Book review: Fortune's Formula

[beth_leonard]

If you are a gambler, or you invest actively in the stock market, read this book! This means you patrissimo and songmonk.

This book is Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street by William Poundstone

It goes into a lot of background and colorful detail about gangsters and betting, mathematicians and economists, and describes why many respected economists don't like the risk structure called the Kelly Criterion.

The Kelly Criterion is basically this: it tells you how much to bet on any given wager to maximize your rate of return on capitol in the long run. How much you should bet is based on how much information you have. It minimizes your chances of going bankrupt. It draws a clear mathematical line between "aggressive" and "insane" betting. In the long run, an aggressive bettor does well. An insane better always bets that 1:1000 will never come up, and one day ends up bankrupt a la Long Term Capitol Management.
...
Take the example of a coin toss that pays you $1 if it comes up heads, and takes your dollar if it comes up tails. Let's say you know you're using a coin that comes up heads 55% of the time, and you have $100 to start with and you can keep making the bet. How much do you bet on each toss?

One way is to bet $1 on each toss. Then your money grows very slowly. Another way is to bet your entire bankroll on every toss -- if the first toss is heads, you now have $200. If the second toss is heads, you now have $400. If the third toss is tails, you now have $0 and can't make any more bets.

The Kelly criteria is to bet edge/odds as a percent of your bankroll, which in this case comes out to 55%. First toss heads, you now have $155. Second toss you bet $85.25, you win and now have $240.25. Third toss you bet $132, you lose and now have $108.25. In the long run, your money will grow exponentially, but it will have it's ups and downs. You can smooth out the curves by making a half-kelly bet.

Kelly gets even better when you're making multiple simultaneous bets. The book gave the example of tossing 100 coins with a 55% chance of coming up heads. In the rare case where only 45 of the coins come up heads in one round of tossing the coins, the Kelly better ends up with $90 of a $100 bankroll at the start of that round. The better who leveraged 30:1 and placed all their money on heads now owes 2 times as much money as they had at the start of the last round.
...

The author claims that LTCM wasn't so much guilty of bad strategies or even making the same bet all over the world, but they were guilty of overbetting, a problem sure to lead to ruin eventually, even if all the bets are "good".


The Kelly criterion is much-maligned in the economic academic circles for reasons the author attributes to prominent academics claiming that real people can't handle the Kelly losses and so mean-variance or other ways of valuing risk are much better. The author is clearly biased in favor of Kelly, and after reading the book I'd have to agree, but Jon and I are still actively reading. The other methods claim to be for investors who are less risk tolerant, but at the same time, those methods can expose them to more risk.

--Beth

Comments

[songmonk.livejournal.com]

Ah, so many books to read!

[patrissimo.livejournal.com]

I think all mathematically inclined gamblers know the kelly criterion. At least, anyone who has played blackjack - it is quite commonly used there. It can be used for poker too, but it is not always easy to know what your mean and variance are. In the stock market, it can be very unclear what your mean and variance are. I mean, when you get "7-sigma moves", like during the LTCM crisis...well...obviously the definition of sigma was wrong!

It is useful in theory, and can be useful in practice.

The book, however...I glanced through it one time, and it looked like a huge over-dramatization of what is a pretty simple mathematical theory. I mean, KC is pretty basic and inarguable. Applying it to the real world is not so straightforward. Casting it as some kind of "outlaw" formula that "beats" casinos and Wall Straight sounds overdramatic. You don't beat the casino with KC, you beat it by counting cards at blackjack. KC just tells you how big to bet so as to maximize the logarithmic growth of your bankroll.

[beth_leonard]

I found it a very fun read, but if you're already aware of it then I suppose it proves his point of "gamblers already know this, why is it taking wall street so long to figure it out?"

it is not always easy to know what your mean and variance are. In the stock market, it can be very unclear what your mean and variance are.

That was one of the author's points -- mean-variance theory isn't right when you're deciding on the size of bets to place. It doesn't look at what it will take to go bankrupt.

The Kelly Criterion is more about information, G[max] = R -- the maximum growth rate of your compounded portfolio is equal to the rate of information where you have an edge.

On blackjack, it's not just counting cards to figure out when the deck is in your favor, but figuring out how much to bet when the deck does get in your favor. If you bet it all because the deck is suddenly "favorable" you can still lose it all. You need both parts: the information AND the bet size.

You already know this. The derivation and history make the book interesting (to me) and I also found the bits about Guiliani and why Thorps Hedge Fund *didn't* blow up the way LTCM did fascinating.

--Beth

[jon-leonard.livejournal.com]

It's quite possible that there's nothing in the book, gambling- or investment-wise that you don't know. But it's largely a (fun!) history book, not an explanation of math or investing.

I'd recommend the book to the investment-minded largely because it provides something of an antidote to the mindset of random-walks, modern portfolio management, and Value-At-Risk. There is, for example, a tendency to assume that mean and variance (if you know them) tell you what you need to know about a distribution. That's only true if the distribution is very close to a Gaussian Normal distribution, and stock movements definitely aren't! 7-sigma moves are a huge deal if the underlying distribution is normal, but not really all that rare for some other distributions (They're moderately rare for stocklike distributions, but don't bet the company that they won't happen).

In terms of applicability, you're right: It's essentially useless in Poker: Announcing that you'll never go all-in is foolish. For that matter, you probably shouldn't be sitting down at a poker table if your utility isn't nearly linear over the range of possible outcomes. (With some exceptions for tournaments.)

The claim of beating the casinos ans Wall Street is a bit more interesting. It's pretty clear that Thorpe could play blackjack profitably, until the casinos either kicked him out or employed cheating dealers (or, eventually, shuffled more decks more often.). Thorpe's fund did beat Wall Street, partially because he had a variant on the Black-Scholes option model before it was published, and was able to do rather a lot of arbitrage as a result. His fund yielded about 15% compounded, for a little less than tripling the S&P 500 over that 19-year run. That particular inefficiency is gone now, though.

Anyway, I think there is a lot in the book that you'd find interesting, but I could be wrong.