Editor,
The article "Secure fractional money management" by Leo J. Zamansky and David C. Stendahl that appeared in the July 1998 issue of STOCKS &
COMMODITIES contains a major error. Optimal f is not the percentage of equity to trade, as stated in the article. Optimal f is used to figure the number of contracts to trade:
number of contracts to trade = (equity) (opt f)/-profit of worst trade
The equations that the authors give are correct, but because of their error in thinking, they misapply the equations.
In the example of the three series given in the article, the authors correctly come up with an optimal f of 1/3. But then to interpret this as being able to buy only three contracts is wrong. Using their interpretation, you could buy only three contracts in each of the series and end up with $101,500, a
terminal wealth relative (TWR) of 101,500/100,000 = 1.015. This is far from the TWR of 1.185 that is seen in their Figure 3. Using the correct
interpretation, you don't buy three but 66 contracts! 66.666 = 100,000 * 1/3/(500)
You always round down the number of contracts. Then, after winning $33,000 in the first, the second series gets 88 contracts: 88 = 133,000/ (1,500). The third series gets 118contracts: 118 = 177,000/(1,500). You end up with $118,000, giving TWR = 1.18.
The authors' calculation of maximum drawdown of $7,500 is dwarfed by the actual maximum drawdown, which is 5/3 of equity, or $220,000, as it occurs in series 2. The correct value of the secure f is 0.01.
Perhaps the authors became confused with equalized optimal f (see page 83 of Ralph Vince's The Mathematics of Money Management). In this
method, you can come up with a number that is a fraction of equity to trade with, but this number is neither f nor optimal f. (As far as I know, Vince does not identify this fraction, but it is evident from his equations.) In this method, you use the percentage loss or gain for each trade. In the authors' example, the trades become ($500/$10,000 = $0.05, 0.05, -0.05) instead of ($500, 500, -500). In this case, the equalized optimal f = 1/3; it is the same as optimal f because the buy price was always $10,000. Now use this equation: fraction of equity = equalized optimal f/ - return of worst losing trade
For the equalized optimal f of 1/3, the fraction of equity is 666.7% = (1/3)/ (0.05). Yes, this does mean you are buying on margin. For the equalized secure f of 0.01, the fraction of equity is 20%.
After getting every other interpretation wrong, the authors do come up with the correct final answer of 20%. This leads me to believe that they do have access to a program that correctly generates these numbers for them.
Despite my criticism, Zamansky and Stendahl are to be congratulated for the idea of secure f. Fixed fractional money management is a wonderful and complex subject that deserves some attention. Too bad this article got the fundamentals wrong.
BRADEN A. BROOKS via E-mail ---
Leo Zamansky and David Stendahl reply:
Thank you for the feedback about our July 1998 article, "Secure fractional money management."
Mr. Brooks is correct in saying that according to Ralph Vince, the formula is number of contracts =(equity) * (optimal f) / (- profit of worst trade). He is also correct that according to this formula, we buy66 contracts, not three. However, as we assumed in the article, one contract price is $10,000. To buy 66contracts, we need to have enough money to buy 66 contracts at $10,000 each. That makes66*$10,000 = $660,000. And that amount should be only one-third of the total capital available, which, as stated, is
$100,000. The question we are answering is: How many contracts can we buy following optimal f? The answer is $33,000 / $10,000 = 3. If the
contract price were $500, then the number of contracts to purchase would be exactly 66. If the contract price were less than $500, then the number of contracts to purchase also would be exactly 66.
Vince, in his book Portfolio Management Formulas, states on page 80, "Margin has nothing to do whatsoever with what is the mathematically optimal number of contracts to have on." We emphasize the word
mathematically. In real trading, if you need to buy one contract, you need to have a certain amount of money in your account, say x, and if you buy n contracts, you need to have the amount of money equal to n*x. In the article's example we call it price, but in reality it is a margin requirement. Inother words, the formula for the number of contracts should be adjusted to the price of the contract and be modified to look as follows:
number of contracts to trade = (equity) (opt f)/max [price of contract, -profit of worst trade]
We agree that we should have specified that. Of course, if you follow this number of contracts purchased, the TWR is not going to be 1.18, because the contract price is too high and does not allow you to buy the number of contracts that would maximize it. However, the lower the price of a
contract, the closer you will come to the calculated maximum of TWR. We develop tools to use in trading futures. We cannot introduce a
calculation that is not based on real trading rules. In real futures trading, the margin requirement per contract always exists and has a very similar meaning to the contract price in the game introduced.
As to the software we use to obtain our results, we use only the software written by us. In fact, readers can download the secure f calculator from our Web site, as mentioned in the article, and run it to obtain values for both secure f and optimal f. This should demonstrate the validity of this approach.
We appreciate the feedback from Mr. Brooks, who makes the important point regarding the difference between optimal f as a mathematical concept and trading using optimal f given the constraints imposed by the reality of futures trading.
Quizlet-Answer: (A: Answer, E: Explanation) A: False:
E: You will never find 2 assets with a satisfying continuous correlation structure (above all at the downside when there usually is a high degree of correlation). You could even trade a single commodity with a portfolio of patterns and models.
It might be easier to breed a custom correlation structure by synthesizing trading models than to find and apply those correlations offered by the markets.
Money Management 10
Money Management, kNOW, DeAmicis
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