Size Your Bets Using the Kelly Criterion with Mathematician Nicholas Yoder
Nicholas Yoder is a mathematician with twelve years of experience in derivatives trading and quantitative finance. He gives lectures to various institutions including The World Bank, Carnegie Mellon, and billion-dollar hedge funds.
Nicholas joins Chris for a conversation on correctly sizing your investments using the Kelly Criterion, the financial alchemy of dollar-cost averaging, and strategies for sticking to your investing principles when you need them most.
See below for audio, resources mentioned and conversation transcript.
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Topics:
(00:00) Intro
(02:07) Sizing your bets in conditions of uncertainty
(04:53) Finding the opportunity in a smorgasbord
(08:55)The phenomenal story of Claude Shannon and Ed Thorp’s first wearable computer
(15:30) What is the Kelly Criterion and how does it work?
(29:45) Ethereum Strategies and deciding a reasonable range for variables
(42:10) Our bets do not live in a vacuum
(46:18) Q&A
Conversation Transcript
Note: transcript is slightly edited for clarity.
Chris: Chris (00:05): Welcome to Forcing Function Hour, a conversation series exploring the boundaries of peak performance. Join me, Chris Sparks, as I interview elite performers to reveal principles, systems, and strategies for achieving a competitive edge in business. If you are an executive or investor ready to take yourself to the next level, download my workbook at experimentwithoutlimits.com. For all episodes and show notes, go to forcingfunctionhour.com.
I'm incredibly excited to introduce our guest for today, Nicholas Yoder. Nick Yoder studied mathematics and finance at Carnegie Mellon University before starting his career as an equity derivatives trader at Morgan Stanley. Nick has over twelve years of experience in derivatives trading and quantitative finance. I don't know how that's possible to have twelve years already. Nick gives lectures on machine learning, quantitative finance, and derivative theory at various institutions including the World Bank and several hedge funds with over twenty-five billion in assets under management. That's a lot. In his spare time (I think he has some) Nick serves on the boards of several public health and education nonprofits in the Tri-state area.
Suffice it to say that Nick is one of the most intelligent but (most importantly) thoughtful humans that I know. You guys are in for a real treat today. If you're joining us live for the first time, you should know that this is going to be a fireside chat followed by a little Q&A, so if you'd like to ask a question, ask your question via the Q&A function at the bottom of the screen, and also I would encourage you to upvote the questions that you'd like to hear so we can make sure we have the conversation that you want to have.
That being said, let's get started. Thanks for joining us, Nick. Awesome to see you, really excited to dig in today.
Nick (01:30): Yeah, Chris, thanks so much for having me on. I really appreciate the opportunity to talk to you for a little bit.
Chris (01:35): So, let's set the stage here. The title for today's talk is, "Sizing Your Bets." What do you consider a bet?
Nick (01:44): So, I would say a bet is any time you take something of value that you have—It could be money, but it could also be time, it could be an opportunity, it could be choosing between a set of different options—And you have to make a decision under uncertainty. So, really classic examples are money and gambling. But it can be in all sorts of life situations.
Chris (02:07): So, high level, when you're choosing where to invest your limited resources, whether that's time, energy, money, in these conditions of uncertainty how do you choose how large to bet?
Nick (02: 24): So, great question. You know, a lot of what I think about are the possible risks which in many cases could be total loss or it could just be, you know, you're liquidating something at a lower level. I think about the upsides and opportunities. Just, where exactly could this go? You have to have some sense of the probability of those two things. And then, really importantly, how does this fit with all of the other bets and decisions that you've made under uncertainty? How does this fit into your portfolio or how does this fit into your life, and then yeah. You decide what the right amount is.
I think, you know, one of the specific examples that we can talk about that's a formula that's known in gambling, in trading, in lots of other environments is the Kelly Criterion. 'Cause that's like the real, real mathematical way of thinking about it. And I love talking about the Kelly Criterion because it gives you a very clean way of thinking about risk and thinking about sizing, and then once you understand this sort of very simple formula, kind of an E=mc^2 type scenario, then you can start progressing a little bit away toward real-world scenarios and saying, "Okay, but do I really know the probability? Do I really know the risk? Do I really know how this fits in with other things?" And getting those questions right is what creates really successful investors, traders, and people who are just managing everything they're doing in life.
Chris (03:50): Yeah. I think you put this really well in the magazine article you had on Kelly Criterion. So I'm gonna just give a direct quote. You said that two keys are needed to unlock success in professional gambling, trading, and investing. The first is finding profitable opportunities, and the second is correctly sizing those investments or bets. And what people from the outside might not realize is that there are an abundance of possibly incredibly profitable opportunities for us, but that the hardest part is understanding how much to commit ourselves to these various opportunities. When we have an abundance of choice, the most important thing is how do we prioritize. And so you talk about how what separates the amateurs from the professionals is this ability to determine, "These are the bets that I want to back up the truck, because there's so many good things we can do—"
I think—Author of Principles—Gonna come to me in a second.
Nick (04:52): Ray Dalio.
Chris (04:53): Ray Dalio puts this very well, it's like, "Life is just a giant smorgasbord. A huge buffet of so many wonderful things that we can taste, but that we need to give up some things that we want in order to increase our chances of getting the things that we want most." Is this a good summary, would you say?
Nick (05:11): Yeah, I think that's a really good point. That smorgasbord of opportunities, most people think when they're getting into investing or when they're getting into trading, poker, or card counting, anything, that the difficulty is in finding that opportunity. And I know that I don't have to tell you this, as a professional poker, you can only have played the game for a relatively short period of time and know whenever you're in a good position. But the question is then how do you play that hand. What sizing do you put in and how do you approach it? And that's truly the difficult part. I mean, I've seen serious people, multi-billion dollar hedge funds, major investors, people who you read about in the news have basically a blow-up, because they're just not able to properly size.
Chris (06:02): So before we get into the formula, let's just set the stage. What do you think Kelly was trying to accomplish when he came up with this formula?
Nick (06:12): So it's really interesting. You know, if we put ourselves back in that time of when the Kelly Criterion came out, this was 1956. He was working at Bell Labs, and for anyone familiar with that Bell Telephone was basically the Google of that day. And they had so much extra capital and they had so many real-world problems they were trying to solve that they were just bringing in mathematicians, physicists, thinkers, and paying them to think and work on problems without even necessarily knowing where it would be applied. A lot like I think what's happening at Google and a few other large firms today. And the results that came out of Bell Labs in the 1930s, '40s, '50s, '60s, benefited the company enormously, and it allowed them to produce better and better communications and the basis of what eventually became the internet, but it also just revolutionized a lot of areas of science.
And so John Kelly was working at Bell Labs at this time, which was just sort of you know, smartest minds in the world all in one building, and just before him, eight years before him, one of his coworkers, a guy named Claude Shannon, who—You know, a lot of people say if you were to say, "Who are the two most intelligent people of the 20th Century," they say it's Einstein and Shannon. And I've also heard a lot of people say that that comparison is unfair to Shannon. Because this absolute brilliant guy, and much like Einstein kind of working in the shadows and then coming out with this fully-developed theory that he launched on the world, or at least, you know, special relativity was fully—Was that area, and then fully developed ten years later—Claude Shannon did the same thing with this theory that is not as well known, but probably more important, called Information Theory.
And in 1948, working on his own (and he never liked to, you know, just publish papers or get anything half out there), he just all at once released this thing that both created a field and completely closed it. He basically solved every problem within the field at once and wrote—Put out this opus. And that was information theory. And that became the basis of pretty much everything that is done today with information and probability and trying to send signals. So cell phones, internet, satellite technology, data compression, MP3s, air correcting code, where you have a bunch of noise and solar flares and background and how can it correct itself and how can you do that in the optimal way—This is all solved in one paper in 1948 called Information Theory.
So you have John Kelly, and he's working in this place at this time and this huge event had just occurred, and as he was reading through it and working with it he realized that Information Theory was perfectly applicable to gambling. And so what he did was he wrote a very specialized version of Information Theory and boiled it down to one formula, and said, "This is the optimal amount that you bet whenever you have a bet under uncertainty." Now there's still a few tricks. You have to know what your probability is of winning, you have to know how much you'll win, the odds, but he was able to boil it down to this one kind of simple formula, and then very quickly after that the world took notice and people started applying it in a lot of interesting areas. In the stock market, in gambling—Chris, you know this. Claude Shannon himself then turns back around, and with another brilliant professor takes this formula and ends up having this sort of amazing escapade that goes on for about ten years in the two of them dabbling in Vegas, and beating the house. And it's a phenomenal story.
Chris (10:05): Let's hear more about that. Obviously, it's personally interesting to me. How did Shannon and Ed Thorp apply this? I think they first tried in Roulette, right? Something that we consider completely random. And then they ended up moving to blackjack, and Thorp's system became the genesis of what we call in the modern day card counting.
Nick (10:26): Yes. So as you mentioned, Claude Shannon's partner in gambling, partner in all of these escapades was a guy named Ed Thorp. He's still alive today, and he's in my opinion the most impressive trader of the last century. The best quantitative mathematician to apply himself to the markets. And for a little bit of context, Ed Thorp was this polymath guy who basically came out of nowhere and without a calculator won a statewide competition in California to then get a free scholarship. He went and he got—He was brilliant in chemistry and then randomly switched to math. Basically finished a PhD in physics, and then in a period of about two months did a PhD in math as well, just because the two fit together. He in that time got access to the old punch card computers, and ended up working it out to create what is now card counting. He figured out optimally the single counting variable, a symmetric variable of high/low, the perfect one variable that a human could remember that would best map over to your implied odds in blackjack, and he founded that industry.
And then Ed Thorp as the you know, real wonks of stock options will know, and people who know the history, Ed Thorp actually invented the Black-Scholes formula. I remember as a young, you know, equity derivatives trader always wondering, "If these two academics, you know, Black, Scholes, and Merton invented this thing, why didn't they publish it and go and win the Nobel Prize? Why didn't they start up a hedge fund and keep it secret, and make money off it? Well, it turns out that the actual inventor actually did that. He found it, he kept it secret, he started a hedge fund, and it was only many, many years later that the guys that we know the name were the first ones to discover it and publish it instead of trying to make money. So we call it Black-Scholes. But Ed Thorp invented that.
So you have this guy that was just sort of a polymath at different areas of trading, and then he comes along and he meets Claude Shannon, who's working at Bell Labs, and is world-famous at this point and can kind of do whatever he wants, and they start working on card counting. Going out to Vegas, plying their money and using these odds and using the Kelly Criterion to take a relatively small amount and just build it into a pretty big fortune.
And then while they were out there, they started looking around at other games, and they started finding where the games were that had some sort of edge in them, and as you mentioned they were looking at roulette. And roulette is interesting because there shouldn't be an edge to it, like blackjack, unless you have some sort of superhuman ability. If you could actually track the ball in the wheel and the rate at which they're counter-spinning and sort of figure out where it's going to land—And they said, "Well, you know, if we had a high-speed counting device, if we had a computer," (and of course computers were the size of rooms), "And we could get this thing to work out, and then If we could get the data back from this computer, really, really quickly, like while the ball's still spinning, while they still let you make bets," (because in roulette after the ball starts you still have a little window where you can makes bets), and they said, "If we could get the information in, calculate it, get it back, and then as a human interpret it fast enough to put the chips on the table, we could have some edge in roulette."
And what they ended up doing in this collaboration in Claude Shannon's garage was they built the world's first wearable computer. And to think about how crazy this thing is, that before this time computers were the size of rooms and they required their own independent power supply and transformers, to where they built something that would sit in their shoe, take in this information, and then send tones to an earpiece in their ear—This world's first computer is now at the museum at MIT, and it's considered just this quantum leap in things.
So these two working together, they built this, they go out, they start getting an edge in roulette. What do they do? They apply the Kelly Criterion. They say, "We have edge, but how much do you bet? How certain are you?" And because the formula is so simple they were able to compress that information into a very short period of time and get that back and make a bet in, like, twelve seconds.
Chris (14:50): Wonderful context, Nick. Thank you for sharing the story. If you guys are interested in this, and I assume if you're here you are, I highly recommend the book, "Fortune's Formula," by William Poundstone. Maybe, in my opinion, one of the most entertaining books that touches on finance out there, so you can kind of learn about how all these different themes, information theory, gambling, investing, really have so many similarities and all the innovations have fed off on each other. Nick, I think you had a couple of slides to share today. Maybe let's open by giving a visual of the formula, and we can break down the component parts.
Nick (15:30): Yeah. Why don't we just talk about what the—Do a little screen share here. Well, I guess first I'll give you just a bit of context, and then I'll show you a couple of graphics. So the idea of the Kelly Criterion is it captures two countervailing forces that happen whenever you place bets, and it figures out how much they interact with each other. So as we said earlier, when you have a winning opportunity, that's actually the relative easy thing to find. The question is how much money do you put in. If you have an opportunity that has some edge—So, let's take a really specific example. Let's say that you are card counting and you have a bet and you know that you have a fifty-two percent chance of winning. Even odds, one to one. You bet a dollar and if you win you'll get your dollar back plus another dollar. And if you have a fifty-two percent chance of winning and a forty-eight percent chance of losing, how much do you bet?
Well, if you just keep increasing your bet size, the bigger your bet size becomes, the more it's going to increase your expected profit on it, that's going to go up linearly, but also you have this problem of if you, say, win a bet and then lose a bet, you start to have this multiplicative problem where you actually end up decreasing your bank roll. Like, if I made ten percent on one bet and then I took all that money and I lost ten percent on another bet, now I'm back to ninety-nine percent of what I originally had. And it doesn't matter the order. If I had lost ten and then made ten, I'd be at ninety-nine percent. If I made a twenty-percent bet and I won, and then a twenty-percent bet and I lost, I would have ninety-six percent of my starting point. It doesn't come back evenly. And then of course the extreme example is if I was betting a hundred percent of my bankroll, if I win and I lose or I win and I lose, either way I'm at zero, and there's no way to come back from being at zero.
So you have this kind of losing amount where the fractions work out such that your losses hurt you more than your wins, and that cumulative effect I call negative geometric drag. Because it's geometric, you're multiplying two fractions together, and then it's going to decrease. But it ends up happening as the square of things. So what I'll do is, I'll show you the formula and then I'll show you what all this looks like graphically and how you actually decide on the best bet. And—Okay. So jumping to the end here, this is the actual formula. This is the answer. And we'll come back to it. But probability P is your chance of winning on a bet. Q is losing, so that's just 1 minus P. B is your bet odds. So in the example I gave it's just one to one, so B is just one. But you could have something where it's a 10x payout or a 100x payout, and so you would include that in here. And whenever you take these bets and you actually apply them out, you get this sort of combination of curves.
So the orange line is as your bets get bigger, and that's going from left to right on the graph, as you're placing more and more of your bankroll into a bet, the amount that you would expect to win, just by saying, "What's my edge," and multiplying it by that, just goes up linearly. Just a straight line. And then the hidden part that you don't see is that effect of losses times wins, that negative geometric drag, is this grey line that starts out—It isn't very impactful, but the bigger you grow, it grows as the square of the size of your bet. And this is the really key point to remember, if you just take one thing with you. It's that whenever you place bigger and bigger trades or you bet more money with your own income or you do whatever, the good part of it grows linearly. Your expected profit grows linearly. But this drag, this how far you're behind the game when you lose, grows as the square of the size of your bet.
So when you add these two together, you get the blue line, your actual profit. And it goes up and up and up and then eventually it levels off, and there's one sort of—I don't even want to say 'optimal' amount to bet, it's just the amount that you would maximize your long-term profit. And then it starts going down. You're still making money, but you're not making it as quickly. And then eventually the blue line goes negative. You'll actually lose money on winning bets. And just for fun I drew out from that card counting example, if you were playing and you had a fifty-two percent chance of winning, I said, "All right. What if you—You know, let's say you had ten thousand dollars. That's your bank roll. What if on every hand you just bet one percent of that?"
And you get this sort of blue line here where you make money and it goes up over time. If you bet two percent, you get orange and that goes up and you have—That's actually pretty profitable, and it's pretty consistent. Orange is probably the amount that I would do personally. If you bet four percent, that's grey, and that's kind of the maximum amount that you can make on this, but it's very erratic. There's one point where you're down more than fifty percent. It's all over the place. You're up over two hundred, you're down below fifty bucks, it's really erratic. And then if you keep going up, eventually at eight percent of your bet roll on each hand you reach a point where you're just not making any progress. You're back to the start after a thousand hands of playing. And then if you go any bigger than that you actually lose money. Even though you're doing the card counting correctly, even though you have the right opportunity, because you're doing too much of a good thing you're just poisoning yourself and your bankroll is going down over time. So I kind of showed this graphically.
So you know, for me I'd probably do something like two percent or four percent. But if I just show this in terms of how big the bets are going from left to right and then how quickly you're compounding your money, the curve basically looks like this. As you use more and more leverage, as you go to bigger sizes, you're going to make more and more and more profit. And what the Kelly formula tells you is actually the peak of this curve. It's the maximum rate that you can grow, but in a very erratic, you know, variant way. And if you go bigger than that you're not making as much money, and if you go really bigger than that you actually start losing money. And so anyone that is a professional trader or a professional investor will tell you that you never actually bet what Kelly Criterion tells you. That's just this limit at the very top of this very, very peak. But even getting close to that limit, you're just like, you're risking more, and more, and more of your capital for barely any more profit. Pennies more. So kind of the happy point is to go about halfway up this curve or three quarters of the way up this curve, where you get really good return, you don't have a ton of variance. This is what most people are aiming for, and then everything beyond that is just, yeah. Silly.
Chris (22:26): Amazing, Nick. I want to make a meta-point here, to just drive it home, how important this is to get right, that you can have most of your investments be profitable and lose money. You can have the majority of your investments be profitable and lose money if you do not size correctly. And as we'll discuss, the way to size is proportional to (A) your conviction on it, right, the size and confidence in your edge, and (2) the odds you are getting. How asymmetric is this bet? But that if you do not understand and internalize this concept that bet sizing is even more important than determining what to bet on, you will lose money. So, yeah. Just want to drive this home, that, "Hey, I personally was like, ooh, math, I don't want to do that. That's icky." Well, I like making money, personally, and the math really isn't all that hard. So even if you aren't going to get it right, and as we'll talk about it in the real world you're not going to get these variables right, you're going to be overconfident and there's a lot of—We are making decisions in conditions of uncertainty. At least modeling it, at least determining what are these invalidated assumptions will allow you to come up with a range of appropriate sizing, and then you can use your own risk tolerance to size within that range.
Nick, maybe this would be a good time to sort of plug some variables into that formula and give a real-world example. I know you have a few.
Nick (24:08): Yeah. Yeah, definitely. Let's do that. And Chris, I think you really hit the nail on the head there, because part of this is as I said you can use the formula to find out an optimal, you can go a little bit less than that to be safe, but something you said is you're going to make wrong assumptions. These numbers that you plug in are not going to be correct. And since we see that if you go a little lower than the optimum you still make a lot of profit and everything is good and there isn't much variance and if you go a little higher it's a very bad scenario, one of the ways of being humble and knowing that your assumptions are wrong is to always skew that to a half of the Kelly size or a third of the Kelly size. Yeah, thank you for making that very clear.
Chris (24:53): I think you shared this with me and it really blew my mind, is that what you get out of the Kelly Criterion is the upper limit of what you should bet. You showed that graphically, that that's—What you receive is the top of the curve, and that you should always be betting a smaller amount than what the full Kelly gives you, and how much smaller you go is a function of how confident you are that the inputs into that equation are correct. Right? The larger the degree of uncertainty, the more you should discount what the formula gives you. And you shared that by—I think it's very counterintuitive that by going smaller, for example, with half Kelly, you really think, "Oh, well I don't want to bet half-size, because I'm only going to get half of the returns." No. You can get—By betting half of the amount that Kelly gives you, you get seventy-five percent of the profit but only twenty-five percent of the variance. This is actually a 3X better adjusted, risk-adjusted return than full Kelly.
So that—We think that going conservative is a bad thing, that we're leaving money on the table, but we're actually capturing a lot of the upside but without all of these ruinous risks of you know, big draw-downs that are so costly when it comes to long-term compounding.
Nick (26:27): And that's a good point. You know, in the example you were showing, by going to half your risk-adjusted return is three times as good. Imagine if you're managing someone else's money. If you're working at a hedge fund, if you're operating with clients. And so it's not just you sitting at home, being totally stoic, and saying, "I'm okay with losing ninety percent of my money and then bouncing back." Well, you can't really do that. If you're managing client money, ten percent down, fifteen percent down might be the end of your business. And so you want to find those places on the Kelly curve where you're improving your risk-adjusted return by three-fold, that's five-fold. Finding somewhere that's the appropriate ratio for your business and for your clients. And that's very important as well, because that's part of winning the long-term game.
Chris (27:14): Awesome, thanks. Sorry to interrupt. Let's hop into an example.
Nick (27:17): Okay. So let's do two examples from this formula. So the one that I started with, if you were card counting or if you had any bet where it was an even bet, pays one to one, but you have a fifty-two percent chance of winning. Then in this formula your B, your odds, would just be one on the top, one on the bottom. And your probability of winning would be fifty-two percent, .52, and of losing is .48. So .52 minus .48, fifty-two percent minus forty-eight percent, you would bet four percent of your net worth, of your bankroll, on each one of those hands. And if we go back to this example, four percent is exactly the one that performed the best.
What you and I are advocating is that going half Kelly and it making almost as much in way less variance, that's the orange line. That's betting two percent. And you can sort of see graphically how consistent it is and how well it does here. But in this formula, the amount you wouldn't go beyond, your benchmark Kelly, would be four percent of your net worth.
All right. What if you had something that was a little harder to think about, intuitively? What if someone came to you and said, "Okay, we're working on this VC investment, if it works out it's going to pay out ten to one on what you invest, but there's only a twenty percent chance of success, eighty percent chance of failure." Well, you could do a little bit of mental math and say, "Okay, well if it's ten to one return and I have a one-fifth chance of winning, it's at least positive expected value. I should expect more than one dollar for every dollar I put in." But how much of my net worth or of my fund should I really be putting into that to be Kelly optimal?
So, if we did the same thing in this formula. So here the B is ten. Ten to one return. The probability of success is twenty percent, and of failure is eighty percent. That's going to be ten times twenty percent, two hundred percent, minus eighty. That's a hundred and twenty percent over ten. So you would want to wager twelve percent of your net worth or twelve percent of your investment fund to be at the Kelly optimal point. To be at that, you know, nothing beyond this. That's the highest profitability.
And in reality, if you were doing this you'd probably do about half Kelly. You'd probably put about six percent of your portfolio into that bet.
Chris (29:45): I think you shared an example about investing in Ethereum as well, which I think would be instructional. Now, a good time for a disclaimer for any lawyers out there, this is not investment advice. We may or may not hold current positions in Ethereum. But I think it's useful to have an example that may be close to the experience of some of the listeners out there as far as many bets in the real world don't have clean odds and probabilities. How do we go about investigating and coming up with a reasonable range for these variables if we want to try to use this equation to model, you know, making a portfolio investment, for example?
Nick (30:33): So, those are a couple of good questions. I guess I'll take two things in turn. One is about the ETH example, which I used in the Lockbox Magazine article, and then some of the more portfolio aspect, how you combine things together. So for the example in the magazine article, I basically just took a historical trend-following ETH strategy. I didn't use one which I personally use in my trading, I just tried to use something that was open source so it would be easy to use as an example. And it did very well historically, so I just started showing, how would it work at different levels of leverage? What would that look like in terms of a Kelly optimal return? And it changes, you know, ETH itself over that time returned about two hundred and thirty percent of IRR, and this version with trend following and the right amount of leverage it comes up to about six hundred percent return. But again, Kelly optimal, way too much risk. But I then demonstrate that if you do about half to three-quarters that size, you will have (historically, big disclaimer, historically, not necessarily in the future), it would have returned about five hundred twenty percent return annually, but at only half the variance.
And that's the example I used. If you were working with that now and you wanted to extrapolate forward, first thing I would do is I would say, "Everything that's happened in this asset, clearly in the past is not indicative of the future. Eth was brand new, no one knew what it was, now it has massive market cap." So I would take all of those assumptions and then cut them by a factor of four or a factor of five. And I would start saying, "Okay, I think there's still edge, I think there's probability that works in my favor, but whatever it is over fare, I'm gonna start right out of the gate and divide it by five. And then plug it in, and say, "Now what is it suggesting that I put as a portion of my net worth into this trading strategy?"
That's kind of the base example if you're doing one asset. Now, the really important thing, and I've touched on this only a little bit in the Kelly article, but I went into, you know, very, very deep depth in the Wall Street University series, is how do you optimize bets whenever you have more than one bet that's happening at once? More than one asset, more than one investment. How do you start thinking about covariance, and the way that pieces fit together, and then how much you're allocating to each one? And you know, it's a little bit too much math to go into in our conversation right now, but I'm sure we can throw the link in the chat, and all the code and fun stuff is in there.
Chris (33:23): Something that was really counterintuitive for me as you were modeling out these Ethereum returns is that you looked at another option which we hear about in traditional investing, and it particularly comes up in Crypto, where someone who's interested in getting into it is always afraid of buying in at the top, and so the advice is always to dollar cost average. To pick an amount and just invest that no matter what without—Each month, and that the nice thing about this strategy is that you tend to buy more shares when things are undervalued. Right? Because you're investing the same amount, but that amount buys you more shares. So you're buying in lower, and the same thing is you're buying less when the price is higher. And you found that just this simple dollar cost averaging strategy actually outperformed a lot of these more complex alpha-driven strategies. Talk to me about that.
Nick (34:34): Yeah. Absolutely. Dollar-cost averaging is, it's one of the really cool sort of parts of financial alchemy, because it's really easy for the average person to do, it very much does work, it's in your favor as the buyer of an asset, and I think most importantly it's tax-efficient. Because you're buying something, it's not like you're trading around and you have to get in and get out. You know, one of my big pet peeves is whenever there are ads online telling everybody to become day traders or try this strategy and it goes in and out. You've already set yourself up where there's such a hurdle to get over, because you've totally changed how you're paying taxes. Everything is short-term trading. And it basically has to be two times as good out of the gate to just end up giving you the same returns.
Dollar-cost averaging is this really cool way of benefiting from the volatility of an asset in a way that is actually profitable to you. And the idea is that if you were to, you know, let's say that you wanted to buy a bunch of S&P 500 stock, or you wanted to buy Ethereum, or any asset that is volatile, how would your returns compare if you bought it in one bullet versus if you spread it out into buying in some regular fashion. You can do every day, but it doesn't even have to be that much. If you go weekly, that's pretty good. And Chris, you said exactly the crux of it, is there are days whenever it's higher, and there are days whenever it's lower, but the number of shares or the number of coins that you get whenever it's lower is so much more impactful than when it's higher.
I sort of imagine, if you had something that cost a hundred dollars. A hundred dollar stock, if you had a thousand dollars you were deploying, if you bought it all in one bullet you would get ten shares. But if you averaged into it—And it was all over the place. Sometimes it was a hundred and fifty, and sometimes it was fifty. Well, on a hundred and fifty days, you would only get a little more than six shares for your thousand, and on the fifty days you would get twenty shares. And so the average would actually work out that with all that variance you would end up buying thirteen and a third shares instead of ten.
And wherever it ends, whatever that final point is where you sell, well now you have thirty-three percent more of it, so by definition you've made thirty-three percent more money.
And you know, just for fun last night I went back and I did a quick comparison of dollar-cost averaging, and I looked at Eth. Since 2015, when it came out, Eth has returned about two hundred and thirty percent annualized. But if you had been doing dollar-cost averaging on Eth, your internal rate of return would have been about two hundred and ninety percent. You would have made about sixty percent more per year on average. And when you talk about that in terms of investing, that's a lot of doublings over the course of your life.
Maybe Eth isn't a great example because it's only been six years and you're not sure if this will happen in the future, but if you take something like stocks, stocks return about eight percent historically. Sometimes they have terrible years, sometimes they have bull markets. But if you were actually doing your long-term planning for your life, and you said, "Okay, this is all over the place but I'm going to save blank number of thousand per year, am I going to return eight percent?" Like, if there was a magic bank account that returned eight percent. And the answer is actually no. If you were really consistent, and you were buying every year, every month the same amount, your return would actually be closer to about ten or eleven percent. Which again, over the course of a lifetime leads to a little more than double your profit.
But you know, the key is just being consistent and not being that person that gets scared whenever prices go down. 'Cause that's the good thing. You know, we saw this in crypto, whenever everybody wanted to buy in that first bubble whenever it was seventeen thousand, but they didn't want to buy Bitcoin when it was two thousand. But with this strategy you would just be in a very disciplined, algorithmic fashion deploying the same amounts, and clearly you would have acquired a lot more Bitcoin during, you know, late 2018 whenever it was at two thousand, three thousand dollars.
Chris (38:52): Brilliant, Nick. It brings up the obvious question for me, is if something that requires this discipline, buying the same, right, especially buying when you really don't want to buy and everyone's telling you to run for the hills, how do you recommend someone stick to a strategy like this that requires this discipline, that requires this consistency? Do you recommend automating? How do you apply these rules so that you can stick to them when you are most inclined not to stick to them?
Nick (39:27): That definitely is the hard part. Self-management is at the end of the day the gate that controls all of these actions. And you know, Chris, you have a huge body of work on self-control and state management in poker playing, in elite performance. I know you work with a lot of notable executives and you help them with planning their state, with exactly that sort of self-management so that you're being consistent over time. I think from an investing point part of it is a set it and forget it. You know, have something that you can automate where money goes over to your blank account and then just every Friday you buy. And it's not even you know, immediately hitting your checking account where you can spend it.
I think for me personally, going back and seeing how this would have worked in the past is what really motivates me to stick with a strategy. You know, doing that little experiment last night where I back-tested Eth dollar cost averaging and it's two hundred and ninety percent versus two-thirty. That's pretty convincing. That's going to make me want to stick it out if, you know, you have another ninety-percent crash, and be committed to something.
You know, the actual—I'll just leave you with this, the actual formula for how much you'll make with dollar-cost averaging, and this is a nice sort of bookend to our conversation, because you know, Kelly talks about variance is really bad and variance destroys long-term wealth, how do you balance profitable opportunities in variance, and then dollar cost averaging is the opposite side of this coin that says variance is very good when you're layering into an investment. How good is it going to be? And the answer is if you have an investment that over the long term returns some rate R, if you are doing DCA the return that you'll get is actually R plus variance over two. Standard deviation squared over two.
So like, in the stock market, stock market returns about eight percent, and it has an annual standard deviation of about twenty percent. Well, .2 times .2 over two is an additional two percent. So that's why you get about a ten percent return. If you look at Eth, if Eth's returning two hundred and thirty percent historically, and Eth has an annual variance of, you know, variance of—Standard deviation's about a hundred twenty or so. If you take 1.2 squared over two, yeah, you're getting a bonus of about sixty or seventy percent because of that variance.
To me, that's pretty motivating to stick with something through the good and the bad.
Chris (42:10): Amazing, Nick. One more question for you, and then we have some good ones in the Q&A. I want to give those an opportunity. So you said at the beginning, and I wanted to—You talked about bookends, I think this is a really good bookend, that our bets do not live in a vacuum. That because we have many opportunities in life, each bet we need to make, we need to consider how this bet fits with our other bets. I think one of the most underrated things from economics is this idea of opportunity cost, when you think about this in terms of productivity in performance is that what you are doing at this very moment comes at the expense of everything else that you could be doing. So the cost of slacking off or going on a Netflix binge isn't that bad until you consider all the things you could have been using that time for instead. And similarly, because we already have all of these investments running, financial and otherwise, when we're thinking about making a new investment, thinking about how does this investment that we're about to make fit, how does it support, how does it complement, how does it synergize with the other things that I am already doing. I would love to hear if you could just share how you think about that when you're considering making a new bet.
Nick (43:35): Yeah. So I guess there's sort of two sides. One is when things are very clear you have a lot of numbers and you have sort of your mathematician's hat on, and then the other is you know, the regular everyday life, how do you make decisions? You know, on the first one, as I mentioned earlier, I wrote a lot about how you do the math of covariance and building portfolios and I think you threw the link out there, and that's all of the Wall Street University series. You can throw it into Python, you can do really cool things. Most of the time, we're not dealing in a situation where we know exactly the way that two things move together. We don't know exactly what the payout's gonna be. But the truth is I think with like a little bit of intuitive thought and a little bit of intelligence you can start finding places in your life where you're making massive mistakes and places where you have opportunities to improve a lot, and kind of go along that Pareto efficient of where you're progressing in.
So, a really simple example. The most common mistake that I see people make: I've had friends that are working at a company. As an example, a really good friend of mine from back in Pennsylvania, works in the gas industry, and he had this abnormally well-paying job. He was making maybe double what he normally would because he was working for this one gas company. And then he got an opportunity to invest in their stock. And they were giving him some mild discount, like ten percent discount on the shares. And he told me, he was like, "Yeah, I think I'm gonna just go all in and put all of my 401k in my company's stock, because we're getting this discount."
And I talked to him and said, "You know, I don't really know that much about what's going to happen with your company's future versus if you put it in Berkshire Hathaway or any other stock, but the one thing I'm absolutely certain is your livelihood and you having this really good job is super, super correlated to that stock price and your company's future." I was saying, "That bonus of ten percent to me just isn't worth combining those two risks and having a really correlated thing. You should if anything keep working your job, making good money, and then keep piling that money into something that's super unrelated to the gas industry. Or even better, if you could pile it into something that's anti-correlated to the gas industry.
And fortunately, that conversation went well, he took my advice, and about two years later, you know, nat gas prices dropped, his company laid off a bunch of people, and he was actually doing really well because he had just made real estate investments instead of buying his company's stock.
Chris (46:18): Let's hand it off to the Q&A. First question comes from SA. Nick, for traders how is B calculated? Is it just .5, or is it your historical win/loss ratio?
Nick (46:33): Good question. So both B and P are always going to be a little bit of a judgment call. Not everything is as clean as card counting where you can figure out the odds and you can figure out the bet odds. I think for me I try to use basic heuristics. Like, if I'm doing an options trade and I say, "All right, I'm buying this straddle and I'm getting it for four dollars, and historically it looks like this thing should be worth six." Okay, that would be a .5 to one bet. Or, maybe in a more extreme example, if I bought it for four dollars and I think it should be worth ten. That's a one and a half to one bet ratio. You know, I would be two and a half X-ing my money.
First thing I'm gonna do is I'm gonna take that bit of edge and really cut it down to see if it still makes sense, to see if, you know, this thing that's worked historically won't work in the future. Because at some point everything stops working. Right? Markets adjust, people catch up, somebody becomes as smart as you, somebody gets the tech that you have and it stops working. So you should always expect that to be in front of you and to average it in. So to answer your question, yes I look at the historical and then I just take, you know, a big haircut of assume that my odds going forward are maybe one third of what they were before.
Chris (47:50): It's pretty rare to underestimate your edge. We are classic thinkers in—We are classic optimists, but not optimists in the good sense of we think everything is going to turn out great and we're gonna be in this utopia, but that we constantly overestimate our own abilities and underestimate just how much noise and confusion exists in the real world. So it's generally better to be conservative with your assumptions and to use the output of Kelly as a rough draft, as a starting point to reveal where you have the most uncertainty. Because it is possible to find more information, to do your own research, to talk to people who have done similar bets before and to reduce your uncertainty, to—I forget. You know, talk about it like, you can do your own modern-day P-Hacking if you're not careful. Like, it's very easy to draw the conclusion that you want to draw. So to maintain objectivity, start with, "What do I know, and what do I not know," and what is the fastest, cheapest experiment you can run to bring things from category B, "I don't quite know," into category A, of like, "I can never actually know, but I have a pretty good sense, I have a lot of conviction around this."
I think that's the part of just life and investing that can't fit into a formula, that is another separator between those who just trust the model and those who are able to internalize and transcend the model.
Nick (49:38): And Chris, just to add something to that, you know, you're so right about the self-assessment bias and that rose-colored glass by which we always sort of overestimate our own abilities or we look at our recent wins and assume that so much of that is something that is innate to us that's going to continue. The other thing that I try to be very careful of is recency bias. Humans are biochemically wired to have a recency bias that if we've recently won we're going to overestimate the odds of winning again, and if we've recently lost we're going to overestimate the probability of something bad happening again. And it's happening at a fundamental level. It is the amount of serotonin and cortisol in your brain. And if you've just taken a really big hit or the market's been falling apart, you're going to have a really high amount of cortisol, and you're going to want to start under-sizing your bets at exactly the time when the opportunities are the best.
Chris (50:37): Yeah. And this is a good time to plug our last Forcing Function Hour with Denise Shull where we get into the psychology of trading and how we are acting out our internal biases and impulses with our trades. So if you're interested in improving your mental game and becoming the trader that you can be, I highly recommend checking out that last episode that we did. It's a really good one.
The next question comes from Justin_Kozlowsi. Nick, on the flip side of dollar cost averaging, how do you feel about adding to your winners on strength, AKA pyramiding up?
Nick (51:14): Wow. That is actually a really tough question. So there's a question of if the thing that you're looking at is going to tend to trend. So, if several wins in a row imply that there are going to be more wins going forward, or if it's something which mean reverts. And there isn't one general answer for this. I would say look at comparable examples, look at what this object is like, and then do your research, see how similar assets. If they tend to trend then pyramiding is a good strategy. If they tend to mean revert, then I would avoid that. A classic example of something which will tend to trend are investments in companies where the company has a real intrinsic value, they've developed a product, they have a good team, and then the world is finding out about them. And the world finds out about them—It's basically that their real value if you had perfect knowledge is so much higher than the price that you invested or other people invested, and it's just about that gap continually closing. And so statistically, it's going to look like those prices are trending and they're going up, and that's a great opportunity for pyramiding.
If you have something that's just sort of a fixed, random asset that's laying around, like commodities, it tends to be the opposite. They tend to have these peaks and valleys, but they really mean revert back out. So at a lot of cases when you're winning at a certain point it makes sense to ask when you should be taking some money off the table.
I won't digress. There's a couple other—There's some fun mathematical ways which you can find out which it is, but I think I'll keep this train on the tracks.
Chris (52:58): Yeah. I'll echo that. There's a really good memo from Howard Marks recently, a classic value investor who's talking about how there is a current paradigm shift in that it used to be historically considered that all investments were mean reverting, so that if something was doing really well it was just a matter of time before it came back to its baseline, but in the modern era—And we're thinking primarily in investments that have a technical component, the moats are much stronger and much longer lasting. So you can have a long-term compounder which compounds for much longer and has fewer threats. Obviously, you know, looking backwards, but this seems to be the case, in it's really hard to find these long-term compounders. So if you have a high conviction that you've found one, you likely should stay in it. So that's just an expansion on Nick's point of, what is the type of investment that you are looking at? Is this something that goes through classic boom/bust cycles, or is this something that is going to—That all of its success is going to plant the seeds for future success? That's up to you to decide.
Okay. Next question is from Samir. Samir is looking to YOLO a little bit. Nick, how would you use this discussion on Kelly towards sizing an options-based spreads position or say a call options on a mean stock?
Nick (54:32): Okay, that's a fun one. So I guess there's sort of two aspects again. There's the probability that it's going to work out, and then there's what your payout would be. Now, if you're talking about an options spread, that's sort of a nice example, because it's a little easier to know exactly what your payouts look like. If your spread doesn't close the money it's total loss, that's pretty straightforward. And then you have a maximum amount where you've gone through both of your strikes, assuming it's a one-to-one spread. So that makes the sizing of your payout a little bit easier to do. And the correct B isn't going to be the maximum amount that you can make, because there's also this whole range of probabilities of it settling between your two strikes and you making money, but not at your maximal amount. So you should take that B and then just bring it back a little bit. And if you wanna know what those probabilities are, one of the best ways to do it would be literally just take the volatility of the stock, take a normal distribution, and see how likely it is to end above your first strike, above your second strike, see if there's any events in there between now and your expiry that would increase that volatility, and then kind of use those as some baselines for your P and for your B.
Then the overlay on top of that, if you're talking about a meme stock that you think there might just be another run-up because it's something that feeds on itself, is you have to make a subjective assessment and say, "What are the chances that I think this will happen between now and blank date?" Because in a lot of cases you might be picking a meme stock that everyone is talking about now but it hasn't gone through the many waves that GameStop or anybody else has, and so you have to take a little bit of transference and go, "Okay, so I'm buying these way out on the money coals, I know what my payout could be." Kind of a classic Black-Scholes is that there is a really small probability of this happening, but I see this social movement over here and I know that whenever Wall Street Bets or whoever starts talking about this, there's a ten percent chance that it turns into a run-up. And you can start playing around with those probabilities. And then again, the final step, just be conservative. Whatever you land on, back it up and then do that.
Chris (56:48): Nick, I wanna lay this one on you a little bit as we begin to wrap. We covered so much ground today, and thank you so much for really making this approachable and applicable. If you had a major takeaway for those who are listening, you know, what would you like them to walk away with? How would you like them to approach this? Is there anything that they should be doing differently?
Nick (57:13): Yes. I would say three serious—Three lessons or thematic things to touch on. And I'll just mention—You know, we spent the majority of the time talking about Kelly. At the end we talked a little about dollar cost averaging, and then there is kind of this idea that in reality most of the world isn't—You don't just get the numbers up front, is how you think about this. So the three lessons that I would say are number one, when you're making investments always side on the side of—You know, err toward more conservative, because it's okay. If you make money, that's great. If you lose money, you always want to make sure you're still in the game. You want to live long enough to live forever. And that's the important thing.
I remember when I first started doing this when I was twenty-two and I was on Wall Street, and like before that when I was trading options in college, and you would just find something great and it was just love at first sight and it felt like the only time you're gonna find this really great trade, and you're just gonna pile into it. And now, two hundred of those later, I can tell you there's always going to be another, and another, and another one after that, and one that you missed 'cause you were back home for Christmas or on vacation or weren't paying attention, and another after that. So like, live long enough to live forever. Undersize.
Number two is, if you really wanna benefit from all the craziness and all the volatility and all the opportunities, be hyper-consistent. Figure out a strategy and just stick to it. Stick to it when it feels great and don't make it bigger, and stick to it when it feels bad and don't turn it off. Just be consistent and it's going to pay out for you.
And then the third thing is use your head. Be logical. You don't have to figure things out to one percent accuracy. You have to get the broad strokes. I used the example of my buddy, he was making good money and I was just talking to him about, "Don't invest in your company's stock." You know, "Diversify yourself out, why are your all your eggs in one basket? Kind of think about this logically." And you just have so many things like that. Just use your head. Warren Buffett made an investment a couple years ago in PetroChina, and people were making fun of him because they were saying, you know, "You didn't do any due diligence, you didn't follow up, you didn't use any spreadsheets." And he said, "Listen, I read the report. It looked like back of the napkin the company should be worth about a hundred billion, I looked them up, and they were only trading at thirty billion." He was like, "So I bought. Like, why would I have to work something out to the nth decimal place or use a calculator? If it's that far off, I do it."
And I think in general if you were making investment decisions where you really need the very last significant digit to work out for it to be a good investment, you're in the wrong field. You're looking at the wrong thing. Stop trying to squeeze bips out of something and find things that are just so stupidly off, and then bet on those.
Chris (01:00:15): What a great way to end it. Nick, I know that there are so many things that you have compressed into what you have said today. If someone's looking to dive deeper, they'd like to learn about some of these things we've talked about, where would you direct them?
Nick (01:00:28): Gosh. You know, you threw the link in for the book, "Fortune's Formula." I think that's a fantastic book. It's just a fun narrative of all this stuff, and spoiler alert it's math, it's gambling, it's the history of the American mafia, it's Wall Street, it's politics, it's just such a fun read and how all these things are interlaced. I would recommend Ed Thorp's book called, "A Man For All Markets." And it goes into a lot more of him discovering options theory and him discovering stat arb, and just a lot of how things play in the market. I think that's a fun one. And yeah, beyond that I think if you read those two you'll start finding other fun things to read.
Chris (01:01:11): Awesome. Thanks Nick, let's bring it home here. So for those of you guys listening who do not know, at Forcing Function we teach performance architecture, and personally my current offering is one-on-one hands-on training for a select group of twelve executives and investors in meaningful companies. If you're interested, I will show you how to unlock your natural brilliance, so you can perform your roles to the best of your abilities and enjoy a life of freedom and purpose. Unfortunately, all of my one-on-one coaching spots are currently full, but I am offering a group coaching experience, Team Performance Training, which kicks off in February. If you're interested in that, you can go to teamperformancetraining.com. I also recommend that you download my free peak performance workbook at experimentwithoutlimits.com and subscribe to our Forcing Function newsletter so you can find out about upcoming events and start taking yourself to the next level.
Thank you again for joining us, Nick Yoder, and see you all again soon.
Tasha (01:02:08): Thank you for listening to the Forcing Function Hour. At Forcing Function, we teach performance architecture. We work with a select group of twelve executives and investors to teach them how to multiply their output, perform at their peak, and design a life of freedom and purpose. Make sure to subscribe to Forcing Function Hour for more great episodes, or go to forcingfunctionhour.com to sign up for our newsletter so you can join us live.