In this video we'll talk about a very important topic in investments and sports bidding management. Kelly Growth criterion. There's a great book by William Poundstone, it's not just about Kelly Growth, but it's called Fortune's Formula. that details the history of the Kelly Growth. And unfortunately Kelly was mugged on the streets of New York and was killed, he used to work for AT&T or I guess it was Bell Labs at the time. Okay, so let's start with a little puzzle for you. You have two investments in which you can put all your money, and these are your only choices. One investment will earn you 4% every year for sure. The other investment, you'll flip a coin every year. Heads, you'll win 60%, atils, you'll loose 40%. Which investment is better? Okay well, most people say this one because on the average you'll make 10% a year. Half the time you win 60, And half the time, you lose 40. And you will average making 10% per year, but it turns out, eventually you'll go bankrupt with that. I mean why? So that's 80 That's 30 minus 20, so that's 0.1 for the expected value. Okay, why will you go bankrupt? Well think about every other year, basically if this went according to plan. You'd win 60% one year, and you'd lose 60% one year. So if you started with a dollar, you'd go to $1.60. Then you'd lose 60%. You'd have 60% of that, you'd have $0.96. And every two years, on average, you'd lose 4%. It's because of the volatility. So eventually you'd be like that. If every two years on average you're going to lose 4% of your money, you're going to end up with nothing. So Kelly Growth vector is based on trying to maximize your long term growth rate of your capitol. And it's really very simple to apply, so let's take a simple example. And you know the Air Supply song Making Love Out Of Nothing At All, we can show you Kelly Growth can make money out of nothing at all. We'll give you a situation where it doesn't seem like you should be able to make money, but if you apply Kelly Growth in the long run you can grow your capital at 12% a year, so there are two stocks, okay. They're independent, and knowing what happens on one tells you nothing about what will happen to the other and so the annual return is basically, you flip a coin for each stock. Heads you would double your money. Tails, lose half your money. So if you start with a dollar you'd go from a dollar, the next year you might end up with two, you'd lose half, you go down to one. So it doesn't seem like you should make any money in the long run right? because, I mean, if you just put money in one stock, we'll see this, probably in the next video when we do a simulation, but basically, on the average your money wouldn't grow at all if you put all your money in either stock. And that's a true statement. But, if you would put half of your money in each stock, your capital will grow 12% a year, see what I mean? Making money out of nothing at all. Okay and we'll see when we get to sports betting, you never want to bet all of your money on a bet unless it's a 100% sure thing. Like maybe betting where the World Cup would be if you knew who took the bribes. Sorry, I couldn't resist that. Okay so the way Kelly Growth works is You choose investments or bets, To maximize the expected utility of your log return. If you figure out instead of expected return you take the logarithm if you start with a dollar, where you'll end up after one period of maximized expected value of the logarithm. And that will basically maximize your long-term growth rate. I mean, it sort of makes sense, I mean because after one period you would have your initial capital. It's like at the end of two periods, at the end of two periods you'd have like your initial capital. Times your period 1 grow. Times your period 2 growth. And if you want to maximize that, you can maximize the log of this. Which would mean maximize the log of on the average log of period 1 growth. Plus log of period two growth, because the initial growth is a constant. And so if you maximize the expected value of where you'll end up, then the expected value of this maximized the logarithm of where you'd end up and that's basically. What you'd end up trying to do. So let's look at our example here, I've got two investments. Heads, the stock doubles, tails, the stock loses 50%. So there are four outcomes here. I could basically get both stocks go up, both go down, first goes up second down, first down second up. These each have a 0.5 times 0.5 probability. Okay, now here's the fraction of my money I'll put in each stock. And we can figure things out and we can solve it. This happens to be the right answer. Okay, what'll happen to Stock 1? Okay, basically if Stock 1 goes up. Okay, if this says up basically you double the amount in Stock 1. If it says down, you cut it in half. So in the first two cases the money in Stock 1 doubles, otherwise it gets cut in half. In the second situation, if Stock 2 goes up you double what's in Stock 2, otherwise you cut it in half. Okay so this is what would happen. And so what you would end up with is the final Stock 1 plus Stock 2 value, which is $2 here. $1.25 here. $0.50 here. $1.25 here. Then you take the logarithm of these final values, and you want to maximize the expected value of that. Okay, so you take 0.25 times the logarithm of what you end up with etc. Okay so you maximize the expected log return. Now let's suppose you do 70% 30%. Okay this becomes 0.97. So what you want to do is pick the two yellows to maximize the red. And then the long-term growth rate it turns out, Is basically e to the target cell. In other words you take e to the expected value of the log of the return, that's going to be your long-term growth rate. Okay snd we'll see it up. So let's just solve this here. We want to choose the yellow to add to 1 and be non-negative to maximize the red. And then we'll see the long term growth rate can be about 12%, and we can verify that in simulation. And then we can in a third video apply this to sports betting. If you're betting on the NFL where you lose 11 dollars when you lose a bet and you win ten. And you can win 60% of your bets which is favorable, what percentage of your bankroll should you bet on each bet to maximize long-term growth rate. Okay, so what you want to do is you want to maximize the expected lock return changed these bets, and they should add the one. They should be non-negative. And you get 50/50. So, in other words, if you grow, if you invest at 50/50 in each stock, half your money in each stock, the expected logarithm of your return is 11%. And if you take e to that power, you should be able to grow at that rate. Okay, which is 11.8%. So we'll run a little simulation to show you that's true and then we can apply this to basically sports betting. Now Edward Thorpe who you may or may not know of, somebody should write his bi, he's mentioned in Fortune's Formula a lot. But he's professor at MIT. He developed the card counting in black jack that you probably saw in the famous 21 movie. He also embedded black scholes, I believe, before Black and Scholes. He had the formula, he didn't publish it, so he didn't win the Nobel Prize for it, but he didn't care because he ran a hedge fund that did fantastically well. I think it was called Newport Partners. And they would basically allocate their money based on Kelly Growth. And they basically did very, very well. If you allocate your money on Kelly Growth, and you know your probability you're going to do very well. In the long run with your portfolio. Okay in the next video we'll come up with a simulation to verify that you're really making money out of nothing at all. If you put all of your money in each stock, you're not making any money, but if you put half of your money in each stock and readjust your portfolio every period, you will average making 12% a period, which is amazing. And then we'll talk about sports betting applied category.
