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© Ed Seykota, 2003 - 2005 ... © 埃德Seykota , 2003 - 2005 ... Write for permission to reprint. 写转载。

Ed Seykota's 埃德Seykota的
Frequently Asked Questions 常见问题
FAQ Index & Ground Rules ...Tribe Directory - How to Join常见问题指数及基本规则 ...族目录-如何加入
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TTP Workshop ...ResourcesTTP的车间 ...资源
Risk Management 风险管理
(c) Ed Seykota, 2003 (三)教育署Seykota , 2003
Risk 风险
RISK 风险 is the possibility of loss. 是有可能的损失。 That is, if we own some stock, and there is a possibility of a price decline, we are at risk. 也就是说,如果我们自己的一些股票,并有可能价格下跌,我们正处于危险之中。 The stock is not the risk, nor is the loss the risk. 股票是不是风险,也不是损失的风险。 The possibility of loss is the risk. As long as we own the stock, we are at risk. 损失的可能性的危险。只要我们自己的股票,我们正处于危险之中。 The only way to control the risk is to buy or sell stock. 只有这样 , 才能控制风险是买入或卖出股票。 In the matter of owning stocks, and aiming for profit, risk is fundamentally unavoidable and the best we can do is to manage the risk . 在这个问题上持有股票,以期获得利润,风险是不可避免的 , 从根本上最好的 , 我们所能做的是管理风险 。
Risk Management 风险管理
To manage is to direct and control.管理是指挥和控制。 Risk management is to direct and control the possibility of loss. The activities of a risk manager are to measure risk and to increase and decrease risk by buying and selling stock.风险管理是指挥和控制的可能性损失。活动的风险管理是衡量风险,增加和减少风险的购买和出售股票。
The Coin Toss Example 硬币掷为例
Let's say we have a coin that we can toss and that it comes up heads or tails with equal probability.比方说,我们有一个硬币,我们可以抛,它是头或尾巴等概率。 The Coin Toss Example helps to present the concepts of risk management .硬币掷为例有助于本概念的风险管理。
The PROBABILITY of an event is the likelihood of that event, expressing as the ratio of the number of actual occurrences to the number of possible occurrences. 发生某一事件的概率的可能性这一事件,并表示作为比实际发生数的数目可能发生。 So if the coin comes up heads, 50 times out of 100, then the probability of heads is 50%. Notice that a probability has to be between zero (0.0 = 0% = impossible) and one (1.0 = 100% = certain). 所以 , 如果出现硬币元首, 50倍的100 ,然后元首的概率为50 % 。请注意 , 概率要与零( 0.0 = 0 % =不可能)和1名( 1.0 = 100 % =某些) 。
Let's say the rules for the game are: (1) we start with $1,000, (2) we always bet that heads come up, (3) we can bet any amount that we have left, (4) if tails comes up, we lose our bet, (5) if heads comes up, we do not lose our bet; instead, we win twice as much as we bet, and (6) the coin is fair and so the probability of heads is 50%. 比方说 , 规则的游戏是: ( 1 )我们从 1000美元, ( 2 )我们总是投注元首出现, ( 3 )我们可以赌任何款项 , 我们已经离开, ( 4 )如果尾巴来了,我们失去打赌, ( 5 )如果元首出现,我们不会失去我们的赌注;相反,我们就赢了两次 , 尽可能多的赌注,和( 6 )金银纪念币是公平的 , 因此元首的概率是50 % 。 This game is similar to some trading methods. 此游戏类似的一些交易方式。
In this case, our LUCK equals the probability of winning, or 50%; we will be lucky 50% of the time. 在这种情况下,我们的运气等于夺冠的概率,或50 % ;幸运的 , 我们将50 %的时间。 Our PAYOFF equals 2:1 since we win 2 for every 1 we bet. 我们的支付等于2:1 , 因为我们赢得2每1我们的赌注。 Our RISK is the amount of money we wager, and therefore place at risk, on the next toss. 我们的风险是我们的钱下注,因此,在风险,下一抛。 In this example, our luck and our payoff stay constant, and only our bet may change. 在这个例子中,我们的运气和我们的回报保持不变,只有我们的赌注可能会改变。
In more complicated games, such as actual stock trading, luck and payoff may change with changing market conditions.  Traders seem to spend considerable time and effort trying to change their luck and their payoff, generally to no avail, since it is not theirs to change. 在更复杂的游戏,如实际的股票买卖,运气和收益可能会改变与变化的市场条件。投资者似乎花费了大量时间和精力试图改变自己的运气 , 他们的回报,一般无济于事,因为它是不是他们改变。 The risk is the only parameter the risk manager may effectively change to control risk. 风险是唯一的风险管理参数可有效地改变来控制风险。
We might also model more complicated games with a matrix of lucks and payoffs, to see a range of possible outcomes. 我们也有可能会更复杂的游戏模式与矩阵lucks和回报,看到了一系列可能的结果。 See figure 1. 见图1 。
Luck 好运
Payoff 回报
10% 10 %
lose 2失去2
20% 20 %
lose 1输1
30% 30 %
break even盈亏平衡
20% 20 %
win 1赢得1
10% 10 %
win 2赢得2
10% 10 %
win 3赢得3
Figure 1:  A Luck-Payoff matrix, showing six outcomes. 图1 :一种幸运,支付矩阵,显示六项成果。
This matrix might model a put-and-take game 这种矩阵可能把A型和采取游戏
with a six-sided spinning top, or even trading. 六面旋转顶端,甚至交易。
For now, however, we return to our basic coin example, since it has enough dimensions to illustrate many concepts of risk management. 但目前,我们回到我们的基本硬币的例子,因为它有足够的尺寸 , 说明许多概念的风险管理。 We consider more complicated examples later. 我们认为 , 更多复杂的例子之后。
Optimal Betting 优化投注
In our coin toss example, we have constant luck at 50%, constant payoff at 2:1 and we always bet on heads.在我们的硬币掷例如,我们不断运气,在50 % ,持续回报为2:1 ,我们总是赌头。 To find a risk management strategy, we have to find a way to manage the bet.找到一个风险管理战略,我们必须找到一种方法来管理赌注。 This is similar to the problem confronting a risk manager in the business of trading stocks.这是类似的问题面临着风险管理业务的买卖股票。 Good managers realize that there is not much they can do about luck and payoff and that the essential problem is to determine how much to wager on the stock.良好的管理人员认识到,没有什么可以做关于运气和回报,而根本问题是要确定多少赌注的股票。 We begin our game with $1,000.我们开始的比赛为1000美元。
Hunches and Systems 预感和系统
One way to determine a bet size is by HUNCH .其中一个方法来确定一个赌大小的预感 。 We might have a hunch and and bet $100.我们有预感,并和BET 100美元。
Although hunch-centric betting is certainly popular and likely accounts for an enormous proportion of actual real world betting, it has several problems: the bets require the constant attention of an operator to generate hunches, and interpret them into bets, and the bets are likely to rely as much on moods and feelings as on science.虽然预感为中心的博彩当然是受欢迎的,可能占了巨大的比例,实际真正的世界博彩,它有几个问题:投注需要不断关注的运营商产生预感,并解释他们下注,并有可能投注依靠多的情绪和情感上的科学。
To improve on hunch-centric betting, we might come up with a betting SYSTEM .改善预感为中心的投注,我们可以得出一个投注系统 。 A system is a logical method that defines a series of bets.制度是一个合乎逻辑的方法,定义了一系列的投注。 The advantages of a betting system, over a hunch method are (1) we don't need an operator, (2) the betting becomes regular, predictable and consistent and, very importantly, (3) we can perform a historical simulation, on a computer, to OPTIMIZE the betting system.的好处投注系统,在直觉的方法是: ( 1 )我们不需要运营商, ( 2 )投注成为经常的,可预见的和一贯的和非常重要的是, ( 3 )我们可以执行的历史模拟,对一台计算机, 优化投注系统。
Despite almost universal agreement that a system offers clear advantages over hunches, very few risk managers actually have a definition of their own risk management systems that is clear enough to allow a computer to back-test it.尽管几乎普遍一致认为,系统提出了明确的优势预感,很少有风险管理者实际上有一个定义,其本身的风险管理系统,是明确的,足以使一台计算机要重新进行测试。
Our coin-flip game, however is fairly simple and we can come up with some betting systems for it. Furthermore, we can test these systems and optimize the system parameters to find good risk management.我们硬币翻转,但比赛过程相当简单,我们可以拿出一些投注系统的。此外,我们可以测试这些系统和优化系统参数找到良好的风险管理。
Fixed Bet and Fixed-Fraction Bet 固定投注和固定部位博彩
Our betting system must define the bet.我们投注系统必须确定赌注。 One way to define the bet is to make it a constant fixed amount, say $10 each time, no matter how much we win or lose.其中一个方法来确定投注是使它成为一个固定数额不断说,每次10元,无论多少,我们的赢输。 This is a FIXED BET system.这是一个固定下注系统。 In this case, as in fixed-betting systems in general, our $1,000 EQUITY might increase or decrease to the point where the $10 fixed bet becomes proportionately too large or small to be a good bet.在这种情况下,如在固定投注系统,一般情况下,我们1000美元的资本有可能增加或减少到了10美元的情况下赌注成为固定比例过大或小,是一个很好的选择。
To remedy this problem of the equity drifting out of proportion to the fixed bet, we might define the bet as as FIXED-FRACTION of our equity.为了解决这一问题的股权漂流不成比例固定赌注,我们可能会确定投注作为固定部分的股权。 A 1% fixed-fraction bet would, on our original $1,000, also lead to a $10 bet. 1 %的固定部分投注将在我们原来的1000美元,还导致10美元的赌注。 This time, however, as our equity rises and falls, our fixed-fraction bet stays in proportion to our equity.然而这一次,我们的股票上升和下降,我们的固定部分赌注留在我们的股权比例。
One interesting artifact of fixed-fraction betting, is that, since the bet stays proportional to the equity, it is theoretically impossible to go entirely broke so the official risk of total ruin is zero.一个有趣的伪固定部分投注是,因为赌注比例保持公平,这是理论上不可能去完全打破,因此官方的总破产风险是零。 In actual practice, however the disintegration of an enterprise has more to do with the psychological UNCLE POINT ; see below.在实际做法,但解体企业有更多的是心理叔叔点 ;见下文。
Simulations 模拟
In order to test our betting system, we can SIMULATE over a historical record of outcomes.为了测试我们的投注系统,我们可以模拟超过历史记录的结果。 Let's say we toss the coin ten times and we come up with five heads and five tails.比方说,我们抛硬币的10倍,我们拿出5元首和5个尾巴。 We can arrange the simulation in a table such as figure 2.我们可以安排模拟在一个表中,如图2 。
Fixed Bet 固定投注 $10 $ 10
Fixed-Fraction Bet 固定部位博彩 1% 1 %
Start 开始
1000 1000年
1000 1000年
Heads 元首
1020 1020年
1020 1020年
Tails 尾巴
1010 1010年
1009.80 1009.80
Heads 元首
1030 1030年
1030 1030年
Tails 尾巴
1020 1020年
1019.70 1019.70
Heads 元首
1040 1040年
1040.09一千零四十○点零九
Tails 尾巴
1030 1030年
1029.69 1029.69
Heads 元首
1050 1050年
1050.28一千零五十点二八
Tails 尾巴
1040 1040年
1039.78 1039.78
Heads 元首
1060 1060年
1060.58一千○六十零点五八
Tails 尾巴
1050 1050年
1049.97 1049.97
Figure 2:  Simulation of Fixed-Bet and Fixed-Fraction Betting Systems. 图2 :模拟固定博彩和固定部位投注系统。
Notice that both systems make $20.00 (twice the bet) on the first toss, that comes up heads. 请注意 , 这两个系统使$ 20.00 (两次投注)第一掷,这是头。 On the second toss, the fixed bet system loses $10.00 while the fixed-fraction system loses 1% of $1,020.00 or $10.20, leaving $1,009.80. 在第二掷,固定投注系统失去$ 10.00 , 而固定部分系统失去的1 % $ 1,020.00或10.20美元,使$ 1,009.80 。
Note that the results from both these systems are approximately identical. 请注意 , 结果这两种系统的有大约相同。 Over time, however, the fixed-fraction system grows exponentially and surpasses the fixed-bet system that grows linearly. 随着时间的推移,固定部分系统呈指数增长 , 并超过了固定投注系统生长线性。 Also note that the results depend on the numbers of heads and tails and do not at all depend on the order of heads and tails. 还注意到 , 结果取决于号码头和尾巴 , 不依赖于所有的命令头和尾巴。 The reader may prove this result by spreadsheet simulation. 读者可能会证明这一点 , 结果仿真试算表。
Pyramiding and Martingale 聚合和鞅
In the case of a random process, such as coin tosses, streaks of heads or tails do occur, since it would be quite improbable to have a regular alternation of heads and tails.如果是随机过程,如硬币击败,条纹的元首或尾巴确实发生,因为这将是非常不可能有定期轮岗头和尾巴。 There is, however, no way to exploit this phenomenon, which is, itself random.但是,有没有办法利用这种现象,这种现象是,这本身就是随机的。 In non-random processes, such as secular trends in stock prices, pyramiding and other trend-trading techniques may be effective.在非随机过程,如长期趋势的股票价格,聚合和其他趋势交易技术可有效。
Pyramiding is a method for increasing a position, as it becomes profitable.聚合是一种增加的立场,因为它变得有利可图。 While this technique might be useful as a way for a trader to pyramid up to his optimal position, pyramiding on top of an already-optimal position is to invite the disasters of over-trading.尽管该技术可能是有用的方式进行交易,以金字塔了自己的最佳位置,聚合顶部已经最优的立场是邀请灾害过度交易。 In general, such micro-tinkering with executions is far less important than sticking to the system.总而言之,这类微型修补处决远远重要性不亚于坚持这一制度。 To the extent that tinkering allows a window for further interpreting trading signals, it can invite hunch trading and weaken the fabric that supports sticking to the system.以至于修修补补允许的窗口进一步解释交易信号,它可以邀请预感交易和削弱织物,支持坚持这一制度。
The Martingale system is a method for doubling-up on losing bets.鞅系统是一种翻一番行动失去赌注。 In case the doubled bet loses, the method re-doubles and so on.增加一倍的情况下,赌输了,该方法重新双打等。 This method is like trying to take nickels from in front of a steam roller.这种方法就像是试图采取硬币从前面的蒸汽压路机。 Eventually, one losing streak flattens the account.最终,一个连败flattens的帐户。
Optimizing - Using Simulation 优化-利用模拟
Once we select a betting system, say the fixed-fraction betting system, we can then optimize the system by finding the PARAMETERS that yield the best EXPECTED VALUE . In the coin toss case, our only parameter is the fixed-fraction.  Again, we can get our answers by simulation.一旦我们选择投注系统,说的固定部分投注系统,我们可以通过优化系统参数 ,找到最佳的收益率预期值 。在硬币掷情况下,我们唯一的参数是固定部分。同样,我们答案可以从我们的模拟。 See figures 3 and 4.见图3和图4 。
Note: The coin-toss example intends to illuminate some of the elements of risk, and their inter-relationships. 注:硬币掷例如打算照亮的一些因素的风险,他们的相互关系。 It specifically applies to a coin that pays 2:1 with a 50% chance of either heads or tails, in which an equal number of heads and tails appears. 它特别适用于硬币支付2:1的机率为50 %或者元首或尾巴,其中有同样数量的头和尾巴出现。 It does not consider the case in which the numbers of heads and tails are unequal or in which the heads and tails bunch up to create winning and losing streaks. 它不认为在这种情况下的人数头和尾巴是不平等的 , 或在该头和尾巴一群赢得了创建和连败。 It does not suggest any particular risk parameters for trading the markets. 它没有提出任何具体的风险参数的交易市场。
% Bet %博彩
Start 开始
Heads 元首
Tails 尾巴
Heads 元首
Tails 尾巴
Heads 元首
Tails 尾巴
Heads 元首
Tails 尾巴
Heads 元首
Tails 尾巴
0 0
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
1000.00 1000.00
5 5
1000.00 1000.00
1100.00 1100.00
1045.00 1045.00
1149.50 1149.50
1092.03 1092.03
1201.23 1201.23
1141.17 1141.17
1255.28 1255.28
1192.52 1192.52
1311.77 1311.77
1246.18 1246.18
10 10
1000.00 1000.00
1200.00 1200.00
1080.00一千〇八十零点〇 〇
1296.00 1296.00
1166.40 1166.40
1399.68 1399.68
1259.71 1259.71
1511.65 1511.65
1360.49 1360.49
1632.59 1632.59
1469.33 1469.33
15 15
1000.00 1000.00
1300.00 1300.00
1105.00 1105.00
1436.50 1436.50
1221.03 1221.03
1587.33 1587.33
1349.23 1349.23
1754.00 1754.00
1490.90一千四百九十○点九零
1938.17 1938.17
1647.45 1647.45
20 20
1000.00 1000.00
1400.00 1400.00
1120.00一千一百二十零点○○
1568.00 1568.00
1254.40 1254.40
1756.16 1756.16
1404.93 1404.93
1966.90 1966.90
1573.52 1573.52
2202.93 2202.93
1762.34 1762.34
25 25
1000.00 1000.00
1500.00 1500.00
1125.00 1125.00
1687.50 1687.50
1265.63 1265.63
1898.44 1898.44
1423.83 1423.83
2135.74 2135.74
1601.81 1601.81
2402.71 2402.71
1802.03 1802.03
30 30
1000.00 1000.00
1600.00 1600.00
1120.00一千一百二十零点○○
1792.00 1792.00
1254.40 1254.40
2007.04 2007.04
1404.93 1404.93
2247.88 2247.88
1573.52 1573.52
2517.63 2517.63
1762.34 1762.34
35 35
1000.00 1000.00
1700.00 1700.00
1105.00 1105.00
1878.50 1878.50
1221.03 1221.03
2075.74 2075.74
1349.23 1349.23
2293.70 2293.70
1490.90一千四百九十○点九零
2534.53 2534.53
1647.45 1647.45
40 40
1000.00 1000.00
1800.00 1800.00
1080.00一千〇八十零点〇 〇
1944.00 1944.00
1166.40 1166.40
2099.52 2099.52
1259.71 1259.71
2267.48 2267.48
1360.49 1360.49
2448.88 2448.88
1469.33 1469.33
45 45
1000.00 1000.00
1900.00 1900.00
1045.00 1045.00
1985.50 1985.50
1092.03 1092.03
2074.85 2074.85
1141.17 1141.17
2168.22 2168.22
1192.52 1192.52
2265.79 2265.79
1246.18 1246.18
50 50
1000.00 1000.00
2000.00 2000.00
1000.00 1000.00
2000.00 2000.00
1000.00 1000.00
2000.00 2000.00
1000.00 1000.00
2000.00 2000.00
1000.00 1000.00
2000.00 2000.00
1000.00 1000.00
55 55
1000.00 1000.00
2100.00 2100.00
945.00 945.00
1984.50 1984.50
893.03 893.03
1875.35 1875.35
843.91 843.91
1772.21 1772.21
797.49 797.49
1674.74 1674.74
753.63 753.63
60 60
1000.00 1000.00
2200.00 2200.00
880.00八百八十○点零零
1936.00 1936.00
774.40 774.40
1703.68 1703.68
681.47 681.47
1499.24 1499.24
599.70 599.70
1319.33 1319.33
527.73 527.73
65 65
1000.00 1000.00
2300.00 2300.00
805.00 805.00
1851.50 1851.50
648.03 648.03
1490.46 1490.46
521.66 521.66
1199.82 1199.82
419.94 419.94
965.85 965.85
338.05 338.05
70 70
1000.00 1000.00
2400.00 2400.00
720.00七百二十〇点〇 〇
1728.00 1728.00
518.40 518.40
1244.16 1244.16
373.25 373.25
895.80 895.80
268.74 268.74
644.97 644.97
193.49 193.49
75 75
1000.00 1000.00
2500.00 2500.00
625.00 625.00
1562.50 1562.50
390.63 390.63
976.56 976.56
244.14 244.14
610.35 610.35
152.59 152.59
381.47 381.47
95.37 95.37
Figure 3:  Simulation of equity from a fixed-fraction betting system. 图3 :模拟股本由固定部分投注系统。
At a 0% bet there is no change in the equity. 在0 %投注没有任何变化的公平性。 At five percent bet size, we bet 5% of $1,000.00 or $50.00 and make twice that on the first toss (heads) so we have and expected value of $1,100, shown in gray. 在百分之五赌注的大小,我们投注5 % $ 1,000.00或$ 50.00 , 并两次 , 在第一掷(头) , 所以我们必须和预期价值1,100美元,显示灰色。 Then our second bet is 5% of $1,100.00 or $55.00, which we lose, so we then have $1,045.00. 那么 , 我们的第二个赌注是5 % ,为1,100.00或$ 55.00 ,我们输了,所以我们当时已经$ 1,045.00 。 Note that we do the best at a 25% bet size, shown in red.  Note also that the winning parameter (25%) becomes evident after just one head-tail cycle. 请注意 , 我们最好在25 %赌大小,以红色显示。还注意到 , 赢得参数( 25 % )之后 , 显然只有一个头尾周期。 This allows us to simplify the problem of searching for the optimal parameter to the examination of just one head-tail cycle. 这使我们能够简化的问题 , 寻找最佳参数的检查只是一个头尾周期。

Figure 4:  Expected value (ending equity) from ten tosses, versus bet fraction, 图4 :预期值(截至股权)由10击败,与投注分数,
for a constant bet fraction system,  for a 2:1 payoff game, 为不断下注分数系统,回报为2:1的比赛,
from the first and last columns of figure 3. 从第一个和最后一个栏的数字3 。
Notice that the expected value of the system rises from $1000.00 with increasing bet fraction to a maximum value of about $1,800 at a 25% bet fraction. 请注意 , 预计该系统的价值上升到千点零零美元打赌分数的增加 , 最高价值约 1 800美元的25 %投注分数。 Thereafter, with increasing bet fraction, the profitability declines. 此后,越来越多地投注分数,利润下降。 This curve expresses two fundamental principles of risk management: (1) The Timid Trader Rule : if you don't bet very much, you don't make very much, and (2) The Bold Trader Rule : If you bet too much, you go broke. 这个曲线表示的两项基本原则的风险管理: ( 1 ) 怯懦交易规则 :如果你不赌非常,你不十分,以及( 2 ) 大胆的交易规则 :如果您投注太多,你去了。 In portfolios that maintain multiple positions and multiple bets, we refer to the total risk as the portfolio heat . 在保持投资组合多个位置和多种投注,我们指的是总风险的投资组合热 。
Note: Note the chart illustrates the Expected Value / Bet Fraction relationship for a 2:1 payoff game. 注:请注意图表显示期望值/博彩分数关系为2:1回报游戏。 For a graph of this relationship at varying payoffs, see Figure 8. 对于这一关系的图形不同贿赂,见图8 。
Optimizing - Using Calculus 优化-利用微积分
Since our coin flip game is relatively simple, we can also find the optimal bet fraction using calculus. Since we know that the best system becomes apparent after only one head-tail cycle, we can simplify the problem to solving for just one of the head-tail pairs.由于我们的硬币翻转游戏相对简单,我们也可以找到最佳的投注分数使用演算。因为我们知道,最好的系统变得很明显只有一个头后尾周期,我们可以简化这一问题解决的只是其中一个头部尾对。
The stake after one pair of flips: 股份后 , 一双翻转:
S = (1 + b*P) * (1 - b) * S 0 秒= ( 1 + b *规划) * ( 1 -二) *卖0
S - the stake after one pair of flips 县-股权后一双翻转
b - the bet fraction 数b -投注分数
P - the payoff from winning - 2:1 P -赢得了回报- 2:1
S 0 - the stake before the pair of flips 卖0 -股权之前对翻转
(1 + b*P) - the effect of the winning flip ( 1 + b *规划) -的影响 , 赢得倒装
(1 - b) - the effect of the losing flip ( 1 - b )段-的影响 , 失去倒装
So the effective return, R, of one pair of flips is: 因此 , 有效的恢复,俄,中一双翻转是:
R = S / S 0 R =的S /卖0
R = (1 + bP) * (1 - b) R = ( 1 + BP )的* ( 1 -二)
R = 1 - b + bP - b 2 P R = 1 -乙+碱基-乙2个P
R = 1 + b(P-1) - b 2 P R = 1 +乙(个P - 1 ) -乙2个P
Note how for small values of b, R increases with b(P-1) and how for large values of b, R decreases with b 2 P. These are the mathematical formulations of the timid and bold trader rules.注意:如何为小值的B ,研究增加与B (个P - 1 ) ,以及如何对大型价值的B ,俄随买2体育这些数学公式的胆怯和大胆的交易规则。
We can plot R versus b to get a graph that looks similar to the one we get by simulation, above, and just pick out the maximum point by inspection.我们可以阴谋R与B的获得图类似于一个我们得到的模拟,首先,仅仅挑选出的最高点的检查。 We can also notice that at the maximum, the slope is zero, so we can also solve for the maximum by taking the slope and setting it equal to zero.我们也可以看到,在最大的斜坡是零,所以我们也可以解决的,采取的最大的坡度和设置等于零。
Slope = dR/db = (P-1) - 2bP = 0,  therefore:边坡=博士/分贝= (个P - 1 ) - 2bP = 0 ,因此:
b = (P-1)/2P , and, for P = 2:1, b = (个P - 1 ) /双路,并为P = 2:1 ,
b = (2 - 1)/(2 * 2) = .25 b = ( 2 - 1 ) / ( 2 * 2 ) = 0.25
So the optimal bet, as before, is 25% of equity.因此,最佳的选择,如前所言, 25 %的股权。
Optimizing - Using The Kelly Formula 优化-利用凯利公式
JL Kelly's seminal paper, A New Interpretation of Information Rate , 1956, examines ways to send data over telephone lines.巨浪凯利的开创性,提出了一种新的解释信息速率 , 1956年,审查的方式发送数据通过电话线。 One part of his work, The Kelly Formula, also applies to trading, to optimize bet size.一个是他的工作,凯利公式,也适用于贸易,优化赌大小。
The Kelly Formula 凯利公式
K = W - (1-W)/R κ =全体委员会- ( 1 -宽) /住宅
K = Fraction of Capital for Next Trade κ =分数资本下一步贸易 W = Historical Win Ratio (Wins/Total Trials) W =历史赢率(快讯/总试验) R = Winning Payoff Rate R =胜利支付率 ------- -------
For example, say a coin pays 2:1 with 50-50 chance of heads or tails. 例如,假设硬币支付2:1的机会与50-50元首或尾巴。 Then ... 然后...
K = .5 - (1 - .5)/2 = .5 - .25 = .25. κ = 0.5 - ( 1 - 0.5 ) / 2 = 0.5 - 0.25 = 0.25 。
Kelly indicates the optimal fixed-fraction bet is 25%. 凯利指出了最佳固定部分赌注是25 % 。
Figure 5:  The Kelly Formula 图5 :凯利公式
Note that the values of W and R are long-term average values, 请注意 , 价值观的W和R都是长期平均值,
so as time goes by, K might change a little. 因此随着时间的推移,钾可能会改变一些。
Reference:http://www.racing.saratoga.ny.us/kelly.pdf 参考:http://www.racing.saratoga.ny.us/kelly.pdf

Some Graphic Relationships 有些图形关系
Between Luck, Payoff and Optimal Bet Fraction 之间好运,支付和最优下注分数
The Optimal Bet Fraction Increases with Luck and Payoff最优下注分数的增加而运气和支付

Figure 6:  Optimal bet fraction increases linearly with luck, asymptotically to payoff. 图6 :最优下注比例的增加而线性与运气,渐近 , 以回报。
This graph shows the optimal bet fraction for various values of luck (Y) and payoff (X). 该图显示的最佳选择部分的各种价值观的运气( Y )和收益(十) 。 Optimal bet fraction increases with increasing payoff. 优化投注分数增加而增加回报。 For very high payoffs, optimal bet size equals luck. 为非常高的回报,最佳赌注大小相当于运气。 For example, for a 5:1 payoff on a 50-50 coin, the optimal bet approaches about 50% of your stake. 例如,对于5:1回报的50-50硬币,最佳投注方式大约50 %的股份。
The Expected Value of the Process, at the Optimal Bet Fraction期望值的进程,在优化投注分数

Figure 7:  The optimal expected value increases with payoff and luck. 图7 :最佳预期值的增加而收益和运气。
This graph shows optimal expected value for various values of luck and payoff, given betting at the optimal bet fraction. 该图显示最佳预期价值的各种价值观的运气和回报,因为赌博在最佳投注分数。 The higher the payoff (X: 1:1 to 5:1) and the higher the luck (Y: .20 to .70), the higher the expected value. 较高的回报(十: 1:1至5:1 )和更高的运气(是: 0.20至0.70 ) ,较高的预期值。 For example, the highest expected value is for a 70% winning coin that pays 5:1. 例如,最高的预期值为70 %赢得硬币支付5:1 。 The lowest expected value is for a coin that pays 1:1 (even bet). 最低的预期值是一个硬币支付1:1 (甚至赌注) 。
Finding the Optimal Bet Fraction from the Bet Size and Payoff寻求最优下注分数由投注大小和支付

Figure 8:  For high payoff, optimal bet fraction approaches luck. 图8 :对高回报,最佳投注分数办法运气。
This graph shows the expected value of a 50% lucky (balanced) coin for various levels of bet fraction and payoff. 该图显示的预期值50 %的幸运(平衡)硬币为各级投注分数和回报。 The expected value has an optimal bet fraction point for each level of payoff. 预期值最佳投注分数点的每一级别的回报。 In this case, the optimal bet fraction for a 1.5:1 payoff is about 15%; at a 2:1 payoff the optimal bet fraction is about 25%; at a 5:1 payoff, the optimal bet fraction is about 45%. 在这种情况下,最佳的投注比例为1.5:1回报是15 %左右;在2:1回报率的最佳选择是25 %左右;在5:1回报,最佳投注比例约为45 % 。 Note: Figure 4 above is the cross section of figure 8, at the 2:1 payoff level. 注:图4以上是截面图8 ,在2:1回报水平。

Non-Balanced Distributions and High Payoffs 非均衡分布和高回报
So far, we view risk management from the assumption that, over the long run, heads and tails for a 50-50 coin will even out.到目前为止,我们认为风险管理的前提是,从长远来看,头和尾巴为50-50硬币将拉平。 Occasionally, however, a winning streak does occur.有时,但是,连胜发生。 If the payoff is higher than 2:1 for a balanced coin, the expected value, allowing for winning streaks, reaches a maximum for a bet-it-all strategy.如果回报高于2:1的平衡硬币,预期值,使连胜,达到最大的赌注,它所有的战略。
For example, for a 3:1 payoff, each toss yields an expected value of payoff-times-probability or 3/2. Therefore, the expected value for ten tosses is $1,000 x (1.5) 10 or about $57,665.例如,对于3:1回报,每个掷产量预计价值回报倍概率或3 / 2 。因此,预期值为10击败是1000美元× ( 1.5 ) 10或五十七点六六五美元。 This surpasses, by far, the expected value of about $4,200 from optimizing a 3:1 coin to about a 35% bet fraction, with the assumption of an equal distribution of heads and tails.这种超越,到目前为止,预计价值约四点二〇 〇美元从优化3:1硬币约35 %的投注比例,并承担平等的分配头和尾巴。
Almost Certain Death Strategies 几乎可以肯定死策略
Bet-it-all strategies are, by nature, almost-certain-death strategies.赌注,它所有的战略,其性质,几乎一定死战略。 Since the chance of survival, for a 50-50 coin equals (.5) N where N is the number of tosses, after ten tosses, the chance of survival is  (.5) 10 , or about one chance in one thousand.由于生存的机会,为50-50硬币等于( 0.5 ) n其中N是一些击败后, 10击败,存活的机会是( 0.5 ) 10个 ,约占一次机会在1000 。 Since most traders do not wish to go broke, they are unwilling to adopt such a strategy.由于大多数交易商不想打破,他们不愿意采取这样的战略。 Still, the expected value of the process is very attractive, so we would expect to find the system in use in cases where death carries no particular penalty other than loss of assets.不过,预期价值的过程是非常有吸引力,所以我们希望找到该系统在使用中死亡的情况下进行任何特别的罚款以外的其他资产损失。
For example, a general, managing dispensable soldiers, might seek to optimize his overall strategy by sending them all over the hill with instructions to charge forward fully, disregarding personal safety.例如,一般而言,管理可有可无的士兵,可能会寻求优化的整体战略,把他们所有的山的指示收取着完全不顾人身安全。 While the general might expect to lose many of his soldiers by this tactic, the probabilities indicate that one or two of them might be able to reach the target and so maximize the overall expected value of the mission.虽然一般预期失去了他的许多士兵的这种战术,概率表明,一个或两个人也许能达到这一目标,因此最大限度地发挥整体预期价值的使命。
Likewise, a portfolio manager might divide his equity into various sub-accounts.同样,投资组合经理可能分裂成不同的股权分帐户。 He might then risk 100% of each sub account, thinking that while he might lose many of them, a few would win enough so the overall expected value would maximize.他可能会100 %的风险每个子帐户,思想,虽然他可能会失去很多人,有几个会赢得足够的总价值预计将最大限度地发挥。 This, the principle of DIVERSIFICATION , works in cases where the individual payoffs are high.为此,原则多样化 ,工程的情况下,个人的回报是很高的。
Diversification 多样化
Diversification is a strategy t o distribute investments among different securities in order to limit losses in the event of a fall in a particular security.多样化是一个战略吨ö散布在不同的证券投资 , 以限制损失发生在某一特定秋季安全。 The strategy relies on the average security having a profitable expected value, or luck-payoff product. 这一战略依靠的平均安全有盈利预期值,或运气,回报的产品。 Diversification also offers some psychological benefits to single-instrument trading since some of the short-term variation in one instrument may cancel out that from another instrument and result in an overall smoothing of short-term portfolio volatility. 多样化也提供一些心理好处单一工具交易 , 因为一些短期的变化在一个工具可能会取消了 , 从另外一个工具 , 并导致整体平滑的短期投资组合的波动性。
The Uncle Point 大叔点
From the standpoint of a diversified portfolio, the individual component instruments subsume into the overall performance. 站在一个多元化的投资组合,个别部分文书归入到整体性能。 The performance of the fund, then becomes the focus of attention, for the risk manager and for the customers of the fund. The fund performance, then becomes subject to the same kinds of feelings, attitudes and management approaches that investors apply to individual stocks. 基金的业绩,则成为关注的焦点,风险经理和客户的资金。基金业绩,就成为必须遵守同样的种情感,态度和管理方法 , 适用于个人投资者的股票。
In particular, one of the most important, and perhaps under-acknowledged dimensions of fund management is the UNCLE POINT or the amount of draw down that provokes a loss of confidence in either the investors or the fund management. 特别是,其中一个最重要的,也许下承认方面的基金管理是叔叔点或数额削减的挑衅失去了信心 , 无论是投资者或基金的管理。 If either the investors or the managers become demoralized and withdraw from the enterprise, then the fund dies. 如果任何一方的投资者或管理者成为士气和退出企业,那么该基金去世。 Since the circumstances surrounding the Uncle Point are generally disheartening, it seems to receive, unfortunately, little attention in the literature. 由于周围环境的叔叔点通常令人沮丧,似乎接受,不幸的是,很少注意的文献。
In particular, at the initial point of sale of the fund, the Uncle Point typically receives little mention, aside from the requisite and rather obscure notice in associated regulatory documentation. 尤其是,在最初的销售点的基金,通常收到叔叔点很少提到,除了必要的和模糊的通知 , 而相关的管理文件。 This is unfortunate, since a mismatch in the understanding of the Uncle Point between the investors and the management can lead to one or the other giving up, just when the other most needs reassurance and reinforcement of commitment. 这是不幸的,因为错在了解大叔点之间的投资者和管理可能会导致一方或另一方放弃,正当其他最需要安慰和加强承诺。
In times of stress, investors and managers do not access obscure legal agreements, they access their primal gut feelings. 在时间的压力,投资者和管理人员不准入晦涩的法律协定,他们获得的原始肠道感情。 This is particularly important in high-performance, high-volatility trading where draw downs are a frequent aspect of the enterprise. 这一点尤其重要 , 在高性能,高波动的交易情况提请起伏是常客方面的企业。
Without conscious agreement on an Uncle Point, risk managers typically must assume, by default to safety, that the Uncle Point is rather close and so they seek ways to keep the volatility low. 如果没有意识到协议的叔叔点,风险管理者必须承担通常,默认情况下 , 以安全,这大叔点是相当接近 , 所以他们设法保持较低的波动。 As we have seen above, safe, low volatility systems rarely provide the highest returns. 正如我们已经看到上述情况,安全,低波动系统很少提供最高的回报。 Still, the pressures and tensions from the default expectations of low-volatility performance create a demand for measurements to detect and penalize volatility. 不过,压力和紧张的默认预期的低波动性的表现创造需求测量检测和处罚的波动。
Measuring Portfolio Volatility 测量组合波动
Sharpe, VaR, Lake Ratio and Stress Testing 夏普, VaR方法,湖比率和压力测试
From the standpoint of the diversified portfolio, the individual components merge and become part of the overall performance. 从角度的多元化投资组合,每个组成部分合并 , 成为整体性能。 Portfolio managers rely on measurement systems to determine the performance of the aggregate fund, such as the Sharpe Ratio, VaR, Lake Ratio and Stress Testing. 投资组合经理依靠测量系统 , 以确定执行情况的总基金,如夏普比率,风险值,湖比率和压力测试。
William Sharpe, in 1966, creates his "reward-to-variability ratio." 威廉夏普,在1966年,创造了他的“奖励对变异的比例。 ” Over time it comes to be known as the "Sharpe Ratio." 随着时间的推移 , 它被称为“夏普比率。 ” The Sharpe Ratio, S, provides a way to compare instruments with different performances and different volatilities, by adjusting the performances for volatilities. 的夏普比率,硫,提供了一种比较文书不同性能和不同的波动情况,通过调整表演波动。
S = mean(d)/standard_deviation(d)  ... 秒=平均值(四) / standard_deviation (四) ... the Sharpe Ratio, where 的夏普比率,在那里
d = Rf - Rb   ... d =迁移-铷... the differential return, and where 返回的差别,并在
Rf - return from the fund 射频-回报基金
Rb - return from a benchmark 铷-返回基准
Various variations of the Sharpe Ratio appear over time. 各种不同的夏普比率似乎随着时间的推移。 One variation leaves out the benchmark term, or sets it to zero. 离开了一个变异的基准来看,或者其设置为零。 Another, basically the square of the Sharpe Ratio, includes the variance of the returns, rather than the standard deviation. One of the considerations about using the Sharpe ratio is that it does not distinguish between up-side and down-side volatility, so high-leverage / high-performance systems that seek high upside-volatility do not appear favorably. 另一个,基本上平方米的夏普比率,包括差额的回报,而不是标准偏差。一个思考用夏普比率是 , 它不区分乐观的一面看,下游侧波动,如此之高杠杆/高性能的系统 , 寻求高颠倒波动似乎并不看好。
VaR, or Value-at-Risk is another currently popular way to determine portfolio risk. Typically, it measures the highest percentage  draw down, that is expected to occur over a given time period, with 95% chance. VaR的,或风险价值是目前流行的方式来确定投资组合的风险。通常情况下,它的最高百分比措施削减,这是预计将发生在某一时间内,有95 %的机会。 The drawbacks to relying on VaR are that (1) historical computations can produce only rough approximations of forward volatility and (2) there is still a 5% chance that the percentage draw down will still exceed the expectation. 缺点依靠的VaR是 : ( 1 )历史的计算可以只生产粗近似着波动和( 2 )仍然有5 %的机会的百分比削减仍将超过预期。 Since the most severe draw down problems (loss of confidence by investors and managers) occur during these "outlier" events, VaR does not really address or even predict the very scenarios it purports to remedy. 由于削减最严重的问题(失去信心的投资者和管理人员)期间发生的这些“异常”的事件,风险并没有真正解决 , 甚至预测非常情况看来补救。
A rule-of-thumb way to view high volatility accounts, by this author, is the Lake Ratio. 规则的拇指的方式 , 以查看高波动性帐户,此作者的,是湖的比例。 If we display performance as a graph over time, with peaks and valleys, we can visualize rain falling on a mountain range, filling in all the valleys. This produces a series of lakes between peaks. 如果我们显示性能作为图形随着时间的推移,与高峰和低谷,我们可以直观降雨出现在山脉,填写所有的山谷。此生产了一系列的湖泊山峰之间。 In case the portfolio is not at an all-time high, we also erect a dam back up to the all time high, at the far right to collect all the water from the previous high point in a final, artificial lake. 如果在投资组合不是历史最高水平,我们还竖立一座水坝恢复到历史高位,在最右边 , 收集所有水前高点 , 最后一次,人工湖。 The total volume of water represents the integral product of drawdown magnitude and drawdown duration. 总水量代表组成的产品缩编规模和削减期限。
If we divide the total volume of water by the volume of the earth below it, we have the Lake Ratio. 如果我们鸿沟总水量的体积是地球下面,我们有湖比例。 The rate of return divided by the Lake Ratio, gives another measure of volatility-normal return. 的回报率除以湖比,使另一项措施的波动正常返回。 Savings accounts and other instruments that do not present draw downs do not collect lakes so their Lake-adjusted returns can be infinite. 储蓄账户和其他文书不本提请起伏不收集湖湖区 , 使他们调整后的回报率可以无限的。

Figure 9: The Lake Ratio = Blue / Yellow 图9 :湖比率=蓝/黄
Getting a feel for volatility by inspection. 获得感觉波动的检验。
Reference for Sharpe Ratio: 参考夏普比率:
http://www.stanford.edu/~wfsharpe/art/sr/sr.htmhttp://www.stanford.edu/ ? wfsharpe /艺术/简/ sr.htm
Stress Testing 压力测试
Stress Testing is a process of subjecting a model of the trading and risk management system to historical data, and noticing the historical performance, with special attention to the draw downs. 压力测试是一个过程的约束模型的交易和风险管理制度 , 以历史数据,并发现的历史业绩,并特别注意提请起伏。 The difficulty with this approach, is that few risk managers have a conscious model of their systems, so few can translate their actual trading systems to computer code. 很难用这个办法,是管理人员 , 很少有风险意识的模型 , 他们的系统,所以很少有人能够将其实际的交易系统 , 以计算机代码。 Where this is possible, however, it provides three substantial benefits (1) a framework within which to determine optimal bet-sizing strategies, (2) a high level of confidence that the systems are logical, stable and efficacious, and (3) an exhibit to support discussions to bring the risk/reward expectations of the fund managers and the investors into alignment. 如果这是可能的,但是,它提供了三个巨大的利益( 1 )的框架内 , 以确定最佳的赌注规模的战略, ( 2 )高度的信任 , 该系统是符合逻辑的,稳定的和有效的,和( 3 )展览以支持 , 使讨论的风险/回报的期望基金经理和投资者进入对齐。
The length of historical data sample for the test is likely adequate if shortening the length by a third or more has no appreciable effect on the results. 长度的历史数据样本的测试可能是足够的 , 如果缩短了三分之一或更多的已没有明显的影响的结果。
Portfolio Selection 资产组合选择
During market cycles, individual stocks exhibit wide variations in behavior. 在市场周期,个别股票表现出千差万别的行为。 Some rise 100 times while others fall to 1 percent of their peak values. 有些上升100倍 , 而另一些下降到百分之一的高峰值。 Indicators such as the DJIA, The S&P Index, the NASDAQ and the Russell, have wide variations from each other, further indicating the importance of portfolio selection. 指标如道指,标准普尔指数, Nasdaq综合股价指数和罗素,有很大的差异 , 相互,进一步表明了投资组合选择的重要性。 A portfolio of the best performing stocks easily outperforms a portfolio of the worst performing stocks. 阿组合表现最好的股市优于容易组合表现最差的股市。 In this regard, the methods for selecting the trading portfolio contribute critically to overall performance and the methodology to select instruments properly belongs in the back-testing methods. 在这方面,该方法挑选贸易投资促进严重的整体性能和使用的方法 , 选择适当的文书属于后的测试方法。
The number of instruments in a portfolio also effects performance.文书数目在投资组合中还影响性能。 A small number of instruments produces volatile, occasionally very profitable performance while a large number of instruments produces less volatile and more stable, although lower, returns.少数文书产生波动,有时非常有利可图的性能的同时,大量的文书产生波动较小和更稳定的,虽然较低,回报。
Position Sizing 仓量
Some position sizing strategies consider value, others risk.一些立场浆纱战略考虑的价值,其他的风险。 Say a million dollar account intends to trade twenty instruments, and that the investor is willing to risk 10% of the account.说100万美元的帐户打算贸易20文书,而且投资者愿意风险10 %的帐户。
Value-Basis position sizing divides the account into twenty equal sub-accounts of $50,000 each, one for each stock.价值为基础的立场分歧上浆的帐户平等进入20分帐户的各50,000美元,每个股票。 Since stocks have different prices, the number of shares for various stocks varies.由于有不同的股票价格,股票的数量不同的各种股票。
Stock 库存 Price/Share 价格/分享 Shares 股票 Value 价值
A 字母a $50 50美元
1000 1000年 $50,000 $ 50,000
B 乙 $100 $ 100 500 500
$50,000 $ 50,000
C ç $200 $ 200 250 250 $50,000 $ 50,000
Value-Basis Position Sizing 价值为基础的位置浆纱
Dividing $50,000 by $50/share gives 1000 Shares 除以$ 50,000美元50/share使1000股
Risk-Basis position sizing considers the risk for each stock, where risk is the entry price minus the stop-out point.风险基础地位浆纱认为每个股票的风险,而风险是入门价格减去停止了点。 It divides the total risk allowance, say 10% or $100,000 into twenty sub accounts, each risking $5,000.它划分的全部风险津贴,说10 %或10万美元到20分账户,每个冒着5000美元。 Dividing the risk allowance, $5,000 by the risk per share, gives the number of shares.分裂的危险津贴, 5000美元每股的风险,使之股份数目。
Stock 库存 Price/Share 价格/分享 Risk/Share 风险/分享 Shares 股票 Risk 风险 Value 价值
A 字母a $50 50美元 $5 5美元
1000 1000年 $5,000 5,000元 $50,000 $ 50,000
B 乙 $100 $ 100 $10 $ 10 500 500
$5,000 5,000元 $50,000 $ 50,000
C ç $200 $ 200 $5 5美元 1000 1000年 $5,000 5,000元 $200,000 $ 200,000
Risk-Basis Position Sizing 风险基础地位浆纱
Dividing $5,000 by $5 risk/share gives 1000 Shares 除以 5000美元5美元的风险/ 1000股的份额为
Note that since risk per share may not be proportional to price per share (compare stocks B & C), the two methods may not indicate the same number of shares.请注意,风险自每股可能不成正比每股价格(股市比较b和c ) ,这两种方法可能不会显示相同数量的股票。 For very close stops, and for a high risk allowance, the number of shares indicating under Risk-Basis sizing may even exceed the purchasing power of the account.对于非常接近停止,并为高风险津贴,股份数表明根据风险大小的依据,甚至可能超过了购买力的帐户。
Psychological Considerations 心理思考
In actual practice, the most important psychological consideration is ability to stick to the system. 在实践中,最重要的考虑因素是心理能力 , 坚持制度。 To achieve this, it is important (1) to fully understand the system rules, (2) to know how the system behaves and (3) to have clear and supportive agreements between all parties that support sticking to the system. 为了实现这一目标,重要的是( 1 )充分了解系统的规则, ( 2 )知道如何系统的行为和( 3 )有明确和支持各方之间的协议 , 支持继续坚持这一制度。
For example, as we noticed earlier, profits and losses do not likely alternate with smooth regularity; they appear, typically, as winning and losing streaks. 例如,当我们发现较早,利润和亏损并不可能候补顺利规律;出现,通常情况下,为获奖和连败。 When the entire investor-manager team realizes this as natural, it are more likely to stay the course during drawdowns, and also to stay appropriately modest during winning streaks. 在整个投资管理团队意识到 , 这是自然的,它更可能留在提取的过程,同时也适当地谦虚逗留期间连胜。
In addition, seminars, support groups and other forms of attitude maintenance can help keep essential agreements on track, throughout the organization. 此外,讨论会,支持团体和其他形式的态度 , 维修可以保持基本协定轨道上,在整个组织。
Risk Management - Summary 风险管理-摘要
In general, good risk management combines several elements: 一般来说,良好的风险管理相结合的几个要素:
1. Clarifying trading and risk management systems until they can translate to computer code. 1 。澄清交易和风险管理系统 , 直到他们可以转化为计算机编码。
2. Inclusion of diversification and instrument selection into the back-testing process. 2 。列入多样化和工具选择到后台的测试过程。
3. Back-testing and stress-testing to determine trading parameter sensitivity and optimal values. 3 。回测试和压力测试 , 以确定交易参数的敏感性和最优值。
4. 4 。 Clear agreement of all parties on expectation of volatility and return. 明确的协议的所有各方的期望 , 不稳定性和回报。
5. Maintenance of supportive relationships between investors and managers. 5 。维护支持投资者之间的关系和管理人员。
6. 6 。 Above all, stick to the system. 总之,坚持制度。
7. 7 。 See #6, above. 见# 6以上。
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Links to articles on Risk Management: 链接文章风险管理:
http://www.turtletrader.com/money.htmlhttp://www.turtletrader.com/money.html















英语原文:
One interesting artifact of fixed-fraction betting, is that, since the bet stays proportional to the equity, it is theoretically impossible to go entirely broke so the official risk of total ruin is zero.
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