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183 G(/) = /> + ln(l +b*f) + (1~p)* ln(l /) where f is the q)timum fixed fraction P is the probability of awinning bet or frade is the ratio of the average winning retum to the average losing retum In is the natural log function The solution for finding the ctimal fixed fraction to invest uses the geomefric product and geometric mean, which represent the way in which profits and losses accrae. OpU„,a./= max (ft (l f/= 0. to 1.0 where max is the function that selects the maximum value 11 is the product (thissjmbol is 2 vert + lacrossthetop (as in equation above)) laigest loss is the laigest loss of any PLi PLi is the profit or loss for trade i N is the number of frades By testing values off between .01 and 1.0, and finding the geometric mean of all trades (each percentage profit or loss applied to the account value before the current frade), we find the value off that gives the best retum. That fvalue is the ctimal f, the amount of the total account that should be invested for each trade. A simpler way of expressing optimal f the amount invested on a sinele trade, is grven a /=[ ( +1)1]/ where R = the ratio of average profit to average loss p = the probability of a winning frade Therefore, if p = .50, there is an equal chance of a profit or a loss, and the average profit is $400 while the average loss is $200 (giving R = 2.0), then f = (. 50(2 + 1)  1)/2 = .5/2 = .25 or 250o of the available capital. Given an equal chance of a profit or a loss, it is not likely that there would be four losses in a row, each of 25" o; however, the theory of runs shows that, out of every 100 trades, there should be one run of six. Eventually, there will be a run of four or five losses in a row. Ctimal f however, invests a fraction of the current equity; therefore, after a loss of 25>o, the next investment is 25>o of the balance, or 18.75>o of the initial equitj. If there are fiiilher losses, that amount drops to 14.06° o. After three losses in arow, instead of having lost 75>o of the initial equity, the investment has only dropped by 57.810. Over time, with profits twice as large as losses, and winning trades normally alternating with losing frades, the losses will be recovered. Observations of Ctimal f According to Alex Elder, there are some difficulties in using ctimal f Because the value is based on every historic trade, the ideal amount to invest on the next trade will keep changing. In addition, if you frade a position larger than the optimal f, with average results, you can expect to go broke eventually because you are overinvesting. On the other hand, if you invest less than the optimal amount, then your rid; decreases arithmetically, but your profits decrease geomefrically, which is another bad scenario. Because this process can become very complicated, the simple solution for most investors is to keep trading the same amount, leaving a large reserve so you are not caught short. On the positive side. Dr. Elder concludes that the most useful results of ctimal f is that it shows the important
principles: 1. Never average down. 2. Never meet maigin calls. 3. Liquidate yourworst position first. Practically Speaking Evaluatiiig 1 risk and reward atoliiig freram, even u.iiig animiber veaif factual perfomiance, rarely gives re«ilts tb»t are statistically accurate Mlrket: cV  and tiie perf nuance fr. .file .f any tradiiie system, can varv .iguibcantly over l"iig time perio* Detemuiiiiig the am.Uiitto invest on a siii trale based or tiie average n.k and average retiiiu of past performance can lead to a tragic end. \ien investing 25" in each trade, a loss that is twice what is expected, followed by a smaller profit and another laige loss (in a more volatile period), could ediaust your capital, regartUess of the statistics. in general, the less you risk, the safer you are. From time to time there are price shods that produce profits and losses far greater than the averages show. Although the theory of an cptimal fixed fraction may be correct, the numbers used in the calculation are not often reliable. COMPARING EXPECTED AND ACTUAL RESULTS in the development of an economic model or trading sjStem, the final selection is usually the result of a performance comparison of the completed models. Often the results are given in terms of profitiloss ratios, annualized percentage profits, e ected reliability (percentage of profitaUe trades to total trades), and potential risk. Although these statistics are common, their predictive qualities and sometimes their accuracy are not known. On occasion, these results are generated by a sample that is too small, and usually they are not the results of a predictive but an historic test. This does not mean that the model will be unsuccessful, but that the pattem of success might vary far fran the expected profit/loss ratio, reliability, and risk. In actual trading, every speculator experiences a series of losses far exceeding anjthing that was e ected; at that point, it is best to know whether this situation could occur within the realm of the sjstems profile or whether the sjStem has failed. For example, a moving average sjStem is expected to have 1 out of 3 profitable hades with a profif loss ratio of 4:1. But the first 10 hades of the sjstem are losers. Should trading be stopped? Binomial Probability Consider the application of a randomnumber sequence to the trading nxxtel. What is the probability of / losses in n mes when the probability of a loss isp? Most of the work in this area of probability is credited to Bernoulli, whose study of a random walk is called a Bernoulli process. A clear representation of a random walk is shown by the Pascal triangle (Figure 2314), where each box represents the probability of being m a particular position at a specific lime in a forward random walk. The result of this process is called a binomial distribution. The forward random walk has an analogy to price movement, with the far edges of Pascals triangle showing the probability ofa continuous sequence of wins or losses using random numbers. The sequence ( )" is exactly the same as in the discussion of the Theoiy of Runs. The probability of successive losses can be calculated as the likelihood ofa run of the same length, FIGURE 2314 Pascals triangle.
A binomial distribution is usefiil in considering the total number of losses that can occur in any order within a sequence of trades; it is the probability of getting to a specific point atthe base of Pascals triangle when there is a high probability of moving to the left (losses) rather than the right (profits). The formula for the binomial probability is: where 1 is the number of losses n is the total number of tries p is the probability of a loss and the sjmbol "!" is the factorial (e.g., 5!=5x4X3x2xl). Consider the first 15 trades of a sjlem with a Probability of success of 1/3 How many losses should be expected? To answer the question, find the binomial probability for all possibilities and form a disfribution functionLet 1=4. Then, B(4:.667.5) = (.667) {.my = 5 X.0659 = 32954 The binomial probability of having 4 losses out of the first 5 trades is about 33"o. The following table shows the probability of losses for the first 5, 10, and 15 trades of a sjstem, with a 1 /3 predicted reliability Results show the highest probability of occurrence of loss is at the "o point (mean) for each sequence, but the standard deviation gives the range of variance about the mean, so that from 2.3 to 4.4, losses are expected in every 5 trades, 5.2 to 8.2 in 10 frades, and 8.2 to 11.8 losses in 15 frades (Table 237). Note that in the 5trade example, the chances of no losses is only lo, and there is a lio chance of all losses. For the piwpose of evaluation, it is easier to look at the maximum rather than the minimum number of losses. For 15 trades, there is an8°o chance of 13 or more losses; if the sjstem has produced more than 12 losses in that period, there is something wrong with the predicted reliability In addition to the Pascal disfribution, the reader may find the Poisson and various skewed disfribution functions have application to sjstem evaluation.
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