# Rule 1: Positive Expectation in Trading System

A trading system that has a positive expectation is likely to be profitable in the future. The expectation here refers to the dollar profit of the average trade, including all available winning and losing trades. The data may be derived from actual trading or system testing. Some analysts call this your mathematical edge, or simply your “edge” in the markets.

The terms “average trade” and “expectation” represent the same object, so they are freely interchanged in the following discussion. Expectation can be written in many different ways. The following formulations are identical:

Expectation($) = Average Trade($), Expectation($) = Net profit($)/(Tbtal number of trades),

Expectation($) = [(Pwin) x (Average win($))] – (1 – Pwin) x (Average loss($))].

The expectation, measured in dollars, is the profit of the average trade. The net profit, measured in dollars, is the gross profit minus the gross loss over the entire test period. Pwin is the fraction of winning trades, or the probability of winning. The probability of losing trades is given by (1-Pwin). The average win is the average dollar profit of all winning trades. Similarly, the average loss is the average dollar loss of all losing trades.

The expectation must be positive because, on balance, we want the trading system to be profitable. If the expectation is negative, this is a losing system, and money management or risk control cannot overcome its inherent limitations.

Assume that you are using system test results to estimate your av-erage trade. Note that your estimate of the expectation is limited by the available data. If you test your system on another data set, you will get a different estimate of the average trade. If you test your system on different subsets of the same data set, you will find that each subset gives a different result for the average trade. Thus, the expectation of a trading system is not a “hard and fixed” constant. Rather, the expectation changes over time, markets, and data sets. Hence, you should use as long a time period as possible to calculate your expectation.

Since the expectation is not constant, you should stipulate a mini-mum acceptable value for the average trade. For example, the minimum value should cover your trading costs and provide a “risk premium” to make it attractive. Hence, a value such as $250 for the expectation could be used as a threshold for accepting a system. In general, the larger the value of the average trade, the easier it is to tolerate its fluctuations.

Note that the expectation does not provide any measure of the variability of returns. The standard deviation of the profits of all trades is a good measure of system variability, system volatility, or system risk. Thus, the expectation does not fully quantify the amount of risk (read volatility) that must be absorbed to benefit from its profitability.

The expectation is also related to your risk of ruin. You can use statistical theory to calculate the probability that your starting capital will diminish to some small value. These calculations require assumptions about the probability of winning, the payoff ratio, and the bet size. The payoff ratio can be defined as the ratio of the average winning trades to the average losing trades. As your payoff ratio increases, and your Pwin increases, your risk of ruin decreases. The risk of ruin is also governed by bet size, that is, percentage of capital risked on every trade. The smaller your bet size, the lower the risk of ruin. Detailed calculations of risk of ruin are presented in chapter 7.

In summary, it is essential that your system have a positive expectation, that is, a profitable average trade. The value of the average trade is not fixed, but changes over time. Hence, you can specify a threshold value, such as $250, before you will accept a trading system. The expectation is also important because it affects your risk of ruin. Avoid trading systems that have a negative expectation when tested over a long time.

The expectation of your system is determined by its trading rules. The next section examines how the number of trading rules affects your system design.