### Optimal Position Sizing

I had a suspicion, that there must be a way to mathematically determine optimal position sizes. Ralph Vince, the author of the *Handbook of Portfolio Mathematics: Formulas for Optimal Allocation and Leverage* , beat me to the punch. In this volume, he summarizes his previous work, and describes in detail, a mathematical model, which helps calculate optimal position sizes, based on their mathematical values. He goes to one of the core elements of trading. Position sizing significantly affects the amplitude at which returns are generated for a particular position. A good trader needs to have a sense of how much he is willing to risk on a position, and how much he may gain as a result. Transaction costs affect this particular trade-off also, and need to be taken into account. As you cannot be sure that any particular trade will be profitable, how you size your positions can affect your level of success and trading consistency, and also in the extremity of your failure, as for example Long-Term Capital Management (LTCM) learned in the late 1990s. He goes at the topic, though, from the perspective of a trading IT consultant, not as a trader. Even though in theory, he may be right, in practice, I think there are a number of risks which need to be taken account of when position sizing, such as liquidity risk. Most of these factors are hard to model, but necessary to take into consideration when putting on positions.

Vince suggests that every position has an optimal f, where the investor maximizes return over a long time frame. He essentially generalizes the Kelly criterion to the typical situation that a trader faces. The Kelly criterion, developed by engineers at Bell Labs in the late 1940s, in order to resolve the problem of data transmission over long-distance lines, can be reapplied in the context of geometric growth, and how it applies to money-management, as both problems “are the product of an environment of favorable uncertainty”. With some basic back testing, a trader gets a sense of what may happen to a position and whether it is worth putting one on. The values generated during back testing can then be used as inputs into Vince’s formulas. Vince uses an empirical approach based on statistical distributions to calculate the optimal f for a particular position. If trader uses of value below optimal f, he will be generating less returns than he could be, assuming a particular investment is modeled as a time series, generating P&L of a certain mean and standard deviation, i.e. a statistical distribution of returns. If the trader is above his optimal f, he runs the risk of complete wipeout, due to his position sizing. This means that a few losses in a row can completely take him out of the market. This modeling of a particular position as a random walk provides a framework for calculating its optimal size.

With this approach, Vince is essentially talking about the amount of leverage being put on a position. While leverage increases the total amount of return, it can also increase total losses. Assuming a particular position continues to follow a random walk according to certain patterns, patterns which can then be described mathematically, it becomes possible to optimize returns via the size of the position. Because the approach is largely empirical, it is rather simple to also include the effect of transaction and financing costs, and their implications for the position size.

**Key takeaways:**

- traders should strive to maximize geometric, but not arithmetic growth, in positions
- the actual approach to position sizing depends on the total value of your portfolio
- classical folio construction, Markowitz, can be superseded by newer models, which take into account the properties of the new asset classes, including derivatives

While in theory, he is right, in practice, I think he is avoiding a few important issues. Position sizing based on historical performance implies that all of future performance will follow the same distribution. This is not necessarily the case, and can be hazardous to your health. Also, you cannot assume that you’re operating environment will always be the same, particularly when trading in a bull market. For example, just because it is easy to enter into and exit transactions in a bull market, when sentiments go south, you can be stuck with positions in which you know will continue to lose money for you, but that you cannot sell; therefore, more conservative position sizing is in order.

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Maturity is the ability to reap without apology and not complain when things don’t go well. Jim Rohn