The Half Kelly bet

One of the smartest and most numerate commenters on GuruFocus is the one who goes by the pseudonym "Buffetteer17". Here is Buffetteer17 recently expounding on the merits of a modified version of the Kelly formula for optimally sizing bets or investment positions:

The half Kelly bet has some interesting mathematical properties. For risk management purposes, the nice property is that it cuts your risk of temporary loss (i.e., volatility) by a large amount while reducing your return expectation only a little. The other important property of the half Kelly bet is that it gives a large margin of safety in the risk estimate. If you are off by a factor of two on your risk of loss estimate, a full Kelly bet will reduce your return expectation to zero. But a half Kelly bet will leave you with 2/3 of the return expectation. Not surprisingly, underbetting is far, far safer than overbetting.

With the full Kelly bet, your probability of temporary loss is a linear function of the amount of loss. For example, you stand a 90% chance of losing 10%, an 80% chance of losing 20%, a 50% chance of losing 50%, etc. Not many investors are comfortable with the prospect of a 50% probability of losing 50% of their money. With the half Kelly bet, your probability of temporary loss is a quadratic function of the amount of loss. For example, you stand a 81% chance of losing 10%, a 64% chance of losing 20%, a 25% chance of losing 50%, etc.

Your expected gain with the half Kelly bet is reduced by 25%. For example, if your expected gain is 40% with the full Kelly, it is 30% with the half Kelly, if your expected gain is 30% with the full Kelly, it is 22.5% with the half Kelly, and if your expected gain is 10% with the full Kelly, it is 7.5% with the half Kelly.

The quarter Kelly bet is even safer. You cut your volatility by a quartic factor while reducing your return expectation by half. For example, you stand only a 6.25% of losing half your money. If you can find enough uncorrelated bets to get all your money invested, you can still invest 100% of your stake with a high safety factor with multiple quarter Kelly bets. The rub, of course, is that it is hard to find truly uncorrelated bets during a market crash like 2008.

In his brief book The Dhando Investor, Buffett wannabe Mohnish Pabrai devoted a chapter to the Kelly formula before concluding that, rather than follow it exactly, he was motivated by it to aim for a 10x10 portfolio (ten positions each comprising 10% of his portfolio). According to notes on his annual investor meeting earlier this year, Pabrai has moved further away from the Kelly formula, to a 3-5-10 set up, where he will only allocate 3% or 5% to most positions, and only occasionally allocate 10% to one idea under extraordinary circumstances.

A problem I have with this gets to a key difference between investing in stocks and betting. Stocks have their own idiosyncratic risks and potential upsides, and the more of them you own, generally, the less informed you will be about them. Better to dig deep and bet big on a handful of stocks with high upside potential, in my opinion, than to broadly diversify as Pabrai is doing. Even broadly diversified index investors got their heads handed to them last year. Idiosyncratic risks aren't the only ones out there, and in over-diversifying to eliminate them, you also diversify away the idiosyncratic high potential returns associated with individual stocks.