Financial theory tends to focus on a notion of risk limited to what we might be termed ‘‘trivial perils’’, those having to do with price fluctuations liable to appear in a relatively stable overall environment. Doesn’t history frequently show that radical alterations in the environment provoke price variations comparable to mood swings? Maybe we just do not know how to analyze major risks. Maybe we do not know how to prevent them.
Examples have demonstrated that rather than base their expectations on past mathematical averages, investors tend to detect correlations between past events. At the beginning of his chapter on ‘‘the state of long-term expectation’’, Keynes wrote:
It would be foolish, in forming our expectations, to attach great weight to matters which are very uncertain. It is reasonable, therefore, to be guided to a considerable degree by the facts about which we feel somewhat confident, even though they may be less decisively relevant to the issue than other facts about which our knowledge is vague and scanty. For this reason the facts of the existing situation enter, in a sense disproportionately, into the formation of our long-term expectations; our usual practice being to take the existing situation and project it into the future, modified only to the extent that we have more or less definite reason for expecting a change.
How is such a judgment to be formulated? Investors tend to make a fetish out of economic ‘‘factoids’’, such as for how many months, at some time in the past, did the market anticipate and predict economic recovery. And yet they interpret such anecdotal data by connecting the ‘‘dots’’ that get their attention in accordance with the ‘‘paths’’ they map out. As with the Impressionists (and most especially with Georges Seurat), investors spend a great deal of their time connecting details. The brain is organized in order to detect correlations. Numerous studies have shown that investors revise their predictions by overemphasizing new information in relation to pre-existing and long-term data.
Curiously enough, the mechanism of risk analysis is altogether different. Rather than behave as they do when forming expectations, investors often use historical (mathematical) volatility to assess the risk of a possible investment. Experience shows that volatility constitutes an excellent basis for risk evaluation; what skyrockets may plummet just as precipitately.
That said, the relationship between risk and return is not as fluid as theory would have it (we shall return to this point). The connection between risk and return is not a detail. It is not because a portfolio presents high risk that one may reasonably expect to win the ‘‘sweepstakes’’. Moreover, unforeseen correlations do crop up; they merely were not noted in the past. Last but not least, volatility evolves along with time. One may think that the dispersion of returns (additional volatility) would increase in times of economic and political uncertainty. The sensitivity of a financial asset to market variations may be estimated historically, but this should only be the basis of ongoing anticipation. Volatility also must be anticipated.

Untimely surprises are especially dangerous when one is counting on the income drawn from one’s portfolio. Capitalization is based on the principle that all such revenues are reinvested and that values may consequently be compared at several points in time. In reality, investors will want to redeem earnings or liquidate a portion of their investments before the date of termination. One may be (or at least feel) compelled to sell off one’s investments at a time when prices are low, which is a disagreeable surprise. If investors regularly liquidate a portion of their investments in order to remain afloat, they will be especially aware of the risk of doing so at a time when prices are heading downwards. Stock market crises compress the per-share value of a portfolio to such an extent that more shares have got to be sold in order to draw the same revenues. Once prices rise again, it will be that much more difficult to compensate through capitalization for the transfers conceded when prices were low.
Let’s take as an example an investor who retired at the end of 1998 at the age of 60 with e300,000 invested in a stock-indexed fund (a fund that follows a market index; see Chapter 9). Suppose that this recent retiree had the intention of eking out an existence thanks to the e18,000 annually withdrawn from his portfolio. If the latter had provided a regular annual return of 8 percent, it would have produced e24,000 each year in dividends and capital gains, and he would have been able to cope with the withdrawal of e18,000 per year. But from 1999 through 2001, the stock exchange went down by about 40 percent! By the end of 2001 the portfolio would have been worth only e180,000. It would have taken annual returns of 10 percent over the following period to allow for the withdrawals. Quite obviously, everything would have been different had the crisis taken place 20 years later. During that period the portfolio would have been enriched by e24,000 – 18,000 = 6,000 per annum, that is 2 percent per year. It would have increased by 50 percent and a 40 percent loss in 2020 would still have left e270,000 in the portfolio. Average 8 percent returns would have permitted scheduled payments of the required annuities. The one way to survive a stock market crisis that comes too early is to diversify one’s portfolio and employ financial instruments that do not all go down at the same time. Such diversification limits portfolio volatility.
Does historical volatility measure the risk of investment for the future? Fluctuations in the vicinity of the average may give a precise idea of the risk, but this is the case if – and only if – the law of probability remains unchanged. If observations of the past are to prove useful when forecasting the future, it is necessary for the law of probability to be stable in time. Is this the case? Just like the laws of mortality, the law of probability is not a known quantity; it can only be estimated on the basis of past series. Compare the probability of stormy weather. Tomorrow’s skies may be predicted as a function of meteorological parameters, provided that the climate is not fundamentally altered. Consider global warming; weather forecasting is of little avail in the event of a phenomenon rendering history obsolete! And in a stock market crisis, investors may have the impression that everything is crashing. From 1989 through 1999 at the Paris Bourse, 95 percent of reported monthly returns ranged from +12.75 percent to – 11.25 percent, which means that 5 percent of the time, returns were greater than +12.75 percent or lower than – 11.25 percent; hence there had been a monthly loss greater than 11 percent for 2.5 percent of the time (or once every three years). In some cases losses become downright catastrophic – in October 1987 shares plummeted by 22 percent in a month; in September 2001 it took just a week for them to lose 18 percent of their previous value. That is twice as much as the lower limit of the range of confidence over the 10 previous years! Yet up to now, the market has always recovered. Even if stock market performances are approximately akin to a random walk and even if their distribution resembles a normal law, this is not what happens at the extremes. Pronounced monthly highs and lows, bubbles and crashes occur, not as often as the normal law indicates, but more often than the normal law can possibly foresee. As Bernstein states:
At the extremes, the market is not a random walk. At the extremes, the market is more likely to destroy fortunes than to create them. The stock market is a risky place.

In what ways do prices take into account the fundamental economic data? Some say that market price always reflects the state of the real economy. That point of view is a travesty of reality. Others contend that given the permanently observed lack of connection between share prices and the economic basics, to see one is not to believe the other. The one obvious fact is that rather than being influenced by these data, prices are largely determined by the expectations that speculators have of them.
So how does the news affect prices? At a forum at Wharton Business School in October 2001, Professor Richard Marston (who teaches there) stated: ‘‘The economy itself, and expectations about it, are what is driving the stock market right now.’’ Right now? Isn’t this always the case? Expectations continually drive the market. Let’s look at what happened in September 2001. During the first five days of trading after the terrorist attacks on the World Trade Center and the Pentagon, the Dow Jones Industrial Average plummeted 14.25 percent, the greatest weekly loss in 61 years. Since the beginning of 2001, the financial markets had been undergoing a phase of ‘‘bubble’’ bursting, in contrast to the euphoria that had characterized the previous few years. Taking as a reference the US-based computerized network for price quotations known as NASDAQ, we may note that it took 14 months to rise from 2000 to 5000 (11 months for this index to rise from 2000 to 3000 points, 2 months from 3000 to 4000 and 10 weeks from 4000 to 5000) and 22 months to bring it down from its maximum (5048.62 on December 3, 2000) to 1694.27 (on September 10, 2001). This fall may have been masked, but it was cumulatively tantamount to a crash – and it had yet to run its course. The atrocities on September 11 and the closing of American stock markets for the following four days accelerated this pronounced trend; on September 21, 2001, the index stood at 1423.19. Given what had been going on for a year, one could be led to believe that the 16 percent loss in one week would have taken place in any event, but might have been strung out over a period approaching a month. After all, alarming news concerning the American economy had been lowering ongoing expectations. What happened was that the attacks compressed the impact of the negative economic news and tidings. But then, in what ways are expectations usually related to economic evolution? Here again, we anchor ourselves in the past and go on to suppose that previous links between expectations and the real economy will be reiterated. During the Wharton forum, Richard Marston also stated:
Because investors try to anticipate future events, stocks tend to rebound before the economy does. It is hard to make any forecast, especially about the future. But the hardest things to predict are turning points. It’s remarkable how much the market moves after it reaches the bottom.
Marston was putting forward the point that speculators anticipated the ‘‘rebound’’. According to him, from June through October 1990, the Gulf War helped drag the Standard and Poor’s index down by 14.7 percent. And yet over the next six months it rose by no less than 25.6 percent! Fast forward to the summer of 1998, when Russia was in turmoil. During July and August the index registered a 15.4 percent drop, but it rose 30.3 percent over the following six months and 39.8 percent again in the year after the end of the crisis.
Anticipations have similar sources, and it matters little whether the expectations are mathematical or based on probable forecasts. In one case the sequence is the historical mean, in the other historical correlations are used. Psychology is invariably involved and what really matters is the confidence with which we make a forecast – Keynes’s state of confidence, i.e. the risk of our forecast turning out to be wrong. Variance and standard deviation assess the variations in the profitability of a security. Do they constitute measurements of risks incurred when investing in the latter? Not to the extent that they measure pleasant as well as unpleasant surprises. And not to the extent that the future fails to renew the past. Let us examine these two negative answers.
Variability also measures agreeable surprises such as shareholder returns that are higher than expected. Is this risk? In fact risk is limited to disagreeable surprises, but as long as returns remain symmetrically scattered, that is as long as the likelihood of a happy surprise is equal to the likelihood of an unhappy surprise, standard deviation adequately measures the risks incurred. The higher it is, the greater the danger. It is the same with the ‘‘risks’’ of manna from heaven, but prudent investors fear the worst more than they hope for the best.

In 1973 a renowned Princeton professor, Burton Malkiel, published his bestseller A Random Walk Down Wall Street, and the book was reprinted seven times in 25 years. His thesis is as follows. Today’s stock market is so efficient that a blindfolded chimpanzee aiming darts at the stock price pages of the Wall Street Journal could select a stock portfolio that would perform just as well as a fund actively managed by a professional broker. It was Malkiel who popularized the notion of the random walk of stock prices, an idea that owed its inception to Louis Bachelier 80 years earlier.
That said, such an analysis is valid only when applied to a sizable number of stocks, as is the case with the Standard and Poor’s 500-stock index, the gauge that Wall Street uses to track stock performance and for which a portfolio is composed, by definition, of 500 stocks. Curiously enough, it was only in 1952, as we saw earlier, that Harry Markowitz based his reasoning on a portfolio of securities rather than holdings taken one by one. His seminal article that opened the way for the random walk is soberly entitled ‘‘Portfolio selection’’. His aim was to formulate rules of portfolio construction for investors who find expected returns desirable and variance of return (a concept not unrelated to standard deviation, as we saw earlier) undesirable.
The random walk notion may also indicate that it suffices to invest in the stock market and ‘‘go with the flow’’ in order to achieve reasonable monthly gains at least equivalent to the average monthly performance of the index over the most recent 10 years. If one is convinced that ultimately the future will resemble the past and that prices will continue to follow the same law of probability, it stands to reason that average future performances should be altogether comparable to those of the past. As measured in terms of standard deviation, price volatility should likewise be based on long-standing precedents. In fact, shareholder return for a portfolio randomly varies according to a distribution that appears similar to a normal law. From a statistician’s point of view, observation of real profitability rates may be interpreted as a random evocation of a law of probability. If one postulates that the former randomly fluctuates, then past sequences may indeed be interpreted as samples of the law of probability for shareholder return on the portfolio. And if price variations indeed manifest themselves totally at random, past distributions may be used not only in hindsight, but also as a means of accurately forecasting.
Let’ s take as an example the Paris Bourse at the start of the twenty-first century. If monthly price changes are randomly distributed, there is a 68 percent chance that they will vary by no less than – 4.9 percent in any one month or by no more than + 6.6 percent. Since 68 percent represents approximately two-thirds, this means that in two out of every three months, these changes will neither exceed 6.6 percent nor dip under – 4.9 percent. The law of probability does not indicate which rate will be attained, nor does it indicate when. All it does is to specify the percentage chances of profitability’ s reaching the designated level.
A random walk does not mean that stock prices evolve haphazardly as if basic information did not exist. Quite the contrary, a random walk constantly draws on incoming data. The market is exceedingly efficient. Prices go up and down as news and information come in. Nobody is in a position to draw profit on a preliminary basis. Nobody can satisfactorily forecast the upcoming market evolution. It was Bachelier himself who wrote in his 1900 doctoral dissertation: ‘‘The mathematical expectancy of the speculator is zero.’’ When the market is characterized as efficient, this means that no investor can make a lifelong living out of beating the market at its own game. Not a single soul can repeatedly and systematically do it. Markets vary at random. The mathematical expectancy cannot possibly be higher than the indexed average. Speculators think and believe otherwise; their expectation is that they will outdo the market. We might say that they anticipate and draw profit from tomorrow’s news. Bernard Baruch appositely wrote: ‘‘A speculator is a man who observes the future, and acts before it happens.’’ Any investor is a speculator in so far as, seeking to foresee, he or she bets. When doing so, investors may exert influence on prices, which reflect the expectations engendered by the news. And yet what was anticipated does not necessarily come into being; quite the contrary. Even if the speculator was not mistaken concerning the repercussions of the news, more recent events may have affected prices by the time of resale. It is highly likely that the speculator will not outperform the market; in fact his mathematical expectancy is zero. And yet he still hopes and strives to buck the odds.

Bloomberg, the financial services firm, presents a bar graph of monthly performances of equity indexes on most stock markets over a 10-year period. Analysis shows that on the main markets, the 120 monthly variations of the index rather faithfully obey the Gaussian law of probability. Bloomberg even provides the bell curve corresponding to the average and to standard deviation.
For example, over the final 10 years of the twentieth century (1989 – 99), the Paris Bourse rose by an average of 0.85 percent a month, with a standard deviation of 5.75 percent. If the prices indeed follow a normal distribution, it means that in 68 percent of the cases, the monthly variation was neither lower than – 4.9 percent nor higher than + 6.6 percent. Since 68 percent represents approximately two-thirds, this means that in two out of every three months, price variations neither go up by over 6.6 percent nor go down by more than 4.9 percent. Over two-thirds of the observations are to be found in a range from – 4.9 percent to + 6.6 percent. One month out of six, prices drop by more than 4.9 percent. One month out of six, prices soar at a rate of over 6.6 percent. Very similar results appear on other major stock markets. Bernstein analyzes the evolution of Standard and Poor’s index of 500 US stocks for January 1926 through December 1995, that is to say 840 observations of monthly price changes.23 The average monthly variation in New York over 70 years was + 0.6 percent compared with + 0.85 percent in Paris, but over only 10 years. Standard deviation in New York was 5.8 percent compared with 5.75 percent in Paris. You will agree that the difference is minuscule. In New York two-thirds of the 840 monthly price fluctuations observed ranged from – 5.2 percent to + 6.4 percent. In Paris, two-thirds of the 120 observations went from – 4.9 percent to + 6.6 percent. The very different stock markets studied over a protracted period turn out to be closely related.
On both markets, prices behaved as though obeying a normal law. This is likewise the case for the indexes of highly liquid stock markets, for the Dow Jones Industrial Average or Standard and Poor’s in the US, the Stoxx, Euronext and FTSE indexes in Europe. This is highly understandable. On an efficient market, prices reflect the available information. And yet the latest news concerning a firm is often unforeseen (strike, accident, merger, technological breakthrough and so on). These events hinge on chance. Influenced by such news, prices generally take a random walk. The prices of particular shares may follow different distributions, but Laplace’s central limit theorem tells us that the average price must comply with the normal law; and stock indexes are in fact structured as averages. There is also another reason for price indexes being made to take a random walk. If there were a way of forecasting them, if some technique allowed us to think that in days or weeks to come they would rise, then they would rise immediately. Price evolution has no more memory than Pascal’s throw of the dice. Each variation is independent of the past. Price variation regularly changes signs and there are few consecutive months in which the market goes either up or down. Variations in the same direction for five consecutive months occur in only one in ten cases. There is no way to use the price tendency in the past to predict their future direction. A price trend negates itself once it becomes known. This is another fundamental difference between the roulette wheel and the stock market; following the Martingale system in the latter – when you double your stake after a loss – leads to self-destruction.