I think this is probably the most nuanced, insightful and _practically helpful_ piece on rebalancing I've ever read - thank you very much. I've really enjoyed everything you've written so far.
I had a few questions in case you were looking for material for future posts (or want to answer here):
1) As you say in your "Diversifying Wrong" article, using Kelly necessarily implies rebalancing (so useful to see that so clearly expressed!). I completely follow your arguments, which make intuitive sense. Do you have any comments on this 2020 article, which found that increasing the frequency of rebalancing increased performance but at the expense of also increasing risk and (obviously) trading costs?
2) It would be brilliant to look at a practical how-to of position sizing and concentration in an overall portfolio. For a non-specialist, the maths can be rather forbidding. Do you have any heuristics to help here, for example on maximum sizes given the tendency of Kelly portfolios to be highly concentrated?
--- This Github Python code helps construct a Kelly optimal portfolio based on the user's expected returns and percentage of full Kelly desired. It then calculates the historical covariances (assuming these will be the same going forward...) to determine the optimal portfolio weightings. Would be really interested to hear your thoughts on an approach like this.
3) In your "Half Kelly" article you began by pointing out that institutional investors must "find ways to grow their assets under the vast majority of scenarios" and therefore need to avoid lottery tickets as well as Russian Roulette. What would be the circumstances and methods for an unconstrained investor to act differently from an institutional one in order to maximise terminal wealth? Or is the institutional approach "the right way of doing things"? :-)
4) Just spotted a few typos in your articles on it's vs its.
As I said, I'm finding your pieces hugely insightful and am very much looking forward to the next one. If you have any recommended sources, books, articles to better understand this puzzle, those would also be widely appreciated I'm sure!
Wow, I really appreciate the kind words. Practical insights are really what I’m shooting for here, so glad to hear these have been thought provoking!
I have a number of thoughts on all your points above and may indeed devote an entire piece to each. In the interim, here are some sketches:
1. My former PM used to work at Bankers Trust on the Equity Derivatives desk. Options, as you likely know, involve someone pre-paying a price for a contract that can be replicated with dynamic beta-hedging (known in options trading as delta-hedging)—the whole art of option market-making is pricing these options with some margin of safety, and then managing the other risks that arise from path-dependency, liquidity constraints, and market microstructure. At the time there was a researcher at BT (I cannot remember his name for the life of me, but will update this post if I do remember) who spent the majority of his time playing with option math.
One major project he undertook was to look at the impact of delta-hedging frequency. His outcome was that delta hedging frequency doesn’t impact the expected cost of delta hedging, but it does reduce the variance around that cost. However, practically this must be offset with slippage, and so the question becomes how to properly balance the two. Not a super easy question, but there are calculations that can help. I may try to translate some of the option theory into portfolio theory, but a great post outlining the math is here:
2. This is such a great question, and I’ll definitely expand on it in another post.
The short answer is that there are heuristics, but the limiting factor of total portfolio allocation isn’t defined by the Gaussian math, it’s defined by tail correlation—in other words, when liquidity disappears from financial markets, all assets tend to drop together. You can even see this in markets right now with the credit markets, equities, and even bitcoin all selling off at the same time.
I’m sure the actual math behind the github code is correct, but you can’t capture this conditional correlation in a covariance table, which in my mind makes any single-use calculation flawed.
More actual heuristics to come, but for now I would just reiterate a point from the diversification post-- if you are an intuitional investor, the most important decision you make is likely not what stock you buy, but how much beta you take. I would also point out that in 100 years of US Equity data we have, a 1.4x leverage was optimal.
3. This is such a fun philosophical question. Institutional investors are “liability driven” investors—their mandate is to meet a target return. Anything at target or above is considered adequate, and anything below has a real-world negative impact on constituents. They therefore have a fiduciary duty to reduce the volatility in a portfolio, and optimize their median return.
But there is no parallel argument for a personal account. It sort of reminds me of the St. Petersburg Paradox: Imagine a game in which you start with a $1 pot and flip a coin. If the coin is tails, the game ends and you receive the pot, but if you flip heads, you double the pot and flip again. And you keep flipping and doubling until you eventually flip tails. How much should you pay to play this game?
This is one of my favorite thought experiments because the expected value math breaks; your expected upside scales faster than your chance of hitting it reduces, and so the expected value veers towards infinity.
But how many games would you realistically have to play to have that math work out in your favor? Behavioral economists stepped in later with “marginal utility theory” to solve the game, saying that each marginal dollar won has less utility than the previous dollar. They also sort of say that everyone has a wholly unique utility function, implying that the answer to the St. Petersburg Paradox is wholly subjective. This is an unfulfilling answer to me, but likely correct—whether you want to solve for median returns, 5th percentile returns, or average expected returns is entirely up to you!
4. Appreciate the feedback— English has never been my strong suit! Will try to get more eyes on the editing 😊
What an amazing reply - thank you!! Your point about conditional tail correlation is a particularly fascinating one. Would love to read more from you on how best to practically think about that. 😁
Could I ask a few quickish follow ups if that's not too much trouble?
1) Do you have any recommendations to read more about the optimal equity beta being ~1.5 and leverage ~1.4x?
2) I'm not sure whether this is at all your bag, but do you have any preferred way to model individual utility functions at a given point? David Berns in "Modern Asset Allocation for Wealth Management" uses a three-dimensional utility function based on an investor's risk aversion, loss aversion, and reflection (I think the most useful part of his book). Would be very interested if you have another approach though, and how that might tie into a choice of median, average, 5th percentile returns, etc.
3) On the off chance that you'd ever like another pair of eyes to take a read before publishing, I'd be delighted to help out - can DM you on Twitter if that'd be useful.
Hi Hawnk,
I think this is probably the most nuanced, insightful and _practically helpful_ piece on rebalancing I've ever read - thank you very much. I've really enjoyed everything you've written so far.
I had a few questions in case you were looking for material for future posts (or want to answer here):
1) As you say in your "Diversifying Wrong" article, using Kelly necessarily implies rebalancing (so useful to see that so clearly expressed!). I completely follow your arguments, which make intuitive sense. Do you have any comments on this 2020 article, which found that increasing the frequency of rebalancing increased performance but at the expense of also increasing risk and (obviously) trading costs?
--- Practical Implementation of the Kelly Criterion: Optimal Growth Rate, Number of Trades, and Rebalancing Frequency for Equity Portfolios, https://www.frontiersin.org/articles/10.3389/fams.2020.577050/full
2) It would be brilliant to look at a practical how-to of position sizing and concentration in an overall portfolio. For a non-specialist, the maths can be rather forbidding. Do you have any heuristics to help here, for example on maximum sizes given the tendency of Kelly portfolios to be highly concentrated?
--- This Github Python code helps construct a Kelly optimal portfolio based on the user's expected returns and percentage of full Kelly desired. It then calculates the historical covariances (assuming these will be the same going forward...) to determine the optimal portfolio weightings. Would be really interested to hear your thoughts on an approach like this.
https://github.com/thk3421-models/KellyPortfolio
3) In your "Half Kelly" article you began by pointing out that institutional investors must "find ways to grow their assets under the vast majority of scenarios" and therefore need to avoid lottery tickets as well as Russian Roulette. What would be the circumstances and methods for an unconstrained investor to act differently from an institutional one in order to maximise terminal wealth? Or is the institutional approach "the right way of doing things"? :-)
4) Just spotted a few typos in your articles on it's vs its.
As I said, I'm finding your pieces hugely insightful and am very much looking forward to the next one. If you have any recommended sources, books, articles to better understand this puzzle, those would also be widely appreciated I'm sure!
Thanks again
Wow, I really appreciate the kind words. Practical insights are really what I’m shooting for here, so glad to hear these have been thought provoking!
I have a number of thoughts on all your points above and may indeed devote an entire piece to each. In the interim, here are some sketches:
1. My former PM used to work at Bankers Trust on the Equity Derivatives desk. Options, as you likely know, involve someone pre-paying a price for a contract that can be replicated with dynamic beta-hedging (known in options trading as delta-hedging)—the whole art of option market-making is pricing these options with some margin of safety, and then managing the other risks that arise from path-dependency, liquidity constraints, and market microstructure. At the time there was a researcher at BT (I cannot remember his name for the life of me, but will update this post if I do remember) who spent the majority of his time playing with option math.
One major project he undertook was to look at the impact of delta-hedging frequency. His outcome was that delta hedging frequency doesn’t impact the expected cost of delta hedging, but it does reduce the variance around that cost. However, practically this must be offset with slippage, and so the question becomes how to properly balance the two. Not a super easy question, but there are calculations that can help. I may try to translate some of the option theory into portfolio theory, but a great post outlining the math is here:
https://artursepp.com/2017/05/01/how-to-optimize-volatility-trading-and-delta-hedging-strategies-under-the-discrete-hedging-with-transaction-costs/
2. This is such a great question, and I’ll definitely expand on it in another post.
The short answer is that there are heuristics, but the limiting factor of total portfolio allocation isn’t defined by the Gaussian math, it’s defined by tail correlation—in other words, when liquidity disappears from financial markets, all assets tend to drop together. You can even see this in markets right now with the credit markets, equities, and even bitcoin all selling off at the same time.
I’m sure the actual math behind the github code is correct, but you can’t capture this conditional correlation in a covariance table, which in my mind makes any single-use calculation flawed.
More actual heuristics to come, but for now I would just reiterate a point from the diversification post-- if you are an intuitional investor, the most important decision you make is likely not what stock you buy, but how much beta you take. I would also point out that in 100 years of US Equity data we have, a 1.4x leverage was optimal.
3. This is such a fun philosophical question. Institutional investors are “liability driven” investors—their mandate is to meet a target return. Anything at target or above is considered adequate, and anything below has a real-world negative impact on constituents. They therefore have a fiduciary duty to reduce the volatility in a portfolio, and optimize their median return.
But there is no parallel argument for a personal account. It sort of reminds me of the St. Petersburg Paradox: Imagine a game in which you start with a $1 pot and flip a coin. If the coin is tails, the game ends and you receive the pot, but if you flip heads, you double the pot and flip again. And you keep flipping and doubling until you eventually flip tails. How much should you pay to play this game?
This is one of my favorite thought experiments because the expected value math breaks; your expected upside scales faster than your chance of hitting it reduces, and so the expected value veers towards infinity.
But how many games would you realistically have to play to have that math work out in your favor? Behavioral economists stepped in later with “marginal utility theory” to solve the game, saying that each marginal dollar won has less utility than the previous dollar. They also sort of say that everyone has a wholly unique utility function, implying that the answer to the St. Petersburg Paradox is wholly subjective. This is an unfulfilling answer to me, but likely correct—whether you want to solve for median returns, 5th percentile returns, or average expected returns is entirely up to you!
4. Appreciate the feedback— English has never been my strong suit! Will try to get more eyes on the editing 😊
What an amazing reply - thank you!! Your point about conditional tail correlation is a particularly fascinating one. Would love to read more from you on how best to practically think about that. 😁
Could I ask a few quickish follow ups if that's not too much trouble?
1) Do you have any recommendations to read more about the optimal equity beta being ~1.5 and leverage ~1.4x?
2) I'm not sure whether this is at all your bag, but do you have any preferred way to model individual utility functions at a given point? David Berns in "Modern Asset Allocation for Wealth Management" uses a three-dimensional utility function based on an investor's risk aversion, loss aversion, and reflection (I think the most useful part of his book). Would be very interested if you have another approach though, and how that might tie into a choice of median, average, 5th percentile returns, etc.
https://www.amazon.co.uk/Modern-Allocation-Wealth-Management-Finance-ebook/dp/B088B1YM6Y/ref=cm_cr_arp_d_pl_foot_top?ie=UTF8
3) On the off chance that you'd ever like another pair of eyes to take a read before publishing, I'd be delighted to help out - can DM you on Twitter if that'd be useful.
Cheers again!