], 10,000 times, initial wealth = 1 (see Note [3] though) presented first in terms of annualized geometric return time averages (figure1) and then individual wealth paths (figure 2).
Figure 1. Geometric return, annualized, over 500 years
Figure 2. Wealth paths and median wealth over 500 years – no spend
What do we see in these figures?
F1a. Since we can’t escape the gravity of random draws on returns, we see that in F1a: a) the time average paths, any one of which we could be on, are extremely variable at first but do in fact more or less converge over time, b) there is still a fair amount of uncertainty, even at 500 years, about convergence, and c) the geo expectation is, in fact, less than the arithmetic input and approaches both the median and the common estimator eventually.
F1b. We also see that it takes about 140 years for the 25th percentile geo return (arbitrary, plus note3) to get within 20% of the expectation and ~ >500 years to get within10%. The minimum in any year – this is harder to interpret since it is not a “path,” just the minimum for any path in some year – doesn’t even get close. I’m curious how far I’d have to run this to get it closer to some selected and arbitrary threshold.
F2a. Wealth compounds like a 5th force here, like we might expect, and the median goes to the moon, ignoring its note3 optimality question. Of course, we should limit ourselves to maybe 30 years, not 500…and add spending. But that is a different story than the growth optimal story.
F2b. When we look at the minimums – again, be careful with the interpretation here – i.e., the bottom of the grey-path mass in F2a, we see that at least one path was at <13% of initial wealth at ~110 years and the time frame where most or all of the paths are above parity with t(0) is what? A couple hundred years? Ever read “Time Enough for Love” by Heinlein? Imagine living for 800 years and watching this unfold were we to be on an un-fortuitous path. Yes, median wealth would be great (ex spend), and there is no technical “ruin” as such, but still… (note 6)
Now With Spending….
Now, just for fun, let’s add some consumption. With an arbitrary 4% spend added on top of the same already arbitrary portfolio process, this is what it looks like over: a) 500 years, and then b) a more human retirement 30 year interval. The geo return process engine (F1a) is the same so ignored. In Figure 3 We’ll focus on Fig2 transformed with a spend:
Figure 3. Wealth trajectories over 500 years - with consumption, then over 30 years
Some Observations:
F3a. This is absurd but makes my point. We see where median wealth is going. Ignore the question on negative wealth for now. It was easier to code this way.
F3b. This is no more no less than a standard Monte Carlo type model with some kind of fail threshold of x% at or before a goal date. The only difference from F2 is that we have added a spend and zoomed into a human-sized interval.
Growth optimality, which frankly has not really been illustrated since we used arbitrary inputs, wants to focus on maxing median terminal wealth. Retirement, otoh, is uniquely attuned to downside risk and protecting lifestyle. One wants to avoid a crash and/or one wants to immunize lifestyle first. Or at least some do. We retirees[5] are not playing abstract portfolio optimization we are attempting to optimize lifetime consumption which is a different game entirely and in which GMM optimization has a smaller role than the literature below might admit [5].
Were we to have an income floor – TIPS, annuities, pensions, SS, etc – the upside portfolio can play the GMM game as a choice, of course, though Critique #3 might still have a say, idk.
While GMM portfolios can have intermediate horizon risk [note 4], behavioral or otherwise, I think that what we should fear is not the distance between a GMM optimal portfolio and some other suboptimal portfolio but rather the impact of consumption which will usually dominate the conversation entirely.
Notes
[1] This point is not up to the level of a critique by a well known economist but here is one example from my own past of irrational choice being situationally rational: in 1989, giving a presentation to the CFO of Air Canada, we leaned on NPV analysis i.e., very MBA-proper and academically defensible. He leaned back for a minute, drew deeply on his Canadian cigarette (they did that in office back then), and said slowly and carefully: “our industry has been in and out of bankruptcy for a decade; we will use payback period.” Payback it was.
[2] note E[g(r)] --> median[g(r)], asymptotically as time periods get large. The distrb of annualized returns approaches normal which, per Michaud 2003, allows us some leeway in how we analyze this.
[3] This is illustrative only. One has to assume that this was some optimal choice. From what little I know a more realistic illustration would probably have much higher vol which would make the analysis even harsher. This is clearly not an all stock portfolio. In fact we haven’t really said all that much about the relative risk (vs. sub optimal) of GMM optimal portfolios in this post. Maybe next time. Note 4 is the same reflection.
[4] I haven’t really shown this. The time to recover or the extreme downside is a characteristic of any risky portfolio not just GMM optimal portfolios. Maybe what I should have done is show the gain or loss in time between GMM and other. Idk, sounds hard.
[7] My one interlocutor on this stuff hates it when I sandbag but I have no formal background in mathematics, economics, stats, or data science -- whether professional or academic. This is just amateur play for the sake of play and because it is an interesting topic. See Selden (2020) for my attempt at getting a grip on topics implied in the post.
Other Reading and References -- I’ve only read a little less than half of these, btw. Later.
Algoet and Cover (1988), Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment, The Annals of Probability Vol. 16, No. 2 (Apr., 1988), pp. 876-898 (23 pages) Published By: Institute of Mathematical Statistics
Breiman, L (1961) Optimal Gambling Systems for Favorable Games, Fourth Berkeley Symposium on Probability and Statistics.
Christensen, M. 2012. On the history of the growth optimal portfolio. L. Györfi, G. Ottucsák, H. Walk, eds., Machine Learning for Financial Engineering. Imperial College Press, London, 1–80
Dempster et al. (2008), Decision Making with Dempster-Shafer Theory of Evidence using Geometric Operators, World Academy of Science, Engineering and Technology 23 2008 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.307.254&rep=rep1&type=pdf
Dempster, M., I. Evstigneev, K. Schenk-Hopp´e. 2008. Financial markets. The joy of volatility. Quantitative Finance 8(1) 1–3
Doctor J, Wakker P, Wang T (2020), Supplementary Information on: “Economists’s view on the Ergodicity Problem.”
Ethier, S N (2004) The Kelly System Maximizes Median Fortune, Journal of Applied Probability Vol. 41, No. 4 (Dec., 2004), pp. 1230-1236
Estrada, J. 2010. Geometric mean maximization: An overlooked portfolio approach? Journal of Investing 19(4) 134–147
There seems to have been an uptick in the last few years of interest in the growth optimality consideration for portfolios. Ergodicity Economics feels like the new kid on the block, but this topic has a pretty long history, and the general interest bubbling up I find useful because it is an interesting and worthy topic. I won’t recapitulate all the math or notation in this post since it is tedious blogging [7] and one can read it comprehensibly, for the most part, and quite usefully, in the following bulleted references. Or check out the recommended reading list at the end and in particular pay attention to the references inside the various papers:
- Hoffstein, Corey (2017). IMO this is the most accessible quant intro. Also Corey does not buy it hook line and sinker and provides excellent counter points and critiques.
- Michaud, Richard (2003, 2015). I’ve read this 10,000 times. Ok, maybe 20 but still, I think this is an inspired intro to geometric returns and portfolio choice that one can read.
- Markowitz, H. (2016), [Chapter 9 “The Mossin Samuelson Model.”] Everyone thinks of HM as the one-period MPT guy. Here, he comes across as a Kelly partisan and a foil for the Samuelson critique. Good stuff. Volume 3 of his book was not that great but Vol 2 I go back to often.
- Peters, Ole; Ergodicityeconomics.com – ton of great papers TBD most of which I have not read (yet) so take any critiques I have of EE with a wink from me. Ole makes bold claims but I cannot tell if he has internalized the list below…maybe, idk. I think his work will be a strong addition to the lit once it is digested by others.
- Rujeerapaiboon (2016) In particular, gives a great intro to the subject and is a decent counterweight to Ergodicity Economics’ claim to have invented growth optimal portfolio math from scratch (kidding, but you know…)
- Doctor J, Wakker P, Wang T (2020) give a robust defense of Utility theory contra EE and ergodicity. Worth a look. Not sure of its role here.
Seven Random Thoughts on Time Averages and Geometric Mean Maximization
This may be redundant with the critique section below but what the heck.
Thought 1 - Ensemble averages use a large number of parallel scenarios over long intervals to estimate a central tendency while a time average uses one scenario over a large (maybe but not necessarily infinite) number of steps to approach some final expected state.
Thought 2 - The realized geometric mean [2] of a stable, continuously rebalanced(?) portfolio, a time average process, approaches its median expectation at long horizons and is related to the median of terminal wealth. The ensemble average of terminal wealth diverges from the median due to extreme outcomes and therefore becomes less useful in multi period models with large dispersion. This is one reason why growth optimization (ie maximize median terminal wealth) makes a lot of sense to people: why not pick the portfolio with the highest estimated median terminal wealth outcome?
Thought 3 - Maximizing the geometric mean return, and hence median terminal wealth, is a mathematical optimizing framework for portfolios over very long horizons but may be problematic over short horizons. Life is lived neither in an ensemble average, nor the median time-averaged terminus, nor in some infinite future, it is lived on the one and only one relatively short path given to us, and that path can be a rough short ride in the moment given a goal that is less than, say, 30 or 40 years out. See critiques.
Thought 4 - A growth optimal portfolio can’t technically “ruin,” but it can fall very close to zero during lifetime, or after, and so it is perhaps a better fit for academics, very-long-horizon sequence gamblers, non-spenders, and institutions.
Thought 5 – Retirees spend – a choice made jointly with the portfolio risk composition – which means a retirement portfolio can go to and through zero, all else equal, with no adaptation along the way. A growth optimizing framework does not consider spending and therefore a retiree could crash hard following that path. As cool as I think the discussion of growth optimality is re time dynamics, and I do, it is just not a serious good-faith participant in the chat with retirees unless we are talking about an upside-no-touch-for-more-than-x-years portfolio with a floor of income to protect lifestyle.
Thought 6 – A multi-period growth optimizing portfolio choice may exist on the single period MPT efficient frontier but that does not always mean it is always qualitatively best. This Kelly-point may be one choice among many although to the right/up does not necessarily make sense. Markowitz (’59) “does not recommend that an investor choose the [growth optimal] portfolio, [just not one higher on the EF]…since such a portfolio has greater short run volatility but less long run return…”
Thought 7 – While any single geometric return time average path will converge towards the expectation for geometric mean at infinity (and median terminal wealth), even at long horizons, say 500 years, there is a fair amount of dispersion in returns even then. Certainly, the wealth paths diverge immensely.
Acknowledging some Critiques of Growth Optimal in the Literature
While I have not set up the math framework here in this piece, the properties of the geometric mean max (GMM, max growth criterion) have been of keen interest in either the special (Kelly and much of the financial lit on growth optimality) or general sense (EE???) for a long time. On the other hand, there are evidently reasons to be wary, some of which I touched on above. In a superficial canvass of the literature below as well as some of my memory of work I’ve done on this before, this list is what I came up with for now for what people worry about when attempting to select growth optimal portfolios. I reserve the right to adjust this later:
1. Time. a) The median of the geometric mean at intermediate real horizons is not the same as the median of the GM at infinity, a common mistake. Plus, more importantly b) the volatility and path dependency mean that it could take a ton of time for a time-average to approach what one might think it should approach. – See: Rubinstein, Rujeerapaiboon, Markowitz 2016, Michaud.
2. Epistemology, by which I mean: the GMM cannot be truly known or calculated due to the unknowability of the inputs and the instability and evolution of the environment enveloping the process over time, i.e., all the stuff required to get you to the place you thought you were going to get to but maybe never can. – See: Michaud
3. Countervailing optimality frameworks. One can solipsistically perseverate on growth optimality as an ultimate dominating desideratum but there are other ways of seeing reality, with: a) the rationality principles embedded in expected economic utility being one and, say, b) return probability frameworks being at least one other. Behavioral stuff might be a third if different from "a." All of any longer list would have their very own critiques, of course, but they are there. See note [1] – See: Rujeerapaiboon, Samuelson, Doctor, Hoffstein, Michaud.
4. Lack of diversification and extreme volatility especially when it’s terribly inconvenient (e.g., when spending and/or raising children are in play alongside the portfolio). This is tied at the hip to #1 but gets separate attention. LCM and consumption smoothing probably need to come into the chat right here but some growth optimalists seem to reject such things out of hand. Most of the growth optimal literature does little to acknowledge life cycle theory or consumption so #4 is either a very strong critique or a blogger-rube’s non-sequitur. TBD. Even without spending the extreme low wealth outcomes in the near-term could fire up behavioral issues an optimalist would never likely consider. – See: Estrada, Doctor, Michaud, Selden.
Observations on at Least One Critique
I am not going to dilate on everything related to geo mean critiques here. I am just going to take a look at #1, the time critique (see note 3), and maybe a little bit on #4. This objection (#1) seems to come up the most in the material I read on growth optimization. Here is a paragraph from Rujeerapaiboon that makes the point for me. I think this might’ve been where Hoffstein got his comments on Rubinstein.
“Even though the growth-optimal portfolio is guaranteed to dominate any other portfolio with probability 1 in the long run, it tends to be very risky in the short term. Judicious investors might therefore ask how long it will take until the growth-optimal portfolio outperforms a given benchmark with high confidence. Unfortunately, there is evidence that the long run may be long indeed. Rubinstein (1991) demonstrates, for instance, that in a Black Scholes economy it may take 208 years to be 95% sure that the Kelly strategy beats an all-cash strategy and even 4,700 years to be 95% sure that it beats an all-stock strategy. Investors with a finite lifetime may thus be better off pursuing a strategy that is tailored to their individual planning horizon.”This is a recognition that even at long horizons the time average optimum is not really an "answer," it is a “proposition” that might or might not be realized for reasons embedded in #2, #3 and #4. The propositional nature of things can be visualized by mortals like us either by way of: a) simple formulas (see Michaud) for the mean and variance of the geometric mean at N periods (the second formula is the important one here and shows the N period effect on the dispersion of the expectation). According to Michaud these are more pedagogic than strictly accurate.
Or b) one can tap into the same intuition via simulation. That’s pretty easy to do. Here, for example is a simulated geo mean process over a long horizon of 500 years, N[.04,.12], 10,000 times, initial wealth = 1 (see Note [3] though) presented first in terms of annualized geometric return time averages (figure1) and then individual wealth paths (figure 2).
Figure 1. Geometric return, annualized, over 500 years
What do we see in these figures?
F1a. Since we can’t escape the gravity of random draws on returns, we see that in F1a: a) the time average paths, any one of which we could be on, are extremely variable at first but do in fact more or less converge over time, b) there is still a fair amount of uncertainty, even at 500 years, about convergence, and c) the geo expectation is, in fact, less than the arithmetic input and approaches both the median and the common estimator eventually.
F1b. We also see that it takes about 140 years for the 25th percentile geo return (arbitrary, plus note3) to get within 20% of the expectation and ~ >500 years to get within10%. The minimum in any year – this is harder to interpret since it is not a “path,” just the minimum for any path in some year – doesn’t even get close. I’m curious how far I’d have to run this to get it closer to some selected and arbitrary threshold.
F2a. Wealth compounds like a 5th force here, like we might expect, and the median goes to the moon, ignoring its note3 optimality question. Of course, we should limit ourselves to maybe 30 years, not 500…and add spending. But that is a different story than the growth optimal story.
F2b. When we look at the minimums – again, be careful with the interpretation here – i.e., the bottom of the grey-path mass in F2a, we see that at least one path was at <13% of initial wealth at ~110 years and the time frame where most or all of the paths are above parity with t(0) is what? A couple hundred years? Ever read “Time Enough for Love” by Heinlein? Imagine living for 800 years and watching this unfold were we to be on an un-fortuitous path. Yes, median wealth would be great (ex spend), and there is no technical “ruin” as such, but still… (note 6)
Now With Spending….
Now, just for fun, let’s add some consumption. With an arbitrary 4% spend added on top of the same already arbitrary portfolio process, this is what it looks like over: a) 500 years, and then b) a more human retirement 30 year interval. The geo return process engine (F1a) is the same so ignored. In Figure 3 We’ll focus on Fig2 transformed with a spend:
Figure 3. Wealth trajectories over 500 years - with consumption, then over 30 years
Some Observations:
F3a. This is absurd but makes my point. We see where median wealth is going. Ignore the question on negative wealth for now. It was easier to code this way.
F3b. This is no more no less than a standard Monte Carlo type model with some kind of fail threshold of x% at or before a goal date. The only difference from F2 is that we have added a spend and zoomed into a human-sized interval.
Growth optimality, which frankly has not really been illustrated since we used arbitrary inputs, wants to focus on maxing median terminal wealth. Retirement, otoh, is uniquely attuned to downside risk and protecting lifestyle. One wants to avoid a crash and/or one wants to immunize lifestyle first. Or at least some do. We retirees[5] are not playing abstract portfolio optimization we are attempting to optimize lifetime consumption which is a different game entirely and in which GMM optimization has a smaller role than the literature below might admit [5].
Were we to have an income floor – TIPS, annuities, pensions, SS, etc – the upside portfolio can play the GMM game as a choice, of course, though Critique #3 might still have a say, idk.
While GMM portfolios can have intermediate horizon risk [note 4], behavioral or otherwise, I think that what we should fear is not the distance between a GMM optimal portfolio and some other suboptimal portfolio but rather the impact of consumption which will usually dominate the conversation entirely.
Notes
[1] This point is not up to the level of a critique by a well known economist but here is one example from my own past of irrational choice being situationally rational: in 1989, giving a presentation to the CFO of Air Canada, we leaned on NPV analysis i.e., very MBA-proper and academically defensible. He leaned back for a minute, drew deeply on his Canadian cigarette (they did that in office back then), and said slowly and carefully: “our industry has been in and out of bankruptcy for a decade; we will use payback period.” Payback it was.
[2] note E[g(r)] --> median[g(r)], asymptotically as time periods get large. The distrb of annualized returns approaches normal which, per Michaud 2003, allows us some leeway in how we analyze this.
[3] This is illustrative only. One has to assume that this was some optimal choice. From what little I know a more realistic illustration would probably have much higher vol which would make the analysis even harsher. This is clearly not an all stock portfolio. In fact we haven’t really said all that much about the relative risk (vs. sub optimal) of GMM optimal portfolios in this post. Maybe next time. Note 4 is the same reflection.
[4] I haven’t really shown this. The time to recover or the extreme downside is a characteristic of any risky portfolio not just GMM optimal portfolios. Maybe what I should have done is show the gain or loss in time between GMM and other. Idk, sounds hard.
[5] I get that GMM is most often framed in terms of non-consumption portfolio management and institutional investing but this is a retirement blog.
[6] For the interested I went back and looked a 5th percentile of the wealth paths. ~40 years to parity ex-spend. Whatever that means...
[7] My one interlocutor on this stuff hates it when I sandbag but I have no formal background in mathematics, economics, stats, or data science -- whether professional or academic. This is just amateur play for the sake of play and because it is an interesting topic. See Selden (2020) for my attempt at getting a grip on topics implied in the post.
Other Reading and References -- I’ve only read a little less than half of these, btw. Later.
Algoet and Cover (1988), Asymptotic Optimality and Asymptotic Equipartition Properties of Log-Optimum Investment, The Annals of Probability Vol. 16, No. 2 (Apr., 1988), pp. 876-898 (23 pages) Published By: Institute of Mathematical Statistics
Breiman, L (1961) Optimal Gambling Systems for Favorable Games, Fourth Berkeley Symposium on Probability and Statistics.
Christensen, M. 2012. On the history of the growth optimal portfolio. L. Györfi, G. Ottucsák, H. Walk, eds., Machine Learning for Financial Engineering. Imperial College Press, London, 1–80
Dempster et al. (2008), Decision Making with Dempster-Shafer Theory of Evidence using Geometric Operators, World Academy of Science, Engineering and Technology 23 2008 http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.307.254&rep=rep1&type=pdf
Dempster, M., I. Evstigneev, K. Schenk-Hopp´e. 2008. Financial markets. The joy of volatility. Quantitative Finance 8(1) 1–3
Doctor J, Wakker P, Wang T (2020), Supplementary Information on: “Economists’s view on the Ergodicity Problem.”
Ethier, S N (2004) The Kelly System Maximizes Median Fortune, Journal of Applied Probability Vol. 41, No. 4 (Dec., 2004), pp. 1230-1236
Estrada, J. 2010. Geometric mean maximization: An overlooked portfolio approach? Journal of Investing 19(4) 134–147
Goldsten, Adam (20xx), "Did Ergodicity Economics and the Copenhagen Experiment Really Falsify Expected Utility Theory?" Deep Value Capital, Syracuse NY
Hoffstein, Corey (2017) “Growth Optimal Portfolios.” https://blog.thinknewfound.com/2017/07/growth-optimal-portfolios/
Hakansson, N. (1971). Multi-period mean-variance analysis: Toward a general theory of portfolio choice. Journal of Finance 26(4) 857–884.
Hakansson, N H (1971) On Optimal Myopic Portfolio Policies With and Without Serial Correlation of Yields, Journal of Business 44(3): 324-334
Hakansson & Miller (1975) Compound-Return Mean-Variance Efficient Portfolios Never Risk Ruin, Management Science Vol. 22, No. 4 (Dec., 1975), pp. 391-400
Kelly J L, (1956), A New Interpretation of the Information Rate, Bell System Technical Journal, 35: 917-926
Latané, H (1957) Rational Decision Making in Portfolio Management, PhD dissertation, UNC
Latané, H. 1959. Criteria for choice among risky ventures. Journal of Political Economy 67(2) 144–155
MacLean, L., E. Thorp, W. Ziemba. 2010. Good and bad properties of the Kelly criterion. The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific, 563–574.
MacLean, L., E. Thorp, W. Ziemba. (2011) The Kelly Capital Growth Investment Criterion: Theory and Practice [Amazon]
Markowitz, H., Blay, K. (2014) Risk Return Analysis, Volume 1. The Theory and Practice of Rational Investing. McGraw Hill.
Markowitz, H. (2016) Risk Return Analysis, Volume 2. The Theory and Practice of Rational Investing. McGraw Hill.
McCulloch, B, (2003) "Geometric Return and Portfolio Analysis," New Zealand Treasury Working Paper
Meucci, A. (2010) “Quant Nugget 2: Linear vs Compounded Returns, Common Pitfalls in Portfolio Management”
Michaud, Richard (2003, 2015) A Practical Framework for Portfolio Choice
Michaud R, Michaud R (2008), Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation (Hardback) -- Less for GMM and more for estimation error and resampling methods
Mindlin, D (2011) “On the Relationship between Arithmetic and Geometric Returns”
Nekrasov, V (2014), Kelly Criteron for Multivariate Portfolios: A Model Free Approach, University of Duisburg-Essen, Dept of Economics
Lo, Andrew et al, (2017) The growth of relative Wealth and the Kelly Criterion
Merton, R., P. Samuelson. 1974. Fallacy of the log-normal approximation to optimal portfolio decision making over many periods. Journal of Financial Economics 1(1) 67–94.
Peters, Ole, Ergodicityeconomics.com – ton of papers TBD
Poundstone, W (2006) Fortune’s Formula – [Ed Thorp]
Roll, R. 1973. Evidence on the “growth-optimum” model. Journal of Finance 28(3) 551–566.
Rubinstein, M. 1991. Continuously rebalanced investment strategies. Journal of Portfolio Management 18(1) 78–81.
Rujeerapaiboon, Kuhn D, Wiesemann (2016), Robust Growth-Optimal Portfolios, Ecole Polytechnique Fédérale de Lausanne, Switzerland; Imperial College Bus School, London. http://www.optimization-online.org/DB_FILE/2014/05/4366.pdf
Samuelson, P (1971) The “Fallacy” of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling, Proc. Nat. Acad Sci USA Vol 68 No 10 pp 2493-2496 October 2017.
Hoffstein, Corey (2017) “Growth Optimal Portfolios.” https://blog.thinknewfound.com/2017/07/growth-optimal-portfolios/
Hakansson, N. (1971). Multi-period mean-variance analysis: Toward a general theory of portfolio choice. Journal of Finance 26(4) 857–884.
Hakansson, N H (1971) On Optimal Myopic Portfolio Policies With and Without Serial Correlation of Yields, Journal of Business 44(3): 324-334
Hakansson & Miller (1975) Compound-Return Mean-Variance Efficient Portfolios Never Risk Ruin, Management Science Vol. 22, No. 4 (Dec., 1975), pp. 391-400
Kelly J L, (1956), A New Interpretation of the Information Rate, Bell System Technical Journal, 35: 917-926
Latané, H (1957) Rational Decision Making in Portfolio Management, PhD dissertation, UNC
Latané, H. 1959. Criteria for choice among risky ventures. Journal of Political Economy 67(2) 144–155
MacLean, L., E. Thorp, W. Ziemba. 2010. Good and bad properties of the Kelly criterion. The Kelly Capital Growth Investment Criterion: Theory and Practice. World Scientific, 563–574.
MacLean, L., E. Thorp, W. Ziemba. (2011) The Kelly Capital Growth Investment Criterion: Theory and Practice [Amazon]
Markowitz, H., Blay, K. (2014) Risk Return Analysis, Volume 1. The Theory and Practice of Rational Investing. McGraw Hill.
Markowitz, H. (2016) Risk Return Analysis, Volume 2. The Theory and Practice of Rational Investing. McGraw Hill.
McCulloch, B, (2003) "Geometric Return and Portfolio Analysis," New Zealand Treasury Working Paper
Meucci, A. (2010) “Quant Nugget 2: Linear vs Compounded Returns, Common Pitfalls in Portfolio Management”
Michaud, Richard (2003, 2015) A Practical Framework for Portfolio Choice
Michaud R, Michaud R (2008), Efficient Asset Management: A Practical Guide to Stock Portfolio Optimization and Asset Allocation (Hardback) -- Less for GMM and more for estimation error and resampling methods
Mindlin, D (2011) “On the Relationship between Arithmetic and Geometric Returns”
Nekrasov, V (2014), Kelly Criteron for Multivariate Portfolios: A Model Free Approach, University of Duisburg-Essen, Dept of Economics
Lo, Andrew et al, (2017) The growth of relative Wealth and the Kelly Criterion
Merton, R., P. Samuelson. 1974. Fallacy of the log-normal approximation to optimal portfolio decision making over many periods. Journal of Financial Economics 1(1) 67–94.
Peters, Ole, Ergodicityeconomics.com – ton of papers TBD
Poundstone, W (2006) Fortune’s Formula – [Ed Thorp]
Roll, R. 1973. Evidence on the “growth-optimum” model. Journal of Finance 28(3) 551–566.
Rubinstein, M. 1991. Continuously rebalanced investment strategies. Journal of Portfolio Management 18(1) 78–81.
Rujeerapaiboon, Kuhn D, Wiesemann (2016), Robust Growth-Optimal Portfolios, Ecole Polytechnique Fédérale de Lausanne, Switzerland; Imperial College Bus School, London. http://www.optimization-online.org/DB_FILE/2014/05/4366.pdf
Samuelson, P (1971) The “Fallacy” of Maximizing the Geometric Mean in Long Sequences of Investing or Gambling, Proc. Nat. Acad Sci USA Vol 68 No 10 pp 2493-2496 October 2017.
Samuelson, P (1979), "Why we should Not Make Mean Log of Wealth Though Years to Act are Long," Journal of Banking and Finance 3 305-307
Selden W (2020), Five Retirement Processes, On Using Some Fragments of Quantitative Finance and Lifecycle Economics as Methodology in the Context of a Retirement Flow in order to Monitor and Manage Decumulation Risk. https://drive.google.com/file/d/1D7cNhMAS1JAhjtE9wBGwV-bPrMT_W_na/view
Thorp, E. 1971. Portfolio choice and the Kelly criterion. Business and Economics Statistics Section of Proceedings of the American Statistical Association 215–224.
Winselmann, K, (2018) Essays on the Kelly Criterion and Growth Optimal Strategies, Dissertation, WHU { Otto Beisheim School of Management, https://opus4.kobv.de/opus4-whu/files/700/Winselmann_Kai_WHU_Diss_2018.pdf
Advice and Assistance from D Cantor.
Selden W (2020), Five Retirement Processes, On Using Some Fragments of Quantitative Finance and Lifecycle Economics as Methodology in the Context of a Retirement Flow in order to Monitor and Manage Decumulation Risk. https://drive.google.com/file/d/1D7cNhMAS1JAhjtE9wBGwV-bPrMT_W_na/view
Thorp, E. 1971. Portfolio choice and the Kelly criterion. Business and Economics Statistics Section of Proceedings of the American Statistical Association 215–224.
Winselmann, K, (2018) Essays on the Kelly Criterion and Growth Optimal Strategies, Dissertation, WHU { Otto Beisheim School of Management, https://opus4.kobv.de/opus4-whu/files/700/Winselmann_Kai_WHU_Diss_2018.pdf
Advice and Assistance from D Cantor.
No comments:
Post a Comment