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You Can't Become Rich In Your Pocket Until You Become Rich In Your Mind | ||||
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The investment gurus who claimed that they could make your $10,000 grow to more than a million in a years timePonzi and the Irrational Discounting of the Future Before returning to other applications of these financial notions, it may be helpful to take a respite and examine an extreme case of undervaluing the future: pyramids, Ponzi schemes, and chain letters. These differ in their details and colorful storylines. A recent example in California took the form of all-women dinner parties whose new members contributed cash appetizers. Whatever their outward appearance, however, almost all these scams involve collecting money from an initial group of investors by promising them quick and extraordinary returns. The returns come from money contributed by a larger group of people. A still larger group of people contributes to both of the smaller earlier groups. This burgeoning process continues for a while. But the number of people needed to keep the pyramid growing and the money coming in increases exponentially and soon becomes difficult to maintain. People drop out, and the easy marks become scarcer. Participants usually lack a feel for how many people are required to keep the scheme going. If each of the initial group of ten recruits ten more people, for example, the secondary group numbers 100. If each of these 100 recruit ten people, the tertiary group numbers 1,000. Later groups number 10,000, then 100,000, then 1,000,000. The system collapses under its own weight when enough new people can no longer be found. If you enter the scheme early, however, you can make extraordinarily quick returns (or could if such schemes were not illegal). The logic of pyramid schemes is clear, but people generally worry only about what happens one or two steps ahead and anticipate being able to get out before a collapse. Its not irrational to get involved if you are confident of recruiting a bigger sucker to replace you. Some would say that the dot-coms meteoric stock price rises in the late 90s and their subsequent precipitous declines in 2000 and 2001 were attenuated versions of the same general sort of scam. Get in on the initial public offering, hold on as the stock rockets upward, and jump off before it plummets. Although not a dot-com, WorldCom achieved its all-toof l eeting dominance by buying up, often for absurdly inflated prices, many companies that were (and a good number that werent). MCI, MFS, ANS Communication, CAI Wireless, Rhythms, Wireless One, Prime One Cable, Digex, and dozens more companies were acquired by Bernie Ebbers, a pied piper whose song seemed to consist of only one entrancing and repetitive note: acquire, acquire, acquire. The regular drumbeat of WorldCom acquisitions had the hypnotic quality of the tinkling bells that accompany the tiniest wins at casino slot machines. As the stock began its slow descent, Id check the business news every morning and was tranquilized by news of yet another purchase, web hosting agreement, or extension of services. While corporate venality and fraud played a role in (some of) their falls, the collapses of the dot-corns and WorldCom were not the brainchilds of con artists. Even when entrepreneurs and investors recognized the bubble for what it was, most figured incorrectly that theyd be able to find a chair when the mania-inducing IPO/acquisition music stopped. Alas, the journey from have-lots to have-nots was all too frequently by way of have-dots. Maybe our genes are to blame. (They always seem to get the rap.) Natural selection probably favors organisms that respond to local or near-term events and ignore distant or future ones, which are discounted in somewhat the same way that future money is. Even the ravaging of the environment may be seen as a kind of global Ponzi scheme, the early investors doing well, later ones less well, until a catastrophe wipes out all gains. A quite different illustration of our short-sightedness comes courtesy of Robert Louis Stevensons The Imp in the Bottle. The story tells of a genie in a bottle able and willing to satisfy your every romantic whim and financial desire. Youre offered the opportunity to buy this bottle and its amazing denizen at a price of your choice. There is a serious limitation, however. When youve finished with the bottle, you have to sell it to someone else at a price strictly less than what you paid for it. If you dont sell it to someone for a lower price, you will lose everything and will suffer excruciating and unrelenting torment. What would you pay for such a bottle? 2 cents for it either since no one would buy it from you for 1 cent since everyone knows that it must be sold for a price less than the price at which it is bought. The same reasoning shows that you wouldnt pay 3 cents for it since the person to whom you would have to sell it for 2 cents would object to buying it at that price since he wouldnt be able to sell it for 1 cent. Likewise for prices of 4 cents, S cents, 6 cents, and so on. We can use mathematical induction to formalize this argument, which proves conclusively that you wouldnt buy the genie in the bottle for any amount of money. Yet you would almost certainly buy it for $1,000. I know I would. At what point does the argument against buying the bottle cease to be compelling? (Im ignoring the possibility of foreign currencies that have coins worth less than a penny. This is an American genie.) The question is more than academic since in countless situations people prepare exclusively for near-term outcomes and dont look very far ahead. They myopically discount the future at an absurdly steep rate. Average Riches, Likely Poverty Combining time and money can yield unexpected results in a rather different way. Think back again to the incandescent stock market of the late 1990s and the envious feeling many had that everyone else was making money. You might easily have developed that impression from reading about investing in those halcyon days. In every magazine or newspaper you picked up, you were apt to read about IPOs, the initial public offerings of new companies, and the investment gurus who claimed that they could make your $10,000 grow to more than a million in a years time. (All right, Im exaggerating their exaggerations.) But in those same periodicals, even then, you also would have read stories about new companies that were stillborn and naysayers claims that most investors would lose their $10,000 as well as their shirts by investing in such volatile offerings. Heres a scenario that helps to illuminate and reconcile such seemingly contradictory claims. Hang on for the math that follows. It may be a bit counterintuitive, but its not difficult to follow and it illustrates the crucial difference between the arithmetic mean and the geometric mean of a set of returns. (For the record: The arithmetic mean of N different rates of return is what we normally think of as their average; that is, their sum divided by N. The geometric mean of N different rates of return is equal to that rate of return that, if received N times in succession, would be equivalent to receiving the N different rates of return in succession. We can use the formula for compound interest to derive the technical definition. Doing so, we would find that the geometric mean is equal to the Nth root of the product [(1 + first return) x (1 + second return) x (1 + third return) x ... (1 + Nth return)] - 1.) Hundreds of IPOs used to come out each year. (Pity that this is only an illustrative flashback.) Lets assume that the f i rst week after the stock comes out, its price is usually extremely volatile. Its impossible to predict which way the price will move, but well assume that for half of the companies offerings the price will rise 80 percent during the first week and for half of the offerings the price will fall 60 percent during this period. The investing scheme is simple: Buy an IPO each Monday morning and sell it the following Friday afternoon. About half the time youll earn 80 percent in a week and half the time youll lose 60 percent in a week for an average gain of 10 percent per week: [(80 0) + (-60%)]/2, the arithmetic mean. Ten percent a week is an amazing average gain, and its not difficult to determine that after a year of following this strategy, the average worth of an initial $10,000 investment is more than $1.4 million! (Calculation below.) Imagine the newspaper profiles of happy day traders, or week traders in this case, who sold their old cars and turned the proceeds into almost a million and a half dollars in a year. But what is the most likely outcome if you were to adopt this scheme and the assumptions above held? The answer is that your $10,000 would likely be worth all of $1.95 at the end of a year! Half of all investors adopting such a scheme would have less than $1.95 remaining of their $10,000 nest egg. This same $1.95 is the result of your money growing at a rate equal to the geometric mean of 80 percent and -60 percent over the 52 weeks. (In this case thats equal to the square root (the Nth root for N = 2) of the product [(1 + 80%) x (1 + (-60%))] minus 1, which is the square root of [1.8 x .4] minus 1, which is .85 minus 1, or -.15, a loss of approximately 15 percent each week.) Before walking through this calculation, lets ask for the intuitive reason for the huge disparity between $1.4 million and $1.95. The answer is that the typical investor will see his investment rise by 80 percent for approximately 26 weeks and decline by 60 percent for 26 weeks. As shown below, its not difficult to calculate that this results in $1.95 of your money remaining after one year. The lucky investor, by contrast, will see his investment rise by 80 percent for considerably more than 26 weeks. This will result in astronomical returns that pull the average up. The investments of the unlucky investors will decline by 60 percent for considerably more than 26 weeks, but their losses cannot exceed the original $10,000. In other words, the enormous returns associated with disproportionately many weeks of 80 percent growth skew the average way up, while even many weeks of 60 percent shrinkage cant drive an investments value below $0. In this scenario the stock gurus and the naysayers are both right. The average worth of your $10,000 investment after one year is $1.4 million, but its most likely worth is $1.95. Which results are the media likely to focus on? The following example may help clarify matters. Lets examine what happens to the $10,000 in the first two weeks. There are four equally likely possibilities. The investment can increase both weeks, increase the first week and decrease the second, decrease the first week and increase the second, or decrease both weeks. (As we saw in the section on interest theory, an increase of 80 percent is equivalent to multiplying by 1.8. A 60 percent fall is equivalent to multiplying by 0.4.) One-quarter of investors will see their investment increase by a factor of 1.8 x 1.8, or 3.24. Having increased by 80 percent two weeks in a row, their $10,000 will be worth $10,000 x 1.8 x 1.8, or $32,400 in two weeks. One-quarter of investors will see their investment rise by 80 percent the first week and decline by 60 percent the second week. Their investment changes by a factor of 1.8 x 0.4, or 0.72, and will be worth $7,200 after two weeks. Similarly, $7,200 will be the outcome for one-quarter of investors who will see their investment decline the first week and rise the second week, since 0.4 x 1.8 is the same as 1.8 x 0.4. Finally, the unlucky one-quarter of investors whose investment loses 60 percent of its worth for two weeks in a row will have 0.4 x 0.4 x $10,000, or $1,600 after two weeks. Adding $32,400, $7,200, $7,200, and $1,600 and dividing by 4, we get $12,100 as the average worth of the investments after the first two weeks. Thats an average return of 10 percent weekly, since $10,000 x 1.1 x 1.1 = $12,100. More generally, the stock rises an average of 10 percent every week (the average of an 80 percent gain and a 60 percent loss, remember). Thus after 52 weeks, the average value of the investment is $10,000 x (1.10)52, which is $1,420,000. The most likely result is that the companies stock offerings will rise during 26 weeks and fall during 26 weeks. This means that the most likely worth of the investment is $10,000 x (1.8 )26 x (.4)26, which is only $1.95. And the geometric mean of 80 percent and -60 percent? Once again, it is the square root of the product of [(1 + .8) x (1 - .6)] minus 1, which equals approximately -.15. Every week, on average, your portfolio loses 15 percent of its value, and $10,000 x (1 - .15 )52 equals approximately $1.95. Of course, by varying these percentages and time frames, we can get different results, but the principle holds true: The arithmetic mean of the returns far outstrips the geometric mean of the returns, which is also the median (middle) return as well as the most common return. Another example: If half of the time your investment doubles in a week, and half of the time it loses half its value in a week, the most likely outcome is that youll break even. But the arithmetic mean of your returns is 25 percent per week-[100% + (-50%)]/2, which means that your initial stake will be worth $10,000 x 1.2552, or more than a billion dollars! The geometric mean of your returns is the square root of (1 + 1) x (1 - .5) minus 1, which is a 0 percent rate of return, indicating that youll probably end up with the $10,000 with which you began. Although these are extreme and unrealistic rates of return, these example have much more general importance than it might appear. They explain why a majority of investors receive worse-than-average returns and why some mutual fund companies misleadingly stress their average returns. Once again, the reason is that the average or arithmetic mean of different rates of return is always greater than the geometric mean of these rates of return, which is also the median rate of return. |
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