Z-account statistics - what is the trader's strategy?

It is often important for an investor to check whether the Trader trades in the market or signals are subject to a pattern he has found in order to make the right choice in investing. Often, a trader wants to find the most successful money management strategy in order to increase the profitability of his tactics. In both cases Z-Account can help.

Calculation and types of Z-bills

In this particular case it does not matter for us the amount we won or lost on the account, the main factor is the very fact whether the result of this transaction was positive or negative. Thus, we will study the series of profitable (positive) transactions and losing (negative) ones. The alternation of profits and losses may occur differently for each trading systemIf, for example, the expected profit is less than the expected loss, then the number of positive trades will be significantly greater than the negative ones. Therefore we will take the same levels of expected profit and loss for more correct readings. So let's calculate the Z-account itself, below is its formula:

Z=(N*(R-0.5)-P)/((P*(P-N))/(N-1))^(1/2)

Where:
- N - the total number of transactions in the sequence;
- R - the total number of series of winning and losing trades;
- P = 2*W*L;
- W - the total number of winning trades in the sequence;
- L - the total number of losing trades in the sequence.

It is necessary to introduce the concept of series. Under series will be understood as a sequence of successive positive trades or negative ones. Let's take the following sequence of trade results and calculate the Z-count.

+ + + + + + + – – – – – – + – – – + – – – – + + + + + + + +

For the Z-score to be reliable, it is necessary that the sample was at least 30 values. We took 30 deal values. The R-value is calculated as the number of series, the first series was a sequence of 7 positive trades, the second series was a sequence of six negative trades, etc. The R parameter equals 7 (7 series). To calculate the parameter P, let's calculate the total number of positive deals (W), which equals 17, and negative deals - 13. As the result R=2*13*17=442. Let's put everything together in the formula:

Z= (30*(7-0.5)-442)/((442*(442-30)/(30-1))^(1/2)=-247/79=-3,12

After substituting all the values, we got the Z-count value equal to -3.12. What does this value indicate? In essence, the Z-score is interpreted as the probability of a random variable distributed according to the law of the standard normal distribution deviating from zero. If Z-account is close to zeroIf you have a sequence of consecutive trades, this indicates that there is a correlation between the sequences of consecutive trades. If Z-count is greater than 3that is more than 3 sigmas (standard deviations), then the probability of falling into this range is 99.74%.

Z-account sign indicates the type of dependence, if the sign positive, then a profitable trade is followed by a losing trade, and a losing trade by a profitable trade (positive or negative series). If the sign of the Z-account negativeIf the Z-scores are not in the same order as the Z-scores, then it means that a win is followed by a win and a loss by a loss. Table 1 shows the values of the Z-score, the probability of dependence and its type.

Table 1

Examples of Positive and Negative Z-Bill Applications

Let's take two examples and consider money management techniques for different values of the Z-account. For the first example let's take a trading system which gives the percentage of profitable transactions 53% and 47% unprofitable and Z-account is 2.15. The confidence interval will be 96%. The presence of a positive Z-score indicates that there were positive trades after losing trades and vice versa. Using this statistical regularity we can use such method as martingale method. That is, after a losing trade we double the size of the position. In our case the probability of winning will not be equal to 50%, but more because of the dependence between the series of losses and wins.

When the Z-account is negative, traders usually use aggressive position volume strategies. Consider another situation when the percentage of losses is greater than gains and is 49% and respectively 51%. Z-score for this trading system is -2.21. The confidence intervals are 97%. As we know, a negative Z-score signals that a positive trade is followed by a positive trade. Therefore, it will be effective to use Parlays method (pyramiding)which is to increase the volume of the contract with each positive transaction.