Once in the hands of the trader appears ready trading system, once the question arises - how to trade with the risk of how lot open positions? In the trading books and on the internet a lot of information on this subject, but often it comes down to either the reasoning of psychology ("risk so as not to be nervous"), or to the rigid rules, such as "risk 2% per trade." Naturally, the question arises: why the 2 per cent instead of 5? Or 2.2%?
Questions of psychology in this article we will not deal with and consider the question of the optimal risk on the transaction from the point of view of mathematics, ie, with maximum precision and rigor. Hereinafter, means that we are trading on a "constant risk in the transaction» (constant fraction), ie in every transaction we may lose a fixed percentage of the amount of R our current account. Accordingly, for the same amount of stop-loss points, the position size is directly proportional to the balance of the trading account. On the advantages and disadvantages of this method, as well as the comparison of different types of money management will be discussed in another article, but now dwell on this, probably the most popular method for calculating the size of the position.
For simplicity and clarity, we assume that we deal with the same and equal to the values of stop-loss and take-profit, and each transaction is terminated or the stop-loss or take-profit on. In other words, in every transaction we either lose R percent of the current account, or acquire the same R percent. I have to say that such a strong simplification of the conditions does not hurt the value of our further calculations and will not change the main conclusions to which we will come after solving the problem. For options with a variable take-profit or trailing stop calculations will be similar, but longer and more boring.
To solve the problem used numerical Monte Carlo method, which is also called the method of statistical tests. The method consists in the fact that the computer creates a history of changing the balance of the account, randomly playing out a deal for a deal. Creating a lot of similar stories and averaging the results, we obtain the probability of occurrence of certain events, depending on the input parameters. The reader may wonder why the transaction played by chance and what does all this have to do with reality, because we are a trade according to certain rules, on certain financial instruments? And the curve of our balance sheet, we hope to grow with time, and not be a "total chaos". The answer is. First, no matter how we traded in terms of mathematical statistics sequence of transactions it is always a random process, and the outcome of each individual transaction is unpredictable. Therefore, the results of the calculations were equally valid and "impartial" for all trading systems, and for any market, it is used for the calculation a random number generator, and not any historical quotes data. Secondly, the balance of the general direction of the curve can be controlled by changing the account when creating history winning percentage X. As is well known, successful traders show from 50 to 75% profitable trades (provided that the stop loss is equal to the take-profit). Since the value of X of its trade forecast is very difficult (lack of statistics, the unpredictability of the market), the calculations are carried out for different values of X. As we shall see, winning percentage does not affect the determination of the optimal risk R.
In the process of solving the problem it turns out that for the unique solutions it is necessary to introduce three more additional conditions - three new parameter T, N, and P. parameter definitions as follows:
T - how many times will increase the initial deposit.
N - the maximum number of transactions for which it is planned to increase the initial deposit in T time.
F - the maximum allowable drawdown in the trade.
Now we can formulate the problem more precisely: What is the risk R is optimal for increasing the deposit in time T for N transactions with the loss of not more than P? Computer, armed with the Monte Carlo method, solves this problem in a few minutes. As a result of calculations we get the graphics probability of a favorable outcome, depending on the risk in the transaction R. favorable outcome in this case - an increase in the initial balance in the T or more. Below are the probability plots for the parameters T = 1.5; N = 200; P = 30% (ie 50% yield for 200 transactions not more than 30% loss).
For simplicity and clarity, we assume that we deal with the same and equal to the values of stop-loss and take-profit, and each transaction is terminated or the stop-loss or take-profit on. In other words, in every transaction we either lose R percent of the current account, or acquire the same R percent. I have to say that such a strong simplification of the conditions does not hurt the value of our further calculations and will not change the main conclusions to which we will come after solving the problem. For options with a variable take-profit or trailing stop calculations will be similar, but longer and more boring.
To solve the problem used numerical Monte Carlo method, which is also called the method of statistical tests. The method consists in the fact that the computer creates a history of changing the balance of the account, randomly playing out a deal for a deal. Creating a lot of similar stories and averaging the results, we obtain the probability of occurrence of certain events, depending on the input parameters. The reader may wonder why the transaction played by chance and what does all this have to do with reality, because we are a trade according to certain rules, on certain financial instruments? And the curve of our balance sheet, we hope to grow with time, and not be a "total chaos". The answer is. First, no matter how we traded in terms of mathematical statistics sequence of transactions it is always a random process, and the outcome of each individual transaction is unpredictable. Therefore, the results of the calculations were equally valid and "impartial" for all trading systems, and for any market, it is used for the calculation a random number generator, and not any historical quotes data. Secondly, the balance of the general direction of the curve can be controlled by changing the account when creating history winning percentage X. As is well known, successful traders show from 50 to 75% profitable trades (provided that the stop loss is equal to the take-profit). Since the value of X of its trade forecast is very difficult (lack of statistics, the unpredictability of the market), the calculations are carried out for different values of X. As we shall see, winning percentage does not affect the determination of the optimal risk R.
In the process of solving the problem it turns out that for the unique solutions it is necessary to introduce three more additional conditions - three new parameter T, N, and P. parameter definitions as follows:
T - how many times will increase the initial deposit.
N - the maximum number of transactions for which it is planned to increase the initial deposit in T time.
F - the maximum allowable drawdown in the trade.
Now we can formulate the problem more precisely: What is the risk R is optimal for increasing the deposit in time T for N transactions with the loss of not more than P? Computer, armed with the Monte Carlo method, solves this problem in a few minutes. As a result of calculations we get the graphics probability of a favorable outcome, depending on the risk in the transaction R. favorable outcome in this case - an increase in the initial balance in the T or more. Below are the probability plots for the parameters T = 1.5; N = 200; P = 30% (ie 50% yield for 200 transactions not more than 30% loss).
Calculations were carried out for the four values of share trades (X = 50% (trade "at random"), X = 55% x 60% = and X = 70%). All four curves have a maximum at R = 3-5%. This area is highlighted in gray rectangle - optimal risk area.
Why have the maximum graphics? Obviously, when the risk is too low R (0-2%), we simply do not have enough transactions to achieve the desired balance level. Too much risk (R> 6%) increases the likelihood of strong subsidence, which is also not part of our plans.
It is seen that increasing the probability of 50% the balance is highly dependent on the quality of our trade, but the value itself for optimal risk transaction approximately the same for different X. For X = 70% (professional level), the probability becomes almost equal to one in the interval from 1% to 6% - this means that with such a good trade we can take less risk in the deal, providing the same returns. Or vice versa, you can increase the risk of not being afraid of strong subsidence.
Since we do not know exactly the proportion of profitable transactions of the future of our trade, then for a specific number of R is best to take the average of the "optimal" range of 3-5%, namely R = 4%. This is the most profitable, optimal risk per trade for the task. Similar calculations can be (and should) be carried out for any other conditions: the ratio of the take / stop, alternating item, trailing stop, Martingale, etc. You can also set different tasks, for example, only look at the drawdown or just for yield.
Let's sum up. Firstly, when trading system permanent risk to the transaction there is an optimal risk value R, which clearly and objectively can be found by calculation.
Secondly, the unambiguous definition of the optimal risk requires a precise statement of the problem. Wording such as "the more the better", "minimal risk" or "long-term and stable" will be useless until there are no concrete numbers (in this example, the parameters T, N and P).
And third, mathematics and numerical methods - a powerful tool in the hands of the trader. In contrast to the nature of the market, risk and capital management technique - entirely in our hands, and well-conducted evaluation and calculations will help to avoid unnecessary losses in the real trading.
Why have the maximum graphics? Obviously, when the risk is too low R (0-2%), we simply do not have enough transactions to achieve the desired balance level. Too much risk (R> 6%) increases the likelihood of strong subsidence, which is also not part of our plans.
It is seen that increasing the probability of 50% the balance is highly dependent on the quality of our trade, but the value itself for optimal risk transaction approximately the same for different X. For X = 70% (professional level), the probability becomes almost equal to one in the interval from 1% to 6% - this means that with such a good trade we can take less risk in the deal, providing the same returns. Or vice versa, you can increase the risk of not being afraid of strong subsidence.
Since we do not know exactly the proportion of profitable transactions of the future of our trade, then for a specific number of R is best to take the average of the "optimal" range of 3-5%, namely R = 4%. This is the most profitable, optimal risk per trade for the task. Similar calculations can be (and should) be carried out for any other conditions: the ratio of the take / stop, alternating item, trailing stop, Martingale, etc. You can also set different tasks, for example, only look at the drawdown or just for yield.
Let's sum up. Firstly, when trading system permanent risk to the transaction there is an optimal risk value R, which clearly and objectively can be found by calculation.
Secondly, the unambiguous definition of the optimal risk requires a precise statement of the problem. Wording such as "the more the better", "minimal risk" or "long-term and stable" will be useless until there are no concrete numbers (in this example, the parameters T, N and P).
And third, mathematics and numerical methods - a powerful tool in the hands of the trader. In contrast to the nature of the market, risk and capital management technique - entirely in our hands, and well-conducted evaluation and calculations will help to avoid unnecessary losses in the real trading.
