Just curious, but how can you apply Kelly-bet system if you follow, say, all the S&P500 stocks with an average of 2 in-and-out’s a day. Now I use 5.000 USD as fixed bet (paper trading …. for the moment)

Bye,

Francis.

Pabrai also mentions as does Poundstone in “Fortune’s Formula” that Edward Thorp published a paper “The Kelly Criterion in Blackjack, Sports Betting, and the Stock Market.” Given Thorp’s investing record as mentioned in “Fortune’s Formula” this should prove to be a interesting read for those looking to further dig into the subject. Thorp was a spectacularly successful investor with very low volatility of returns. I have not read this paper, but I might this afternoon.

Futhermore, applying the Kelly Criterion either using the website mentioned, Excel, or numerical methods (science phrase for pen, paper, and calculator), you gain some further incite as to why the current breakdown in the commercial paper markets is occurring. With such low potential returns and such high potential losses, any investor whose probability of default on their commerical paper deviates from 0% should sprint to the door. This is just what is happening. Fear is powerful.

Regarding portfolio construction and the Kelly Criterion. Theoretically, any investment decision that you make would have the construciton of a Kelly bet. I think that includes working from a top down or bottom up perspective on your investments. The decision outcomes generally always include a comparison of a risky investment to the certainty of cash. See past inflation here if you will. Buying a house? Benefits of home ownership versus certainty of cash. Allocating assets to bonds, stocks, or real estate versus cash. When you start constructing an equity portfolio, you are weighing probabilities of cash versus the expected outcomes for your equity positions. I’m pretty sure you could rollup your expectations from individual risky assets selections into to an bankroll optimization for your entire asset base. You would do this instead of using Markowitz’s mean-varriance optimization. I think Poundstone talks about this.

Finally on overbetting the Kelly, there’s a great graph in “Fortune’s Formula” that shows the pitfall of overbetting the Kelly. The graph shows the geometric mean (what the Kelly maximizes) versus the proportional size of the Kelly bet. I don’t have my copy of the book handy but here is what I remember.

The parabolic shaped graphed with the wide end pointing down. The geometric mean is increasing approaching a full Kelly bet where it is maximized. Past a full Kelly, the geometric mean starts decreasing. At twice the Kelly bet, the geometric mean is zero and beyond 2-Kelly it is negative.

Given the fuzziness of estimating probabilities (especially in investments) input into the Kelly Criterion, taking a full Kelly bet could actually be a 2-Kelly bet or more fairly innocently. Thus you would always want to take less then a full Kelly bet. This does not mean that you wouldn’t use leverage when applying the Kelly Criterion.

Sorry, but that would fail the logic test. When you make a bet in your portfolio using the Kelly Criterion, how does the Kelly formula know if you are making a single bet itself or a single bet consisting of a “basket” of securities?

I think Tim was trying to say that in order for it to work in the stock market, one must use the REAL Kelly Formula, not the simplified Edge/Odds version. The Edge/Odds works with simple bets like a coin toss or horse race (as shown in Fortune’s Formula). This is because it requires two things: 1) There are only two outcomes – “win” and “lose”; and 2) In the losing outcome, 100% of your bet is lost. So, you really couldn’t use it for stock situations that have many possible outcomes with varying gains/losses.

Also – the Kelly Formula used for gambling assumes you are only placing one bet at a time, not a portfolio of diversified bets. By using the actual Kelly Criterion (maximizing the logarithm of the portfolio as a whole) you can see what fraction of the bankroll should go into many stocks with many outcomes. I think the reason the Edge/Odds formula is seen in Fortune’s Formula and Pabrai’s book (and this blog) is because it’s much easier to understand and gets the very basic concept right.

Using a percentage (like 25% of bankroll) will ensure you never go broke because it is a percentage of total bankroll – not starting bankroll. If you invest $250 from a $1,000 bankroll, and you lose it all, you’ll have $750 left. Your next bet would be 25% of $750 – or $187.50.

The smaller your bankroll gets, the less you bet on a dollar basis (even though the percentage remains constant). Eventually, if you get down to a $0.01 bankroll, you have to stop betting because you can’t divide it any more.

Of course, that assumes you place one bet each time and wait for the result. If you bet 25% of your bankroll on 4 bets – all at the same time – you can lose it all. Otherwise, if you lost 25% at a time – on a $1,000 bankroll – you’d have to stop betting when you got down to $0.03 some 38 bets later.

Even still, as you have said, the Kelly Formula is a guide – not an exact science – in the world of investing. Then again, there is no exact science in investing.

Of course, as other posters have also pointed out, one’s expected value (or edge) can only be estimated and is not known precisely for a common stock investment. This is another reason to use a conservative fraction of the Kelly number for each purchase.

Fortune’s Formula by William Poundstone is an excellent book, which I recommend to anyone interested in investing, trading or playing games of chance with a positive expectation.

http://en.wikipedia.org/wiki/John_Larry_Kelly,_Jr

I think Parbrai in his book uses it as a general illustration of the virtues of concentrating rather than saying it is mathematically correct.

Everything we do is a guess of the future. We look at the past for consistency. Substitute to your heart’s content. Just make sure that your numbers are reasonable and that your reasoning is sound.

Check out Do The Math In Your Head. It is all about false precision and guesstimating.