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You Can't Become Rich In Your Pocket Until You Become Rich In Your Mind | ||||
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Investors can play the same game on a larger scale by buying and selling puts on the S&P 500Let me start with the original puzzle. Which of the following two situations would you prefer to be in? In the first one youre given a fair coin to flip and are told that you will receive $1,000 if it lands heads and lose $1,000 if it lands tails. In the second youre given a very biased coin to flip and must decide whether to bet on heads or tails. If it lands the way you predict you win $1,000 and, if not, you lose $1,000. Although most people prefer to flip the fair coin, your chances of winning are 1/2 in both situations, since youre as likely to pick the biased coins good side as its bad side. Consider now a similar pair of situations. In the first one you are told you must pick a ball at random from an urn containing 10 green balls and 10 red balls. If you pick a green one, you win $1,000, whereas if you pick a red one, you lose $1,000. In the second, someone you thoroughly distrust places an indeterminate number of green and red balls in the urn. You must decide whether to bet on green or red and then choose a ball at random. If you choose the color you bet on, you win $1,000 and, if not, you lose $1,000. Again, your chances of winning are 1/2 in both situations. Finally, consider a third pair of similar situations. In the f i rst one you buy a stock that is being sold in a perfectly efficient market and your earnings are $1,000 if it rises the next day and -$1,000 if it falls. (Assume that in the short run it moves up with probability 1/2 and down with the same probability.) In the second there is insider trading and manipulation and the stock is very likely to rise or fall the next day as a result of these illegal actions. You must decide whether to buy or sell the stock. If you guess correctly, your earnings are $1,000 and, if not, -$1,000. Once again your chances of winning are 1/2 in both situations. (They may even be slightly higher in the second situation since you might have knowledge of the insiders motivations.) In each of these pairs, the unfairness of the second situation is only apparent. You have the same chance of winning that you do in the first situation. I do not by any means defend insider trading and stock manipulation, which are wrong for many other reasons, but I do suggest that they are, in a sense, simply two among many unpredictable factors affecting the price of a stock. I suspect that more than a few cases of insider trading and stock manipulation result in the miscreant guessing wrong about how the market will respond to his illegal actions. This must be depressing for the perpetrators (and funny for everyone else). Expected Value, Not Value Expected What can we anticipate? What should we expect? Whats the likely high, low, and average value? Whether the quantity in question is height, weather, or personal income, extremes are more likely to make it into the headlines than are more informative averages. Who makes the most money, for example, is generally more attention-grabbing than what is the average income (although both terms are always suspect because-surprise-like companies, people lie about how much money they make). Even more informative than averages, however, are distributions. What, for example, is the distribution of all incomes and how spread out are they about the average? If the average income in a community is $100,000, this might reflect the fact that almost everyone makes somewhere between $80,000 and $120,000, or it might mean that a big majority earns less than $30,000 and shops at Kmart, whose spokesperson, the (too) maligned Martha Stewart, also lives in town and brings the average up to $100,000. Expected value and standard deviation are two mathematical notions that help clarify these issues. An expected value is a special sort of average. Specifically, the expected value of a quantity is the average of its values, but weighted according to their probabilities. If, for example, based on analysts recommendations, our own assessment, a mathematical model, or some other source of information, we assume that 1/2 of the time a stock will have a 6 percent rate of return, that 1/3 of the time it will have a -2 percent rate of return, and that the remaining 1/6 of the time it will have a 28 percent rate of return, then, on average, the stocks rate of return over any given six periods will be 6 percent three times, -2 percent twice, and 28 percent once. The expected value of its return is simply this probabilistically weighted average(6% + 6% + 6% + (-2%) + (-2%) + 28%)/6,or7%. Rather than averaging directly, one generally obtains the expected value of a quantity by multiplying its possible values by their probabilities and then adding up these products. Thus .06 x 1/2 + (-.02) x 1/3 + .28 x 1/6 = .07, or 7%, the expected value of the above stocks return. Note that the term mean and the Greek letter p (mu) are used interchangeably with expected value, so 7% is also the mean return, p. The notion of expected value clarifies a minor investing mystery. An analyst may simultaneously and without contradiction believe that a stock is very likely to do well but that, on average, its a loser. Perhaps she estimates that the stock will rise 1 percent in the next month with probability 95 percent and that it will fall 60 percent in the same time period with probability 5 percent. (The probabilities might come, for example, from an appraisal of the likely outcome of an impending court decision.) The expected value of its price change is thus (.01 x .95) + (-.60) x .05), which equals -.021 or an expected loss of 2.1%. The lesson is that the expected value, -2.1 %, is not the value expected, which is 1 %. The same probabilities and price changes can also be used to illustrate two complementary trading strategies, one that usually results in small gains but sometimes in big losses, and one that usually results in small losses but sometimes in big gains. An investor whos willing to take a risk to regularly make some easy money might sell puts on the above stock, puts that expire in a month and whose strike price is a little under the present price. In effect, hes betting that the stock wont decline in the next month. Ninety-five percent of the time hell be right, and hell keep the put premiums and make a little money. Correspondingly, the buyer of the puts will lose a little money (the put premiums) 95 percent of the time. Assuming the probabilities are accurate, however, when the stock declines, it declines by 60 percent, and so the puts (the right to sell the stock at a little under the original price) become very valuable 5 percent of the time. The buyer of the puts then makes a lot of money and the seller loses a lot. Investors can play the same game on a larger scale by buying and selling puts on the S&P 500, for example, rather than on any particular stock. The key to playing is coming up with reasonable probabilities for the possible returns, numbers about which people are as likely to differ as they are in their preferences for the above two strategies. Two exemplars of these two types of investor are Victor Niederhoffer, a wellknown futures trader and author of The Education of a Speculator, who lost a fortune by selling puts a few years ago, and Nassim Taleb, another trader and the author of Fooled by Randomness, who makes his living by buying them. For a more pedestrian illustration, consider an insurance company. From past experience, it has good reason to believe that each year, on average, one out of every 10,000 of its homeowners policies will result in a claim of $400,000, one out of 1,000 policies will result in a claim of $60,000, one out of 50 will result in a claim of $4,000, and the remainder will result in a claim of $0. The insurance company would like to know what its average payout will be per policy written. The answer is the expected value, which in this case is ($400,000 x 1/10,000) + ($60,000 x 1/1,000) + ($4,000 x 1/50) + ($0 x 9,979/10,000) = $40 + $60 + $80 + $0 = $180. The premium the insurance company charges the homeowners will no doubt be at least $181. Combining the techniques of probability theory with the definition of expected value allows for the calculation of more interesting quantities. The rules for the World Series of baseball, for example, stipulate that the series ends when one team wins four games. The rules further stipulate that team A plays in its home stadium for games 1 and 2 and however many of games 6 and 7 are necessary, whereas team B plays in its home stadium for games 3, 4, and, if necessary, game 5. If the teams are evenly matched, you might be interested in the expected number of games that will be played in each teams stadium. Skipping the calculation, Ill simply note that team A can expect to play 2.9375 games and team B 2.875 games in their respective home stadiums. Almost any situation in which one can calculate (or reasonably estimate) the probabilities of the values of a quantity allows us to determine the expected value of that quantity. An example more tractable than the baseball problem concerns the decision whether to park in a lot or illegally on the street. If you park in a lot, the rate is $10 or $14, depending upon whether you stay for less than an hour, the probability of which you estimate to be 25 percent. You may, however, decide to park illegally on the street and have reason to believe that 20 percent of the time you will receive a simple parking ticket for $30, 5 percent of the time you will receive an obstruction of traffic citation for $100, and 75 percent of the time you will get off for free. The expected value of parking in the lot is ($10 x .25) + ($14 x .75), which equals $13. The expected value of parking on the street is ($100 x .05) + ($30 x .20) + ($0 x .75), which equals $11. For those to whom this is not already Greek, we might say that PL, the mean costs of parking in the lot, and Ps, the mean cost of parking on the street, are $13 and $11, respectively. Even though parking in the street is cheaper on average (assuming money was your only consideration), the variability of what youll have to pay there is much greater than it is with the lot. This brings us to the notion of standard deviation and stock risk. Whats Normal? Not Six Sigma Risk in general is frightening, and the fear it engenders explains part of the appeal of quantifying it. Naming bogeymen tends to tame them, and chance is one of the most terrifying bogeyman around, at least for adults. So how might one get at the notion of risk mathematically? Lets start with variance, one of several mathematical terms for variability. Any chance-dependent quantity varies and deviates from its mean or average; its sometimes more than the average, sometimes less. The actual temperature, for example, is sometimes warmer than the mean temperature, sometimes cooler. These deviations from the mean constitute risk and are what we want to quantify. They can be positive or negative, just as the actual temperature minus the mean temperature can be positive or negative, and hence they tend to cancel out. If we square them, however, the deviations are all positive, and we come to the definition: the variance of a chancedependent quantity is the expected value of all its squared deviations from the mean. Before I numerically illustrate this, note the etymological/psychological association of risk with deviation from the mean. This is a testament, I suspect, to our fear not only of risk but of anything unusual, peculiar, or deviant. Be that as it may, lets switch from temperature back to our parking scenario. Recall that the mean cost of parking in the lot is $13, and so ($10 - $13)2 and ($14 - $13)2, which equal $9 and $1, respectively, are the squares of the deviations of the two possible costs from the mean. They dont occur equally frequently, however. The first occurs with probability 25%, and the second occurs with probability 75%, and so the variance, the expected value of these numbers, is ($9 x .25) + ($1 x .75), or $3. More commonly used in statistical applications in finance and elsewhere is the square root of the variance, which is usually symbolized by the Greek letter r (sigma). Termed the standard deviation, it is in this case the square root of $3, or approximately $1.73. The standard deviation is (not exactly, but can be thought of as) the average deviation from the mean, and it is the most common mathematical measure of risk. Forget the numerical examples if you like, but remember that, for any quantity, the larger the standard deviation, the more spread out its possible values are about the mean; the smaller it is, the more tightly the possible values cluster around the mean. Thus, if you read that in Japan the standard deviation of personal incomes is much less than it is in the United States, you should infer that Japanese incomes vary considerably less than U.S. incomes. Returning to the street, you may wonder what the variance and standard deviation are of your parking costs there. The mean cost of parking in the street is $11, and the squares of the deviations of the three possible costs from the mean are ($100 - $11)2, ($30 - $11)2, and ($0 - $11)2, or $7,921, $361, and $121, respectively. The first occurs with probability 5%, the second with probability 20%, and the third with probability 75%, and so the variance, the expected values of these numbers, is ($7,921 x .05) + ($361 x .20) + ($0 x .75), or $468.25. The square root of this gives us the standard deviation of $21.64, more than twelve times the standard deviation of parking in the lot. Despite this blizzard of numbers, I reiterate that all we have done is quantify the obvious fact that the possible outcomes of parking on the street are much more varied and unpredictable than those of parking in the lot. Even though the average cost of parking in the street ($11) is less than that of parking in the lot ($13), most would prefer to incur less risk and would therefore park in the lot for prudential reasons, if not moral ones. This brings us to the markets use of standard deviation (sigma) to measure a stocks volatility. Lets use the same approach to calculate the variance of the returns for our stock that yields a rate of 6% about 1/2 the time, -2% about 1/3 of the time, and 28% the remaining 1/6 of the time. The mean or expected value of its returns is 7%, and so the squares of the deviations from the mean are (.06 - .07)2, (-.02 - .07)2, and (.28 - .07)2 or .0001, .0081, and .0441, respectively. These occur with probabilities 1/2, 1/3, and 1/6, and so the variance, the expected value of the squares of these deviations from the mean, is (.0001 x 1/2) + (.0081 x 1/3) + (.0441 x 1/6), which is .01. The square root of .01 is .10 or 10%, and this is the standard deviation of the returns for this stock. The Greek lesson again: The expected value of a quantity is its (probabilistically weighted) average and is symbolized by the letter p (mu), and the standard deviation of a quantity is a measure of its variability and is symbolized by the letter 6 (sigma). If the quantity in question is the rate of return on a stock price, its volatility is generally taken to be the standard deviation. If there are only two or three possible values a quantity might assume, the standard deviation is not that helpful a notion. It becomes very useful, however, when a quantity can assume many different values and these values, as they often do, have an approximately normal bell-shaped distribution-high in the middle and tapering off on the sides. In this case, the expected value is the high point of the distribution. Moreover, approximately 2/3 of the values (68 percent) lie within one standard deviation of the expected value, and 95 percent of the values lie within two standard deviations of the expected value. Before we go on, lets list a few of the quantities that have a normal distribution: age-specific heights and weights, natural gas consumption in a city for any given winter day, water use between 2 A.M. and 3 A.M. in a given city, thicknesses of a particular machined part coming off an assembly line, I.Q.s (whatever it is that they measure), the number of admissions to a large hospital on any given day, distances of darts from a bulls-eye, leaf sizes, nose sizes, the number of raisins in boxes of breakfast cereal, and possible rates of return for a stock. If we were to graph any of these quantities, we would obtain bell-shaped curves whose values are clustered about the mean. Take as an example the number of raisins in a large box of cereal. If the expected number of raisins is 142 and the standard deviation is 8, then the high point of the bell-shaped graph would be at 142. About two-thirds of the boxes would contain between 134 and 150 raisins, and 95 percent of the boxes would contain between 126 and 158 raisins. Or consider the rate of return of a conservative stock. If the possible rates are normally distributed with an expected value of 5.4 percent and a volatility (standard deviation, that is) of only 3.2 percent, then about two-thirds of the time, the rate of return will be between 2.2 percent and 8.6 percent, and 95 percent of the time the rate will be between -1 percent and 11.8 percent. You might prefer this stock to a more risky one with the same expected value but a volatility of, say, 20.2 percent. About two-thirds of the time, the rate of return of this more volatile stock will be between -14.8 percent and 25.6 percent, and 95 percent of the time it will be between -35 percent and 45.8 percent. In all cases, the more standard deviations from the expected value, the more unusual the result. This fact helps account for the many popular books on management and quality control having the words six sigma in their titles. The covers of many of these books suggest that by following their precepts, you can attain results that are six standard deviations above the norm, leading, for example, to a minuscule number of product defects. A six-sigma performance is, in fact, so unlikely that the tables in most statistics texts dont even include values for it. If you look into the books on management, however, you learn that Sigma is usually capitalized and means something other than sigma, the standard deviation of a chance-dependent quantity. A new oxymoron: minor capital offense. Whether they are defects, nose sizes, raisins, or water use in a city, almost all normally distributed quantities can be thought of as the average or sum of many factors (genetic, physical, social, or financial). This is not an accident: The socalled Central Limit Theorem states that averages and sums of a sufficient number of chance-dependent quantities are always normally distributed. As well see in chapter 8, however, not everyone believes that stocks rates of return are normally distributed. |
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