The optimal fraction - in search of the perfect money management
Greetings, friends forex traders!
It is believed that Optimal F is the only best money management method. The concept of “optimal f” is probably known to every currency speculator who is at least seriously interested in money management and money management. The main promoter of this idea, American author Ralph Vince, is well acquainted with Larry Williams - the legendary futures trader. Vince himself is not a practicing trader, and many reproach him. Today we will delve into this topic in detail, we will analyze the pros and cons of the optimal fraction method, calculation, and also consider modifications of the approach.
Ralph noticed in Kelly the error that the Kelly criterion formula was originally intended to determine the direction of flow of electronic particles, and then used for blackjack. The trouble is that blackjack is not at all the same as trading stocks or currencies. In blackjack, your potential loss at each bet is limited to the chips that you bet, while your potential win is always the same in relation to the chips placed.
When working on the market, the size of our wins and losses is constantly changing. Sometimes we get big gains, sometimes tiny. Our losses are subject to the same law - their size is random. Ralph came up with a formula similar to Kelly’s formula and called “optimal F”, but unlike Kelly’s formula, it can be adapted to market trading.
Optimal F (formed from the word "fraction") - the share of the deposit at which we will have maximum profit. Naturally, the optimal f - the value is not constant, and as transactions are completed, the value will change. That is, it is necessary to do a recount.
If you graphically represent the change in the final deposit (TWR) versus the percentage of use of funds (F), then the dependence will be described by the curve:
As you can see, when investing too little of the deposit, we will have a small profit. If risks increase, the value of the final deposit will also increase until a certain point. With a further increase in risks, the final deposit will begin to fall. This very moment when the growth of the deposit is maximum, just corresponds to the optimal f. Thus, it is quite logical to assume that the optimal solution for the trader would be to use for each transaction such a percentage of the deposit, which will be at the point of the upper extremum of this curve.
Well, let's generate a random trading system for our research.
The main statistical indicators of the system are as follows:
When creating the TS, I used the random number generator for the normal distribution:
The mathematical expectation of our trading system came out a little more than 1, and the standard deviation of about 4, which is quite suitable for our purposes.
Now we introduce a money management system - in each transaction we will risk a certain percentage of our capital (in this case 3%):
All that remains for us to do is to find such a percentage of risk at which our final balance will be maximum. To do this, we, as usual, can use the solution search functionality built into Excel, which found the optimal value of 20%:
For the calculation, they usually do not use the amount of final capital, but TWR. This is an indicator characterizing the relative final capital, or, more simply, the fact how many times we increased our initial deposit. And in this example, the maximum TWR was 8159238.337 with a risk of 20%. In other words, the optimal f specifically for a given system is 0.2.
As you saw on the graph at the very beginning, the optimal f is, in fact, an extremum, above and below which there are already non-optimal TWR values:
The graph shows that the optimal F for our system is 0.2 or 20% risk per trade. Moreover, if we lay the risk of 21%, this will give the final result the same as if we risked 18%. Moreover, if we add literally another percent and risk 22%, the account will be merged.
The calculation of the optimal f
Let us dwell in more detail on calculating the optimal f. To calculate the optimal f, you must first calculate the profit for a certain period - HPR.
HPR = 1 + f * (- deal / largest loss), where:
f - risk in each transaction;
transaction - profit or loss in a particular transaction (in case of loss, the expression in parentheses will turn out negative, as well as the final value);
largest loss - the largest loss per trade (negative number).
Then TWR is calculated as the product of all HPRs, that is:TWR = HPR1 * HPR2 * .... * HPRn, where n is the last deal in your sample.
Well, in the end, we calculate the geometric mean HPR (G), which is calculated as a root of degree n from TWR: G = TWR ^ (1 / n), where n is the total number of transactions.
All parameters for the calculation are known, except for the value f. Your task is to step through f from 0.01 to 1 in such a way as to find the maximum G. Moreover, f, at which G is maximum, will be optimal f.
Danger of optimal f
Optimality f is fraught with great danger. You probably noticed that in our example, the optimal f was 20 and at the same time, at a risk of 22%, we simply merge everything clean. Deviation of only 10% from the optimal value leads to fatal consequences for your account.
But the fact is that when we discuss TWR, then we allow the use of fractional lots. For example, you can trade in 5,4789 lots, if that is what is required at any moment. TWR calculation allows for fractional lots, so that its value is always the same for a given set of trading results, regardless of their sequence. You may doubt the correctness of this approach, since in real trading this is impossible. In real life, you cannot trade such fractional lots. This argument is correct. But if we use only the round values of the lots for calculation, the calculation itself will become incorrect. In this case, the closer you are to the optimal f, the better. And on the other hand, having missed a little, you will merge your score.
Obviously, the greater the capitalization of the account, the more accurately you will be able to adhere to the optimal f, since the amount in dollars required for one lot will be a smaller percentage of the total balance.
Many professionals use a fixed share when trading, but this share has never been close to as high as the optimal f. The fact is that Ralph Vince, undoubtedly, is a professional in his field and an excellent theoretician. But one detail is very annoying. The fact is that no matter how we try to predict the size of the maximum unprofitable transaction, there is always a considerable chance that in the future this value may be exceeded. We can more or less well predict average values such as mathematical expectation or average profitable trade, provided that there are enough statistics. But the most unprofitable, the most profitable deal, the maximum drawdown - all these are pretty poorly predicted values. That is why the optimal f is not so much use, because a little mistake in this very maximum loss-making transaction, we will make a mistake in the optimal f. And having made a mistake in the optimal f of just a percentage or two, we get a margin call.
And yet, it cannot be said that this formula is completely useless. Moreover, in some special cases, for example, for binary options or for systems with hard stops and profits (though such systems, in my opinion, are not optimal in themselves), it is accurate. Therefore, if you know exactly your maximum loss, this method of money management is quite suitable for you.
Let's check my point - we will generate another 1000 deals with the same characteristics - an average value of 1 and a standard deviation of 5. In doing so, we will use the same optimal f equal to 20%.
Add 1000 deals:
And look at trading at the same risk of 20%:
As you can see, this level of risk cannot be called optimal. He just changed just.
Suppose the optimal fraction for the previous 100 transactions was 15%, in the next 100 transactions this share may turn out to be 9%. If the percentage of 15% was optimal for the previous 100 transactions and you decided to conduct the next 100 transactions with the same fraction, then you may well be mistaken and easily go beyond the amount in your trading account.
Practical application of the optimal fraction strategy optimizes past trades. Therefore, the next transaction immediately falls into sequence, and the optimal share is re-optimized. And it will be optimized at the conclusion of each transaction. That is, in real trading, after each transaction you will have to re-calculate the optimal fraction.
In addition, trade is completely unpredictable, despite all the indicators that can be calculated on the basis of available statistics. With the help of logic, we can only draw certain conclusions regarding reasonable expectations and probabilities. No mathematical expression can guarantee us that of the N number of transactions, 50% will be profitable, and the remaining 50% will bring losses. Trading strategies are formed on the basis of logic and, to a large extent, market statistics. Market behavior is changing. That which seemed beneficial yesterday could become dangerous today.
There are still hundreds of other and reasonably logical reasons why the optimal fraction method is perfect from a mathematical point of view, but it turns out to be quite dangerous in practical applications. However, some points that I analyzed above show that it makes no sense to continue discussing this topic further. Risk in itself is a strong enough argument against using the optimal fraction method. If you think that you are able to cope with the risk, then make sure that you understand this method well before you apply it in your trading practice.
So, the main problem of the optimal fraction, as you already understood, is its binding to the maximum losing trade. In the case of using hard stop losses, this is not scary, but when the exits from transactions in the unprofitable zone are mainly based on signals from the market, the optimal f becomes not optimal and overvalued, which threatens to drain the deposit or cause serious losses.
Suppose an event occurred during the trading day that caused a shock in the market, and before that shock, volatility was quite low. Of course, under such conditions, your optimal f will be very high and it is very likely that you will enter the market on this most unfortunate day with a risk of thirty percent, which will result in a total 50% loss.
It is for the reasons listed above that they use various modifications of the optimal f method, with which we will now become acquainted.
Diluted Optimum Fraction
In order to avoid deposit drainage with a slight deviation, a method of diluted optimal f was proposed. In fact, the diluted optimal f is a percentage of the optimal f. This technique is used, firstly, so that, as a result of optimization on historical data, the optimal amount of capital is not overestimated and, secondly, so that the trader can regulate his own risk (the amount of capital used in trading) when using the optimal f.
The calculation formula is very simple:
Diluted optimal f = Optimal f * X, where X is the percentage of optimal f you have chosen
You can, for example, set X = 0.5 and be sure that half of the optimal f calculated on the history is unlikely to ever exceed the real optimal f in the future.
The drawbacks here are the same as with the optimal f, but the probability of reassessing the risk, which can lead to a drain, is significantly lower in this case.
Safe faction (Secure f) is part of the capital involved in each transaction while limiting drawdown and maximizing profits. A safe fraction has some advantages over an optimal fraction, since it does not rely on the maximum loss, but on some other factors. The strategy is similar to the optimal f technique, with the only difference being that when using optimal f, your strategy is optimized for profit taking into account the maximum drawdown for the calculation period of historical data, and when using safe f, you yourself limit this drawdown.
The calculation is also quite simple. Instead of the maximum losing trade, we simply use the maximum drawdown in the currency. Working with the safe fraction method is less risky than using the optimal fraction, but capital growth will be much slower, especially on small deposits.
Today we met with such methods of money management as the optimal and safe fraction. From the point of view of logic and mathematics, both of these methods of calculating risk look very attractive. However, as we have seen today, these methods have their drawbacks. A considerable number of traders in the forex market believe that it is necessary to take risks to the maximum and direct their efforts precisely at the maximum deposit growth. In other words, there are a very large number of so-called "deposit accelerators." And for them, such methods of money management as the optimal fraction and the Kelly criterion may seem like an excellent solution.
For the same traders who are also not against high risks, but do not like to lose deposits too often, I can recommend using a lighter safe fraction or diluted optimal fraction, which will eliminate the likelihood of applying too much risk.