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两个世界的尾部：肥尾与投资

2002 · report · 原文约 3510 词
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2002 年 4 月 9 日 第 1 卷，第 7 期

April 9, 2002 Volume 1, Issue 7

两个世界的尾巴：肥尾与投资。“维克多·尼德霍夫把市场看作一座赌场，人们在那里扮演赌徒，而他们的行为可以通过研究赌徒来理解。他基于这一理论，经常在交易中小赚一笔。然而，他的方法存在一个缺陷。一旦浪潮袭来……他可能遭受重创，因为他缺乏有效的失效保护机制。”

A Tail of Two Worlds Fat Tails and Investing “[Victor Niederhoffer] looked at markets as a casino where people act as gamblers and where their behavior can be understood by studying gamblers. He regularly made small amounts of money trading on that theory. There was a flaw in his approach, however. If there is a…tide...he can be seriously hurt because he doesn’t have a proper fail-safe mechanism.”

乔治·索罗斯¹《索罗斯谈索罗斯》（1995 年）

George Soros 1 Soros on Soros (1995)

从统计学角度看，我估算自己交易了约 200 万份合约……每份合约的平均利润为 70 美元。这个平均利润与随机结果的偏离幅度约为 700 个标准差，这种偏离如果纯粹靠运气发生，其概率大概等同于汽车报废场的零配件能自行组装成一家麦当劳餐厅。

“In statistical terms, I figure I have traded about 2 million contracts…with an average profit of $70 per contract. This average profit is approximately 700 standard deviations away from randomness, a departure that that would occur by chance alone about as frequently as the spare parts in an automotive salvage lot might spontaneously assemble themselves into a McDonald’s restaurant.”

维克多·尼德霍夫 《一个投机者的教育》（1997）

Victor Niederhoffer 2 The Education of a Speculator (1997)

周三，尼德霍夫告知其管理的三支对冲基金的投资者，他们的份额已在周一“归零”，损失源于连续三天的股价下跌，再加上今年早些时候在泰国遭遇的重创。

“On Wednesday Niederhoffer told investors in three hedge funds he runs that their stakes had been ‘wiped out’ Monday by losses that culminated from three days of falling stock prices and big hits earlier this year in Thailand.”

戴维·亨利 《今日美国》（1997 年 10 月 30 日）

David Henry USA Today (October 30, 1997)

现实世界中的许多事物，与其说是由分布的平均值或中位数所决定，不如说是受“尾部”事件控制：是例外而非均值，是灾难而非持续点滴，是极富者而非“中产阶级”。我们需要摆脱“平均”思维。

“Much of the real world is controlled as much by the ‘tails’ of distributions as by means or averages: by the exceptional, not the mean; by the catastrophe, not the steady drip; by the very rich, not the ‘middle class.’ We need to free ourselves from ‘average’ thinking.”

菲利普·安德森（Philip Anderson）诺贝尔物理学奖得主

3

关于经济学中分布的几点思考

迈克尔·莫布森（Michael J. Mauboussin）

电话：212-325-3108

邮箱：[email protected]

在他 2001 年的致股东信中，沃伦·巴菲特区分了经验与克里斯滕·巴托尔森（Kristen Bartholdson）风险敞口。虽然巴菲特的评论是在伯克希尔·哈撒韦保险业务的背景下作出的，但他的观点适用于任何涉及主观概率的决策。经验当然是指回顾过去，根据历史事件的发生概率来考虑未来的结果。而风险敞口则考虑的是历史（尤其是近期历史）可能未揭示的事件的可能性——以及潜在风险。巴菲特认为，2001 年保险业在没有收取相应保费的情况下承担了巨大的恐怖主义风险，原因就在于他们关注的是经验，而非风险敞口。

Philip Anderson Nobel Prize Recipient, Physics 3 Some Thoughts About Distribution in Economics Michael J. Mauboussin 212-325-3108 [email protected] In his 2001 letter to shareholders, Warren Buffett distinguishes between experience and Kristen Bartholdson exposure. Although Buffett’s comments are in the context of Berkshire Hathaway’s insurance 212-325-2788 business, his point is valid for any exercise with subjective probabilities. Experience, of course, looks to the past and considers the probability of future outcomes based on occurrence of historical events. Exposure, on the other hand, considers the likelihood—and potential risk—of an event that history (especially recent history) may not reveal. Buffett argues that in 2001 the insurance industry assumed huge terrorism risk without commensurate premium because it was focused on experience, not exposure.

投资者同样必须分辨经验与经历之间的区别。长期资本管理公司和维克托·尼德霍夫的高调失败，正印证了这一点。然而值得注意的是，标准金融理论并不容易容纳极端事件。金融经济学家通常假定股价变动是随机的，类似于花粉在水中被分子撞击的运动。

Investors, too, must discern between experience and exposure. The high-profile failures of Long Term Capital Management and Victor Niederhoffer give witness to this point. Remarkably, however, standard finance theory does not easily accommodate extreme events. Financial economists generally assume that stock price changes are random, akin to the motion of pollen 4 in water as molecules bombard it.

金融理论为了模型便利而牺牲实证结果，将价格变化视为独立同分布变量，并普遍假设收益率呈正态分布或对数正态分布。这些假设的优点在于，投资者可以运用概率计算来理解分布的均值与方差，从而以统计精度预判各种百分比的价格变动。好消息是，这些假设在大多数情况下是合理的。坏消息则如物理学家菲尔·安德森所言，分布的尾部往往支配着世界。

In a triumph of modeling convenience over empirical results, finance theory treats prices changes as independent, identically distributed variables and generally assumes that the distribution of returns is normal, or lognormal. The virtue of these assumptions is that investors can use probability calculus to understand the distribution’s mean and variance, and can therefore anticipate various percentage price changes with statistical accuracy. The good news is that these assumptions are reasonable for the most part. The bad news, as physicist Phil Anderson notes above, is that the tails of the distribution often control the world.

厚尾分布。正态分布是金融理论的基石，包括随机游走、资本资产定价模型、风险价值模型和布莱克-斯科尔斯模型。以风险价值（VaR）模型为例，它试图量化一个投资组合在给定概率下可能遭受的损失。尽管 VaR 模型形式多样，但基础版本都依赖标准差作为衡量风险的指标。在正态分布下，计算标准差进而衡量风险相对直接。然而，如果价格变动并非正态分布，标准差就可能成为非常误导人的风险代理指标。

Tell Tail Normal distributions are the bedrock of finance, including the random walk, capital asset pricing, value-at-risk, and Black-Scholes models. Value-at-risk (VaR) models, for example, attempt to quantify how much loss a portfolio may suffer with a given probability. While there are various forms of VaR models, a basic version relies on standard deviation as a measure of risk. Given a normal distribution, it is relatively straightforward to measure standard deviation, and hence risk. However, if price changes are not normally distributed, standard deviation can be a very 5 misleading proxy for risk.

事实上，研究（其中一些早在 40 年前就已进行）表明，价格变化并不遵循正态分布。图 1 展示了 1979 年 1 月至 2002 年 3 月期间标普 500 指数日回报率的频率分布，以及从该数据推导出的正态分布。图 2 凸显了实际回报率与正态分布之间的差异。对不同资产类别和时间跨度的分析也得出类似结果。这些图表表明：

In fact the research, some done as long as 40 years ago, shows that price changes do not follow a normal distribution. Figure 1 shows the frequency distribution of S&P 500 daily returns from January 1979 to March 2002 and a normal distribution derived from the data. Figure 2 highlights the difference between the actual returns and 6 the normal distribution. Analysis of different asset classes and time horizons yield similar results. The figures show that:

• 微小的变化比正态分布预测的更为频繁地出现。

• Small changes appear more frequently than the normal distribution predicts.

• 中等程度的变化（大约 0.5 到 2.0 个标准差）发生的频率比模型所暗示的要低。

• There are less medium-sized changes than the model implies (roughly 0.5 to 2.0 standard deviations).

• 尾部比标准模型所展示的更厚。这意味着，大幅变动的发生次数会超出预期。

• There are fatter tails than what the standard model suggests. This means that there are a greater-than-expected number of large changes.

图 1：标普 500 指数日收益率频数分布（1979–2002 年）

Figure 1: Frequency Distribution of the S&P 500 Daily Returns (1979-2002)

标普 500 指数每日涨跌幅频率分布，1979 年 1 月 1 日至 2002 年 3 月 22 日 6.0%

Frequency Distribution of S&P 500 Daily Returns January 1, 1979 - March 22, 2002 6.0%

5.0%

5.0%

4.0%

4.0%

原件此处是表格，PDF 抽取时列结构已丢失，下面只剩按列读出的数字，行列对应关系无法还原。核对数据请打开来源正文。

频率
   3.0%
   2.0%
   1.0%
   0.0%
   -10   -9   -8   -7   -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10
   标准差
Frequency
   3.0%
   2.0%
   1.0%
   0.0%
   -10   -9   -8   -7   -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10
   Standard Deviation

图 2：频次差异

Figure 2: Frequency Difference

频率差异：正常日回报率与实际日回报率对比 1979 年 1 月 1 日至 2002 年 3 月 22 日 3.50%

Frequency Difference: Normal versus Actual Daily Returns January 1, 1979 - March 22, 2002 3.50%

3.00%

3.00%

2.50%

2.50%

原件此处是表格，PDF 抽取时列结构已丢失，下面只剩按列读出的数字，行列对应关系无法还原。核对数据请打开来源正文。

频率差异
   2.00%
   1.50%
   1.00%
   0.50%
   0.00%
   -10   -9   -8   -7   -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10
   -0.50%
   -1.00%
   标准差
Difference in Frequency
   2.00%
   1.50%
   1.00%
   0.50%
   0.00%
   -10   -9   -8   -7   -6   -5   -4   -3   -2   -1   0   1   2   3   4   5   6   7   8   9   10
   -0.50%
   -1.00%
   Standard Deviation

厚尾分布尤其值得多说几句。这种极端价值波动的发生频率远高于标准模型所预示的水平，并且可能对投资组合的表现产生重大影响——尤其是对杠杆投资组合。例如，在我们为展示图表而剔除数据的 1987 年 10 月股灾中，标普 500 指数暴跌超过 20%，这一变动偏离均值达 19 个标准差。罗杰·洛温斯坦指出：

The fat tails, in particular, warrants additional comment. These extreme value changes happen considerably more frequently than the standard model suggests, and can have a substantial influence on portfolio performance— especially for leveraged portfolios. For example, during the October 1987 crash, which we excluded from our figures for presentation purposes, the S&P 500 plummeted over 20%, a change that is 19 standard deviations from the mean. Roger Lowenstein notes:

经济学家后来推算，基于市场历史波动性，就算从宇宙诞生之日起市场每天都开盘，那么出现单日如此巨大跌幅的概率依然微乎其微。事实上，即便宇宙的生命历程重复十亿七次，从理论上看，这样的崩盘仍然“不太可能”发生。

“Economists later figured that, on the basis of the market’s historical volatility, had the market been open every day since the creation of the Universe, the odds would still have been against its falling that much in a single day. In fact, had the life of the Universe been repeated one billion 7 times, such a crash would still have been theoretically ‘unlikely’.”

许多小事件和少数大事件并存的现象，并非资产价格独有。事实上，这正是“自组织临界态”系统的一个标志性特征。自组织是个体代理人（在本文中即投资者）之间相互作用的产物，不需要任何领导。临界态是指微小的扰动就可能引发多种类型事件的状态。自组织临界性存在于形形色色的系统中，从地震、物种灭绝事件，到交通堵塞，莫不如此。

The pattern of many small events and few large events is not unique to asset prices. Indeed it is a signature of systems in the state of “self-organized criticality.” Self-organization is the result of interaction between individual agents (in this case investors) and requires no leadership. A critical state is one where small perturbations can lead to events of many types. Self-organized criticality marks systems as varied as earthquakes, extinction events, 8 and traffic jams.

有没有什么机制可以帮助解释这些间歇性的猛冲？我们认为有。正如我们在之前的报告中指出的，当足够多不同类型的投资者相互交易时，市场往往运行良好。反之，当这种多样性被打破，投资者行为趋同时（也可能是部分投资者离场所致），市场往往会变得脆弱。关于羊群效应的文献正在大量涌现，专门研究这一现象。羊群效应指的是，一大群投资者基于对他人行为的观察而做出相同的选择，而不依赖自身所知的信息。信息级联是自组织临界系统的另一个绝佳例证，它与羊群效应密切相关。

Is there a mechanism that can help explain these episodic lunges? We think so. As we have noted in previous 9 reports, markets tend to function well when a sufficient number of diverse investors interact. Conversely, markets tend to become fragile when this diversity breaks down and investors act in a similar way (this can also result from some investors withdrawing). A burgeoning literature on herding addresses this phenomenon. Herding is when a large group of investors make the same choice based on the observations of others, independent of the own knowledge. Information cascades, another good illustration of a self-organized critical system, are closely linked to 10 herding.

肥尾效应对投资者意味着什么好。价格的大幅变动比理论上应该出现的频率更高。这对投资者从实际角度意味着什么？我们认为有几个重要的启示：

What Fat Tails Mean for Investors O.K. Big changes in prices appear more frequently than they are supposed to. What does this mean for investors from a practical standpoint? We believe there are a few important implications:

• 因果思维。自组织临界系统的一个关键特征在于：扰动的大小与最终事件之间可能不存在线性关系。有时小规模输入会引发大规模事件。这粉碎了为所有结果找到原因的希望。例如，在一篇 1989 年被广泛引用的论文中，卡特勒、波特巴和萨默斯回顾了标普 500 指数在战后 50 次最大波动及其《纽约时报》次日报道的“原因”，他们的总结是：

• Cause and effect thinking. One of the essential features of self-organized critical systems is that the size of the perturbation and resulting event may not be linearly linked. Sometimes small-scale inputs can lead to large-scale events. This dashes the hope of finding causes for all effects. For example, in a widely cited 1989 paper, Cutler, Poterba, and Summers review the 50 largest post-war moves in the S&P 500 Index and the “causes”, as reported by the New York Times the subsequent day. They summarize:

“在大多数大幅波动的交易日里……媒体引述为市场变动原因的信息，其实并不特别重要。后续几天的媒体报道，也同样未能给出任何令人信服的解释，说明未来的利润或折现率为什么会发生变化。”

“On most of the sizable return days…the information that the press cites as the cause of the market move is not particularly important. Press reports on subsequent days also fail to reveal any convincing accounts 11 of why future profits or discount rates might have changed.”

• 风险与回报。评估风险的标准模型——资本资产定价模型——假定风险与回报呈线性关系。相比之下，在股市这类自组织临界系统中，非线性是内生的。投资者必须记住，金融理论是对现实世界数据的理想化处理。学术界和投资界如此频繁地谈论偏离均值五个或更多标准差的事件，这本身就足以说明，那些广泛使用的统计指标并不适用于市场。

• Risk and reward. The standard model for assessing risk, the capital asset pricing model, assumes a linear relationship between risk and reward. In contrast, nonlinearity is endogenous to self-organized critical systems like the stock market. Investors must bear in mind that finance theory stylizes the real world data. That the academic and investment community so frequently talk about events five or more standard deviations from the mean should be a sufficient indication that the widely used statistical measures are inappropriate for the markets.

• 投资组合构建。那些使用标准统计方法构建投资组合的投资者，可能会低估风险（实际经验与风险敞口之间存在偏差）。对于通过杠杆放大回报的投资组合，这种担忧尤为突出。对冲基金界最引人瞩目的失败案例，许多都直接源于肥尾事件。投资者在构建投资组合时必须将这些事件纳入考量。

• Portfolio construction. Investors that design portfolios using standard statistical measures may understate risk (experience versus exposure). This concern is especially pronounced for portfolios that use leverage to enhance returns. Many of the most spectacular failures in the hedge fund world have been the direct result of fat tail events. Investors need to take these events into consideration when constructing portfolios.

穿越肥尾世界的一个实用方法是：先衡量资产当前价格所蕴含的市场预期，然后思考各种可能的价值区间及其对应的概率。这个过程能让投资者对潜在的肥尾事件给予一定的权重。

A useful means to navigate a fat-tailed world is to first measure the current expectations underlying an asset price, and then contemplate various ranges of value outcomes and their associated probabilities. This 12 process allows investors to give some weight to potential fat tail events.

标准金融理论极大地提升了我们对市场的理解。但该理论的某些基础假设并未得到市场事实的印证。投资者必须认识到理论与现实之间的这些差距，并相应调整自己的思维方式（以及投资组合）。

Standard finance theory has advanced our understanding of markets immensely. But some of the theory’s foundational assumptions are not borne out by market facts. Investors must be aware of the discrepancies between the theory and reality and adjust their thinking (and portfolios) accordingly.

注瑞士信贷第一波士顿公司可能在最近三年内担任过上述任何一家或多家公司证券公开发行的主承销商或联席主承销商，或为其证券创造一级市场。

N.B.CREDIT SUISSE FIRST BOSTON CORPORATION may have, within the last three years, served as a manager or co-manager of a public offering of securities for or makes a primary market in issues of any or all of the companies mentioned.

_____________________________ 1 乔治·索罗斯，《索罗斯谈索罗斯》（纽约：约翰·威利父子出版公司，1995 年），第 17 页。

_____________________________ 1 George Soros, Soros on Soros (New York: John Wiley & Sons, 1995), 17.

2 维克多·尼德霍夫，《投机者的教育》（纽约：约翰·威利父子出版公司，1997 年），第 ix 页。

2 Victor Niederhoffer, The Education of a Speculator (New York: John Wiley & Sons, 1997), ix.

3 Philip W. Anderson，“关于经济学中分布的一些思考”，载于 W. B. Arthur、S. N. Durlaf 和 D. A. Lane 编，《作为演化复杂系统的经济体 II》（马萨诸塞州雷丁：艾迪生-韦斯利出版社，1997 年），第 566 页。

3 Philip W. Anderson, “Some Thoughts About Distribution in Economics,” in W. B. Arthur, S. N. Durlaf and D.A. Lane, eds., The Economy as an Evolving Complex System II (Reading, MA: Addison-Wesley, 1997), 566.

4 这个过程被称为布朗运动。阿尔伯特·爱因斯坦指出，这种运动是由受热激发的水分子对花粉的随机撞击引起的。

4 This process is know as Brownian motion. Albert Einstein pointed out that this motion is caused by random bombardment of heat excited water molecules on the pollen.

5 参见 http://www.gloriamundi.org/var/varintro.htm。

5 See http://www.gloriamundi.org/var/varintro.htm.

6 埃德加·E·彼得斯，《分形市场分析》（纽约：约翰·威利父子出版公司，1994 年），第 21-27 页。

6 Edgar E. Peters, Fractal Market Analysis (New York: John Wiley & Sons, 1994), 21-27.

7 Roger Lowenstein，《天才的失败：长期资本管理公司的兴衰》（纽约：兰登书屋，2000 年），

7 Roger Lowenstein, When Genius Failed: The Rise and Fall of Long-Term Capital Management (New York: Random House, 2000),

72. 洛温斯坦引用了延斯·卡斯滕·雅克沃斯与马克·鲁宾斯坦的论文《从期权价格中恢复概率分布》（Recovering Probability Distributions from Option Prices），刊于《金融学期刊》第 51 卷第 5 期，1996 年 12 月，第 1612 页。雅克沃斯与鲁宾斯坦指出，假设市场年化波动率为 20% 且服从对数正态分布，标普 500 指数期货 29% 的跌幅是一个 27 标准差事件，概率为 10 的 -160 次方。⁸ Per Bak，《自然如何运作》（How Nature Works）（纽约：施普林格出版社，1996 年）。

72. Lowenstein is quoting Jens Carsten Jackwerth and Mark Rubinstein, “Recovering Probability Distributions from Option Prices,” The Journal of Finance, 51, no. 5, December 1996, 1612. Jackwerth and Rubinstein note that assuming annualized volatility of 20% for the market and a lognormal distribution, the 29% drop in the S&P 500 futures was a 27 standard deviation event, with a probability of 10-160. 8 Per Bak, How Nature Works (New York: Springer-Verlag, 1996).

9 Michael J. Mauboussin 和 Kristen Bartholdson，《超越市场的流程》（A Process for Outperformance），载于《融会贯通观察家》（The Consilient Observer），第 1 卷，第 6 期，瑞士信贷第一波士顿股票研究，2002 年 3 月 26 日。

9 Michael J. Mauboussin and Kristen Bartholdson, ’’A Process for Outperformance,” The Consilient Observer, 1, no. 6, Credit Suisse First Boston Equity Research, March 26, 2002.

10. Sushil Bikhchandani 与 Sunil Sharma，《金融市场中的从众行为》，国际货币基金组织职员论文，第 47 卷，第 3 期，2001 年。参见 http://www.imf.org/External/Pubs/FT/staffp/2001/01/bikhchan.htm。

10 Sushil Bikhchandani and Sunil Sharma, “Herd Behavior in Financial Markets,” IMF Staff Papers, 47, no. 3, 2001. See http://www.imf.org/External/Pubs/FT/staffp/2001/01/bikhchan.htm.

11 David M. Cutler、James M. Poterba 与 Lawrence H. Summers，《什么在推动股价？》，《投资组合管理期刊》，1989 年春季刊。

11 David M. Cutler, James M. Poterba, and Lawrence H. Summers, “What Moves Stock Prices?” The Journal of Portfolio Management, Spring 1989.

12 Michael S. Gibson, “Incorporating Event Risk into Value-at-Risk” 美联储金融与经济讨论系列，2001-17 号，2001 年 2 月。参见 http://www.federalreserve.gov/pubs/feds/2001/200117/200117abs.html。

12 Michael S. Gibson, “Incorporating Event Risk into Value-at-Risk” The Federal Reserve Board Finance and Economics Discussion Series, 2001-17, February 2001. See http://www.federalreserve.gov/pubs/feds/2001/200117/200117abs.html.

原件此处是表格，PDF 抽取时列结构已丢失，下面只剩按列读出的数字，行列对应关系无法还原。核对数据请打开来源正文。

城市电话城市电话城市电话
阿姆斯特丹31 20 5754 890吉隆坡603 2143 0366旧金山1 415 836 7600
亚特兰大1 404 656 9500伦敦44 20 7888 8888圣保罗55 11 3841 6000
奥克兰64 9 302 5500马德里34 91 423 16 00首尔82 2 3707 3700
巴尔的摩1 410 223 3000墨尔本61 3 9280 1888上海86 21 6881 8418
曼谷62 614 6000墨西哥城52 5 283 89 00新加坡65 212 2000
北京86 10 6410 6611米兰39 02 7702 1悉尼61 2 8205 4433
波士顿1 617 556 5500莫斯科7 501 967 8200台北886 2 2715 6388
布达佩斯36 1 202 2188孟买91 22 230 6333东京81 3 5404 9000
布宜诺斯艾利斯54 11 4394 3100纽约1 212 325 2000多伦多1 416 352 4500
芝加哥1 312 750 3000帕洛阿尔托1 650 614 5000华沙48 22 695 0050
法兰克福49 69 75 38 0巴黎33 1 53 75 85 00华盛顿1 202 354 2600
休斯敦1 713 220 6700帕萨迪纳1 626 395 5100惠灵顿64 4 474 4400
香港852 2101 6000费城1 215 851 1000苏黎世41 1 333 55 55
约翰内斯堡27 11 343 2200布拉格420 2 210 83111
 AMSTERDAM............... 31 20 5754 890   KUALA LUMPUR.........603 2143 0366   SAN FRANCISCO...... 1 415 836 7600
 ATLANTA ......................1 404 656 9500   LONDON ...................44 20 7888 8888   SÃO PAULO ............ 55 11 3841 6000
 AUCKLAND.....................64 9 302 5500   MADRID .....................34 91 423 16 00   SEOUL ....................... 82 2 3707 3700
 BALTIMORE..................1 410 223 3000   MELBOURNE .............61 3 9280 1888   SHANGHAI............... 86 21 6881 8418
 BANGKOK..........................62 614 6000   MEXICO CITY ..............52 5 283 89 00   SINGAPORE ................... 65 212 2000
 BEIJING.......................86 10 6410 6611   MILAN .............................39 02 7702 1   SYDNEY ..................... 61 2 8205 4433
 BOSTON........................1 617 556 5500   MOSCOW....................7 501 967 8200   TAIPEI ...................... 886 2 2715 6388
 BUDAPEST .....................36 1 202 2188   MUMBAI......................91 22 230 6333   TOKYO ....................... 81 3 5404 9000
 BUENOS AIRES..........54 11 4394 3100   NEW YORK.................1 212 325 2000   TORONTO.................. 1 416 352 4500
 CHICAGO ......................1 312 750 3000   PALO ALTO................1 650 614 5000   WARSAW................... 48 22 695 0050
 FRANKFURT ....................49 69 75 38 0   PARIS........................33 1 53 75 85 00   WASHINGTON........... 1 202 354 2600
 HOUSTON .....................1 713 220 6700   PASADENA ................1 626 395 5100   WELLINGTON.............. 64 4 474 4400
 HONG KONG..................852 2101 6000   PHILADELPHIA ..........1 215 851 1000   ZURICH ....................... 41 1 333 55 55
JOHANNESBURG..........27 11 343 2200   PRAGUE ...................420 2 210 83111

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