The Value Perspective Podcast – with ‘10-K Diver’


Juan Torres Rodriguez

Juan Torres Rodriguez

Fund Manager, Equity Value

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

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Hi, everyone. If you are a big fan of FinanceTwitter, you are going to love this episode as we are having a masterclass with 10-K Diver today. 10-K runs a popular and anonymous Twitter account that beautifully explains complex financial concepts in digestible and engaging ways. We have been rabid followers of the Twitter account for the past two years and we were thrilled when he agreed to come on to the pod and discuss his passion for education, how to construct ‘thought problems’ and what tools and theories he has learned in his self-directed finance education Andy and 10k discuss ergodicty quite a bit in this episode and  if you want to learn a lot more about this concept, please check out our episode with Taylor Pearson released in September 2020. Enjoy.

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Chapter headings for ‘10-K Diver’ on The Value Perspective Podcast

Please click on the link below to jump straight to a chapter

* ‘10-K Diver’, welcome ...

* Simplifying complex ideas with mental models

* Thinking in probabilities

* Shannon’s Demon and the volatility tax

* Explaining game theory

* Book recommendations and one big mistake


‘10-K Diver’, welcome ...

JTR: 10-K Diver, welcome to The Value Perspective podcast. It is a pleasure to have you here. How are you?

10-KD: I am doing well, thank you. I am happy to be here. Thank you so much for inviting me on.

JTR: I have to say both Andy and I are so happy to be able to talk to you today – although I am a little bit sad as well because I thought we were going to be the first ones to have your on a podcast. And then Jim O’Shaughnessy from Infinite Loops, beat us to it! So for those who do not know you very well or are not following you on Twitter, can you provide us with a little bit of your background?

10-KD: Sure. I am a computer scientist. I do not have very much formal training in finance or business or investing or anything like that but I did get started investing in 2011. And a lot of the concepts that fundamental investors need that are just so basic to investing – I had to figure them out on my own. And it took me an enormous amount of time to learn these concepts – simple things like operating leverage or depreciation or the fundamentals of accounting or why do you need to diversify a portfolio and how do you do that? Simple things like this, which an investor needs – it took me an enormous amount of time to figure out for myself.

So I thought I might be able to add some value by sharing these things so that others, who are in the same boat as me and who maybe have not had formal training in business or finance or anything like that, may be able to learn these concepts as well. And they may be able to learn it in an easier way than what I had to go through. So that is why I started my Twitter account two years back – and I was completely unprepared for how quickly this account would grow and things like now being on a podcast with you guys. I have several followers on Twitter and things like that but I was completely unprepared for all this growth. I am just trying to help people understand how to make better financial decisions. That is it basically.

JTR: I would say ‘several followers’ is a very humble way to describe your 2,000 and counting followers within only 18 months of going live with your Twitter account?

10-KD: Yes. I have been on Twitter for about two years now and I believe it is 200,000 – not 2000 – but who’s counting! With a lot of the followers, luck plays a big role. On Twitter, you never know what piece of content will be picked up by somebody big and retweeted and things like that. So luck definitely played a big role in my account becoming so popular. But I am just trying to have fun with it.

AE: You have expanded a bit away from the thought problems you started out with and towards some more collaborative pieces with other people. Could you talk about that in a bit more detail?

10-KD: Right. A guy like me, with no formal training in finance and things like that, can only have so much knowledge! And very often I want to write about something – I want to learn more about a particular topic in finance – and I find I do not know enough about it. But, because I have this Twitter account, I can reach out to a few people who may be experts in that particular area and then I may be able to talk to them and get their ideas and try to understand them better and then write a thread about it.

Recently, for example, I wrote a thread about something called the ‘cockroach portfolio’. For a long time, I did not fully appreciate the importance of things like having negative correlations between the assets in a portfolio. So I was talking to these guys, Jason Buck and Taylor Pearson, who run this fund called the Mutiny fund. They have an approach that is based on an approach by a guy called Harry Brown, who was a financial adviser. Well, he was many things but, among those things, he was a financial adviser and he wrote a book on planning out your portfolio, how to allocate assets in your portfolio and so on.

He called his approach the ‘permanent portfolio’ – and the whole idea behind a permanent portfolio is you are not just looking at the absolute returns you get from the portfolio, you are also trying to minimise drawdowns as much as possible and minimise the probability of ruin. So you construct a portfolio for maximum survival – even if you might give up one or two points in terms of absolute returns – because, in the long run, survival is the is the name of the game in investing.

I did not know enough about these topics but I was very interested in them and so I contacted these guys, Taylor Pearson and Jason Buck. They are experts on this kind of thing and they run a fund that is based around these ideas. We got together and we wrote a thread about the cockroach portfolio – they shared some of their writings with me and I read through them and so on. So, when I find I am out of my depth in a particular area, I like to find somebody who is an expert in that, then maybe talk to them, try to understand it and then explain it to my audience.

JTR: And you are doing that through a new platform? It is not only on your Twitter account but through a different app. Is that correct?

10-KD: That is correct. I have something called a ‘social podcast’ – named Money Concepts – through this app called Callin. So it is not on YouTube, it is not Twitter Spaces – it is a separate app. And the neat thing about this is the social aspect. I can talk for maybe 15 minutes or half an hour about a particular topic and then people can call in and ask me questions. I really liked the interactivity of this podcast – it is not just me droning on and on for hours about something nobody else is interested in! It is an interactive format and I like that a lot.

So I am doing that on the side and sometimes we have some very nice guests on. Recently, we had Nick Maggiulli come on the show and talk about his book and this week, we are going to have some professors of probability, who are going to talk about how to learn probability – things like that. So having a show like that lets me invite people I admire onto the show and it also gives a broad audience access to these people to ask questions. So, for example, we had Chris Blumstran on the show just a few days ago and it was the same week as the Berkshire Hathaway annual meeting. So a lot of people were able to tune in and ask Chris Blumstran questions about Berkshire and what he thinks Berkshire will do with the Alleghany acquisition and things like that. So it' is nice to have a platform like this.

JTR: I have to ask – why keep your account so anonymous. That makes it so interesting but what is the idea behind it?

10-KD: That is a great question. I work for an employer who has some strict policies about social media. They do not force me to be anonymous or anything like that – they would let me use my name if I wanted to – but they have a very stringent set of rules and it is just much easier to abide by all these rules, if I am anonymous. So that is why I am anonymous.

JTR: That is fantastic – and it keeps things very interesting, I have to say. That is really cool.

10-KD: If being interesting is a by-product of that, I’ll take it!

Simplifying complex ideas with mental models

JTR: Your mindset and mental models definitely go down the more mathematical and engineering path but you have successfully simplified some very complex subjects and applied these mental models to explore other areas – not just finance but many other topics. Can you talk about your approach to decision-making using these mental models? Is there a particular framework you like to follow?

10-KD: Right, let’s take the first part about finance versus non-finance mental models. I find that, if you try to understand the fundamentals of one subject, there are some fundamental concepts you can learn and then apply to a variety of different subjects. I will give you a simple example, just this whole idea of ‘state quantity’ versus ‘flow quantity’, where a state is something you have at a particular point in time, while a flow tells you how that state changes between two points in time. Just that.

So the state at a particular time could be the number of humans on Earth and the flow is then how many new humans are born and how many humans die. So between two points in time, if you know how many humans existed at a particular point and then how many were born and how many died, you can predict what the future state of the system is going to look like, based on the flow – assuming people do not go off to Mars and things like that!

This is such a fundamental idea – and the same idea applies to analysing financial statements. The balance sheet, for example, gives you the state of the company and the income statement and the cash flow statement tell you the flows that are happening with the company. The same thing applies to analysing population dynamics or analysing electrical circuits, where the state is all the voltages in the system and the flow is all the currents in the system and so on.

So, if you acquaint yourself with a set of basic ideas, it turns out a lot of disciplines use exactly the same ideas and mental models. They may be couched in slightly different language and so on but, at a fundamental level, if you are the kind of person who tries to get into things at a deep level, try to understand them from first principles and then you will find a large number of quantitative fields are very similar in the kinds of concepts they adopt.

There is no applied math! The math is the same for everybody. There is no real demarcation saying this particular math should have only these applications and not those applications. The same math that applies to decision-making in a casino, say, may also apply to diversifying a portfolio of assets or to take some decision about how to construct a bridge or something like that. So once you understand the fundamentals, the same mental models generally apply to a wide range of different disciplines – at least in my experience. Over the years, I have found that a lot of concepts I learned in engineering classes and circuit-design classes and things like that, I am able to apply fairly straightforwardly to investing.

JTR: Is there any area of mathematics or any specific framework that you find yourself referring to more often than anything else?

10-KD: I talk a lot about probability, which is partly because I love probability as a subject – it is so delightful to me. I use a fair bit of game theory. I do basic simulations – the ‘state’ versus ‘flow’ models – to build a set of financial projections or financial models. There is a whole set of frameworks and models that come with something like accounting – so, if you want to understand financial statements and how to invest in companies, it is a good idea to learn the fundamentals of accounting. So there is a set of models and frameworks that accountants use to think about companies and so on and I have found them to be pretty useful as well.

Then there is psychology – psychology is endlessly fascinating to me. If you look at the work of Dan Kahneman or some of these people, it is just amazing what kinds of biases we all have – even if you know all about probability and so on. Your ‘system one’ thinking versus ‘system two’ thinking, as Kahneman calls it, for example – the ‘fast thinking’ versus the ‘slow thinking’. When you are thinking with your emotions or instincts, you do not have time to map out all the probabilities and the decision trees and so on so your mind has a quick way of arriving at an approximate decision – and sometimes that approximation can be way off. So this is endlessly fascinating to me – how people have so many biases that they do not even realise, say, or the madness of crowds. We see financial markets get into bubbles all the time. So all these different concepts are so interesting to me and they play a role in how I think about the world, how I think about investing, how I make decisions and so on.

Thinking in probabilities

AE: Your followers – of whom I am very much one – definitely appreciate the way you boil things down to first principles. Then you turn it into a little bit of a game and then, from there, you kind of explain something that, at its heart, should be quite straightforward but lots of people do not really understand it in the depth you manage to explain it. We will definitely go into a couple of those games a little bit later on but you have already mentioned one of the big topics that comes up time and time again is the idea of thinking in probabilities. You talk about outcomes as a range of outcomes and you think about the probabilities and the probabilistic outcome. So why is it so important to think in probabilities?

10-KD: Absolutely. Jim O’Shaughnessy has this wonderful saying that we are deterministic thinkers living in a probabilistic world – and then he adds that hilarity or tragedy often ensue. I think he is absolutely right. There are no certainties in life. This is a sad fact of life – or maybe a happy fact! Even if you buy a stock that you think has a very high chance of delivering a good return, there is always something that can show up and mess up your results.

Whenever you make any kind of decision, you cannot predict the future of the world perfectly and there are always any number of possible outcomes that can happen in the future. So probability is just a systematic way to reason about a multitude of possible outcomes that can happen in the future. So, whenever you make a decision and that decision could play out in a number of different ways and you do not really know what you will get at the end of it – you do not know your pay-off ahead of time – probability applies there. That is because you could have a number of possible outcomes and you have to reason about them.

Take the example of Warren Buffett selling airline stock at the beginning of the pandemic – when the pandemic was just getting very severe, he decided he was going to sell all his airline stocks, right? Now, the question is – was that a good decision or a bad decision? Look at those stocks today and you can see the stocks are much higher than when he sold so, on the face of it, it looks like Buffett made a bad decision. But one of the important concepts in probability is you don’t just do a back-testing – you cannot tell whether something is a good decision or a bad decision based solely on what happened.

Take somebody playing Russian roulette for $1m, for example, where the type of gun means there is a five-sixths probability they will survive to get the $1m and one-sixth probability of death. Suppose somebody does this and gets the $1m, you cannot say this guy made a good decision, just based on the outcome, because in an alternative universe he could have lost his life, right? So this whole idea that you look at the current state of the world and try to predict whether some past action was a good decision or not – that whole idea breaks down because the current state of the world is not the only state the world could have been in. The world could have gone to any number of different states and, if you look at all the states the world could be in today, you have to have some idea of that, to tell whether a past decision was good or not.

This is a such a simple probabilistic concept but it is not natural for us to think like this. To tell whether somebody made a good decision or not, people usually just look at the result and then say, if the result was good, yes, they made a good decision. And, if the result is bad, they made a bad decision. This is how we normally think. But if we think a little deeper, we may conclude that, even if the outcome was good, the process used means the decision may have been the wrong decision to make. This is a probabilistic concept.

Simple things like this are so important to understand – in any kind of decision-making, not just investment – so that is one simple application of probability or probabilistic thinking. There are several other applications. You can you can talk about diversification, for example. We all know, intuitively, not to put all your eggs in one basket. We know that. So the next question becomes, OK, fine. Do not put all your eggs in one basket. That is great advice – but how many baskets should I have? And how many eggs should I put in each basket?

If I have 100 eggs, how many baskets should I take? And how many eggs in each basket? How should I watch those baskets? And so on. And that depends a lot on things like whether the baskets are correlated or not. If you put all your baskets in one place where a thief can easily come and steal all the baskets at once, then it does not really matter whether you put all the eggs in one basket or you put all the eggs in multiple different baskets – they are all exposed to the same kind of risk, right?

These are all probabilistic concepts and there is this wonderful book, Fooled by Randomness, by Nassim Taleb, where he touches upon all these different concepts in detail. A third concept, for example, is the whole idea of ‘skewness’. Say you have a strategy that makes money 95% of the time but loses money 5% of the time, is that a good strategy or a bad strategy? You cannot really tell because you do not know how much money it makes when it succeeds and how much money loses when it fails. If, for example, you make a 10% return every time it succeeds but you lose all your money when it fails, then that is a very skewed set of probabilities – even if you have a 95% chance of success, there is a 5% chance you will lose everything. So you would not put all your money behind a bet that looks like that.

This is a very simple probabilistic concept and it does not matter if the chance of success is 99.99% – if there is a 0.01% chance that you will lose it all, then you do not go and put all your money behind it. Simple concepts like this – these are all products of probabilistic thinking and humans are generally not good at this kind of thinking. So I use probability as a tool to help me systematically think about all the possible outcomes that can happen, my pay-offs in each situation and the odds of different outcomes happening – just so I can reason about how to back and so on.

AE: That makes a lot of sense – and your probabilistic thinking is something we come back to time and time again on this podcast. It is a recurring theme. Actually, this podcast was started after we had a conversation with Annie Duke, who wrote Thinking in Bets and, obviously, that topic is at the heart of her book. One thing we always ask our guests who think in this way is, if you are not mathematically minded, how can you get people to think in probabilities? It is not intuitive, as you said, so how can you encourage them to think in this way?

10-KD: That is a wonderful question. The kinds of tools we have to think probabilistically today – people did not have many years ago. So it actually does not require a whole lot of math for you to just understand the basics of probability – for example, today, you can run Monte Carlo simulations in Excel or Python or anything like that. You do not have to know how to calculate a whole bunch of probabilities or when to add probabilities or when to multiply probabilities, things like that – you can simply simulate a set of outcomes on a computer.

You can say, OK, run a million simulations and tell me what the odds look like and, very frequently, those odds will look something like reality. Of course, if there are some very rare events or something like that, the computer may or may not catch them but, for most cases, if you know how to get a computer to run a simple simulation of all the possible outcomes, you can very easily use that to think probabilistically. So we should take advantage of the tools we have.

If you needed mathematical knowledge to understand the ideas of Nassim Taleb, say, then he definitely would not have sold all the books he has because not that many people have the mathematical knowledge. If you take Taleb’s books – Fooled by Randomness, say, or The Black Swan – these are such intuitive ideas. It does not matter how many white swans you have seen – you may have seen a million – you still cannot say all swans are white. Because who knows, there may be a black swan around the corner you have not seen. This is such a simple idea – it does not require a whole lot of math or anything like that to understand. And if the idea required math to understand, Taleb would not have sold as many copies of The Black Swan!

So there are things you can do to think more probabilistically – for example, reading non-technical books like those by Nassim Taleb or Annie Duke. These are authors who have done a wonderful job at taking fundamental probabilistic notions – the ideas of risk and skewness, the importance of diversification and betting in such a way that you do not risk total ruin and always live to fight another day. There are these authors who have made ideas like this accessible to a non-technical audience and, even if you are not mathematically inclined, you can read these books and learn the basics. And then, if you want to learn things in greater depth, you can always learn the mathematics – the fundamentals are not actually so complicated.

The science of probability probably began with the mathematicians Fermat and Pascal, who pretty much figured out hundreds of years ago that there are only a few fundamental rules of probability. And it is not that hard to understand the basic rules that were laid down by Fermat and Pascal and to apply them to solving simple puzzles. I love solving puzzles as a way to get into ideas and to wrestle with them and to think about how to take decisions and so on. This whole idea of taking a decision by drawing a decision tree, for example – just figure out all the possible outcomes, draw a tree with your pay-offs in each case and try to see if you are comfortable with them or not. It does not require a whole lot of math to draw a simple decision tree.

So I would say the math is useful to know but it is not a deal-breaker if you do not know the math – if you have decent intuition, you can understand a lot of the fundamental concepts. And if you can slow down and find the decision -making process that lets you get into the system 2 thinking that Dan Kahneman mentioned, you do not need a whole lot of math. You should not just make all decisions off-the-cuff because your intuition can lead you astray. But if you just slow down, think about all the possible things that can happen and take a decision based on that, you are perhaps 80% of the way there without knowing too much of the maths.

JTR: Even if the maths is not that complicated, at a theoretical level, people find it quite counterintuitive to think in terms of probabilities. If that is the case, then, how can you fight against that human weakness?

10-KD: Well, we are all subject to that weakness because we are all humans. I would say that, in the normal course of life, we make so many terrible decisions but, thankfully, most of those decisions do not really have adverse consequences. So, when we are making a decision that could potentially have an adverse consequence, we should slow down – write things down, try to think about it carefully, maybe build some models in Excel or Python or something like that – and just try to ensure that, for the most important decisions in our life, we use a systematic way of thinking, rather than just our intuition.

Dan Kahneman and Richard Taylor have this example of going to the movies. A lot of people, if they have spent a lot of money on a movie ticket and they see the movie is terrible in the first five minutes, they let the cost of the ticket affect whether they stay and watch the rest of the movie or not. There is this sunk-cost fallacy – and the sunk-cost fallacy is a human weakness. Just because we have already sunk some money into buying a ticket for a movie, say, we feel the money will be wasted if we walk out of the movie. But that is really not the right way to think about it because that money is gone. We are not going to get that money back – it is already sunk.

Instead, the best thing to do is to think, OK, how do I maximise my enjoyment over the next two hours? Do I want to sit in this miserable movie? Or do I want to just get out and use that two hours for something else? That is the right way to think about it – but, even so, most of us fall prey to the sunk-cost fallacy. We throw good money, good effort, good time and so on after bad. But thankfully, in a lot of these cases, even though we are subject to all these biases, the consequences are not that severe. Sure, we may lose a couple of hours and we may not have a good time at the movie but, in this particular case, the sunk-cost fallacy has not really hurt us a whole lot.

So we do not have to worry too much about our weaknesses in situations where the consequences are not very adverse. But if we are going to take, say, 50% of our net worth and put it into a stock or something like that – now there the consequences could be terrible. And so we have to be very careful about making those kinds of decisions. So maybe 95% of the time, we can cheerfully live with our weaknesses – we do not really care because the consequences are not that severe. But the 5% of the time we are taking important decisions, there we should try to be conscious of our biases.

And there are wonderful books written about the biases we have – Charlie Munger has this wonderful speech called The Psychology of Human Misjudgement; Dan Kahneman has his excellent book; Robert Cialdini has a wonderful book called Influence. So there are all these tools that let us build a checklist of biases and, if we just go through that checklist, say, and reason while bearing in mind we are subject to these biases, for the most part we can get away with a reasonably comfortable life – even though we have all these weaknesses.

JTR: We had Andrew Elliott as a guest on this podcast – I don’t know if you have heard of him?

10-KD: Not before I received your email inviting me along but now I’ve heard of him – thanks to you!

JTR: OK. He has written two wonderful books – the latest being What are the Chances of That?, the whole aim of which is to try to help people understand probabilistic thinking and be able to calculate some probabilities. I am going to read you a quote from it that caught my attention. It goes: “Outside the casino, probabilities are very often of this kind – degrees of belief, either well or ill-founded, supported by reasoning or not. We apply the rules of frequentist probability to subjective measures and they will make sense.” People push often back against adopting probabilistic thinking on events that will not repeat themselves – for example, a judgement on the outcome of an election or whether or not Russia would invade Ukraine and what that might look like – because we cannot go back in time and model the outcome of those specific events thousands of times. So people believe, to a certain extent, the rules of frequentist probabilities do not apply there. How do you think about this and what would you say to people who follow this way of thinking?

10-KD: That is a wonderful question. Here is what I would say. It is absolutely true that many of the famous results of probability – things like expected value, the central limit theorem and so on – apply only when you can play the same game many, many times. So if you toss a coin and say 0 is heads and 1 is tails, the expected value of that outcome is 0.5. But if you toss a coin once, you are going to get either 0 or 1 – you will never get 0.5! So you are absolutely right – you can then ask, OK, if I am going to toss this coin only once, how does the 0.5 help me? Because the 0.5 is a long-run average – if I toss the coin a million times and then take the average of those million, I will get something close to 0.5. But how does it help me in this particular situation where I am going to toss the coin only once? And that is a perfectly reasonable question.

I would say, in situations like this, the best application of probability theory is to position yourself in such a way that you will do reasonably well regardless of whether you get a 0 or a 1. Mark Spitnagel, who has worked with Nassim Taleb, has this wonderful saying that, in life, you will have one path that is followed – you do not get the ensemble average of all possible paths. There is one path that will be taken in the future. You do not know which path that is ahead of time but one thing is for certain – you will not get the average of all possible paths.

There are people who think very deeply about these questions – for example, Ole Peters and his theory of ergodicity. He has this wonderful example where he says, if you didn’t have a choice and you had to play Russian roulette, would you like to be one of six people playing Russian roulette once each or would you like to be the one playing Russian roulette six times? The odds are very different in the two cases! If you play Russian roulette six times yourself and your gun has one bullet and six chambers, there is 100% chance that you will die. But if you if you are one of six people playing Russian roulette, there is one-sixth of a chance that you will die. Ergodicity is the branch of science that tries to grapple with questions like this.

And I would say that Mark Spitznagel has some wonderful ideas here. Basically, his ideas boil down to, if I can run an event only once – like the examples you mentioned – then I should try and make sure that, no matter what the outcome is, my losses are limited. I should not bet enormous amounts of money on the outcome of one thing, which could go either way. So even if I think the odds of somebody winning an election are 95%, I should still make sure that, if the other 5% were to happen, I will still be reasonably comfortable – I will not lose the whole lot. That, I would say, is one of the fundamental ways in which probabilistic thinking applies – even to events where there is only one likely outcome and you do not get to repeat the event multiple times.

Shannon’s Demon and the volatility tax

AE: This would be a good point to delve into some of your own thought pieces because they tie in nicely with some of the things you have just discussed. My own personal favourites, which are very closely linked – and are probably the most specific for investing – are Shannon’s Demon and the volatility tax. They are very different thought pieces – but also very connected – and they tie in with ergodicity. Can we spend some time delving into those in a bit more detail? Maybe you could start by explaining the problem you set, which is fairly straightforward, but then we can talk through its ramifications?

10-KD: Absolutely. Shannon’s Demon is one of my favourite examples while volatility tax is such an important concept – but it is very often misunderstood. Let’s start with Shannon’s Demon, which is this idea where you have an extremely volatile stock or asset or something like that. Let’s say you have a stock that is extremely volatile and, every day, the stock either doubles or it halves – so, if the stock is at $100 today, it may be at $200 tomorrow or it may be at $50 – and let’s also say there is a 50% chance of either outcome. Now, the question is, can you make money from the stock over a long period of time? If you assume the stock is either going to double or halve every single day and each day is independent of all the others that came before, there is a 50/50 chance – regardless of what the stock did yesterday – that today, it is going to either double or halve.

Now, if you just think about this, there are several interesting concepts that come out of this example – the first of which is you can calculate the expected outcome. If the stock is at $100 today, what is the expected value of tomorrow’s price? It could be $200 or it could be $50. So if you just take $200 plus $50 divided by two that is the expected value – and that is $125. So today the stock is at $100; tomorrow, the expected value of the stock is $125 – and so your expected return is 25%, in one day, which is a phenomenal expected return. This is called the ‘arithmetic return’ and it looks like this is a positive expectation back.

So it looks like, OK, you have to go all-in on this bet because it is a positive expectation – you have to bet all your money on it. But the idea is you should not bet all your money on this because you need to think about what happens over the long term. Let’s say you put all your money into this stock, and then you let it double and halve and so on. Over a long period of time, what is going to happen is the number of doublings and the number of halvings are going to be approximately equal to each other because there is a 50/50 chance of each. And the thing is, each doubling is going to cancel out the halving. If the stock doubles and then it halves or it halves and then it doubles, you are left exactly where you started.

So if you put whatever money you have into the stock and you wait for a very, very long time, what is the most likely result you are going to get? Well, the most likely result, if you play this game 1,000 times, is you will get 500 doublings and 500 halvings in some order. And if you get 500 doublings and 500 halvings, you are left with exactly the same amount of money you started with. So the return over these 1000 days is 0% – or the most likely return that you will get is 0%.

So we had a positive arithmetic expectation of 25% – this was a positive expectation bet. But if you try to compound a sequence of such bets, what you end up getting is a 0% return, which is called the ‘geometric expectation’. So the arithmetic expectation of this bet is 25% whereas the geometric expectation is 0%. And in situations where you let your winners ride – where each bet works on the previous bet successively – the bets all compound together. In situations like that, the geometric return is far more representative of what you will be getting than the arithmetic return. So over a long period of time, if you go all-in on the stock, the most likely outcome is you will not make any money. Of course, there are some situations where you will make a lot of money but the most likely outcome is you will make 0%.

AE: The beauty of this set-up is, to some degree, there is actually a solution, which the US mathematician Claude Shannon came up with. Could you talk us through that?

10-KD: Right. Claude Shannon – working with another mathematician, John Kelly – came up with a beautiful solution to this problem, where he said, no, you can actually make money out of the stock. Even though it seems like, over a long period of time, you will get a 0% return, it does not have to be 0% because you do not have to put all your money into the stock each time. Shannon’s policy for betting on the stock is very simple – at each turn, you rebalance your portfolio. You have a certain amount of your money in cash and you put a certain amount of your money into the stock – in this particular case, you keep 50% of your portfolio in cash and you put 50% of the portfolio into the stock.

And the next day, if the stock were to double, the stock part of the portfolio would increase and the cash part of the portfolio would remain exactly the same. Then what you do is rebalance – you sell some of the stock and get the money back into cash. On the other hand, if the stock were to halve, then, in order to bring the stock portion of the portfolio back to 50%, you would have to spend some of your cash to buy the stock. So on days the stock halves, you go and buy more of it; and on days the stock doubles, you sell.

And this idea of rebalancing is super-powerful because now it lets you escape this difference between the arithmetic return and the geometric return. Remember – the arithmetic return is 25% and the geometric return is 0% but Claude Shannon’s strategy gets you something like 6% per day. If you follow this particular strategy, which Shannon discovered, the most likely return you will get over a long period of time is 6% per day. So you can’t do as well as the arithmetic return – you can’t compound at 25% per day over a long period of time – but you do not have to settle for 0%. You can actually get 6% and the way you do that is through rebalancing.

That is the whole idea of the volatility tax as well. The volatility tax is not a tax imposed by a government or something like that – it is essentially the difference between the arithmetic return and the geometric return. Over a long period of time, you will get the geometric return, if you go all-in on a stock, but the geometric return will be less than the arithmetic return. Even if you have a positive expectation bet, the geometric expectation of that bet could be zero or even negative – and the difference between these two is called ‘the volatility tax’.

For investors, this has particular implications because stocks are volatile. Let’s say you have the choice between two stocks – Stock A offers a 10% return every year, like clockwork, while Stock B could return either 8% or 12%. The average return for Stock B is still 10% a year but, over a long period of time, there is actually a very high chance that Stock A will outperform Stock B – simply because of this volatility tax.

JTR: When you published your piece on the volatility tax, you started the thread by making a reference to how some investors – especially value investors who follow Warren Buffett and Charlie Munger – state they do not care about the volatility of a stock because they are fundamental investors or, over the long term, the only thing that matters is the fundamentals of the company. And you made the point that people should be very careful who they listen to – even Warren Buffett or Charlie Munger – because things like volatility do matter, even if you are highly focused on fundamentals.

10-KD: That is a great point. The idea is that, when somebody like Warren Buffett or Charlie Munger makes a statement, they are very careful in choosing their words. If you read one of Warren Buffett’s letters, he might say something like, ‘we prefer a lumpy 12% to a steady 10%’ or something like that. And what a lot of people take away from this is there is no need to care about volatility or volatility does not matter – what matters is the return you get.

And it is true: you would prefer a lumpy 12% to a steady 10% – but that lumpy 12% has to be a geometric average, not an arithmetic average. And a lot of people fail to realise this. You know – if I had a stock that could double or halve, is that a lumpy 25% return? No, that is not a lumpy 25% return that is 0% return because the geometric average is zero – the arithmetic average is 25%. So this is not a lumpy 25% return. This is a 0% return. So when Warren Buffett says he is OK with a lumpy return, if it is higher, what he means is, after accounting for the volatility tax, he wants a higher return – and if that return happens to be lumpy, he can live with that.

The second point is about being solely focused on the fundamentals – where you believe that, no, stocks are not really instruments that just go up and down; instead, they are based on the earnings and cashflows of the underlying company and, as such, you should not care about short-term volatility in the market price of the stock, you should only care about the fundamentals of the company. And yes, that is true – over a long period of time, stock prices usually follow earnings and cashflows and the fundamental quantities of the company.

But there is not a single company on earth that can deliver the same return on capital every single year – so there is going to be volatility in the fundamentals as well. So you may not be worried about the volatility in the market price of a stock but you still have to be worried about the volatility in the fundamentals. So, if some years a company earns 10% on capital and in other years the company earns 20% on capital, then over a long period of time, you will not get a 15% return. If you believe that, over a long period of time, the market price of this company will follow the business performance, that is fine – but you still have to care about the volatility. That is because, over a long period of time, you are not going to get the arithmetic average, you will get the geometric average of 10% and 20% – at least, that is the most likely outcome.

JTR: Have you met or come across Jake Taylor? He was a guest on the pod and he made this powerful point in relation to Buffett’s preference for a ‘lumpy’ higher return – observing that some people do not realise how difficult it is to endure the volatility that likely comes alongside that. So, for some investors, it will actually be better to stick with the steadier lower return because volatility is just so very difficult to endure. I guess that goes hand-in-hand with what you were just saying.

10-KD: Yes, absolutely. A lot of people do not have the stomach to see their stocks drop by a lot in a short period of time. Now, there are several ways this can come and hurt a person. One way a sudden drop in your stocks can hurt you is if seeing the drop increases your chances of making a bad decision – at that point, this becomes kind of a game with feedback, instead of just a game where you get a lumpy return over time, right? Because, based on what you are getting, you are going to change your strategy – maybe you get too scared and you sell out at exactly the wrong time, say. So, if you make bad decisions as a result of the volatility you see, that can really hurt you over a long period of time.

The second way this can hurt you is if you have your money in the market, or some volatile instrument, and then you have a sudden expense. If you have to withdraw money out of the market at exactly the wrong time, volatility can actually hurt you. Let’s say you are going to have an emergency expense – you do not really know when this emergency expense is going to strike you. If you have some situation like that, then, it may be rational not to take the lumpy higher return but to settle for a slightly lower return that is not as lumpy because you do not know whether you may have to withdraw money at exactly the wrong time if this emergency hits.

Warren Buffett does not have to care about that kind of thing. Even if his stocks go down 50%, say, he has plenty of money on the side so he can afford to play the long game and let his stocks ride over the long term. People like us, however, may be concerned about the maximum drawdown in our portfolio because – who knows? – we may be hit with an emergency. You know – usually, when there is a crisis, the stockmarket is not doing well and people also lose their jobs. So the time they have to take money out of the portfolio is likely to coincide with a time when stocks are down.

So if you can afford to wait and let your stocks ride for a long period of time, you may not care much about volatility. Even in those cases, however, if you believe in fundamental investing, you have to care about the volatility of the fundamentals of the company’s earnings and cashflows and things like that. But if you are in a situation where you could have a need for emergency cash, and you would have to satisfy that need by liquidating a portion of your portfolio, then you may still prefer to have a portfolio that does not exhibit too many drawdowns and too much volatility.

JTR: Andy, you and I were talking about the volatility tax the other day on the back of 10K’s piece and you made the point that, to a certain extent, it could be mitigated when you build a portfolio.

AE: That is definitely something we have explored. 10K, we were talking about it before – in terms of, if you have the same situation but with uncorrelated stocks, you could actually get around the volatility tax – but we would love to hear your thoughts.

10-KD: Absolutely. If you have uncorrelated stocks – and it works even better if you have negatively correlated stocks – you can actually generate an even higher return. So in the case of Shannon’s Demon, there were two possible instruments you could put your money in – there was cash and there was this extremely volatile stock, which was doubling or halving every single day. But suppose cash was not an option. Suppose the only options were Stock 1 and Stock 2 – how does this play out? Each stock may be volatile but, if the two stocks are sufficiently negatively correlated, say, then it is actually possible to get an even higher return than the 6% per day. So you can take advantage of negative correlations in a big way. If you can find two assets that are reliably negatively correlated, you can build a portfolio out of them that achieves even higher returns than what each individual asset can possibly deliver over a long period of time.

JTR: For the benefit of our listeners who may not be so technically driven, could you define ‘uncorrelated’ and ‘negative correlation’?

10-KD: Oh, absolutely. To explain ‘negative correlation’, if you have two stocks and, a lot of the time, their prices move in opposite directions on a particular day, then you can say the stocks are negatively correlated. That is in non-technical terms – of course, there is a precise meaning of negative correlation and it is not just important to see how often they move in opposite directions, you should also see the size of those movements. So if Stock 1 moves up 5% and Stock 2 has a very high chance of moving down, but only by 0.005% or something like that, then the negative correlation is not that much. So generally, if one asset moves up and the other asset has a higher chance of moving down in some similar amount – if this relationship holds, then you have two negatively correlated assets.

‘Uncorrelated’ meanwhile means the movement of Stock 1 tells you absolutely nothing about Stock 2. It is like if Stock 1 is a coin and Stock 2 is another coin, just because the first coin lands ‘heads’, it does not change the likelihood of the second coin landing ‘heads’ or ‘tails’ – they are two completely independent, uncorrelated events. Whereas, if coin one could somehow influence coin two, in such a way that if coin one lands  ‘heads’, then coin two is more likely to land ‘tails’, something like that, then you have negative correlation between the two coins.

Explaining game theory

JTR: Thank you very much for that. I believe you are a big fan of game theory but I have not seen that many threads on your Twitter account elaborating on the subject. Either way, how can you apply game theory concepts to the world of investing? Or, to put it in another way, what tools and concepts from game theory can be useful in helping us improve our decision making and deal better with uncertainty?

10-KD: Absolutely. I do have one thread that is focused exclusively on game theory, where I talk about ‘Nash equilibrium’. In some of my other threads, however, I reference concepts that I learned in game theory class and I will give you an example shortly. At the end of the day, though, if you look at companies, they are all competing against each other – Visa and MasterCard, Coca Cola and Pepsi and so on. So we have a competitive situation and let’s say Company A is competing against Company B.

Now, Company B has a strategy it is following and Company A also has a strategy it is following but, at the end of the day, the profits these companies make are not dependent only on the strategy they follow – they are also dependent on the strategy their competitor is following. So, if Company A’s strategy is to slash its prices, say, while Company B’s strategy is to keep its prices high, then people may opt to buy Company A's product because it is cheaper – and so Companies B’s profits may come down. In this situation, Company B’s profits depend not just on Company B’s strategy but also on what Company A is doing.

A lot of life is like this: you have to make decisions but there are also other people who are making decisions – and the pay-offs you get depend, not just on the decisions you take, but also on the decisions other people around you are taking. And game theory is just a systematic way to analyse situations like this. If you have a situation where your pay-off does not depend exclusively on your strategy but also on the strategies other people choose to adopt, then game theory applies in those situations.

Nobody operates in a vacuum. Investors are concerned about businesses, businesses operate in markets, markets have multiple players and so on. So the profits businesses make depend on the actions of others – for example, on the actions of the US Federal Reserve or on the regulatory environment. Again, the profit a business earns is not completely within its control – it has to deal with suppliers, it has to deal with customers, it has to deal with employees. So businesses deal with all these different entities and they all take decisions in a way to maximise their own self-interest but that ends up affecting the outcome for all of these different people. So game theory definitely applies in a systematic analysis of these kinds of situations.

AE: That is great, thank you. We focus heavily on probabilities in this podcast and we know that, over the past two years, you have put loads of decision-making tools on your Twitter handle that can help people with this. If you could pick just one from all those you have looked at, which would it be?

10-KD: Oh, that is a hard one! As Charlie Munger likes to say, to a man with a hammer, everything looks like a nail. So if you have just the one tool, there is a good chance you will abuse it and use it for all kinds of things it is not to be used for. All the concepts we have talked about today – game theory and probabilistic thinking and so on – they generally apply to decision-making but you really need to have a large number of different mental models because real life is complicated. Just having one tool may not be the right answer as it may force you to do things you are not supposed to do and so on. So I don’t know what one tool I would pick! I would say this – build an entire toolbox.

Book recommendations and one big mistake

JTR: 10-K, we are coming to the end of our conversation, which has been fascinating. At this point, we always ask our guests two questions – the first of which is for a book recommendation. I have a suspicion you are going to recommend something on probabilities maybe?

10-KD: I love to read books but, when people ask me for book recommendations, I am always a little wary about suggesting one because what you take out of a book ... ‘no man crosses the same river twice’, as the saying goes. So what you get out of a book depends on how much you know already about its subject and also on what your other experiences are. Whether you find value in a book or not depends strongly on these factors.

Generally, the books by Nassim Taleb are great if you want to familiarise yourself with basic ideas of probability. If you want to learn about investing, meanwhile, I would suggest you read Warren Buffett’s letters, which have also been summarised in many books. Larry Cunningham has written a lot of different books on Buffett but his The Essays of Warren Buffett is great for understanding the basics of investing as laid out by Buffett. Another great Buffett book is Roger Lowenstein’s Buffett: The Making of an American Capitalist.

If you are interested in reading business biographies to learn more about specific kinds of businesses, Brad Stone’s The Everything Store about Amazon is just a wonderful book to read. My friend Jimmy Soni recently wrote a book about the history of PayPal, The Founders, which I really love. He has managed to take a book about a Silicon Valley tech company and make it read almost like a mystery thriller. It is a page-turner. But it is hard for me to recommend just one book – I think all these different books are great.

JTR: That is a great list. I have to say I did take the recommendation from your Twitter account on David Barash’s The Survival Game.

10-KD: That was my first introduction to game theory – and, oh yes, I said I would give you an example of where I use game theory in some other threads. There is this wonderful technique called ‘backwards induction’ that I learned in one of my game theory classes and recently I used it to solve this little problem called ‘The Devil’s Card Game’. So you will find game theory ideas sprinkled throughout my threads. The Survival Game is a non-technical book about game theory so you do not have to know any math or anything like that to read it. It is a very good book about the fundamentals of game theory and goes through a lot of different games, noting what is rational and what is not rational for each player and each game and things like that. It is a good read. Sylvia Nasar’s A Beautiful Mind is also a good book about game theory.

JTR: We very much enjoyed your thread on The Devil’s Card Game and discussed it among different team members only the other day. That was a great thread. Before we let you go, we also always ask for an example of a decision you have made where the poor outcome was down to bad process not bad luck.

10-KD: Oh, I think I am an expert on bad process simply because I have so much experience in making bad decisions! When I first started out investing, I had absolutely no idea what I was doing so I made a lot of bad decisions – and that all comes down to bad process. In fact, when I first started out investing, any good decision I made was purely by luck!

One of the biggest mistakes I made as an investor early on is I was perhaps influenced a little too much by Benjamin Graham and Warren Buffett and I started looking at price above business quality. First of all, I did not even know when I looked at a set of financial statements whether these were the financial statements of a wonderful business or a terrible business. Now, I did not pay any attention to this and I did not know even how to read the financial statements to try and get an answer to this question.

I was buying stocks that were trading at a low price/earnings (P/E) ratio and I was focused very much on value, when it turned out that maybe I should have bought some higher-quality companies – so not paid as much attention to the P/E ratio as to the quality of the business. I first started investing in 2011, when there were some very high-quality companies that were trading at very reasonable prices and multiples – for example, Visa and MasterCard were trading at something like 15x or 16x earnings.

I could have just bought those companies but, instead, I bought companies that were trading at a P/E ratio of 5x, 6x, 7x, 8x – things like that – and I lost a lot of money because they turned out to be not-great companies, not great investments. Soon after I bought a company, the earnings would drop or something like that and the stock would crash. And I would be left wondering – where did I go wrong? If I bought the company at a P/E ratio of 8x, how can I possibly go wrong? So I made a lot of mistakes and could fill an entire podcast with all the decisions I made that went wrong because of bad process and not bad luck!

JTR: As deep-value investors, we can totally relate to those experiences! 10-K, thank you very much for your time. It was absolutely fascinating.

10-KD: It was a delight. Thank you so much for having me.


Juan Torres Rodriguez

Juan Torres Rodriguez

Fund Manager, Equity Value

I joined Schroders in January 2017 as a member of the Global Value Investment team and manage Emerging Market Value. Prior to joining Schroders I worked for the Global Emerging Markets value and income funds at Pictet Asset Management with responsibility over different sectors, among those Consumer, Telecoms and Utilities. Before joining Pictet, I was a member of the Customs Solution Group at HOLT Credit Suisse.  

Andrew Evans

Andrew Evans

Fund Manager, Equity Value

I joined Schroders in 2015 as a member of the Value Investment team and manage the European Value and European Yield funds. Prior to joining Schroders, I was responsible for the UK research process at Threadneedle. I began my investment career in 2001 at Dresdner Kleinwort as a Pan-European transport analyst and hold a Economics degree.

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