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賭博、風險與預測

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賭博、風險與預測

對未來的希望可以說是人類的一個特徵,不斷地在尋找掌握自己命運的方式。在大多數的人類歷史上,當人們面對未來不確定的時候,只有訴求神諭,依直覺行事。如果時間的箭頭是單向的,那麼未來是否可以通過量化,用數學的方法做出預測?自文藝復興、歐洲啟蒙運動以來,人們進行了不懈的探索。

  不確定的事務往往隱藏著更深的哲學思考。和華人一樣,古代希腊人和羅馬人都相信“運氣”、“命運”的說法。基督教在歐洲傳播開了以後,歐洲人接受猶太人的觀點,相信人在地球上的未來是一個謎,受一個更高意志的支配。在其他地區,神的願望有無限的形式,決定著未來。當命運,或神,或其他強大力量橫跨在人們面前的時候,人便無能為力。
  
 但問題是,人有自由意志。人在多大程度上受上帝的支配,在多大程度上是由自己的意志來決定?從文藝復興到啟蒙運動,人們改變了對自己未來的認識,推動了人類史上第一次嚴肅的爭論,結果發現:人可以選擇自己的命運。
  
  與此同時,數學的方法也用來預測未來,這是從研究賭博問題入手的,在十七世紀法國數學家帕斯卡和費馬那里開創了先河。1601年出生的業余數學家費馬,以簡捷的“費馬大定理”而出名。他和帕斯卡從1654年7月到10月之間,藉著通信探討賭博中的數學問題,為創立古典概率論做出了貢獻。
  
  柏拉圖相信知識在人發現它以前就存在著,就像幾何學產生於尼羅河的沖洗一樣,概率論來自於賭博。當數學與現實發生碰撞的時候,就顯示出了它的威力。
  
  賭注合理分配問題被認為是概率論的起源:如果一場賭博因故中斷,現在知道了兩個賭徒當時的積分,以及贏得賭博所需要的點數,問賭金如何分配?這個問題困擾亂數學家一百多年。
  
  在實際中帕斯卡和費馬研究了這樣一個問題:兩個賭徒在五局賽中約定先贏三局為勝。如果這時甲贏了二局,乙贏了一局,而你有100元,應該拿出多少賭金押在乙的最終獲勝上才比較合理?
  
  根據帕斯卡和費馬的計算,乙這時獲勝的可能性只有25%。因此,你應該拿出25元來下注。如果低於25元,對你有利,但對莊家不利,他不會和你打賭;如果高於25元,按概率來講也會對你不利。
  
  如果賭博還在進行,風險則依然存在。數學可以告訴你成敗的概率,但並不能消除風險。賭徒往往認為他們在與幾率打賭,而實際上是和時間在打賭。時間停止了,風險也就化為零。不確定性也是人類的朋友,要不然生活就像一部看過的電影,你總知道它的結果,生活變得枯燥乏味。
  
  早在1494年,意大利數學家帕喬利就提出過類似的賭博問題。十七世紀中葉,當法國的一個賭徒梅雷(Chevalier Mere)向帕斯卡重提這類問題時,才引起帕斯卡與費馬發明概率的方法來解決這個問題。這是一個智慧的霹靂,預測未來、掌握未來也許是人類歷史上最大膽的設想(2000年4月25英國《金融時報》)。
  
  無論古代埃及、中國、希腊、阿拉伯和羅馬的數學家,還是中世紀的數學家都不覺得風險管理值得他們花時間來研究,不確定性是自然而然的現象。賭博,這個世界上最古老的職業之一,其規則不過是根據最簡單的或然率。古人發明了原始的骰子卻沒有發現或然率。
  
  彼得·伯恩斯坦在《與天為敵》里說:“人類對賭博著迷,因為它讓我們跟命運當面抗衡,我們投身這種令人膽寒的戰鬥,只因自以為有個強大有力的盟友:運氣站在我們這邊,勝算握在我們手中。”
  
  華人向來認為命運是不可抗爭的,把賭博的勝負往往放在“手氣”上,麻將的複雜組合似乎曲折地反映這種宿命傾向:一方面不斷錘煉技巧,另一方面又對“運氣”無可奈何。就像樸克反映了中世紀歐洲的社會等級關係一樣,麻將也反映了華人的社會關係。小賭可以加深友誼,把家庭成員、親戚朋友廣泛地聯系在一起;只有少部分西方人固定打樸克消磨時光。華人也許不屑去研究麻將裡的數學問題,不像西方人對賭博裡的概率問題津津樂道。
  
  十七世紀以後,當概率論不斷完善,在經濟中研究效用、回歸、分散經營的時候,不可避免地出現了人們用概率來研究賭博,挑戰在骰子、樸克和輪盤賭裡的勝負幾率問題,樂此不疲地在不確定中尋找確定。
  
  概率研究的發展推動了賭博業的發展,世界上不少國家推行一種累積賭注(Jackpot)的彩票。獲獎的機會就像從一迭幾百米高的彩票中任意抽出一張來,幾率比卡西洛賭場還小,沒有人可以預測誰可以獲獎,對於那些暴富的人也感到“公平”。對於每一個人來講,則是小的風險可能獲得大利的游戲。
  
  根據概率的不可能定理,可以編造這樣一個故事:一個猴子在打字機上任意敲打鍵盤,結果打出了一本《莎士比亞全集》,而且沒有錯誤。這一理論用於取代計算的的蒙特卡羅抽樣法(Monte Carlo methods)。
在《魔鬼辭典》裡,賭博被定義為一種游戲,“通常是因為一時的手癢而引起,而結果往往是一世的後悔。”如果把買彩票、打麻將也包括在內,全世界可以說沒有人不賭博。賭城拉斯維加斯、澳門和蒙特卡羅就專門為人提供這樣的場地。賭徒相信運氣,夜總會相信數學,而數學揭示的風險可以打敗最有運氣的賭徒。一個賭徒曾經問帕斯卡,為什麼他總是輸,帕斯卡的回答很簡單:“你在賭桌旁邊的時間太長了。”賭場並不對賭徒按照“平均律”設賭局,數學幾率一旦確定有利於賭場,賭徒投機不可能改變勝負的百分比。如果一個賭徒賭的時間足夠長,必將輸得精光無遺。
  
  為什麼明明知道贏的機會比較小,人們還要去賭?答案不在數學,而在心理學。對某些人來講,賭博不過是一種刺激的游戲、一種不同於看電影的娛樂;但對另一些人來講則沉溺上癮,賭博可能發財,也可能讓他們“後悔一世”。概率論可以預測隨機事件的可能性,但不可能準確地預測某個人的命運。
  
  概率的方法往往與直覺相對,可以揭示一些表面上看不到的東西。不少職業賭徒成為業余數學家,一些數學家專注賭博問題。例如,1657年荷蘭科學家惠更斯就完成了《論賭博中的計算》一書;笛卡爾年輕時應用數學,曾是一個獲利的賭徒;據說,華盛頓在獨立戰爭期間,還在自己的帳蓬裡聚賭。
  
  樸克牌產生於埃及,為中世紀的歐洲引入阿拉伯數學扮演過關鍵角色,現在則是世界上通用的賭博工具。從樸克牌的洗法到玩法,都有人作概率分析。美國加州有一名數學專家,1991年發明了“紙牌的二十一點”理論,因而成為賭徒崇拜的偶像。他聲稱這是對內華達州賭場的“報復”。
  
  數學提供了一個工具,把不確定性轉化為風險分析,以便控制風險。保險業根據未來可能發生也可能不發生的概率來索取保險費,推動了早期遠洋貿易,迎合了歐洲人的海洋擴張。企業中出現的風險管理、風險投資、決策理論等等,則催化了經濟的發展。
  
  和投資股票一樣,過去人們認為炒賣期貨是純粹的賭博,它的複雜性遠遠超過了數學的解釋範圍。期貨以預先固定的價格購買存貨,如果將來存貨昇值,你將獲利;如果存貨貶值,你將虧本。如何確定那個預先固定的價格?過去人們憑著信心和勇敢,現在人們則可以用數學來預測,1997年哈佛和斯坦福大學的兩名教授因此而獲得諾貝爾經濟學獎。
  
  人自己也許是最複雜的,暴露了數學預見力的有限。就像一個看上去自信的賭徒也知道扔出去骰子可能把他的財富擲掉,諸多有關博弈的預測就象蹩腳的 “天氣預報”。對於戰略游戲來講,不確定的因素不在於自然,而在人自己。正如莎士比亞所說的:“過錯不在我們的星辰,而在我們自己。”現代複雜性理論(Complexity theory)揭示:人們之間的相互作用從本質上來講是無法預測的。
  
  因此,在人類共存的社會裡,風險無法完全被戰勝,利益不可以獨享而必須和其他人形成妥協。這不是擲骰子或旋轉彩票輪盤的簡單游戲,而有些類似於橋牌或象棋裡的變化。前者純靠幾率,或者在一定程度上靠智謀。在商業裡,客戶、職員、投資者都是決策的變量。每個人都在為其他人創造不確定的紊亂。玩這類風險游戲的方法就必須合理地回應由他人引起的不確定性。
  
  概率論的奠基者也給人們留下了一些疑難雜症。帕斯卡晚年戲劇性地提出:上帝存在或者不存在,不可以用推理來回答。但可以把這個問題看作是一個硬幣的兩面,各有二分之一的概率。人可以有選擇,假如拒絕,而教義又是假的,你沒有什麼損失;但教義如果是真的,你就會在地獄裡受苦。假如接受,而教義又是騙人的,你可能什麼也得不到;而如果教義是真實的,你就可以進天堂。因此,他主張接受是上策。這是對後世的一種賭博。
  
  這裡至少可以提出兩個問題。一是佛教、伊斯蘭教裡也有天堂和地獄,帕斯卡是否同意有兩種以上的信仰?二是這樣的推論有濫用“中立原則”之嫌。經濟學家約翰·凱恩斯認為:如果沒有充分的理由來說明某件事的真偽,就選對等的概率來確定每件事物的真實性,他稱之為“中立原則”。但應用不當就會鬧出笑話。
  
  例如,“立方體悖論”。如果櫃子裡藏著一個立方體。你知道它的邊長從2米到4米,根據“中立原則”,最好的估計是它的邊長為3米。根據體積,這個立方體的體積從8立方米到64立方米之間,同樣根據“中立原則”,最好的估計是它的體積為36立方米。顯然,這個估計的體積的邊長不可能是3米。這就是使用“中立原則”不當造成的。
  
  同樣,按照帕斯卡的計算方法,上帝存在有二分之一的概率。如果進一步推下去,上帝創造宇宙的概率是二分之一,上帝創造人類的概率也是二分之一。因此,上帝既創造宇宙也創造人類的概率就是四分之一,而上帝不存在的概率就是四分之三,不定的因素越多,上帝存在的可能性就越小。反之也是一樣,上帝存在的可能性就越大。人將如何選擇?回答這個問題很簡單,“中立原則”不適合解決這個問題。
  
  上帝無情地拒絕了我們知道未來,但人類至少邁出了一大步。古人也許並沒有認識到變化無常的背後,天氣往往是影響他們收成的唯一變量,除了祈禱沒有別的辦法。隨著時代的變遷,愛因斯坦相信上帝不玩擲骰子的游戲,但也有人斷言,上帝在任何場合無不擲骰子。但具有操縱風險觀念的現代人,至少在一定程度上知道了人類的行為可能導致與願望相反的概率分布,可以用文明的大腦來擺脫危厄的命運,在過去的五百年裡,我們正是這樣做的。

盯住止損

 盯住止損,止損是自己控制的;不考慮利潤,因為利潤是有市場控制的!波動不用看,更不用關注盤面,因為有90%的虧損都是因為短線的波動造成的,大部分的虧損是來自自己對市場的主觀預測,預測在模擬交易里有效,因為心態是非常平穩的,但自己做交易時,短線的波動就會引起情緒的極大變化,看盤很仔細,不光沒有必要,而且出錯的概率就越大,因為對盤面的細致的觀察也必然引起心理的劇烈波動,做一波趨勢,盤是不用看的,只是需要大致看一下,是不是到了我的止損。我們需要關注的就是盯住止損。
   現在是很多人不怕虧,怕贏。表面上是大家都怕虧,本質上是都怕贏!一虧損很平靜,很冷靜。一贏利,比誰跑的都快,只有把根留住,樹才會張大,不能一發芽就砍了,拿去當柴火燒了!贏利是有小到大累積的,讓你的贏利充分擴大,做交易時最可怕的就時贏利了不堅持,讓你的贏利張成大樹。在價格上不必關注高低,只關注漲跌。
   做交易的本質,就是處理正在發生的情況,而不是處理將要發生的情況。我們應該把主要精力用在處理正在發生的情況,而不要對未來做無謂的預測,因為將要發生的情況,沒有人會知道。所以,根本不用考慮未來會怎樣,知道自己現在應該怎麼做就足夠了。價格的漲跌我們不能控制,把握未來是不能控制的,對於這種東西,只能放棄。人總是很”善良“,而“善良”的人都是會虧錢的,多頭已經獲利很豐厚了,應該回吐一點了,或者空頭已經損失很多,可以做空了。而我們這種心理必然會把自己的交易陷入泥沼,會去不斷的猜頂猜底,一旦一種趨勢形成後,往往會把對方殺的片甲不留。一定要讓自己的交易系統告訴你怎麼做,不要靠本能去操作,不要靠自己的分析和思考去做交易,你的分析和思考只是你自己的,因為你掌握的資訊是有限的,要考慮這種分析和思考的價值何在,而你按照自己的系統自己的紀律自己的原則去做,你才能去贏利!
   期貨市場是一個很奇特的市場,是一個集中的利益的碰撞,一定要有自己的原則和方法,如果你一定要人告訴你,明天會跌或者會漲,你干脆不要去做交易,若做了那麼你一定會虧,如果市場給你信號,而與你得到的資訊相互矛盾,只能說明你所得到的資訊已經沒有價值,我們需要時刻關注的時價格,以及自己的止損。

點評  一個好的交易系統會有很明確的買賣信號!止損以自己所能承受的風險為限,引出資金管理的問題!不同的交易方法和投資理念對止損的設置各有不同,但終究一點,止損是貫穿交易的始終的必不可少的程序,當一個頭寸贏利了解時,止損就變成了止贏,其實質是一樣的。止損的目的是使虧損限於小額,讓利潤充分增長,所以止損也必須是不斷移動的,這樣才能保住既得得利潤。一個適合交易的點位首先必須是一個便於止損的點位!好的交易系統要和投資理念和交易技巧融合一體形成一種交易的信念,用信念交易,使各種紀律和交易守則成為一種交易的習慣! 

我的止損理念

自從炒股以來,最大的困惑就是止損的問題。
    當初買入第一支股票後,正趕上大勢不好,一路下行。面對日甚一日的虧損,實在狠不下心來割肉,於是用心學習。
    有一天突然想起一個道理:佛說三心不可得,因此你說現在的時候已經是過去了,這樣每一天都是一個新的現在和新的開始。由是我轉變觀念,我想,根本沒有必要念念不忘自己的成本價,只要你當自己是昨天才買入的股票就好。其實這一點以前看書和一些文章也明白這個道理,但那一剎那才覺得自己是真正的領悟了。從此慢慢地鍛煉自己不再計較買入價(當然目前還處於理入行未入的階段),只是以技術形態來判斷下一步的行動。
     隨著技術上的日臻完善,操作理念上慢慢地轉向了短線。因此我一般設立如下幾個止損位:
    1、買入時點盡量選擇在下午收盤前(特殊形態的股票除外)。因為目前的“T+2”操作下,如果當天買入價格過高,萬一出現跳空的情況就沖掉若是無資券無法出貨的。因此在收盤前買入一般來說比較保險,如果第二天發現不對,可以立刻出局了事。因此當天買入的止損只能是盡量買在低價,沒有其他辦法。最好的情況是當天收盤能夠高於買入價。
    2、第二天開盤時要特別注意集合競價的高低和量的大小,一般來說我做短線都是追的強勢股,因此在第二天即以自己的成本價做為止損。我做了一個公式在主圖顯示來提示自己,原碼:{保本價:買入價*0.015+買入價;};只要當天跌破保本價且有效,則立刻出局,不能抱一絲幻想。當然如果股價沒有站在保本價之上時則以買入價做為止損的依據。
    在下跌中我自己認為是絕不應該參與調整或洗盤的。寧可看錯,絕不做錯。即使賣出後立刻上漲也不後悔,因為你避免了可能的大跌和資金被套牢的慘劇。
    3、如果以後的行情價格能夠站在自己的保本價之上說明帳戶就不會虧損了。這樣在未來的時間裡我主要以“未來指標”判斷,我做的該類指標組中的公式大概意思就是看股票未來的走勢是否依然能夠保持強勁。在論談中就有類似理念的公式免費下載。如果判斷當天或明天不能夠站在一些關鍵點之上則賣出,介入新的強勢股。
    4、注意,我以上說的都是針對個股而言。應該同時注意大盤的止損條件。比如你可以用均線判斷,如果大盤的三日均線跌破五日均線,可以確認大盤已經到了止損的時候。除非手裡的股票非常強,否則一定要先出來再說。
    以上說的都只是大概,其實真正的操作方法與技巧不在外而在內,就在你自己的腦子裡。通過你對自己的了解掌握適合你自己的操作方法,然後才是止損的細節方面的問題。

Ed Seykota談資金管理

Risk Management

 

(c) Ed Seykota, 2003

 

 

Risk

 

RISK is the possibility of loss. That is, if we own some stock, and there is a possibility of a price decline, we are at risk. The stock is not the risk, nor is the loss the risk. The possibility of loss is the risk. As long as we own the stock, we are at risk. The only way to control the risk is to buy or sell stock. 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.

 

Risk Management

 

To manage is to direct and control. 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.

 

The Coin Toss Example

 

Let's say we have a coin that we can toss and that it comes up heads or tails with equal probability. The Coin Toss Example helps to present the concepts of risk management .

 

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. 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).

 

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%. This game is similar to some trading methods.

 

In this case, our LUCK equals the probability of winning, or 50%; we will be lucky 50% of the time. Our PAYOFF equals 2:1 since we win 2 for every 1 we bet. Our RISK is the amount of money we wager, and therefore place at risk, on the next toss. In this example, our luck and our payoff stay constant, and only our bet may change.

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. The risk is the only parameter the risk manager may effectively change to control risk.

We might also model more complicated games with a matrix of lucks and payoffs, to see a range of possible outcomes. See figure 1.

 

Luck

Payoff

10%

lose 2

20%

lose 1

30%

break even

20%

win 1

10%

win 2

10%

win 3

Fixed Bet

$10

Fixed-Fraction Bet

1%

Start

1000

1000

Heads

1020

1020

Tails

1010

1009.80

Heads

1030

1030

Tails

1020

1019.70

Heads

1040

1040.09

Tails

1030

1029.69

Heads

1050

1050.28

Tails

1040

1039.78

Heads

1060

1060.58

Tails

1050

1049.97

Notice that both systems make $20.00 (twice the bet) on the first toss, that comes up heads. 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. 

Note that the results from both these systems are approximately identical. Over time, however, the fixed-fraction system grows exponentially and surpasses the fixed-bet system that grows linearly. 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. The reader may prove this result by spreadsheet simulation.

% Bet

Start

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

Heads

Tails

0

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

1000.00

5

1000.00

1100.00

1045.00

1149.50

1092.03

1201.23

1141.17

1255.28

1192.52

1311.77

1246.18

10

1000.00

1200.00

1080.00

1296.00

1166.40

1399.68

1259.71

1511.65

1360.49

1632.59

1469.33

15

1000.00

1300.00

1105.00

1436.50

1221.03

1587.33

1349.23

1754.00

1490.90

1938.17

1647.45

20

1000.00

1400.00

1120.00

1568.00

1254.40

1756.16

1404.93

1966.90

1573.52

2202.93

1762.34

25

1000.00

1500.00

1125.00

1687.50

1265.63

1898.44

1423.83

2135.74

1601.81

2402.71

1802.03

30

1000.00

1600.00

1120.00

1792.00

1254.40

2007.04

1404.93

2247.88

1573.52

2517.63

1762.34

35

1000.00

1700.00

1105.00

1878.50

1221.03

2075.74

1349.23

2293.70

1490.90

2534.53

1647.45

40

1000.00

1800.00

1080.00

1944.00

1166.40

2099.52

1259.71

2267.48

1360.49

2448.88

1469.33

45

1000.00

1900.00

1045.00

1985.50

1092.03

2074.85

1141.17

2168.22

1192.52

2265.79

1246.18

50

1000.00

2000.00

1000.00

2000.00

1000.00

2000.00

1000.00

2000.00

1000.00

2000.00

1000.00

55

1000.00

2100.00

945.00

1984.50

893.03

1875.35

843.91

1772.21

797.49

1674.74

753.63

60

1000.00

2200.00

880.00

1936.00

774.40

1703.68

681.47

1499.24

599.70

1319.33

527.73

65

1000.00

2300.00

805.00

1851.50

648.03

1490.46

521.66

1199.82

419.94

965.85

338.05

70

1000.00

2400.00

720.00

1728.00

518.40

1244.16

373.25

895.80

268.74

644.97

193.49

75

1000.00

2500.00

625.00

1562.50

390.63

976.56

244.14

610.35

152.59

381.47

95.37

At a 0% bet there is no change in the equity. 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. Then our second bet is 5% of $1,100.00 or $55.00, which we lose, so we then have $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. This allows us to simplify the problem of searching for the optimal parameter to the examination of just one head-tail cycle.
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. Thereafter, with increasing bet fraction, the profitability declines. 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. In portfolios that maintain multiple positions and multiple bets, we refer to the total risk as the portfolio heat.

Note: Note the chart illustrates the Expected Value / Bet Fraction relationship for a 2:1 payoff game. For a graph of this relationship at varying payoffs, see Figure 8.

The Kelly Formula 

K = W - (1-W)/R

K = Fraction of Capital for Next Trade

W = Historical Win Ratio (Wins/Total Trials)

R = Winning Payoff Rate

-------

For example, say a coin pays 2:1 with 50-50 chance of heads or tails. Then ...

K = .5 - (1 - .5)/2 = .5 - .25 = .25.

Kelly indicates the optimal fixed-fraction bet is 25%.

This graph shows the optimal bet fraction for various values of luck (Y) and payoff (X). Optimal bet fraction increases with increasing payoff. For very high payoffs, optimal bet size equals luck. For example, for a 5:1 payoff on a 50-50 coin, the optimal bet approaches about 50% of your stake.

This graph shows optimal expected value for various values of luck and payoff, given betting at the optimal bet fraction. The higher the payoff (X: 1:1 to 5:1) and the higher the luck (Y: .20 to .70), the higher the expected value. For example, the highest expected value is for a 70% winning coin that pays 5:1. The lowest expected value is for a coin that pays 1:1 (even bet).

This graph shows the expected value of a 50% lucky (balanced) coin for various levels of bet fraction and payoff. The expected value has an optimal bet fraction point for each level of payoff. 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%. Note: Figure 4 above is the cross section of figure 8, at the 2:1 payoff level.

StockPrice/ShareSharesValue
A
B$100
C$200250$50,000
StockPrice/ShareRisk/ShareSharesRiskValue
A
B$100$10
C$200

Pyramiding and Martingale

 

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. There is, however, no way to exploit this phenomenon, which is, itself random. In non-random processes, such as secular trends in stock prices, pyramiding and other trend-trading techniques may be effective.

 

Pyramiding is a method for increasing a position, as it becomes profitable. 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. In general, such micro-tinkering with executions is far less important than sticking to the system. 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.

 

The Martingale system is a method for doubling-up on losing bets. In case the doubled bet loses, the method re-doubles and so on. This method is like trying to take nickels from in front of a steam roller. Eventually, one losing streak flattens the account.

 

Optimizing - Using Simulation

 

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. See figures 3 and 4.

 

Note: The coin-toss example intends to illuminate some of the elements of risk, and their inter-relationships. 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. 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. It does not suggest any particular risk parameters for trading the markets.

 

 

Figure 3:  Simulation of equity from a fixed-fraction betting system.

Figure 4:  Expected value (ending equity) from ten tosses, versus bet fraction,

for a constant bet fraction system,  for a 2:1 payoff game,

from the first and last columns of figure 3.

 

 

 

Optimizing - Using Calculus

 

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.

 

The stake after one pair of flips:

S = (1 + b*P) * (1 - b) * S0

S - the stake after one pair of flips

b - the bet fraction

P - the payoff from winning - 2:1

S0 - the stake before the pair of flips

(1 + b*P) - the effect of the winning flip

(1 - b) - the effect of the losing flip

So the effective return, R, of one pair of flips is:

R = S / S0

R = (1 + bP) * (1 - b)

R = 1 - b + bP - b2

R = 1 + b(P-1) - b2P

 

Note how for small values of b, R increases with b(P-1) and how for large values of b, R decreases with b2P. These are the mathematical formulations of the timid and bold trader rules. 

 

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. 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.

 

Slope = dR/db = (P-1) - 2bP = 0,  therefore:

b = (P-1)/2P , and, for P = 2:1,

b = (2 - 1)/(2 * 2) = .25

 

So the optimal bet, as before, is 25% of equity.

Optimizing - Using The Kelly Formula

 

J. L. Kelly's seminal paper, A New Interpretation of Information Rate, 1956, examines ways to send data over telephone lines. One part of his work, The Kelly Formula, also applies to trading, to optimize bet size.

 

 

Figure 5:  The Kelly Formula

 

Note that the values of W and R are long-term average values,

so as time goes by, K might change a little.


Figure 6:  Optimal bet fraction increases linearly with luck, asymptotically to payoff.

The Expected Value of the Process, at the Optimal Bet Fraction

Figure 7:  The optimal expected value increases with payoff and luck.

 

 

 

Finding the Optimal Bet Fraction from the Bet Size and Payoff


Figure 8:  For high payoff, optimal bet fraction approaches luck.

 

Non-Balanced Distributions and High Payoffs

 

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. Occasionally, however, a winning streak does occur. 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.

 

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. 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.

 

Almost Certain Death Strategies

 

Bet-it-all strategies are, by nature, almost-certain-death strategies. 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. Since most traders do not wish to go broke, they are unwilling to adopt such a strategy. 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.

 

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. 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.

 

Likewise, a portfolio manager might divide his equity into various sub-accounts. 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. This, the principle of DIVERSIFICATION, works in cases where the individual payoffs are high.

 

Diversification

 

Diversification is a strategy to distribute investments among different securities in order to limit losses in the event of a fall in a particular security. The strategy relies on the average security having a profitable expected value, or luck-payoff product. 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.

 

The Uncle Point

 

From the standpoint of a diversified portfolio, the individual component instruments subsume into the overall performance. 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.

 

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. If either the investors or the managers become demoralized and withdraw from the enterprise, then the fund dies. Since the circumstances surrounding the Uncle Point are generally disheartening, it seems to receive, unfortunately, little attention in the literature.

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. 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.

 

In times of stress, investors and managers do not access obscure legal agreements, they access their primal gut feelings. This is particularly important in high-performance, high-volatility trading where draw downs are a frequent aspect of the enterprise.

 

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. As we have seen above, safe, low volatility systems rarely provide the highest returns. Still, the pressures and tensions from the default expectations of low-volatility performance create a demand for measurements to detect and penalize volatility.

 

Measuring Portfolio Volatility

Sharpe, VaR, Lake Ratio and Stress Testing

 

From the standpoint of the diversified portfolio, the individual components merge and become part of the overall performance. 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.

 

William Sharpe, in 1966, creates his "reward-to-variability ratio." Over time it comes to be known as the "Sharpe Ratio." The Sharpe Ratio, S, provides a way to compare instruments with different performances and different volatilities, by adjusting the performances for volatilities.

 

S = mean(d)/standard_deviation(d)  ... the Sharpe Ratio, where

 

d = Rf - Rb   ... the differential return, and where

Rf - return from the fund

Rb - return from a benchmark

 

Various variations of the Sharpe Ratio appear over time. One variation leaves out the benchmark term, or sets it to zero. 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.

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. 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. 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.

 

A rule-of-thumb way to view high volatility accounts, by this author, is the Lake Ratio. 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. 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. The total volume of water represents the integral product of drawdown magnitude and drawdown duration.

 

If we divide the total volume of water by the volume of the earth below it, we have the Lake Ratio. The rate of return divided by the Lake Ratio, gives another measure of volatility-normal return. Savings accounts and other instruments that do not present draw downs do not collect lakes so their Lake-adjusted returns can be infinite.


Figure 9: The Lake Ratio = Blue / Yellow

 

Getting a feel for volatility by inspection.

 

 

 

Stress Testing

 

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. 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. 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.

 

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.

 

Portfolio Selection

 

During market cycles, individual stocks exhibit wide variations in behavior. Some rise 100 times while others fall to 1 percent of their peak values. 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. A portfolio of the best performing stocks easily outperforms a portfolio of the worst performing stocks. 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.

 

The number of instruments in a portfolio also effects performance. 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.

 

Position Sizing

 

Some position sizing strategies consider value, others risk. Say a million dollar account intends to trade twenty instruments, and that the investor is willing to risk 10% of the account.

 

Value-Basis position sizing divides the account into twenty equal sub-accounts of $50,000 each, one for each stock. Since stocks have different prices, the number of shares for various stocks varies.

 

$50

1000 $50,000

500

$50,000

 

Value-Basis Position Sizing

Dividing $50,000 by $50/share gives 1000 Shares

 

Risk-Basis position sizing considers the risk for each stock, where risk is the entry price minus the stop-out point. It divides the total risk allowance, say 10% or $100,000 into twenty sub accounts, each risking $5,000. Dividing the risk allowance, $5,000 by the risk per share, gives the number of shares.

 

 

$50

$5

1000 $5,000 $50,000

500

$5,000 $50,000

 

Risk-Basis Position Sizing

Dividing $5,000 by $5 risk/share gives 1000 Shares

 

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. 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.

 

Psychological Considerations

 

In actual practice, the most important psychological consideration is ability to stick to the system. 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.

 

For example, as we noticed earlier, profits and losses do not likely alternate with smooth regularity; they appear, typically, as winning and losing streaks. 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.

 

In addition, seminars, support groups and other forms of attitude maintenance can help keep essential agreements on track, throughout the organization.

 

 

Risk Management - Summary

 

In general, good risk management combines several elements:

 

1. Clarifying trading and risk management systems until they can translate to computer code.

2. Inclusion of diversification and instrument selection into the back-testing process.

3. Back-testing and stress-testing to determine trading parameter sensitivity and optimal values.

4. Clear agreement of all parties on expectation of volatility and return.

5. Maintenance of supportive relationships between investors and managers.

6. Above all, stick to the system.

7. See #6, above.

 

止損的設定

2009/01/05 00:07
在股票市場上,為數不少的投資者是沒有止損概念的。遇上被套就套到哪裡算哪裡,套到什麼時候算什麼時候,十分被動。造成這種局面的主要原因是沒有一個既定的投資計劃,又或有計劃而沒有切實執行。
  有計劃而沒執行是屬於紀律的問題,和投資人對市場的認識程度有關,也和其性格有關,這裡就不討論了。本文想著重談談投資計劃中關於“止損的設定”的問題。
  投資計劃是一項系統工程。由於不同的投資人在投資喜好、投資取向、風險承受能力、用於投資的錢的性質等各方面有所不同,故而會有不同的選擇。比如:在投資喜好方面,有人喜歡短線搏擊,有人熱衷中線炒波段,也有人醉心於長線投資;在投資取向方面,有的人喜歡進取些、刺激些,有的人則喜歡穩健些、踏實些;有的人家庭比較富裕,有穩定而豐厚的收入來源,甚至是腰纏萬貫的大款,用於投資的又都是自己的閑錢,其風險承受能力自然強些;有的人家庭經濟拮據,甚至失業,用於投資的是養命錢又或是借來的錢,風險承受能力自然低。所以,投資計劃不能一概而論,只有是適合自己的,才是最好的。
  由於有不同的選擇,因而在止損的設定方面也應該有不同的宗旨。比如:立足做中、短線的,大都是依據中、短線的技術分析之類定進出。然而,中、短線的技術指標多有騙線,反映到個股層面更是如此,而且具體到每個人對中、短線技術分析的理解也未必那麼透徹,一旦發生與自己預期相反的情形,就應該嚴格執行預先設定的止損計劃,止損點的設定也多以一些中、短線的支撐位(如:中短期的上昇趨勢線、移動平均線、平台等)為主。以損失若干個百分點為止損界限也是一個常見的做法。由於做中、短線炒作大都不以個股內在的投資價值為入市依據,更多的是追逐市場當時的熱點和概念,如果套住以後不止損,變中、短線為長線,持股的風險更高。立足長線,在選股方面自然應該以投資價值為依歸,在具備相當安全邊際的價位上介入,當然可以忽略中、短期的股價波動,但也不等於就可以不設止損,只是止損的設定不同而已。在技術層面上,雖然我們也會以長線的技術分析方法定止損,但由於是在具備相當安全邊際的價位上介入,買在長期頭部區域的機會幾乎為零,所以,長線止損更多的是在發現選錯了投資對象,或所選的上市公司經營變壞,又或其所處的行業景氣度變差的情況下執行

對止損的一些想法

止損的原則無論是在股票、期貨或其他行業中都是很重要的。但從我收集和理解的關於止損的一些文章看,各有各的觀點,也不能說誰對誰錯。因為他們同樣有著輝煌的實戰成績。由此可見,所謂的“止損”並不是贏利的必要條件,即它可以幫助我們在某些時候杜絕輕微的虧損,但未必一定能幫助我們實現贏利。
    但是為什麼還有那麼多的前輩提倡止損的原則呢?我覺得原因是------每個人對股票的理解程度不同。
    其實大家都知道,在股市中的人不外乎“一賺二平七虧”這個結果。因此這個現象在個人自身的操作結果也是如此表現的,即每個人大概十次的操作中可能有一次是賺的,二次是平的,七次是虧損的。可以說初學者的絕大多數想買股票的想法是錯誤的。所以我一直也在強調,炒股其實並不僅僅是炒股票,它其實是炒你的心。應該時常和你自己鬧別扭,明明想做的事,一定要強迫自己不要做。反之對不想做的事情也要強迫自己去嘗試一下。佛經上說這樣可以改化自己的習氣,培養自己的氣質。因此對於初學者來說,只要能夠少犯錯誤就可以保證基本不虧損。然而實際上做到“不犯錯”已經是聖人了。所以在操作中為了保證能夠長久在股市中生存下去,前人提出了“止損”、“割肉”的概念。其實不過就是為了讓自己在錯誤的操作中能夠少賠一些錢。因為十次的操作中有七次是虧損的,既然無法讓你做到成功率次數的提高,就只好讓你做到金額上的少虧。
    因此高手和庸手的區別就只有從二點上衡量:操作的成功次數和贏利的百分比。高手的成功是來源於上漲的高概率,而普通人無法把握概率。高手可以保證在十次操作中贏利七次以上,而且能夠讓贏利最大化,讓虧損最小化。普通人能夠做到三次以上的成功率就不錯了,而且多數是不明白為什麼會贏利和虧損的原因。因此如果能夠讓這三成的成功操作達到利潤的最大化和讓七次的失敗做到虧損的最小化,也基本可以保證不虧本或少虧。這樣慢慢地在股市學的時間長了,自然會有成為高手的一天。到那時所謂的“止損”自然已經不在話下了。
      如果我們自認為沒有到達高手的境界,“止損”還是在股市生存下去不可或缺的原則。我想,只有到了對股票的感覺已經完全了解、絕對把握時才可以不再止損吧。




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