Money management: the best kept secret of professional investors

Average log gain as a function of bet size

Watching CNBC you will constantly be bombarded with interviews of traders and investors. List of stock picks and expert opinions dominate the theme.

Putting the correctness of such expert’s opinion aside, the sheer single minded focus on stock picking is already very misleading. In fact, most successful professional traders and investors will tell you money management is an important component of their trading systems.

To understand why let us try this question: Can one lose money in a game in which one has a favorable probability of winning? The answer is, absolutely yes. For example suppose that a skillful gambler is, on average, right 9 out of 10 of his bets. For each right bet he will double his money and wrong bet, loss all the bet. The following table created with a spread sheet is thought provoking. We assume that the gambler plays 10 games and the sole losing one is game 7.

Bet Size and the Gambling Outcome
Game/Bet Size 20% 40% 60% 80% 90% 100%
1        Win 120 140 160 180 190 200
2        Win 144 196 256 324 361 400
3        Win 173 274 410 583 686 800
4        Win 207 384 655 1050 1303 1600
5        Win 249 538 1049 1890 2476 3200
6        Win 299 753 1678 3401 4705 6400
7        Lose 239 452 671 680 470 0
8        Win 287 632 1073 1224 894 0
9        Win 344 885 1718 2204 1698 0
10      Win 413 1240 2749 3967 3227 0

Columns of table corresponding to the size of the gambler’s bets as a percentage of his account balance and the rows are his balance after corresponding number of games.

Clearly betting all one has the worst outcome since this will certainly lead to loss of all capital in the event of one single losing bet despite the gambler’s high success rate. By examining the last row of the table we will find that the best betting size in this situation is 80%. A bit of thought will convince us that, in fact, when this one losing bet happens does not influence the outcome. The question of how to find the best betting size in general was first carefully analyzed by John Kelly, a scientist in AT&T Bell Lab. Here is Kelly’s reasoning.  Suppose that the gambler has a winning probability of p, so that the losing probability is q=1-p. Then the long term asymptotical average gain per play is

p*log(1+f)+q*log(1-f).

A typical graph of the log return as a function of  using the data from our gambling example is featured at the beginning of this blog.

This simple graph contains rich information that has important ramifications for investment decisions. First, we see that the curve has a unique peak at 0.8 which coincides with the maximum in the table above validating 80% is the best betting size. Moreover, it is not hard to calculate the critical point of the log return function and derive that the general best betting size is at p-q. This is the famous Kelly’s formula.  Furthermore, when f approaches 1 the curve dips below the horizontal axis indicating that even with a huge advantage in the game one could still lose money if he bets inappropriately. Similarly, inappropriate investment allocation could significantly hurt the performance even for a skilled stock picker. Finally, the curve on the two sides of the peak is asymmetric. The right hand side of the peak looks like a cliff which drops into losing territory in no time. Thus, in practice one should never go near the peak. In fact, Professor Ed Thorpe, the pioneer who extended the Kelly formula and used it in stock and option trading, said in a recent interview that he is more comfortable using only a fraction of the allocation that the Kelly formula suggests.

We restrict ourselves on analyzing a simple gambling problem mainly to avoid technical complications. The methods and the qualitative principle indicated by the results equally apply to investment problems. The take home from this discussion is that how to appropriately allocation your investment capital is at least as important as how to pick your investment. The technical details involved in applying such capital allocation in practice also explains why this is one of the best kept secret among the professionals – it is hard to convey in 30 second sound bites. This and similar other in-depth knowledge is exactly how serious investors distinguish themselves from the rest of the pack.

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