In our blog “Money Management: the best kept secret of professional investors,” we have emphasized the importance of Kelly’s formula in money management. In the meantime, we have also pointed out that the capital allocation suggested by Kelly’s formula is too risky.

So what should we do in practice? Here we shed some lights on this issue by giving a graphic illustration of the recent analysis of Vince and Zhu [1].

Consider a simple game betting on a flipping of a biased coin. Kelly’s formula is well illustrated by the graph of the average log return function per play below where the unique peak corresponds to the Kelly optimal betting size.

Nevertheless, two key components pertinent to the real applications are absent in Kelly’s formula: risk aversion and a finite investment horizon. Adding these practical considerations the picture changes dramatically. It turns out that, when playing the game for only a finite number of times, the total return as a function of the betting size becomes, in general, a bell shaped curve. Moreover, the risk measured by drawdown is approximately proportional to the bet size. Thus, the goal of a player is then to maximize the ratio of the total return and the bet size. Graphically, for any given point on the graph of the total return curve this ratio is exactly the slope of the line connecting this point and the origin. Three typical lines are illustrated in the featured graph in the beginning of the blog. It is clear that the top line that tangents to the return curve indicates the bet size maximizing the return / bet size ratio. Comparing to the middle line that passes the peak (which can be shown to be very close to the Kelly optimal bet size) of the return curve we see a theoretical justification for needing to be more conservative than the Kelly optimal bet size in practice. The lower line is also significant. This line passes through the inflection point, the boundary of increasing or decreasing marginal return when the bet size increases. This is the most conservative of the three points. When bet size increases beyond this inflection point while the return / bet size ratio may still increase the marginal increase diminishes. This makes the inflection point also a reasonable conservative choice.

Empirical analysis of a realistic Blackjack game in [1] shows that in practice the reasonable bet size should only be one quarter to one third of that the Kelly best bet size suggests. The idea and qualitative conclusion discussed here also apply to investment capital allocation. Of course, when dealing with problems with multiple investment assets or strategies, the analysis is much more involved technically. Detailed discussion of related theory and implementations can be found in Lopez de Prado, Vince and Zhu [2]. The clear message regarding to the problem of asset allocation is when in doubt be conservative.

References:

[1] Vince, R. and Zhu, Q. J., Optimal Betting Sizes for the Game of Blackjack, (2013) SSRN 2324852.

[2] Lopez de Prado, M., Vince, R. and Zhu, Q. J., Optimal Risk Budgeting under a Finite Investment Horizon, (2013) SSRN 2364092.