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Trades from the S&P 500 Data Sample on Which the Lunar Model Was Verified
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Enby Date Exit Dale 850207 850221 950309 950323 950407 950421 850508 850523 850806 850620 850704 850719 850603 850816 850901 950918 851002 851017 851031 951114 951128 951214 851228 860112 860128 860213 960227
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ProfiliLoss Cumulative 650 88825 66325 950223 -2500 950323 92350 6025 950324 -2500 89850 850221 950419 850424 850516 950524 850609 050622 850718 950725 950618 950901 850816 950829 951003 851016 951114 951116 951214 851228 860109 8601 I7 860213 860213 960227 -2500 -2500 -2500 -25W -2500 -2500 4400 -2500 2575 25 10475 -2600 -2500 -2550 3150 -2500 6760 5250 -2500 -2500 18700 -2500 -2500 a7350 84850 82350 79850 77350 74050 79250 76750 79325 78350 89825 87325 84625 a2275 85425 82925 89675 94925 92425 69925 108625 106125 103 325
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Analyses of Mean Profit/Loss 47.0000 Devlatlon 974.4681 6091.1028 088.4787 1.0868 0.1392 0.2120 1.4301 0.1572 16.0000 0.3404 0.5318 0.1702
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Sample Size Sample Mean Sample SIandard
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Expected SD of Mean T Statislic (PiL > 0) Probability (Siiniflcance) Serial CorrelaIion (lag=l) Associated T Statistic Probability (Significance) Number Of Wlns
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hD ntaQe Of Wins
Upper 98% Bound Lower 89% Bound
Additional rows follow but are not shown in the table.
(Microsoft s Excel spreadsheet provides a function to obtain probabilities based on the t-distribution. Numen cal Recipes in C provides the incomplete beta function, which is very easily used to calculate probabilities based on a variety of distributions, including Student s t.) The cumulative t-distribution calculation yields a figure that represents the probability that the results obtained from the trading system were due to chance. Since this figure was small, it is unlikely that the results were due to capitalization on random features of the sample. The smaller the number, the more likely the system performed the way it did for reasons other than chance. In this instance, the probability was 0.1392; i.e., if a system with a true (population) profit
F I G U R E
Frequency and Cumulative Distribution for In-Sample Trades
of $0 was repeatedly tested on independent samples, only about 14% of the time would it show a profit as high as that actually observed.
Although the t-test was, in this example, calculated for a sample of trade profit/loss figures, it could just as easily have been computed for a sample of daily returns. Daily returns were employed in this way to calculate the probabilities referred to in discussions of the substantitive tests that appear in later chapters. In fact, the annualized risk-to-reward ratio (ARRR) that appears in many of the tables and discussions is nothing more than a resealed t-statistic based on daily returns. Finally, a con$dence interval on the probability of winning is estimated. In the example, there were 16 wins in a sample of 47 trades, which yielded a percentage of wins equal to 0.3404. Using a particular inverse of the cumulative binomial distribution, upper 99% and lower 99% boundaries are calculated. There is a 99% probability that the percentage of wins in the population as a whole is between 0.1702 and 0.5319. In Excel, the CRITBINOM function may be used in the calculation of confidence intervals on percentages. The various statistics and probabilities computed above should provide the system developer with important information regarding the behavior of the trading model-that is, if the assumptions of normality and independence are met and
CHAPTER
Statistics
if the sample is representative. Most likely, however, the assumptions underlying the t-tests and other statistics are violated; market data deviates seriously from the normal distribution, and trades are usually not independent. In addition, the sample might not be representative. Does this mean that the statistical evaluation just discussed is worthless Let s consider the cases. What if the Distribution Is Not Normal An assumption in the t-test is that the underlying distribution of the data is normal. However, the distribution of profit/loss figures of a trading system is anything but normal, especially if there are stops and profit targets, as can be seen in Figure 4- 1, which shows the distribution of profits and losses for trades taken by the lunar system. Think of it for a moment. Rarely will a profit greater than the profit target occur. In fact, a lot of trades are going to bunch up with a profit equal to that of the profit target. Other trades are going to bunch up where the stop loss is set, with losses equal to that; and there will be trades that will fall somewhere in between, depending on the exit method. The shape of the distribution will not be that of the bell curve that describes the normal distribution. This is a violation of one of the assumptions underlying the t-test. In this case, however, the Central Limit Theorem comes to the rescue. It states that as the number of cases in the sample increases, the distribution of the sample mean approaches normal. By the time there is a sample size of 10, the errors resulting from the violation of the normality assumption will be small, and with sample sizes greater than 20 or 30, they will have little practical significance for inferences regarding the mean. Consequently, many statistics can be applied with reasonable assurance that the results will be meaningful, as long as the sample size is adequate, as was the case in the example above, which had an n of 47. What if There Is Serial Dependence.3 A more serious violation, which makes the above-described application of the t-test not quite cricket, is serial dependence, which is when cases constituting a sample (e.g., trades) are not statistically independent of one another. Trades come from a time series. When a series of trades that occurred over a given span of dates is used as a sample, it is not quite a random sample. A truly random sample would mean that the 100 trades were randomly taken from the period when the contract for the market started (e.g., 1983 for the S&P 500) to far into the future; such a sample would not only be less likely to suffer from serial dependence, but be more representative of the population from which it was drawn. However, when developing trading systems, sampling is usually done from one narrow point in time; consequently, each trade may be correlated with those adjacent to it and so would not be independent, The practical effect of this statistically is to reduce the eflective sample size. When trying to make inferences, if there is substantial serial dependence, it may be as if the sample contained only half or even one-fourth of the actual number of trades or data points observed. To top it off, the extent of serial dependence cannot definitively be determined. A rough guestimate, however, can be made. One
such guestimate may be obtained by computing a simple lag/lead serial correlation: A correlation is computed between the profit and loss for Trade i and the profit and loss for Trade i + I, with i ranging from 1 to n - 1. In the example, the serial correlation was 0.2120, not very high, but a lower number would be preferable. An associated t-statistic may then be calculated along with a statistical significance for the correlation In the current case, these statistics reveal that if there really were no serial correlation in the population, a correlation as large as the one obtained from the sample would only occur in about 16% of such tests. Serial dependence is a serious problem. If there is a substantial amount of it, it would need to be compensated for by treating the sample as if it were smaller than it actually is. Another way to deal with the effect of serial dependence is to draw a random sample of trades from a larger sample of trades computed over a longer period of time. This would also tend to make the sample of trades more representative of the population, What ifthe Markets Change When developing trading systems, a third assumption of the t-test may be inadvertently violated. There are no precautions that can be taken to prevent it from happening or to compensate for its occurrence. The reason is that the population from which the development or verification sample was drawn may be different from the population from which future trades may be taken. This would happen if the market underwent some real structural or other change. As mentioned before, the population of trades of a system operating on the S&P 500 before 1983 would be different from the population after that year since, in 1983, the options and futures started trading on the S&P 500 and the market changed. This sort of thing can devastate any method of evaluating a trading system. No matter how much a system is back-tested, if the market changes before trading begins, the trades will not be taken from the same market for which the system was developed and tested; the system will fall apart. All systems, even currently profitable ones, will eventually succumb to market change. Regardless of the market, change is inevitable. It is just a question of when it will happen. Despite this grim fact, the use of statistics to evaluate systems remains essential, because if the market does not change substantially shortly after trading of the system commences, or if the change is not sufficient to grossly affect the system s performance, then a reasonable estimate of expected probabilities and returns can be calculated,
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