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173 not limited to computerized sjstems. Suppose there is a choice of sjstems to trade, and the daily equity of each is available, if only one can be selected, which should it be? There are some characteristics of performance that are universal: 3 1. Laiger profits are better than smaller profits. 2. Small, shortterm fluctuations are better than large, shortterm fluctuations. 3. Upward equity surges fprofitsl are better than downward surges flossesl. Naive Performance Criteria The use of a single piece of information is not sufficient to satisfy the basic criteria stated previously. The sjstem with the num profit is not necessarily a good choice. There might have been a run of losses greater than the investment before profitability was achieved. Similarly, the rid; alone is not sensible. The sjstem with the smallest risk is one that never trades. A valid performance measurement technique must include a comparison of risk and reward The Sha e Ratio The classic measurement of performance is the Sha e Ratio (SR), expressed as where E is the expected retum, I is the rid;fiee interest rate, and a is the standard deviation, or fluctuation, of the periodic retums (for example, the yearly retums). For practical piwposes, E should be an annualized rate of retum in percent, if you that retum to continue, and I is often omitted. The inclusion of I is important when a large part of the performance is a contribution of interest on the investment, or when the equity streams being compared are inconsistent in their use of interest income. The standard deviation of the equity is a standard way of measuring rid;. The Sha e Ratio satisfies the first criterion, that all else being equal, higher profits are better It does not satisfy either of the other requirements, because it cannot distinguish between: 1. Consecutive small losses (System B) and alternating small losses (System A) 2. Large surges of profits and large losses (Tigure 233 Clearly, System A is best in both cases. Average Maximum Retracement Schwager" has presented a comprehensive study of evaluation techniques. Although each method may emphasize a particular equity trait, he seems to favor the average num retracement (AMR). This method answers the question: "For each day, what would be the retracement if one had started trading the system on the worst possible trade entry date?" The AMR is the average of the daily maximum rdracement values. AMR = POSiMCE,  TEt)
and MCE = closedout equity (realized profits) on any trade entry date prior to i , = total equity on day i N = total number of dajs of equity data 4jackD ScLwaser, A O.mpletejifle to tiie Futures Markets () Wiley & Sons, New ¥ tk. 19S4) FIGURE233 Two cases in which the 8 Ratio fails, (a) The order in which profits and losses occur, (b) Surges in profits versus evenly dishibuted losses. \ien TEi := MCE, all traders will have a profit on day i, regartUess of when they began. Schwager suggests that a much simpler computation would use only the low total equity day of each month; it would give a rough bul good spproximation. Largest Loss Measurements such as Schwagers AMR, and even the basic standard deviation, are good techniques for comparing the longterm performance of one sjstem against another: they lack a certain reality of simply looking at the largest loss seen over the test periods. John Sweeney calls this the maximum atlverse excursion. claiming that traders should minimize the size of their largest loss Consider the standard deviation of equity changes, showing that in any month there is a 68° chance that your retums will be between 15>o and 5>o (a mean of 5">o and a standard deviation of 10°o). There is only a 2.5>o chance that you will lose more than 15>o in 1 month (2 standard deviations); therefore, there is a 50°o chance you will lose that 150oin 1 of the first 20 months (20 .2.5). Yet probability shows that you should lose more, if you keep hading, or less if you stop sooner. The largest historic loss, called the maximum tlrawdown, is a practical alternative. It simply states that the tratUng program did, for example, lose 15>o during 1 month of a 3year test. While it is possible, and even likely, that the program will have a larger loss in the future, you must be prepared for a 15" loss in a single month. ions for Lb T.S Tecimical Analysis .f cts & uuodities (April 19S7.
The measurement of rid; may include other safety factors. Once a large loss has occurred, it is likely that a laiger one may follow. It is unreasonable to think that all future losses will be smaller than the ma.ximum alreadj experienced. This potential for loss can be expressed as a probability by calculating the standard deviation of all equity drops, measured fran lows to previous high equity points, and creating an equity drop ratio (EDR): EDR = ~ 0{ED) = £7@5tdev(£D) where E is the annualized retum (equity), and o(ED) is the standard deviation of the equity drops. Although this is not far from Schwagers spproach, it satisfies all three of the original criteria: higher profits are favored, the order of profits and losses will result in larger and smaller net equity drops, and large gains are not penalized because only the drops are used to measure risk. The conservative investors may want to include some additional simple considerations of potential rid;. All else being equal, sjstems with greater rid; will be: 1. Those tested with samples that are too small 2. Those that have not shown any (or few) equity drops 3. Those that concentrate on fewer product groups (not property diversified) 4. Those that compound positions Efficient Frontier It is alwajs helpful to visualize the relative risk of how one sjstem performs with respect to others. A plot of each sjdem using its retum and rid; as coordinates will appear as in Figure 234. Clearly, the best sjstem would be the one that had the highest retums (C) if all of them had the same ride or A, the one with the lowest risk, if it also had the highest profit. The three sjdems. A, B, and C, are similar because they have the highest retums for their level of rid;. However, each used a different degree of leverage, or reserves, and therefore had close to the same profit to risk ratio but in different magnitudes. The choice between sjstems A, B, and is a personal preference discussed earlier in this chapter. Theoretically, the best choice is found by drawing a straight line fran the rid;fiee rate of rdum (on the left scale) to the tangent point on the efficient fiontier. This selection avoids the combinations in which rid; increases more quickly than reward FIGURE 234 Efficient frontier. Efficient f rwitier Risk measurement Rid; Charaderistics of Sjstems
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