You Still Have Made a Choice: Things that Matter #2

Drummers are really nothing more than time-keepers. They’re the time of the band. I don’t consider I should have as much recognition as say a brilliant guitar player. I think the best thing a drummer can have is restraint when he’s playing — and so few have today. They think playing loud is playing best. Of course, I don’t think I’ve reached my best yet. The day I don’t move on I stop playing. I don’t practice ever. I can only play with other people, I need to feel them around me.

— Ginger Baker (founder of Cream), from a 1970 interview with Disc Magazine

La cuisine, c’est quand les choses ont le goût de ce qu’elles sont.
(Good cooking is when things taste of what they are.)

— Maurice Edmond Sailland (Curnonsky) — 1872-1956

There are those who think that life
Has nothing left to chance
A host of holy horrors
To direct our aimless dance

A planet of playthings
We dance on the strings
Of powers we cannot perceive
The stars aren’t aligned
Or the gods are malign
Blame is better to give than receive

You can choose a ready guide
In some celestial voice
If you choose not to decide
You still have made a choice

 — Rush, “Freewill”, Permanent Waves (1980)

For the kingdom of heaven is like a man traveling to a far country, who called his own servants and delivered his goods to them. And to one he gave five talents, to another two, and to another one, to each according to his own ability; and immediately he went on a journey. Then he who had received the five talents went and traded with them, and made another five talents. And likewise, he who had received two gained two more also. But he who had received one went and dug in the ground, and hid his lord’s money. After a long time the lord of those servants came and settled accounts with them.

So he who had received five talents came and brought five other talents, saying, ‘Lord, you delivered to me five talents; look, I have gained five more talents besides them.’ His lord said to him, ‘Well done, good and faithful servant; you were faithful over a few things, I will make you ruler over many things. Enter into the joy of your lord.’ He also who had received two talents came and said, ‘Lord, you delivered to me two talents; look, I have gained two more talents besides them.’ His lord said to him, ‘Well done, good and faithful servant; you have been faithful over a few things, I will make you ruler over many things. Enter into the joy of your lord.’

Then he who had received the one talent came and said, ‘Lord, I knew you to be a hard man, reaping where you have not sown, and gathering where you have not scattered seed. And I was afraid, and went and hid your talent in the ground. Look, there you have what is yours.’

But his lord answered and said to him, ‘You wicked and lazy servant, you knew that I reap where I have not sown, and gather where I have not scattered seed. So you ought to have deposited my money with the bankers, and at my coming I would have received back my own with interest. Therefore, take the talent from him, and give it to him who has ten talents.

For to everyone who has, more will be given, and he will have abundance; but from him who does not have, even what he has will be taken away. And cast the unprofitable servant into the outer darkness. There will be weeping and gnashing of teeth.

The Bible, The Gospel of Matthew 25:14-30

This note was featured in Meb Faber’s book The Best Investment Writing – Volume 2, alongside another Epsilon Theory note from Ben Hunt. Click here to get a copy.

I will never understand why more people don’t revere Rush.

With the possible exception of Led Zeppelin[1], I’m not sure there has been another band with such extraordinary instrumentalists across the board, such synergy between those members and their musical style and such a consistent approach to both lyrical and melodic construction. And yet they were only inducted into the Rock & Roll Hall of Fame in 2013. A short list of bands and singers the selection committee thought were more deserving: ABBA, Madonna, Jackson Browne, the Moonglows, Run DMC. At least they got in when Randy Newman did. I remember the first time I heard YYZ, the Rush tune named after the IATA airport code for Toronto’s Pearson International Airport, pronounced “Why Why Zed” in the charming manner of the Commonwealth. It was then that I decided I would be a drummer. I did play for a while, and reached what I would describe as just above a baseline threshold of competence.

That’s not a throwaway line.

There’s a clear, explicit line that every drummer (hopefully) crosses at one point. A step-change in his understanding of the role of the instrument. The true novice drummer always picks up the sticks and plays the same thing. Common time. Somewhere between 90-100 beats per minute. Eighth note closed hi-hat throughout. Bass drum on the down and upbeat of the first beat. Snare on second down beat. And then it’s all jazzy up-beat doodling on the snare for the rest of that bar until the down beat of four. Same thing for three measures, and on the fourth measure it’s time for that awesome fill he’s been practicing. I don’t know how many subscribers are drummers, but I assure you, literally couples of you are nodding your heads.

The fills and off-beat snare hits are all superfluous and not necessary to the principal role of a drummer in rock and roll: to keep the damned beat. But there are a number of reasons why every neophyte does these same things. Mimicry of more advanced players who can do the creative and interesting things without losing the beat, for one. We see Tony Williams, John Bonham, or Bill Bruford and do what it is we think they are doing to make the music sound good. The amateur often also thinks that these are the necessary things to be perceived as a more advanced player, for another. He doesn’t just imagine that his mimicry will make him sound more like the excellent players, but imagines himself looking like them to others. More than anything, the amateur does these things because he hasn’t quite figured out that keeping a good beat is so much more important than anything else he will do that he’s willing to sacrifice it for what he thinks is impressive.

This thought process dominates so many other fields as well. Consider the number of amateur cooks who hit every sauce or piece of meat with a handful of garlic powder, onion powder, oregano, salt, pepper and cayenne, when the simplicity of salt as seasoning dominates most of the world’s great cuisine. There is an instinct to think that complexity and depth must come from a huge range of ingredients[2] or from complexity in preparation, but most extraordinary cooking begins from an understanding of a small number of methods for heating, seasoning and establishing bases for sauces. Inventiveness, creativity and passion can take cuisine in millions of directions from there, but many home cooks see the celebrity chef’s flamboyant recipe and internalize that the creative flourishes are what matters to the dish, and not the fact that he cooked a high-quality piece of meat at the right heat for the right amount of time.

If you’re not much of a cook, consider instead the 30-handicap golfer who wouldn’t be caught dead without a full complement of four lob wedges in his bag. You know, so that he can address every possible situation on the course. The trilling singer of the national anthem who can’t hold a pitch but sees every word of the song as an opportunity to sing an entire scale’s worth of notes. The karate novice who addresses his opponent with a convoluted stance. The writer who doesn’t know when to stop giving examples to an audience who understood what he was getting at half-way through the one about cooking.

I’m guessing at least one of these things pisses you off, or at the very least makes you do an internal eye roll. And yet, as investors we are guilty of doing this kind of thing all the time, any time the topic of diversification comes up.

It comes from a good place. We know from what we’ve been taught (and from watching the experts) that we should diversify, but we don’t have a particularly good way of knowing what that means. And so we fill our portfolios with multiple flavors of funds, accounts and individual securities. Three international equity funds with different strategies. Multiple different styles in emerging markets. Some value. Some growth. Some minimum volatility. Some call writing strategies. Some sector funds. Maybe some long/short hedge funds. Some passively managed index funds, some actively managed funds. Definitely some sexy stock picks. And in the end, the portfolio that we end up with looks very much like the global equity market, maybe with a tilt here or there to express uniqueness — that flashy extra little hit on the snare drum to look impressive.

This piece isn’t about the time we waste on these things. I already wrote a piece about that a few weeks ago. This is about the harm we do to our portfolios when we play at diversifying instead of actually doing it.

The Parable of the Two FA’s

So what does actually diversifying look like?

There are lot of not-very-useful definitions out there. The eggs-in-one-basket definition we’re all familiar with benefits from simplicity, which is not nothing. In addition, it does work if people have a good concept of what the basket is in the analogy. Most people don’t. Say you have $100, and you decide that a basket is an advisor or a fund. So you split the money between the two, and they invest in the same thing. You have not diversified[3]. The other definitions for diversification tend to be more complicated, more quantitative in nature. That doesn’t make them bad, and we’ll be leaning on some of them. But we need a rule of thumb, some heuristic for describing what diversification ought to look like so that we know it when we see it. For the overwhelming majority of investors, that rule of thumb should go something like this:

Diversification is reducing how much you expect to lose when risky assets do poorly or very poorly without necessarily reducing how much total return you expect to generate.

Now, this is not exactly true, and it’s very obviously not the whole definition. But by and large it is the part of the definition that matters most. The more nuanced way to think about diversification, of course, is to describe it as all the benefits you get from the fact that things in your portfolio don’t always move together, even if they’re both generally going up in value. But most investors are so concentrated in general exposure to risky assets — securities whose value rises and falls with the fortunes and profitability of companies, and how other investors perceive those fortunes — that this distinction is mostly an academic one. Investors live and die by home country equity risk. Period. Most investors understand this to one degree or another, but the way they respond in their portfolios doesn’t reflect it.

I want to describe this to you in a parable.

There was once a rich lord who held $10 million in a S&P 500 ETF. He knew that he would be occupied with his growing business over the next year. Before he left, he met with his two financial advisors and gave them $1 million of his wealth and told them to “diversify his holdings.”

He returned after a year and came before the first financial advisor. “My lord, I put the $1 million you gave me in a Russell 1000 Value ETF. Here is your $1.1 million.” The rich man replied, “Dude, that’s almost exactly what my other ETF did over the same period. What if the market had crashed? I wasn’t diversified at all!” And the financial advisor was ashamed.

Furious and frustrated, the rich man then summoned his second financial advisor. “Sir, I put your $1 million in a Short-Duration Fixed Income mutual fund of impeccable reputation. Here’s your $1 million back.”

“Oh my God,” the lord replied, “Are you being serious right now? If I wanted to reduce my risk by stuffing my money in a mattress I could have done that without paying you a 65bp wrap fee. How do you sleep at night? I’m going to open a robo-advisor account.”

Most of us know we shouldn’t just hold a local equity index. We usually buy something else to diversify, because that’s what you do. But what we usually do falls short either because (1) the thing we buy to diversify isn’t actually all that different from what we already owned, or (2) the thing we buy to diversify reduces our risk and our return, which defeats the purpose. There’s nothing novel in what I’m saying here. Modern portfolio theory’s fundamental formula helps us to isolate how much of the variation in our portfolio’s returns comes from the riskiness of the stuff we invested in vs. the fact that this stuff doesn’t always move together.

Source: Salient 2017 For illustrative purposes only.

The Free Lunch Effect

So assuming we didn’t have any special knowledge about what assets would generate the highest risk-adjusted returns over the year our rich client was away on business, what answer would have made us the good guy in the parable? Maximizing how much benefit we get from that second expression above — the fact that this stuff doesn’t always move together.

Before we jump into the math on this, it’s important to reinforce the caveat above: we’re assuming we don’t have any knowledge about risk-adjusted returns, which isn’t always true. Stay with me, because we will get back to that. For the time being, however, let’s take as a given that we don’t know what the future holds. Let’s also assume that, like the Parable of the Two FA’s, our client holds $10 million in S&P 500 ETFs. Also like the parable, we have been asked to reallocate $1 million of those assets to what will be most diversifying. In other words, it’s a marginal analysis.

The measure we’re looking to maximize is the Free Lunch Effect, which we define as the difference between the portfolio’s volatility after our change at the margin and the raw weighted average volatility of the underlying components. If the two assets both had volatility of 10%, for example, and the resulting portfolio volatility was 9%, the Free Lunch Effect would be 1%.

If maximizing the Free Lunch Effect is the goal, here’s the relative attractiveness of various things the two FA’s could have allocated to (based on characteristics of these markets between January 2000 and July 2017).

Volatility Reduction from Diversification — Adding 10% to a Portfolio of S&P 500

Source: Salient 2017. For illustrative purposes only. Past performance is not indicative of how the index will perform in the future. The index reflects the reinvestment of dividends and income and does not reflect deductions for fees, expenses or taxes. The index is unmanaged and is not available for direct investment.

The two FA’s failed for two different reasons. The first failed because he selected an asset which was too similar. The second failed because he selected an asset which was not risky enough for its differentness to matter. The first concept is intuitive to most of us, but the second is a bit more esoteric. I think it’s best thought of by considering how much the risk of a portfolio is reduced by adding an asset with varying levels of correlation and volatility. To stop playing at diversification, this is where you start.

Volatility Reduction by Correlation and Volatility of Diversifying Asset

Source: Salient 2017. For illustrative purposes only. Past performance is not indicative of how the index will perform in the future. The index reflects the reinvestment of dividends and income and does not reflect deductions for fees, expenses or taxes. The index is unmanaged and is not available for direct investment.

If You Choose not to Decide

If there are some complaints that can be leveled against this approach, two of them, I think, are valid and worthy of exploration.

The first is that diversification cannot be fully captured in measures of correlation. If you read Whom Fortune Favors, you’ll know that our code recognizes that we live in a behaviorally-influenced, non-ergodic world. While I think we’d all recognize that U.S. value stocks are almost always going to be a poor diversifier against global equities (and vice versa), clearly there are events outside of the historical record or what we know today that could completely change that. And so the proper reading of this should always be in context of an adaptive portfolio management process.

The second complaint, as I alluded to earlier, is the fact that we are not always indifferent in our risk-adjusted return expectations for different assets. I’m sure many of you looked at the above chart and said to yourself, “Yeah, I’m not piling into commodities.” I don’t blame you (I’m still not satisfied with explanations for why I ought to be paid for being long contracts on many commodities), but that is the point. Not owning commodities or MLPs because you don’t get them isn’t the same as not expressing an opinion. If you choose not to decide, you still have made a choice.

When investors choose to forgo diversification, on any basis, they are implicitly betting that decisions that they make will outperform what diversification would have yielded them. It may not be optimal to own the most diversified portfolio you can possibly own, because anti-diversifying decisions might, in fact, be worth it. But it is exactly that thought process that must become part of our code as investors. It’s OK to turn down a free lunch, but you’d damn well better know that what you’re going to spend your money on is better.

So how do you quantify that implicit bet? Again, the Free Lunch Effect gives us our easiest answer. Consider the following case: let’s assume we had two investment options, both with similar risk of around 15%. For simplicity’s sake we’ll start from our naïve assumption that our assets produce, say, 0.5 units of return for every unit of risk we take. If the two assets are perfectly uncorrelated, how much more return would we need to demand from Asset 1 vs. Asset 2 to own more of it than the other? To own 100% Asset 1?

Well, the chart below shows it. In the case above, if you invest 100% of your portfolio in Asset 1, an investor who thinks about his portfolio in risk-adjusted terms is implicitly betting that Asset 1 will generate more than 3% more return per year, or an incremental 0.21 in return/risk units. If the assets are less similar, this implicit view grows exponentially.

Implied Incremental Return Expectation from Overweighted Asset

Source: Salient 2017. For illustrative purposes only. Past performance is not indicative of how the index will perform in the future. The index reflects the reinvestment of dividends and income and does not reflect deductions for fees, expenses or taxes. The index is unmanaged and is not available for direct investment.

A Chain of Linked Engagements

If we do not learn to regard a war, and the separate campaigns of which it is composed, as a chain of linked engagements each leading to the next, but instead succumb to the idea that the capture of certain geographical points or the seizure of undefended provinces are of value in themselves, we are liable to regard them as windfall profits.

— On War, Carl von Clausewitz

The point of this note isn’t to try to convince you to focus your portfolio construction efforts on higher volatility diversifiers like those highlighted earlier (although many of you should). It’s also not to argue that maximizing diversification should be your first objective (although most of us are so far from the optimum that moving in this direction wouldn’t hurt). It is to emphasize that portfolio construction and the decisions we make are a chain of linked engagements. It is to give you pause when you or your client asks for a ‘best new investment idea’. If your experiences are like mine, the question is nearly always expressed in isolation — recommend me a stock, a mutual fund, a hedge fund. These questions can never be answered in isolation. If you really must tinker with your allocation, sure, I can give you my view, but only if I know what else you own, and only if I know what you intend to sell in order to buy the thing.

Anyone who will make a recommendation to you without knowing those things is an idiot, a charlatan, or both.

Most of us, whether we are entrenched in financial markets or not, think about our decisions not in a vacuum but in terms of opportunity cost. If we buy A, we’re giving up B. If we invest in A, we’re giving up on B. If we do A, we won’t have time for B. Opportunity cost is fundamental to thinking about nearly every aspect of human endeavor but for some reason is completely absent from the way many investors typically think about building portfolios.

Look, if you didn’t completely follow where I was going with Whom Fortune Favors, I get it. Telling you to think about risk and diversification separately is more than a little bit arcane. But here’s where it comes together: an investor can only make wise decisions about asset allocation, about selecting fund managers, about tactical bets and about individual investments when he has an objective opportunity cost to assess those decisions against that allows him to make his portfolio decisions intentionally, not implicitly. That opportunity cost is the free lunch provided by diversification.

If we take this way of thinking to its natural extreme, we must recognize that we can, at any point, identify the portfolio that would have provided the maximum diversification, at least using the tools we’ve outlined here. For most periods, if you run through that analysis, you are very likely to find that a portfolio of those assets in which every investment contributes a comparable amount of risk to the whole — a risk parity portfolio, in other words — typically provides something near to that maximum level of diversification. I am not suggesting that your portfolio be the maximum diversification portfolio or risk parity. But I am suggesting that a risk parity portfolio of your investable universe is an excellent place to use as an anchor for this necessary analysis.

If you don’t favor it for various reasons (e.g. using volatility as a proxy for risk is the devil, it’s just levered bonds, etc.), then find your home portfolio that accomplishes similar goals in a way that is rules-based and sensible. Maybe it’s the true market portfolio we highlight in I am Spartacus. If you’re conservative, maybe it’s the tangency portfolio from the efficient frontier. And if you’re more aggressive, maybe it is something closer to the Kelly Optimal portfolio we discussed in Whom Fortune Favors. From there, your portfolio construction exercise becomes relatively simple: does the benefit I expect from this action exceed its diversification opportunity cost?

How do you measure it? If you have capital markets assumptions or projections, feel free to use them. Perhaps simpler, assume a particular Sharpe Ratio, say 0.25 or 0.30, and multiply it times the drop in diversification impact from the action you’re taking. Are you confident that the change you’re making to the portfolio is going to have more of an impact than that? That’s…really it. Now the shrewd among you might be saying, “Rusty, isn’t that kind of like what a mean-variance optimization model would do?” It isn’t kind of like that, it’s literally that. And so what? We’re not reinventing portfolio science here, we’re trying to unpack it so that we can use it more effectively as investors.

Recognize that this isn’t just a relevant approach to scenarios where you’re changing things around because you think it will improve returns dramatically. This is also a useful construct for understanding whether all the shenanigans in search of diversification, all that Chili P you’re adding, are really worth the headache. Is that fifth emerging markets manager really adding something? Is sub-dividing your regions to add country managers really worth the time?

In the end, it’s all about being intentional. With as many decisions as we have to manage, the worst thing we can do is let our portfolios make our decisions for us. Given the benefits of diversification, investors ought to put the burden of proof on anything that makes a portfolio less diversified. In doing so, they will recognize why this code recognizes the intentional pursuit of real diversification as the #2 Thing that Matters.


[1] I don’t want to hear it from the “but they stole people’s music and weren’t super nice about it” crowd. Zep played better rock and roll music than anyone before or after, and it’s not even close.

[2] And it can. Pueblan and Oaxacan cuisine feature moles with extraordinary complexity that does come from the melding of a range of seasonings and ingredients. Traditional American chilis, South Asian curries and soups from around the world often do as well. Dishes en croute (e.g. pate en croute, coulibiac, etc.) are notoriously tricky, too.

[3] Cue the fund-of-funds due diligence analyst pointing out that we would have, in fact, diversified our fraud risk. Die on that hill if you want to, friend.


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Whom Fortune Favors: Things that Matter #1, Pt. 2


Click here to read Part 1 of Whom Fortune Favors


Fook: There really is an answer?

Deep Thought: Yes. There really is one.

Fook: Oh!

Lunkwill: Can you tell us what it is?

Deep Thought: Yes. Though I don’t think you’re going to like it.

Fook: Doesn’t matter! We must know it!

Deep Thought: You’re really not going to like it!

Fook: Tell us!

Deep Thought: Alright. The answer to the ultimate question…of Life, the Universe, and Everything…is… “42”. I checked it thoroughly. It would have been simpler, of course, to have known what the actual question was.

— Douglas Adams, Hitchhiker’s Guide to the Galaxy

As investors, our process is usually to start from the answer and work our way back to the question. Unfortunately, the answers we are provided are usually pre-baked products, vehicle types or persistent industry conventions, which means that the answers we get when we actually focus on the questions that matter may be counterintuitive and jarring. The entire point of developing a personal code for investing is knowing which questions matter and ought to be asked first, before a single product, vehicle or style box gets thrown into the mix.

The purpose you undertake is dangerous.’ Why, that’s certain. ‘Tis dangerous to take a cold, to sleep, to drink; but I tell you, my lord fool, out of this nettle, danger, we pluck this flower, safety.

William Shakespeare, Henry IV, Part 1, Act 2, Scene 3, Hotspur

Thomasina: When you stir your rice pudding, Septimus, the spoonful of jam spreads itself round making red trails like the picture of a meteor in my astronomical atlas. But if you stir backwards, the jam will not come together again. Indeed, the pudding does not notice and continues to turn pink just as before. Do you think this is odd?

Septimus: No.

Thomasina: Well, I do. You cannot stir things apart.

Septimus: No more you can, time must needs run backward, and since it will not, we must stir our way onward mixing as we go, disorder out of disorder into disorder until pink is complete, unchanging and unchangeable, and we are done with it forever. This is known as free will or self-determination.

Thomasina: Septimus, do you think God is a Newtonian?

Septimus: An Etonian? Almost certainly, I’m afraid. We must ask your brother to make it his first enquiry.

Thomasina: No, Septimus, a Newtonian. Septimus! Am I the first person to have thought of this?

Septimus: No.

Thomasina: I have not said yet.

Septimus: “If everything from the furthest planet to the smallest atom of our brain acts according to Newton’s law of motion, what becomes of free will?”

Thomasina: No.

Septimus: God’s will.

Thomasina: No

Septimus: Sin.

Thomasina (derisively): No!

Septimus: Very well.

Thomasina: If you could stop every atom in its position and direction, and if your mind could comprehend all the actions thus suspended, then if you were really, really good at algebra you could write the formula for all the future; and although nobody can be so clever as to do it, the formula must exist just as if one could.

Septimus (after a pause): Yes. Yes, as far as I know, you are the first person to have thought of this.

— Tom Stoppard, Arcadia, (1993)

On this most important question of risk, we and our advisors often default to approaches which rely on the expectation that the past and present give us profound and utterly reliable insights into what we ought to expect going forward. As a result, we end up with portfolios and, more importantly, portfolio construction frameworks which don’t respect the way in which capital actually grows over time and can’t adapt to changing environments. That’s not good enough.

Most of these notes tend to stand on their own, but this one (being a Part 2) borrows a lot from the thinking in Part 1. If you’re going to get the most out of this note, I recommend you start there. But if you’re pressed for time or just lazy, I wanted you to take away two basic ideas:

  • That the risk decision dominates all other decisions you make.
  • That the risk decision is not exactly the same as the asset class decision.

Children of a Lazier God

Before I dive into the weeds on those ideas, however, I want to tell you about a dream I have. It’s a recurring dream. In this dream, I have discovered the secret to making the most possible money with the least possible effort.

Hey, I never said it was a unique dream.

It is, however, a unique investing case. Imagine for a moment that we had perfect omniscience into returns, but also that we were profoundly lazy – a sort of Jeffersonian version of God. We live in a world of stocks, bonds and commodities, and we want to set a fixed proportion of our wealth to invest in each of those assets. We want to hold that portfolio for 50+ years, sit on a beach watching dolphins or whatever it is people do on beach vacations, and maximize our returns. What do we hold? The portfolio only needs to satisfy one explicit and one implicit objective. The explicit objective is to maximize how much money we have at the end of the period. The implicit objective is the small matter of not going bankrupt in the process.

This rather curious portfolio is noteworthy for another reason, too: it is a static and rather cheeky case of an optimal portfolio under the Kelly Criterion. Named after John Kelly, Jr., a Bell Labs researcher in the 1950s, the eponymous criterion was formally proposed in 1956 before being expanded and given its name by Edward O. Thorp in the 1960s. As applied by Thorp and many others, the Kelly Criterion is a mechanism for translating assessments about risk and edge into both trading and betting decisions.

Thorp himself has written several must-reads for any investor. Beat the Dealer, Beat the Market and A Man for All Markets are all on my team’s mandatory reading list. His story and that of the Kelly Criterion were updated and expanded in William Poundstone’s similarly excellent 2005 book, Fortune’s Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street.  The criterion itself has long been part of the parlance of the professional and would-be professional gambler, and has also been the subject of various finance papers for the better part of 60 years. For the less prone to the twin vices of gambling and authoring finance papers, Kelly translates those assessments about risk and edge into position sizes. In other words, it’s a guide to sizing bets. The objective is to maximize the geometric growth rate of your bankroll — or the expected value of your final bankroll — but with zero probability of going broke along the way. It is popular because it is simple and because, when applied to games with known payoffs, it works.

When we moonlight as non-deities and seek to determine how much we ought to bet/invest, Kelly requires knowing only three facts: the size of your bankroll, your odds of winning and the payout of a winning and losing bet. For the simplest kind of friendly bet, where a wager of $1 wins $1, the calculation is simple: Kelly says that you should bet the difference between your odds of winning and your odds of losing. If you have a 55-to-45 edge against your friend, you should bet 10% of your bankroll. Your expected compounded return of doing so is provably optimal once you have bet against him enough to prove out the stated edge — although should you manage to reach this point, you are a provably suboptimal friend.

Most of the finance papers that apply this thinking to markets have focused on individual trades that look more or less like bets we’d make at a casino. These are usually things with at least a kinda-sorta knowable payoff and a discrete event where that payoff is determined: a single hand of blackjack, an exercise of an option, or a predicted corporate action taking place (or not taking place). It’s a lot harder to get your head around what “bet” we’re making and what “edge” we have when we, say, buy an S&P 500 ETF instead of holding cash. Unless you really are omniscient or carry around a copy of Grays Sports Almanac, you’re going to find estimating the range of potential outcomes for an investment or portfolio of investments pretty tricky. Not that it stops anyone from trying.

Since I don’t want to assume that any of us is quite so good at algebra as to write the formula for all the future, at a minimum what I’m trying to do is get us to think about risk unanchored to the arbitrarily determined characteristics and traits of asset classes. In other words, I want to establish an outside bound on the amount of risk a person could theoretically take in a portfolio if his only goal was maximizing return. Doing that requires us to think in geometric space, which is just a fancy way of saying that we want to know how the realization of returns over time ends up differing from a more abstract return assumption. It’s easy enough to get a feel for this yourself by opening Excel and calculating what the return would be if your portfolio went up 5% in one year and down 5% in the next (works for any such pair of numbers). Your simple average will always be zero, but your geometric mean will always be less than zero, by an increasing amount as the volatility increases.

So, if we knew exactly what stocks, bond and commodities would do between 1961 and 2016, what portfolio would we have bought? The blend of assets if we went Full Kelly would have looked like this:

Source: Salient 2017. For illustrative purposes only.

Only there’s a catch. Yes, we would have bought this portfolio, but we would have bought it more than six times. With perfect information about odds and payoffs, the optimal bet would have been to buy a portfolio with 634% (!) exposure, consisting of $2.00 in stocks, $3.21 in bonds and $1.13 in commodities for every dollar in capital we had. After all was said and done, if we looked back on the annualized volatility of this portfolio over those 50 years, what would we have found? What was the answer to life, the universe and everything?

44. Sorry, Deep Thought, you were off by two.

Perhaps the only characteristic of this portfolio more prominent than its rather remarkable level of exposure and leverage, is its hale and hearty annualized volatility of 44.1%. This result means if all you cared about was having the most money over a 50+ year period that ended last year, you would have bought a portfolio of stocks, bonds and commodities that had annualized volatility of 44.1%, roughly three times the long-term average for most equity markets[1], and probably five times that of the typical HNW investor’s portfolio.

And before you go running off to tell my lovely, charming, well-dressed and distressingly unsusceptible-to-flattery compliance officer that I told you to buy a 44% volatility super-portfolio, allow me to acknowledge that this requires some… uh… qualification. Most of these qualifications are pretty self-explanatory, since the whole exercise isn’t intended to tell you what you should buy going forward, or even the right amount of risk for you. This portfolio, this leverage and that level of risk worked over the last 50 years. Would they be optimal over the next 50?

Of course not. In real life, we’re not omniscient. Whereas a skilled card counter can estimate his mathematical edge fairly readily, it’s a lot harder for those of us in markets who are deciding what our asset allocation ought to look like. Largely for this reason, even Thorp himself advised betting “half-Kelly” or less, whether at the blackjack table or in the market. When asked why, Thorp told Jack Schwager in Hedge Fund Market Wizards, “We are not able to calculate exact probabilities… there are things that are going on that are not part of one’s knowledge at the time that affect the probabilities. So you need to scale back to a certain extent.”

Said another way, going Full Kelly on a presumption of precise certainty about outcomes in markets is a surefire way to over-bet, potentially leading to a complete loss of capital. Now, scaling back is easy if we are starting from an explicit calculation of our edge as in a game of blackjack. It’s not as easy to think about scaling down to, say, a Half Kelly portfolio. There is, however, another fascinating (but intuitive) feature of the Kelly Optimal Portfolio that allows us to scale back this portfolio in a way that may be more familiar: the Kelly Optimal Portfolio can be generalized as the highest return case of a set of portfolios generating geometric returns that are most efficient relative to the risk they take[2].

This may sound familiar. In a way, it’s very much like a presentation of Markowitz’s efficient frontier. Markowitz plots the portfolios that generate the most return for a given unit of risk, but his is a single-period calculation. It isn’t a geometric approach like Kelly, but rather reflects a return expectation that doesn’t incorporate how volatility and non-linearities impact the path and the resulting compound return. There have been a variety of academic pieces over the years covering the application of geometric returns to this framework, but most have focused on either identifying a single optimal geometric portfolio or on utility. Bernstein and Wilkinson went a bit further, developing a geometric efficient frontier.

All of these analyses are instructive and useful to the investor who wants to take path into account, but because the efficient frontier is heavily constrained by the assumed constraint on leverage, it’s not as useful for us. What we want is to take the most efficient portfolio in geometric terms, and take up or down the risk of that portfolio to reflect our tolerance for capital loss. In other words, we want a geometric capital market line. The intuitive outcome of doing this is that we can plot the highest point on this line as the Full Kelly portfolio. The second, and perhaps more satisfying outcome, is that we can retrospectively identify that scaling back from Full Kelly just looks like delevering on this geometric capital market line.

The below figure plots each of these items, including a Half Kelly portfolio that defines ruin as any scenario in the path in which losses exceed 50%, rather than full bankruptcy. The Half Kelly portfolio delivers the highest total return over this period without ever experiencing a drawdown of 50%.

Source: Salient, as of December 31, 2016. For illustrative purposes only.

When we de-lever from the Full Kelly to Half Kelly portfolio, we drop from a terrifying 44% annualized volatility number (which experiences an 80% drawdown at one point) to 18.5%, closer to but still materially higher in risk than most aggressive portfolios available from financial advisors or institutional investors.

This can be thought of in drawdown space as well for investors or advisors who have difficulty thinking in more arcane volatility terms. The below exhibit maps annualized volatility to maximum loss of capital over the analysis period. As mentioned, the 50% maximum drawdown portfolio historically looks like about 18.5% in volatility units.

Source: Salient, as of December 31, 2016. For illustrative purposes only.

For many investors, their true risk tolerance and investment horizon makes this whole discussion irrelevant. Traditional methods of thinking about risk and return are probably serving more conservative investors quite well. And there are some realities that anyone thinking about taking more risk needs to come to terms with, a lot of which I’m going to talk about in a moment — there’s a reason we wanted to talk about this in geometric terms, and it’s all about risk. But for those with a 30, 40 or 50-year horizon, for the permanent institutions with limited cash flow needs, it’s reasonable to ask the question: is the amount of risk in the S&P 500 Index or in a blend of that with the Bloomberg Barclays Aggregate Bond Index the right amount of risk to take? Or can we be taking more? Should we be taking more?

Did you think that was rhetorical? Nope.

Many investors can – and if they are acting as fiduciaries probably ought to — take more risk.

If every hedge fund manager jumped off a bridge…

This may not be a message you hear every day, but I’m not telling you anything novel. Don’t just listen to what your advisors, fund managers and institutional peers are telling you. They’re as motivated and influenced by career risk concerns as the rest of us. Instead, look at what they’re doing.

The next time you have a conversation with a sophisticated money manager you work with, ask them where they typically put their money. Yes, many of them will invest alongside you because that is right and appropriate (and also expected of them). But many more, when they are being honest, will tell you that they have a personal account or an internal-only strategy operated for staff, that operates at a significantly higher level of risk than almost anything they offer to clients. Vehicles with 20%, 25% or even 30% volatility are not uncommon. Yes, some of this is hubris, but some of it is also the realization on the part of professional investors that maximizing portfolio returns — if that is indeed your objective — can only be done if we strip back the conventions that tell us that the natural amount of risk in an unlevered investment in broad asset classes is always the right amount of risk.

Same thing with the widely admired investors, entrepreneurs and business operators. The individual stocks that represent their wealth are risky in a way that dwarfs most of what we would be willing to tolerate in individual portfolios. We explain it away with the notion that they are very skilled, or that they have control over the outcomes of the company — which may be true in doses — but in reality, they are typically equally subject to many of the uncontrollable whims that drive broader macroeconomic and financial market outcomes.

Then observe your institutional peers who are increasing their allocations to private equity and private real estate. They’re not just increasing because hedge funds have had lower absolute returns in a strong equity environment, although that is one very stupid reason why this is happening. It’s also happening because institutions are increasingly aware that they have limited alternatives to meet their target returns. While few will admit it explicitly, they use private equity because it’s the easiest way to lever their portfolios in a way that won’t look like leverage. In a true sense of uncertainty or portfolio level risk, when the risk of private portfolios is appropriately accounted for, I believe many pools of institutional capital are taking risk well beyond that of traditional equity benchmarks.

Many of the investors we all respect the most are already taking more risk than they let on, but explain it away because it’s not considered “right thinking.”

To Whom Much is Given

When we make the decision to take more risk, however, our tools and frameworks for managing uncertainty must occupy more of the stage. This isn’t only about our inability to build accurate forecasts, or even our inability to build mostly accurate stochastic frameworks based on return and volatility, like the Monte Carlo simulations many of us build for clients to simulate their growth in wealth over time. It’s also because the kinds of portfolios that a Full Kelly framework will lead you to are usually pretty risky. Their risk constraint is avoiding complete bankruptcy, and that’s not a very high bar. The things we have to do to capture such a high level of risk and return also usually disproportionately increase our exposure to big, unpredictable events. If you increase the risk of a portfolio by 20%, most of the ways you would do so will increase the exposure to these kinds of events by a lot more than 20%.

Taken together, all these things create that famous gap between our realized experience and what we expected going in. This is a because most financial and economic models assume that the world is ergodic. And it ain’t. I know that’s a ten-dollar word, but it’s important. My favorite explanation of ergodicity comes from Nassim Nicholas Taleb, who claims to have stolen it from mathematician Yakov Sinai, who in turns claims to have stolen it from Israel Gelfand:

Suppose you want to buy a pair of shoes and you live in a house that has a shoe store. There are two different strategies: one is that you go to the store in your house every day to check out the shoes and eventually you find the best pair; another is to take your car and to spend a whole day searching for footwear all over town to find a place where they have the best shoes and you buy them immediately. The system is ergodic if the result of these two strategies is the same.

There are infinite examples of investors making this mistake. My mind wanders to the fund manager who offers up the fashionable but not-very-practical “permanent loss of capital” definition of risk, a stupid definition that is the last refuge of the fund manager with lousy long-term performance. “Sure, it’s down 65%, but that’s a non-permanent impairment!” Invariably, the PM will grumble and call this a 7-standard deviation event because he assumed a world of ergodicity. Because of the impact of a loss like this on the path of our wealth, we’ll now have to vastly exceed the average expected return we put in our scenario models in Excel just to break even on it.


“It’s not a permanent impairment of capital!”

It matters what path our portfolios follow through time. It matters that our big gains and losses may come all at once. It matters to how we should bet and it matters to how we invest. You cannot stir things apart!

So if you’ve decided to take risk as an investor, how we do avoid this pitfall? Consider again the case of the entrepreneur.

The entrepreneur’s portfolio is concentrated, which means that much of his risk has not been diversified away. A lot of that is going to be reflected in the risk and return measures we would use if we were to plot him on the efficient frontier. That doesn’t necessarily mean his risk of ruin will appear high, and his analysis might, in fact, inform the entrepreneur that he ought to borrow and hold this business as his sole investment. He’s done the work, performed business plan SWOT analyses, competitor analyses, etc., and concluded that he has a pretty good grasp of what his range of outcomes and risks look like.

In an ergodic world, this makes us feel all warm and fuzzy, and we give ourselves due diligence gold stars for asking all the right questions. In a non-ergodic world, the guy dies using his own product. A competitor comes out of nowhere with a product that immediately invalidates his business model. A bigger player in a related industry decides they want to dominate his industry, too. And these are just your usual tail events, not even caused the complexity of a system we can’t understand but by sheer happenstance. For the entrepreneur, all sorts of non-tail events over time may materially and permanently change any probabilistic assessment going forward. How do we address this?

The first line of defense as we take more risk must be diversification. After all, there is a reason why the Kelly Portfolios distribute the risk fairly evenly across the constituent asset classes.[3]

Even that isn’t enough. Consider also the case of the leveraged investor in multiple investments with some measure of diversification, for example a risk parity investor, Berkshire Hathaway[4], or the guy who went Full Kelly per our earlier example, but without the whole perfect information thing. This investor has taken the opposite approach, which is to diversify heavily across different asset classes and/or company investments. His return expectation is driven not so much by his ability to create an outcome but by the exploitation of diversification. As he increases his leverage, his sensitivity to the correctness of his point-in-time probabilistic estimates of risk, return and correlations between his holdings will increase as well. In an ergodic world, this is fine and dandy. In a non-ergodic world, while he has largely mitigated the risk of idiosyncratic tails, he is relying on relationships which are based on a complex system and human behaviors that can change rapidly.

Thus, the second line of defense as we take more risk must be adaptive investing. Sometimes the only answer to a complex system is not to play the game, or at least to play less of it. Frameworks which adapt to changing relationships between markets and changing levels of risk are critical. But even they can only do so much.

Liquidity, leverage and concentration limits are your rearguard. These three things are also the only three ways you’ll be able to take more risk than asset classes give you. They are also the three horsemen of the apocalypse. They must be monitored and tightly managed if you want to have an investment program that takes more risk.

It’s not my intent to end on a fearful note, because that isn’t the point at all. More than asset class selection, more than diversification, more than fees, more than any source of alpha you believe in, nothing will matter to your portfolio and the returns it generates more than risk. And the more you take, the more it must occupy your attention. That doesn’t mean that we as investors ought to cower in fear.

On the contrary, my friends, fortune favors the bold.


[1] Back in 1989, Grauer and Hakansson undertook a somewhat similar analysis on a finite, pre-determined set of weightings among different assets with directionally similar results. Over most windows the optimal backward-looking levered portfolio tends to come out with a mid-30s level of annualized volatility.

[2] For this and the other exhibits and simulations presented here, I’m very grateful to my brilliant colleague and our head of quantitative strategies at Salient, Dr. Roberto Croce.

[3] And that reason isn’t just “we’re at the end of a 30-year bond rally,” if you’re thinking about being that guy.

[4] One suspects Mr. Buffett would be less than thrilled by the company we’re assigning him, but to misquote Milton Friedman, we are all levered derivatives users now.


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