Do Professional Investors Understand Fees?

Fees Are In Focus


Giant fund manager/brokerages like Vanguard and Fidelity have made fees front and center. Like Walmart, if you are the lowest cost provider and wield blue whale scale, you are going to compete on price. Competition has spurred a race to the bottom on fees. With many investment choices commoditized, the focus on fees has served customers well. 

If I wanted to nit-pick, I might say investors don’t fully account for more opaque fees when choosing funds. These can swamp the management fees. Turnover, slippage costs, borrowing costs and abysmal sweep account rates all have significant impacts on net performance. These hidden costs are not easily reduced to a number that can be compared to a management fee. Hint: it’s a good place to search for how managers are able to drive fees to zero. But that’s a digression. I’m not especially interested in retail. Their financial advisors are doing a good job using steak and wine to box out the fund managers. There’s only so much fee to go around.


Allocators have a more difficult job. They devote teams to parsing alternative investments. A sea of private investments and complex hedge fund strategies. Within that context the allocators must construct portfolios that trade-off between tolerable risks and the probability of meeting their mandates. 

The allocators rummage through a diverse mix of strategies each with their own mandates. Growth, wealth preservation, defensive, hedged alpha. A fund can be thought of as a payoff profile with an associated risk profile. A thoughtful allocator is crafting a portfolio like a builder. They want to know how the pieces interlock so the final product is useful and can withstand the eventual earthquake. 

A builder cannot think of materials without considering cost. Wood might make for a better floor than vinyl but at what price would you accept the inferior material? When builders estimate their costs they must consider not only the materials, but transportation costs and how the cost of labor may vary with the time required to install the material. 

So let’s go back to the allocators. If the menu they were choosing from wasn’t complicated enough, they must also evaluate the costs. This is a daunting topic. They face all the opaque costs the retail investors face. But since they are often investing in niche or custom strategies that are not necessarily under a public spotlight they have additional concerns. A basic due diligence process would review:

  • Which costs are allocated to the GPs vs the LPs
  • Liquidity schedules
  • Fund bylaws
  • Specific clauses like “most-favored-nations”
  • Netting risks1

Unlike their retail counterparts, the professional investor’s day job is devoted to more than just investments but terms. Like our builder, this cannot be done faithfully without understanding the costs. Mutual funds sport fixed fees but complex investments often have incentive fees (a fee that is charged as a percentage of performance, sometimes with a hurdle) making them harder to evaluate. Regretfully, I suspect a meaningful segment of pros do not have a strong grasp on how fees affect their investments. 

Understanding Fees

While it is challenging to price many of the features embedded in funds’ offering documents, there is little excuse for not understanding fees whether they are fixed or performance-based.  After all, if you are an investor this is one of the most basic levers that affect your net performance and does not rely on having skills. It’s a classic high impact, easy to achieve objective. It’s the best box in that prioritization matrix that floats around consulting circles. 

Let’s take a quick test. 

You have a choice to invest in 2 funds that have identical strategies.

They have the same Sharpe ratio of .5

There are 2 differences between the funds. The fee structure and volatility.

  Fund A Fund B
Expected Return 5% 15%
Annual Volatility 10% 30%
Annual Fee 1% 2%

Let’s assume the excess volatility is simply a result of leverage and that the leverage is free.

Which fund do you choose?

Normalizing Fees By Volatility

The correct way to think about this is to adjust the fee for volatility.

  • Fund A’s fee is 10% of its volatility (1% / 10%).
  • Fund B’s fee is 6.7% of it volatility (2% / 30%)

If you doubt that Fund B is cheaper from this reasoning you could simply sell Fund A and buy 1/3 as much of Fund B.

Let’s use real numbers. Suppose to had a $300,000 investment in Fund A. You would be paying 1% or $3,000 in fees. 

Instead, invest $100,000 in fund B. Your expected annual return and volatility would remain the same, but you would only pay 2% of $100k in fees or $2,000. Same risk/reward for 2/3rd the price. Compound that.

I am not alone in this observation. From his book Leveraged Returns, Rob Carver echoes that a fund’s fees can only be discussed in context with its volatility:

I calculate all costs in risk-adjusted terms: as an annual proportion of target risk. For target risk of 15%/year and costs of 1.5%/year, your risk-adjusted costs are 1.5%/15% = 0.10. “This is how much of your gross Sharpe ratio will get eaten up by costs.


A Clue That Some Allocators Get This Wrong

Allocators will often target lower vol products for the same fee when a higher vol fund would do. To be fee-efficient they should prefer that managers ran their strategies at a prudent maximum volatility. Optimally some point before they were overlevered or introduced possible path problems. There are many funds and CTAs that would just as easily target higher volatility for the same fee. Investors would be better off for 2 reasons:

  • Allocators could reduce their allocations

As we saw in the Fund B example, it is more fee-efficient for vol targeting to be done at the allocation level not the fund level.

  • Limit cash drag.

They would stop paying excess fees for a fund that had been forced to maintain large cash reserves since it was targeting a sub-optimal volatility. Why would an allocator be ok with paying fees for funds that are holding excessive t-bills?

If you are not convinced that investors’ preference for lower vol versions of strategies demonstrates a lack of fee numeracy then check out this podcast with allocator Chris Schindler.  As an investor at the highly sophisticated Ontario Teachers Pension he witnessed firsthand the folly of his contemporaries’ thinking around fees. While mingling at conferences he would hear other investors bragging that they never pay fees above a certain threshold.

As we saw from our example, these brags are self-skewers, revealing how poorly these managers understood the relationships between fees and volatility. Not surprisingly, these very same managers would be invested in bond funds and paying optically low nominal fees. Sadly, once normalized for volatility, these fees proved to be punitively high. 

This brings us to our next section. How would you like to pay for low volatility or defensive investments?

Tests to Compare Fixed Fee Funds with Incentive Fee Funds

A Low Volatility Example

Let’s choose between 2 identical funds which only vary by the fee structure.

Both funds expect to return 5% and have a 5% volatility. Yes, a Sharpe ratio of 1.

  • Fund A charges a fixed .75%
  • Fund B charges 10% of performance from when you invest. Fund B has a high watermark that crystallizes 2 annually.

Which fund do you choose?

A Large Cap Equity Example

This time let’s choose between funds that have SPX-like features

Both funds expect to return 7% and have a 16% volatility.

  • Fund A again charges a fixed .75%
  • Fund B again charges 10% of performance from when you invest. Fund B has a high watermark that crystallizes < annually.

Which fund do you choose?

Studying The Impact Of Fee Structure

I wrote simulations to study the impact of fees on the test examples.

The universal setup:

  • Each fund holds the exact same reference portfolio
  • 10 years simulation using monthly returns
  • Random monthly returns drawn from normal distribution 
  • 1000 trials
  • Fixed Fee Fund charges .75% per year deducted quarterly
  • Incentive Fee Fund charges 10% of profits crystallized annually

Case 1: Low-volatility 

Simulation parameters:

  • Monthly mean return of .42% (5% annual)
  • Monthly standard deviation of 1.44% (5% annually)3

This chart plots the outperformance of the fixed fee return vs incentive fee return fund annually vs the return of the portfolio which they both own. The relative performance of the 2 funds is due to fees alone. 


  • It takes a return of about 7% or higher for the fixed fee fund to outperform.
  • This makes sense. A 75 bp fee is difficult to overcome for a 5% vol asset.
  • If the asset returns 5% the performance fee would only be 50bps and we can see how the difference in fees approximates the underperformance of the fixed fee fund for 5% level of returns.

Case 2: Large Cap Equity Example

The universal setup remains the same. 

We modify the simulation parameters:

  • Monthly mean return of .58% (7% annual)
  • Monthly standard deviation of 4.62% (16% annually)


  • Most of the time the fixed fee fund outperforms. So long as the return is north of about 4% this is true.
  • The most the fixed fee fund can underperform is by the amount of the fixed fee. Consider the case in which both portfolios lose value every year. The incentive fee fund will never charge a fee, while you will get hit by the 75bps charge in the fixed fee fund. You can see these cases in the negative points on the left of the chart where the portfolio realizes an annual CAGR of -5%.
  • Conversely, the incentive fee can be very expensive since it captures a percentage of the upside. In cases where the underlying portfolio enjoys +20% CAGRs, the simple fixed fee fund is outperforming by about 150 bps per year. 

Bonus Case: The High Volatility Fund

Finally I will show the output for a low Sharpe, high volatility fund.

The universal setup remains the same. 

We modify the simulation parameters:

  • Monthly mean return of .42% (5% annual)
  • Monthly standard deviation of 10.10% (35% annually)


  • This case demonstrates how complicated the interactions of fees and volatility are. The fixed fee fund will massively outperform by even as much as 200bps per year when the portfolio compounds at 20% annually.
  • The fixed fee fund even outperforms at low to mid single-digit returns albeit modestly. 
  • The high volatility nature of the strategy means lots of negative simulations, thanks to geometric compounding (for further explanation I discuss it here). When a fund performs poorly you pay less incentive fees so it’s not surprising that in many of these case the fixed fee fund underperforms by nearly the entire amount of the management fee. 


Fixed Fees

  • Best when the volatility of the strategy is high and the returns are strong (again you are warned: most high volatility strategies don’t have strong returns because of geometric compounding).
  • The most a fixed fee investor can underperform an incentive fee investor is by the amount of the fixed fee.

Incentive Fees

  • Best when the strategy is low volatility or returns are negative. Or the asset is defensive in nature. For hedges or insurance like funds, you may prefer to pay a performance fee to minimize bleed.
  • The amount an incentive fee investor can underperform is technically unbounded since it’s a straight percent of profits.


  • Fee structures must be considered relative to the volatility and goals of the strategy. There are no absolutes. 
  • By dividing fixed fees by the fund’s volatility you can normalize and therefore compare fund fees on an apples-to-apples basis. Even seemingly low fixed fees can be very expensive when charged on low volatility funds. 
  • Incentive fees look like long options to the manager (which implies the investor is short this option). The investor has unbounded potential to underperform a fixed fee solution and can only outperform by the amount of the fixed fee (the left hand side of those charts). To further study the embedded optionality of incentive fees see Citigroup’s presentation.
  • Incentive fees are meant to align investors and management. Who can argue with “eat what you kill”? But they can also create bad incentives. If trapped below the high watermark, the manager has nothing to lose and may swing for the fences irresponsibly. In addition, a staff working at a fund that is underwater might be dusting off their resumes instead of focusing on getting back on track knowing that they need to work through uncompensated p/l before they see another bonus. 
  • Fixed fees can encourage management to diversify or hold more cash to lower the fund volatility. These maneuvers can be combined with heavy marketing in a strategy more colloquially known as “asset-gathering”.


Fees need to be considered in light of the strategy. This requires being thoughtful to understand the levers. Unless you are comparing 2 SP500 index funds, it’s rarely as simple as comparing the headline fees. If we all agree that fees are not only critical components of long-term performance, while being one of the few things an allocator can control, then misunderstanding them is just negligent. A one size fee doesn’t fit all  alternative investments so a one size rule for judging fees cannot also make sense. Compared to the difficulty of sourcing investments and crafting portfolios getting smart about fees is low-hanging fruit. 

“Avoid Boring People”…Ok, sure thing.

“Avoid boring people”

This warning is the title of scientist James D. Watson’s memoir. (That’s Watson of Watson & Crick. DNA. Double helix. Adenine-Thymine base pairs. Bueller?)

Don’t worry about high school bio. Focus on the title. First, note the double meaning. It’s an admonition about who to avoid as well as a prescription for yourself. Two pieces of advice in one punchy aphorism. What a value. I can see how people might use the advice as a compass to find their orientation when they feel life has spun them around a few times.

You know me too well by now to know I can’t leave a perfectly fine adage with its ankles unbitten. Let’s deconstruct.

What does it even mean to be boring?

Feels a bit rubbery when you try to pin it down. Like that time the Supreme Court tried to define porn by appealing to common sense — “I know it when I see it”.

Maybe it’s easier to answer “what’s not boring?”

  • Ordinary people with extraordinary life paths. Extreme example: Louie Zamperini from Unbroken whose life was Forrest Gump-esque.
  • Very rich or famous people. They have access to exclusive life experiences. And their day-to-day life is hard to recognize to us normies. Chrissy Teigen did a candid AMA last December on Twitter if you want proof. (Link)
  • People with dangerous jobs. CIA agents. Navy Seals. Lion tamers. Orca trainers. If you hang out with these people, you won’t even want to talk about yourself. You will naturally ask questions and listen.
  • Excellence. This one might be personal, but I happen to find anyone who is elite at anything to be interesting. Maybe it’s a byproduct of the “unusual life path” definition. I’m strangely fascinated by the idea of spending 10,000 hours of shuffling cards to develop sleight of hand.

Even after listing what’s not boring, it’s hard to shake the sense that I’m just projecting my own preferences rather than honing in on some universal acceptance of “interesting”.

Why being less boring might not make you happier?

Tyler Cowen brought my attention to a 4-line short story:

We know only four boring people. The rest of our friends we find very interesting. However, most of the friends we find interesting find us boring: the most interesting find us the most boring. The few who are somewhere in the middle, with whom there is reciprocal interest, we distrust: at any moment, we feel, they may become too interesting for us, or we too interesting for them.  — excerpted from Lydia Davis’ Samuel Johnson is Indignant (Link)

Couple thoughts:

1) Within the dynamics of a relationship the spread of boringness between 2 individuals moderates trust. If your friend became famous and dropped into a different life petri dish of experience is it reasonable to expect your relationship to change? Probably.

2) Maybe boringness, like status, is zero-sum. If nobody is boring the word’s meaning calibrates to once again set a fixed proportion of people of and below the bar. If everyone shoots like Steph Curry the 3-point line will just get pushed back. Mental model fanboys and fangirls will recall the “red queen effect”.

“Avoid boring people” as a North Star is easily contradicted

1) Boringness is relative.

The pretty, rich brat teen Astrid in the Politician runs away from home because she’s bored of being part of the Santa Barbara super-affluent crowd. (I’m sure there are better examples but I couldn’t resist a dog whistle for fans of the Netflix series). She wants to be broke and feel risk, believing it will recharge her deadened senses.

Speculating aloud, if “boring” is indeed relative then our novelty-seeking instinct may be a vestige that outlived its usefulness. Boring may be nothing more than a perspective.

2) Dutiful grandmas

This is how I describe my mother and mother-in-law. I am thinking most people would look at these women’s lives and think “boring”. But they are tremendous inspirations because of how committed they are to their duties. I don’t need to expand on this. We all know these people. Unheralded heroes in our life stories. Nobody writes a Wikipedia page for them. They got a dance when their sons married for all they endured. Big whoop.

“Avoid boring people” feels like elitist advice when you put it under a microscope.

It’s just hustleporn 

Venkat Rao, as he does, stirred the pot with a Twitter thread. (Link)

Here’s how I understand it. Self-described “lifelong learners” and the growth mindsetters are just following a religion they think will make them rich. These people are actually boring and if they ever fell into FU money, personal growth would cease being an altar to worship at. Being one of these self-described lifelong learners I had to take the bait and read the thread. Ouch.

So then I started wondering: is the “avoid boring people” crowd the same as the “lifelong learner” crowd? Is everyone in this Venn diagram just trying to get to any combo that includes “rich”? If you ever figure it out let me know.


If you have the time to actually be bored, sure shake it up. Get a pet. Do a local Skillshare or class. Find a rabbit hole. Get arrested. But organizing your life so as to not be boring is hardly a proven recipe for anything valuable. It might work for you. But remember “Avoid Boring People” is just a book title. Lots of them can work for you, I’m not convinced this one is special.

Nobody Is Bigger Than The Market

2 brief things this week.

1)  Remember nobody is bigger than the market.

He pulled this from Michael Batnick’s post about how when you were born dominates your investment performance. (Link)

Be humble about what is actually in your control and structure your life as best you can within that understanding. This thing we call the market is a tyrant. Nobody can live outside what it allows for. This is true of all markets. Consider most businesses. A dumb realtor in a bull market vs a smart realtor in the recession. The market is the biggest factor. This gives credence to the strategy of trying to put yourself in front of the wave of a growing industry. I can see you smart people stewarding “run-off” industries solemnly nodding.

How about the labor market? This is bonus season on Wall Street so lots of little violins playing for people who felt they got screwed. But screw you once, shame on them. Screw you twice? Shame on you. If you don’t leave a crappy employer then you have validated your place in the market. Management is not bigger than the market. If they really screwed you, you can appeal to the labor market. Your boss only gets one swing at your pinata. If you let them take more then perhaps you’re at the only party in town.

2) Morgan Housel’s latest about the psychology of the country leading up to the Great Depression is instructive. (Link)

A timeless takeaway:

The problem when studying historical events is that you know how the story ends, and it’s impossible to un-remember what you know today when thinking about the past. It’s hard to imagine alternative paths of history when the actual path is already known. So things always look more inevitable than they were.

The post is worth a full read. Learn how the collective mindset of the country changed in the decade after WWI and the Spanish Flu. American despair gave way to prosperity before lapsing into the worst economic disaster in US history. He relies on newspaper clippings to provide the real-time perspective countering our hindsight view.

Here’s the nuggets that seem to have burrowed into my long-term memory since reading it:

  • The stock market fell 89% from the peak and took 25 years and another world war before it got back to it’s 1929 level. Birth rates fell 17%. Sobering. Especially when you consider what the path of something like that looks like. If you had bought stocks down 80% from the highs, by the time they got to the lows you were down 50% on your money.
  • The 1920s seemed to be the cradle of how American’s think of prosperity even today. I take it for granted when apparently it has roots that are less than 100 years old.

Investing lessons written in blood shouldn’t fade. This post will stay with me.

A Simple Rule For Giving

I just wanted to share a personal simplifying rule for life: when a friend is raising money for charity always give. Whether it’s a cause, a GoFund Me, or the school fundraiser. Why? If you are like many other people with good intentions but little free time, you aren’t hanging out in the comments section of effective altruism blogs splitting hairs about which charities are maximizing return per dollar. Some people may even use this as an excuse. “I haven’t done the homework, so I’m not ready to give”. Guess what aspiring philanthropist? You’re not going to do the homework.

Instead, trust that your friends have done the work since they are putting the effort to raise money. By offering you a chance to give, they are actually doing you a favor. You get to act instead of just meaning to do good. Even if you accidentally give to, over the course of your life you will have maximized area under the curve.

I didn’t grow up with money. Not poor but no real savings. The way the remaining middle class lives today. My mother then and now always gives. Small amounts, what she feels comfortable with. But it’s a thing that just sort of stuck with me in watching her. I’m not a political or activist type person. Those people tend to be more wired to give and ask for money from what I’ve seen firsthand. But I think a willingness to give comes from observing others. So if it’s something you value and you have kids, show them that you do this. It probably matters.

A practical note: if you deduct your giving from your taxes, it’s easy to create a label in Gmail for every tax year. When you get the receipt emailed to you just file it under the year’s label. Your receipts will be in one place when you sit down to prepare your taxes.

Notes From Invest Like the Best: Brian Christian


About Brian: Author covering humans’ relationship with technology and AI

Q: What advice would you give to people, building careers. We’re in a political cycle now where things like basic income are being discussed. In your view, what are the most defensible areas of human activity, whether that’s some sort of creativity or asking great questions coming up with the objective functions that you then feed the machines? What would you recommend people focus on as they think about either early or late in their career, adding value?

A: There are sort of two ways that I can approach this question. My second book is called the Algorithms to Live By and it looks at things like career decisions from an explicitly algorithmic perspective.

1) Explore/Exploit Trade-off


There’s this paradigm, called the “explore/exploit” trade-off, which is: How much of your energy do you spend gathering information vs how much do you spend committing based on the information? There’s a number of decisions that we face throughout life, that take the form of a tension or a balance between trying new things and committing to the things that seem to be the best. Where to go out to eat, go to our favorite restaurant and we try a new restaurant. Reach out to a new acquaintance we’d like to get to know better or spend time with our close family or best friend. The same thing is true in investing, the same thing is true in managing your time and your career.

Generalizing the Problem

The structure of this problem is an iterated decision that you get to make over and over again. Do you continue to put energy into the things that seem promising, or do you spend your energy trying new things? A clinical trial can have that same structure, and indeed the FDA has been increasingly interested in looking over the disciplinary fence at the computer scientists and saying, maybe those algorithms that you’re using to optimize ads, could also be used to optimize human lives. The way a computer scientist, approaches this question is through something that’s called the multi-armed bandit problem.

The Multi-armed Bandit Problem


In the multi-armed bandit problem you walk into a casino that has all these different slot machines. Some of them pay out with a higher probability than others, but you don’t know which are which. What strategy do you employ to try to make as much money in the casino as you can. It’s going to necessarily involve some amount of exploration trying out different machines to see which ones appear to pay out more than others, and exploitation, which to a computer scientist doesn’t have the negative connotation that it has you know in regular English exploitation meaning, but just leveraging the information you’ve gained so far to crank away on those machines that do seem to be the best. Intuitively I think most of us would recognize that you need to do some amount of both, but it’s not totally obvious what that balance should look like in practice, and indeed for much of the 20th century, this was considered not only an unsolved problem but an unsolvable problem, and sort of career suicide to think about. During WWII, the British mathematicians joked about dropping the multi armed bandit problem over Germany in the ultimate intellectual sabotage. Just waste the brainpower and nerd snipe all of the German mathematicians. To the field’s own surprise, there came a series of breakthroughs on the multi-armed bandit problem through the second half of the 20th century.


Now we have a pretty good idea of what exact solutions look like given a number of constraints, but also what sort of more general flexible algorithms look like. The critical insight into thinking about this problem is that your strategy should depend entirely on how long you plan to be in the casino. If you feel that you have a long time ahead of you, then it’s worth it to invest in exploration, because if you do find something great, it has a long horizon to pay out. On the other hand, if you feel that you are about to leave the casino, then the return that you would get on making a great new discovery is going to be much smaller, because you have fewer opportunities to crank away on that handle once you find it. We should naturally transition from being more exploratory at the beginning of a process to more exploitative at the end. I think that’s an intuition that makes sense, but the math bears that out very concretely.

Observation of “Explore/Exploit” Trade-Off in Real Life


It’s interesting to see this idea that emerges in computer science in the late 50s through the 70s getting picked up by psychologists and cognitive scientists who are interested in human decision making. For example, Alison Gopnik at UC Berkeley who studies infant cognition, has been thinking about the “explore/exploit” trade-off as a framework for how the infant mind works. If you think about how children behave, we have all these stereotypes about children are just kind of random, they’re generally incompetent at things, and there’s a huge literature that shows that they have what’s called a “novelty bias”. They’re relentlessly interested in the next thing and the next thing and the next thing. Rather than viewing that as a kind of low willpower or attentional control issue, you can view it as the optimal strategy. It’s as if you’ve just burst through the doors of life’s casino and you have 80 years ahead of you. It really does make a lot of sense to just run around wildly pulling handles at random. The same is true for being in the later years of one’s life. We have a lot of stereotypes about older people being set in their ways and resistant to change. There’s a psychology literature that shows that older adults, maintain fewer social connections than younger people, and it’s tempting to view that pessimistically. In fact if you build an argument from the mathematics, you can see that older adults are simply in the exploit phase of their life and they are again doing the optimal thing, given where they are in that interval of time. You have psychologists like Stanford’s Laura Carstensen appealing to the “explore/exploit” trade off to make this argument that older adults know exactly what they’re doing and they’re very rationally choosing a strategy that makes sense given where they are. They have a lifetime’s exploration behind them, they know what they really like, they know the people and the connections that matter to them, and they have a finite amount of time left to reap the fruits of some new connection or new discoveries so they’re very deliberately enacting the strategy. The math should predict that, on average, older adults are happier than young people. Despite our preconceptions, and her research bears this out, that appears to be the case.


In business, the problem is very dynamic, which will classify it in the domain of the “restless bandit problem”. Since the research here is cloudier, researchers can invert the thinking to infer the conditions that lead to the business strategies we can observe.

Q: Interesting how this maps on to the life cycles of businesses. In the business context, “explore” might be innovation and “exploit” might be to run the same playbook to earn high returns on capital or something you know works. It seems like you always want to be handing off to a next batch of exploration or innovation, while thoughtfully maintaining something that you know works if you want to survive for very long time.

A: There’s a couple of things that I think are interesting in a business context. One is that implicitly the casino framing that I’ve described assumes that those probabilities are stable and fixed. Of course, we know that the world is not stable and not fixed that things change over time. This is true in our personal lives as well. Your favorite restaurant gets a new line cook and the burgers are not as good. These things shift. This is known as the “restless bandit problem”. How do you play this game when these probabilities are drifting on a random walk?

This is a very interesting case where the theory is not yet consolidated but humans, in practice, seem to have no problem. If you put people in a lab and give them a restless bandit problem, they have no trouble making choices within that environment but we don’t yet know what the mathematics of the optimal solution looks like. So here’s the case where the computer scientists and the mathematicians are asking the cognitive scientists, what are your models for how humans are actually approaching this because there may be some insight that we can use from the theory side. One of the implications of thinking in this way that is particularly relevant in a business setting is if the interval of time you perceive yourself to be on determines the strategy that you should employ, then it should be the case that if you observe someone else’s strategy, you can infer the interval that they’re optimizing over.

Inferring The Explore/Exploit Strategy in a Restless Bandit Problem

Let’s give an example from Hollywood. Most people have noticed, it feels like we’re living through this deluge of sequels, such as Marvel movies. It turns out that this is objectively true. There’s a sea change in Hollywood. In 1982, 2 of the top 10 grossing films were sequels. By 1990 it was six. By the year 2000, it was eight, and I think most recently it was all ten. From that, we can infer that Hollywood has taken a very hard turn towards an exploitative strategy. They are milking their existing franchises, rather than investing money speculatively to try to develop new franchises that will last them into the next few decades. From that, it’s reasonable to infer that movie ticket sales are declining, which turns out to be the case. Hollywood correctly perceives itself to be at the waning time of the golden era of cinema-going. If that’s true, then they really should invest all of their money into just squeezing everything they can out of the existing franchises. More broadly, so you can look at different industries and different corporations to see if they cut their r&d budget. If they’ve given that money to marketing that’d be an indication that they feel that the area has matured or plateaued.

My thoughts

    1. Ahem, asset management, cough
    2. Reminds me of a great Peter Chernin interview where he suggests that every business must be trying to grow new opportunities faster than the the old ones die out. While you must do your best to milk the old, it’s imperative to develop the new.

2) Predicting the Impact of Automation

The second avenue is totally different from this way of thinking, which is just what will the impacts of something like AI or UBI be on the economy. I’m reminded of a McKinsey report on which jobs they thought would be the most robust. The big picture thing that was interesting to me is that it cuts across the traditional class lines. It is not a white-collar versus blue-collar thing. It’s not an upper middle class versus lower middle class thing. It’s very sector dependent. The most resilient or robust jobs at the top end was gardener, legislator, and psychotherapist. I thought that was very fascinating that it’s this eclectic mixture of things. I don’t think of myself as a prognosticator about these sorts of things but my way of thinking about it is that there’s a lot of kind of human machinery around how capital moves and how laws get made. How licensing and permitting happen. It’s still done at a human negotiation level. “I know a guy. I’ll talk to Joe and we’ll sort it out.” I think humans will maintain oversight of these kind of flows of power and capital, even if the actual value is being created by software. So position yourself closer to the flow of that value than the actual creation of the value, which may be counterintuitive.

As far as the question of UBI, I don’t have a great intuition for that. There is already a restlessness in the labor force. A lot of the careers that employ some of the most numbers of people are the most vulnerable. People who drive cars or trucks, people who work in warehouses. A lot of those jobs are just one innovation away, and it’s not clear to me that there’s going to be a political response as well as just a pure economic response. I grew up in New Jersey where there was a robust toll collector union yet they had machines where you could toss your change in a bin and it would automatically sort your change and give you whatever you needed back from that. There was an effective effort to unionize the toll collectors so that you still had a human being in the booth counting out your quarters. That’s an example where it’s not for lack of technology. We had a coin sorting machine, but there was a political process that was directing the actual level of implementation. People will fight to use licensing requirements and regulations to maintain those things. Despite the actual technological capability having radically changed, it’s very hard to know which areas will look shockingly different than the world looks today. Which things will be in some ways shockingly backwards for their time because we’ve had for political reasons to hold the line.

(Reminds me of how rent flows to the owner of a relationship in a competitive market that has been flattened by technology)

Algorithms to make other types of decisions

The mathematics is very instructive, both in a specific way but also has a broader set of principles.

Optimal Stopping Problem

Difference from “explore/exploit” trade-off

One thing that comes to mind is the idea called “optimal stopping”. The multi-armed bandit problem in the “explore/exploit trade off” presumes framing that’s highly iterative. You can pull the handles again and again and again. You can go from one machine to another and back. There are many decisions in life where you are forced to make a single binding commitment that could be anything as banal as pulling into a parking space. It could be something like purchasing a house or signing a lease. It could be something like marrying your spouse. There’s a separate mathematics of cases where you need to find the right moment in time to go all-in, commit to an option, and no longer gather any further information.

37% Rule

There’s this very famous result called the “37% rule”. Let’s say you’re looking for an apartment. And it’s a really competitive marketplace. You’re in a situation where you encounter a series of options one by one. And at each point in time, you must either immediately commit, and then never know what else might have been out there, or decide to walk away and keep exploring your options but lose that opportunity forever. What do you do to try to end up with the best thing possible, even though you, you won’t necessarily know at the time, whether you found the best option that might be out there? There’s this beautifully elegant result that says that you should spend the first 37% of your search non-committally exploring your options. Don’t bring your checkbook, don’t commit to anything No matter how good it seems you’re just purely setting a baseline. After that 37%, whether it’s 37% of the time that you’ve given yourself to make the decision or 37% of the way through the pool of options, be prepared to immediately commit to the very first thing you see that’s better than what you saw in that first 37%. This is not just an intuitively satisfying balance between looking and leaping, this is the mathematically optimal result.

Broader insights on algorithms

Elegant solutions under a range of narrow assumptions about goals and acceptable risks

There are strategies like that that I think are wonderfully crisp in the recommendation they give, but they, of course, rest on this bed of many different assumptions about exactly how the problem is structured and exactly what your goals are. This rule, presumes that your entire goal is to maximize the chance that you get the very best thing in the entire pool, but it comes with a 37% chance of course that you have nothing at all, because you’ve passed. Many people would find that unacceptable. We can go down the rabbit hole of how do you modify this and the solutions get less and less clean as you wiggle the assumptions around.

Intuition for how complex decision-making is can be strangely comforting

More broadly, one of the highest level takeaways for me, from working on the book and just thinking in computational terms about decisions in my own life, is some decisions are just hard. The classical optimal stopping problem, due to a weird mathematical symmetry, is that if you follow the 37% rule you will only succeed 37% of the time. The other 63% of the time you’ll fail, and that is the best possible strategy you could enact in that situation. In a weird way, that’s some measure of consolation because often, in real life, we find ourselves not getting the outcome we wanted. While we can rake ourselves over the coals or try to reconstruct our entire thought process, I think it’s some comfort that computer science and mathematics can, in effect, certify that you were just up against a hard problem. There is some measure of comfort that if you have the kind of the vocabulary to understand the type of problem that you’re facing, and you have some intuitions about the general shape of what optimal solutions look like, then even when you don’t get the outcome that you wanted you can in some sense rest easy because you knew that you followed the appropriate procedure or the appropriate process for dealing with that situation.

A Few Books and Why They Influenced Me

I thought I’d share my response to a new reader who asked me about my “favorite books regarding the grander questions?”

As far as the metaphysical questions I don’t have a great answer. You can demerit me all you want for intellectual softness but I know I’m lacking in my reading of philosophy. So here’s my picks. Eye roll away.

The 5 Love Languages

  • The lesson is that connecting with others starts with recognizing that people are so different, especially in how they communicate. But there’s a much broader lesson embedded in this book if you read between the lines. Putting yourself in another’s shoes, as well-intentioned as you may be, is actually arrogant. Because the truth is, you can’t. If empathy through your own eyes is a prerequisite for getting along in this world then we are doomed. You must be humble enough to appreciate that you can’t fathom how another feels. Once you choose to care about a person then you must acknowledge their terms even if you can’t understand them.

The Road

  • This is a sad, depressing love story of a father to his son. It’s pure sacrifice. The narrator’s melancholy mindset and recognition of impossible odds never derails his duty to his boy. It’s the pinnacle of grace.

The Fountainhead

  • Naysayers will have a lot to object about Rand. But Roark embodied integrity. Integrity is more complicated than Rand presents. I don’t think the hardest questions are in here. But Roark is a good guide for many decisions life will throw at you even if he is an incomplete (or possibly just unfinished) hero.

The Postmortal

  • Imagine a pill that can give us all immortality. If you read this book that idea will fill you with dread. But you can take the lesson up a few levels of abstraction to my general conclusion: if you don’t think things through, the world will screw you by giving you what you want. Don’t make requests from a literal genie.

How Tails Constrain Investment Allocations

You would need to be living under a rock to not know about the importance of small probabilities on asset distributions. By 2020, every investor has been Talebed to death by his golden hammer. But knowing and understanding are not the same. I know it’s painful to give birth. But if I claimed more than that I’d end up only understanding what it felt like to be slapped in the face.

I’m hoping the above discussion of the devilish nature of small probabilities makes the seemingly academic topic of fat-tails more visceral. But if it didn’t I’m going to try to drive it home in the context of a real-life investing decision.

Step 1: Understand the impact of fat tails

I ran a simple monte carlo assuming the SPX has a 7% annual return (or “drift” if you prefer to sound annoying). I assume a 16% annual vol or standard deviation and ran a lognormal process since we care about geometric returns. We’ll call this model the “naive simulation”. It does not have fat tails.

Based on these parameters, if you invest on January 1st:

  • You have a 5% chance of being down 23% at some point during the year.
  • You have a 50% chance of being down 7% at some point during the year.

Now be careful. These are not peak-to-trough drawdowns. They are actually a subset of drawdown since they are measured only with respect to your Jan 1st allocation. The chance of experiencing peak-to-trough drawdown of those sizes is actually higher, but these are the chances of your account being X% in the red.

That’s the naive simulation. To estimate the odds in a fat-tailed distribution we can turn to the options market which implies negative skewness and excess kurtosis (ie fat tails). I used 1-year option prices on SPY. Option prices answer the question, “what are the chances of expiring at different prices?” not “what are the chance of returning X at any point in the next year?”. To estimate what we want we will need to use the pricing from strikes that correspond to the equivalent one-touch option. Walking through that is overkill for this purpose but hit me offline if you want to see how I kluged it.

Let’s cut to the market-implied odds.

  • You have a 5% chance of being down 39% at some point during the year.
  • You have a 50% chance of being down 11% during the year.

Now you can see the impact of fat-tails: the gap between 23% and 39%. This is the impact of kurtosis in the options. Meanwhile, in the heart of the distribution, the downside moves from 7% to 11%. Not as dramatic and attributable to market skew.

When we shift probabilities in the tails of distribution vs the meat the impact on the payoffs is significant.

Repeating this insight in a different way may help your understanding. Consider tossing a pair of dice. Imagine playing a game that pays the fair odds for a roll (i.e. craps).

Now let’s chip the dice to change the probability of how they land.

  • In scenario 1, add 1% to the “7” and shave .5% from each tail.
  • In scenario 2, add 1% to the “7” and shave .5% from the meat, the “6” and “8”

By shaving from the tails we take a fair game and turn it into a negative 30% expected value per toss. This is far worse than almost any casino game you might play. By changing the tail probabilities the effect on the game is magnified because the odds are multiplied across an inversely proportional payoff!

Step 2: How should tail sensitivity affect allocations?

By now, the danger of poorly estimating should be a bit more clear. How do we use this when making allocation decisions? After all, most of the time whether they are 1% or 2% events, huge moves are usually not in play. But we must care because when these events hit the impact is huge.

Tail outcomes should dictate constraints based on what you can tolerate. I’ll work through a conservative framework so you can see the impact of naive tail probabilities versus market-implied tail probabilities. The exact answers don’t matter but I’m hopefully offering a way to make tail-thinking relevant to your allocation decisions.

Reasoning through sizing decisions

Suppose things are going well and you are able to save $50,000 per year after paying expenses. You decide that losing $50,000 in the stock market is the largest loss you can accept, reasoning that it’s a year’s worth of savings and that you could make up the lost sum next year. If you impose a restraint like that, well, the most you can allocate to stocks is $50,000. That’s too conservative especially if you have accumulated several hundred thousand dollars in savings.

So you must relax your tolerance. You decide you are willing to accept a $50,000 loss 5% of the time or 1 in 20 years. Roughly a generation. If we use the naive model’s output that we lose 23% of our investment with 5% likelihood then the maximum we can allocate to stocks is $50,000/.23 = $217,000.

The naive model says we can allocate $217k to stocks and satisfy our tolerance of losing $50k with 5% probability. But if the market’s fat-tails are implied more accurately by the option skew, then our max allocation can only be $128k ($50,000/.39).

If we constrain our allocation by our sensitivity to extreme losses, the max allocation is extremely sensitive to tail probabilities. In this example, we simply varied the tail probability between a naive model using a mean and variance to a market-implied model which adjusted for skew and kurtosis. The recommended allocation based on our tolerance dropped a whopping 42% from $217k to $128k.

Many will point out that this approach is extremely conservative. Constraining your max loss tolerance to the amount of money you can save in a year seems timid. But the probabilities we used here did understate the risk. Again these were not peak-to-trough drawdown probabilities but the narrower chance of incurring losses on your start of year allocation. If we are thinking about the true experience of investing and how you actually feel it, you probably want to consider the higher drawdown probabilities which are out of scope for a piece like this. I know many financial advisors read this letter, I’m curious how allocation models reason through risk tolerance.

Current examples to consider in context of small probabilities

1) Bernie

There are market watchers who believe that electing Bernie Sanders would send us back to living in caves. Democrats are trading for about 40% to win the election. Bernie is trading at about 45% to win the nomination, implying an 18% chance to win the election. Market watchers who fear a Bernie presidency are either totally overstating his alleged market impact or the market is already discounting his odds. If the latter is true and the market is efficient, math dictates that it should shoot much higher in the event he loses.

At 18%, Bernie is no longer in the tail of the distribution. So you could argue that as he went from single-digit probability to his current chances, the market strongly re-calibrated either his impact or the sustained rally in the meantime would have been much larger. One of these things must have happened by the necessity of math as odds shifting from a few percents to 18%.

Or there is a third option. The market never really believed that Bernie’s impact would be as deep as his detractors contend.

2) Tesla

We have all seen this stock double in the past month. There has been a lot of talk about far out-of-the-money call options trading on the stock. These are bets on the upside tails of the stock over relatively short time frames. I won’t comment too much on that other than to point out a different tail in the matter. All the credit for this observation goes to a friend who keenly remembered that a year ago the Saudi’s collared their position in TSLA. That means they bought puts and financed by calls sold on the stock. Given the size of the move, the calls they sold are definitely deep in the money. This hedge likely cost them over 3 billion dollars. Billion with a “b”. That’s 6% of there projected government deficit. Their investment in TSLA stock was supposed to be a tail hedge against electric cars destroying demand for oil permanently. In the meantime, they got smoked hedging the hedge. The other tail in this story is going to be that of the official who recommended the hedge. This is a government that nearly executed a 13-year old for protesting. Fair warning to anyone looking to be an execution trader for the kingdom. You are probably short the mother of all puts. Make sure you are getting paid at least as much as a logger.

And one last TSLA note. This keen observation by Professor Bakshi.

Sometimes Keynes’ beauty contest doesn’t just judge beauty. It can create it.

Tails Explained

Left brain issue today, but stick with me. Let’s start with a puzzle.

“You have 100kg of potatoes, which are 99% water by weight. You let them dehydrate until they’re 98% water. How much do they weigh now?”

[Jeopardy music]
[Keep trying…]
[Here’s a hint if you’re stuck]

Here’s the answer.

Congrats, you just solved what is known as the Martian Potato Paradox. I came across it on Twitter and I re-tweeted it to explain why I think the problem’s general lesson is important.

Small Probabilities Are Devilish

The potato problem is tricky is because small percents are tricky. The jump between 1% and 2% feels insignificant. I suspect that is an artifact of our additive thinking. But once we point out that the 2% is 100% larger than 1% we get closer to the correct intuition. We can see that a jump from 1% to 2% is more significant than the jump from say 50% to 60%. The solution to the potato problem holds the key…you need to look at the significance in the payoff space not the probability space.

Let’s consider a bet. In 2014, the betting odds of Donald Trump being elected was say 1%. 1% corresponds to 99-1 odds. If his probability increased from 1% to 2% the new odds are 49-1. A person who was long Trump just doubled their equity in the position. 1 to 2 feels small. But 99 to 49, similar to the potato problem, shows how significant that extra 1% truly is. (I have a friend, a world-class gambler actually, who lost $400k betting against Trump before he was even a nominee. Highlights how dangerous it is to be wrong about the tails of a distribution.)

Meanwhile, a bet that has 33% chance, has fair odds of 2-1. A bet with a 25% chance has fair odds of 3-1. While these are large differences in odds, being wrong about them is less likely to be catastrophic. Beware the small probabilities.

I made this chart as a visual reminder that being miscalibrated by just 1% leads to massive error in payoff space when dealing with probabilities or payoffs below 10%. Again the difference between 1% and 2% is a 100% error in payoff space while the difference between 50% and 51% is a mere 2% error in payoff space.

I Don’t Gamble. Why Should I Care?

If you are a taxpayer, commuter, ever been a patient, or live near natural disasters you are gambling. Whether you want to be an ostrich about that is up to you, but modern society is constantly forced to handicap small probabilities. And miscalibration can have catastrophic payoffs. Let’s move to examples.

Natural Disasters

I’ve covered basic earthquake math before. If the odds of “the big one” in any given year is 1% it’s a 1 in 100-year event. If it’s a 3% event, it’s a 1 in 33-year event. That seemingly small delta is the difference between ignoring the risk and being prepared for one in your homeownership years.

My commute is a good demonstration of how the government’s assessment of risk affects all of us. Last year, SF undertook a multi-year retrofit of the Transbay Tube to withstand a once in 1000 years earthquake. They believe it can currently withstand a 1 in 100 years event. As you can imagine this is an expensive use of public funds predicated on their ability to estimate small probabilities. The direct costs are obvious. A dollar for this project is a taxpayer dollar. Which means it’s also a police force, firefighter, or teacher opportunity cost dollar. Never mind the opaque costs. The retrofit required them to start the trains later every day. I know because I used to take the first train which is now 30 minutes later. This forced many people whose work hours are extremely rigid to now drive to work. BART claims this later start will save 4 months of construction time and $15mm+ in costs. Well, how many lives can be lost to car accidents before that measly $15mm is offset? That leads us right to our next topic…

Tort Damages

Remember Ed Norton’s job in Fight Club? He would compute the expected value of lives lost vs the cost of recalling a faulty vehicle. While this sounds callous, this calculation, known as the “value of a statistical life” or VSL, is not the strict domain of evil corporate calculus. It’s the basis of medical damages, workers comp, and various forms of insurance. If you look online you can find ranges of these values based on various countries’ legislation but on average you are talking about a 7-figure sum ascribed to human life.

There’s a rich economic and legal literature dealing with these calculations. We can make inferences by what people pay for insurance or how much they say they be willing to pay to reduce their risk of injury. No method is perfect but pragmatism is such that human life is very much not “priceless”. When I was in college I wrote a paper for an Econ and Law course that tackled this problem by way of revealed preferences. Let’s pretend 2 occupations were the same qualitatively but differed solely in their risk. The riskier job would pay more. The difference in pay can be used to imply how people put a price on their life.

After a quick search, I found that a cab driver’s chance of death on the job was about 2 in 10,000. A logger’s occupational fatality rate is 5x higher, but still just 10 in 10,000 or 1/10th of a percent. Making up numbers now, if a logger makes $70k per year and the cab driver $60k, then we can imply a value of $12.5mm for the value of human life based on this revealed preference method.

Put your distaste for this approach away for a moment and note how sensitive the value of human life is for seemingly small changes in “perceived death risk”. If the logger thought his actual death risk was 1% instead of .10% and he only accepted a $10k per year premium he’s valuing his life 12x cheaper. Even though we are talking about a seemingly small change in probability, in percent terms we increased the risk by 10x and this is obvious when we see the result in payoff space. The logger is valuing his life at a mere $1mm.

The broader policy takeaway is that torts and damages are built up from small probabilities making them swing wildly or prone to large deviation based on optically small differences in risk assessments.


Without math, we can see that the decision to do Lasik or eat blowfish is sensitive to very small probabilities. Lasik you would likely only do once in your life but the difference between a bad outcome being 1 in 10,000 or 1 in a 1,000 might matter if your livelihood depended heavily on your vision.

And for a repeated risk, and I’m sorry but taking a helicopter every day falls in this category, the math deserves a visit.

No doubt, accidents are exceedingly rare. In 2019, less than half the accidents were fatal which is even more comforting. Kobe’s fate was awful luck even considering how frequently he flew. And we can see how flying frequently certainly compounds the cumulative risk. But I want to point out that tripling the accident rate shows up proportionally in the “payoffs”, while in probability space it remains invisible. If I told you the accident rate was 1 per 100,000 flight hours or 3 per 100,000 flight hours you probably wouldn’t bat an eye.

The lesson:

You need to look at the payoffs of small probabilities to appreciate the differences.

…To apply this understanding in your investing check out my post How Tails Constrain Investment Allocations 


It’s Super Bowl weekend but as a NY Giants fan I am ambivalent. But I’m going to do my best to override my indifference and cheer for the Niners. Let me explain.

I’m well aware that rooting for a team is basically rooting for a uniform. Over the years the players come and go while general managers and coaches play musical chairs. The only constant is the colors. And even those get updated permanently or color-rushed temporarily. Comedian Greg Giraldo used to joke that fandom was the opiate of the masses. It was a tool to keep a person distracted from the fact that they live in Cleveland.

As expected, hipsters would adopt this view. The “sportsball” attitude (life tip: Urban Dictionary is your friend if you don’t know a reference) uses a tinge of truth to supply justification to this contrarian-for-the-sake-of-it crowd. A few weeks ago when I asked a bartender to change the channel to the Patriots playoff game, he said he didn’t even know what sport that was. Pinch me but are people using willful ignorance to signal enlightenment now? (Insert “blinking white guy” gif)

The irony of his elitist irony is that it’s faux-intellectual. It’s a clumsy violation of lindy wisdom. The dismissal of sport and competition as unimportant is a buzzkill that defies centuries of human experience.  I remember exactly where I was in 1990 when LT stripped Roger Craig in the NFC Championship game leading to Matt Bahr’s game-winning field goal. Weeks later I remember jumping in euphoria in a friend’s basement and scratching my suddenly bloodied knuckles on the ceiling when Scott Norwood’s kick sailed wide right. The emotion in those seconds encodes those memories forever. And we become lifelong fans. From childhood.

The highs and lows in sports mirrors life itself. And that’s fun. I can’t explain it but I know this. My 6 yr old likes flag football. He recently got the hang of throwing a spiral and wants to play catch constantly. And he was born in San Francisco. That city’s team is playing for a ring today. This could be the start of his lifelong affection. I’m all for magic we can’t explain.

Go Niners!

As you all prepare for the game today, I’ll leave you with a 20-year-old movie clip I’ve watched a hundred times. (Link)

If you are wondering what the awesome guitar instrumental that builds in the background, it’s Peace by Paul Kelly. You’re welcome.

Negotiating Pay

Never throw out the first number.

You’ll recognize this as the cardinal rule of salary negotiation. Thought it would be useful to share the best posts I’ve found on the topic. They are a good read if you are a hiring manager or job seeker.

  • Patrick McKenzie starts by emphasizing just how important it is to negotiate seriously and how to not think like a “peasant”. His comprehensive advice is targeted to engineers but the scripts can be repurposed for other fields. At the end of the post, there are links to more negotiation reads. (Link with my highlights)
  • For a shorter read, check this recent Twitter thread with tips for negotiating an offer. (Link)