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You Can't Become Rich In Your Pocket Until You Become Rich In Your Mind | ||||
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But what if we dont choose one or the other to invest in, but split our investment funds and buy half as much of each stock?Whats important is the utility to you of the dollars that you receive, and this utility drops off as you receive more of them. (Note that this is not irrelevant to the rationale for progressive taxation.) For this reason people consider not the dollar amount involved in any investment (or game), but the utility of the dollar amount for the individual involved. The St. Petersburg paradox disappears, for example, if we consider a so-called logarithmic utility function, which attempts to reflect the slowly diminishing satisfaction of having more money and which results in the expected value of the game above being finite. Other versions of the game, in which the payoffs increase even faster, require even slower-growing utility functions so that the expected value remains finite. People do differ in their utility assignments. Some are so acquisitive that the 741,783,219th dollar is almost as dear to them as the first; others are so laid back that their 25,000th dollar is almost worthless to them. There are probably relatively few of the latter, although my father in his later years came close. His attitude suggests that utility functions vary not only across people but also over time. Furthermore, utility may not be so easily described by simple functions since, for example, there may be variations in the utility of money as one approaches a certain age or reaches some financial milestone such as X million dollars. And were back to Virginia Woolfs essay. Portfolios: Benefiting from the Hatfields and McCoys Maynard Keynes wrote, Practical men, who believe themselves to be quite exempt from any intellectual influences, are usually the slaves of some defunct economist. Madmen in authority, who hear voices in the air, are distilling their frenzy from some academic scribbler of a few years back. A corollary of this is that fund managers and stock gurus, who slickly dispense their investment ideas and advice, generally derive them from a previous generations Nobel prize-winning f i nance professor. To get a taste of what a couple more of these Nobelists have written, assume youre a fund manager intent on measuring the expected return and volatility (risk) of a portfolio. In stock market contexts a portfolio is simply a collection of different stocks-a mutual fund, for example, or Uncle Jakes ragbag of mysterious picks, or a nightmare inheritance containing a bunch of different stocks, all in telecommunications. Portfolios like the latter that are so lacking in diversification often become portfolios lacking in dollars. How can you more judiciously choose stocks to maximize a portfolios returns and minimize its risks? Lets first envision a simple portfolio consisting of only three stocks, Abbey Roads, Barkley Hoops, and Consolidated Fragments. Lets further assume that 40 percent (or $40,000) of a $100,000 portfolio is in Abbey, 25 percent in Barkley, and the remaining 35 percent in Consolidated. Assume further that the expected rate of return from Abbey is 8 percent, from Barkley is 13 percent, and from Consolidated is 7 percent. Using these weights, we compute that the expected return from the portfolio as a whole is (.40 x .08) + (.25 x .13) + (.35 x.07), which is .089 or 8.9 percent. Why not put all our money in Barkley Hoops since its expected rate of return is the highest of the three stocks? The answer has to do with volatility and the risk of not diversifying, of putting all ones proverbial eggs in one basket. (The result, as was the case with my WorldCom misadventure, may well be egg on ones face and the transformation of ones nest egg into a scrambled egg if not a goose egg. Sorry, but thought of the stock even now sometimes momentarily unhinges me.) If you were indifferent to risk, however, and simply wanted to maximize your returns, you might well put all your money in Barkley Hoops. So how does one determine the volatility-that is, sigma, the standard deviation-of a portfolio? Does one just weight the volatilities of the companies stocks as we weighted their returns to get the volatility of the portfolio? In general, we cant do this because the stocks performances are sometimes not independent of each other. When one goes up in response to some news, the others chances of going up or down may be affected and this in turn affects their joint volatility. Let me illustrate with an even simpler portfolio consisting of only two stocks, Hatfield Enterprises and McCoy Productions. They both produce thingamajigs, but history tells us that when one does well, the other suffers and vice versa, and that overall dominance seems to shift regularly back and forth between them. Perhaps Hatfield produces snow shovels and McCoy makes tanning lotion. To be specific, lets say that half the time Hatfields rate of return is 40 percent and half the time it is -20 percent, so its expected rate of return is (.50 x .40) + (.50 x (-.20)), which is .10 or 10 percent. McCoys returns are the same, but again it does well when Hatfield does poorly and vice versa. The volatility of each company is the same too. Recalling the definition, we first find the squares of the deviations from the mean of 10 percent, or . These squares are (.40 - .10)2 and (-.20 - .10)2 or .09 and .09. Since they each occur half the time, the variance is (.50 x .09) + (.50 x .09), which is .09. The square root of this is .3 or 30 percent, which is the standard deviation or volatility of each companys returns. But what if we dont choose one or the other to invest in, but split our investment funds and buy half as much of each stock? Then were always earning 40 percent from half our investment and losing 20 percent on the other half, and our expected return is still 10 percent. But notice that this 10 percent return is constant. The volatility of the portfolio is zero! The reason is that the returns of these two stocks are not independent, but are perfectly negatively correlated. We get the same average return as if we bought either the Hatfield or the McCoy stock, but with no risk. This is a good thing; we get richer and dont have to worry about whos winning the battle between the Hatfields and the McCoys. Of course, its difficult to find stocks that are perfectly negatively correlated, but that is not required. As long as they arent perfectly positively correlated, the stocks in a portfolio will decrease volatility somewhat. Even a portfolio of stocks from the same sector will be less volatile than the individual stocks in it, while a portfolio consisting of Wal-Mart, Pfizer, General Electric, Exxon, and Citigroup, the biggest stocks in their respective sectors, will provide considerably more protection against volatility. To find the volatility of a portfolio in general, we need what is called the covariance (closely related to the correlation coefficient) between any pair of stocks X and Y in the portfolio. The covariance between two stocks is roughly the degree to which they vary together-the degree, that is, to which a change in one is proportional to a change in the other. Note that unlike many other contexts in which the distinction between covariance (or, more familiarly, correlation) and causation is underlined, the market generally doesnt care much about it. If an increase in the price of ice cream stocks is correlated to an increase in the price of lawn mower stocks, few ask whether the association is causal or not. The aim is to use the association, not understand it-to be right about the market, not necessarily to be right for the right reasons. Given the above distinction, some of you may wish to skip the next three paragraphs on the calculation of covariance. Go directly to For example, if we let H be the cost ... . Technically, the covariance is the expected value of the product of the deviation from the mean of one of the stocks and the deviation from the mean of the other stock. That is, the covariance is the expected value of the product [(X - x) x (Y - py)], where Nx and p y are the means of X and Y, respectively. Thus, if the stocks vary together, when the price of one is up, the price of the other is likely to be up too, so both deviations from the mean will be positive, and their product will be positive. And when the price of one is down, the price of the other is likely to be down too, so both deviations will be negative, and their product will again be positive. If the stocks vary inversely, however, when the price of one is up (or down), the price of the other is likely to be down (or up), so when the deviation of one stock is positive, that of the other is negative, and the product will be negative. In general and in short, we want negative covariance. We may now use this notion of covariance to find the variance of a two-equity portfolio, p percent of which is in stock X and q percent in stock Y. The mathematics involves nothing more than squaring the sum of two terms. (Remember, however, that (A + B)2 = A2 + B2 + 2AB.) By definition, the variance of the portfolio, (pX + qY), is the expected value of the squares of its deviations from its mean, ppx + qpy. That is, the variance of (pX + qY) is the expected value of [(pX + qY) (ppx + gpy)]2, which, upon rewriting, is the expected value of [(pX - ppx) + (qY - gpy)]2, which, using the algebra rule cited above, is the expected value of [(pX - ppX)2 + (qY - qpy)2 +2X the expected value of [(pX - ppx) x (qY - qpy)]. Minding (that is, factoring out) our ps and qs, we find that the variance of the portfolio, (pX + qY), equals [(p2 x the variance of X) + (q2 x the variance of Y) + (2pq x the covariance of X and Y)]. If the stocks vary negatively (that is, have negative covariance), the variance of the portfolio is reduced by the last factor. (In the case of the Hatfield and McCoy stocks, the variance was reduced to zero.) And when they vary positively (that is, have positive covariance), the variance of the portfolio is increased by the last factor, a situation we want to avoid, volatility and risk being bad for our peace of mind and stomach. For example, if we let H be the cost of a randomly selected homeowners house in a given community and I be his or her household income, then the variance of (H + I) is greater than the variance of H plus the variance of I. People who live in expensive houses generally have higher incomes than people who dont, so the extremes of the sum, house cost plus personal income, are going to be considerably greater than they would be if house cost and personal income did not have a positive covariance. Likewise, if C is the number of classes skipped during the year by a randomly selected student in a large lecture and S is his score on the final exam, then the variance of (C + S) is smaller than the variance of C plus the variance of S. Students who miss a lot of classes generally (although certainly not always) achieve a lower score, so the extremes of the sum, number of classes missed plus exam scores, are going to be considerably less that they would be if number of classes missed and exam scores did not have a negative covariance. When choosing stocks for a diversified portfolio, investors, as noted, generally look for negative covariances. They want to own equities like the Hatfield and the McCoy stocks and not like WCOM, say, and some other telecommunications stock. With three or more stocks in a portfolio, one uses the stocks weights in the portfolio as well as the definitions just discussed to compute the portfolios variance and standard deviation. (The algebra is tedious, but easy.) Unfortunately, the covariances between all possible pairs of stocks in the portfolio are needed for the computation, but good software, troves of stock data, and fast computers allow investors to determine a portfolios risk (volatility, standard deviation) fairly quickly. With care, you can minimize the risk of a portfolio without hurting its expected rate of return. Diversification and Politically Incorrect Funds There are countless mutual funds, and many commentators have noted that there are more funds than there are stocks, as if this were a surprising fact. It isnt. In mathematical terms a fund is simply a set of stocks, so, theoretically at least, there are vastly more possible funds than there are stocks. Any set of n stocks (people, books, CDs) has 2N subsets. Thus, if there were only 20 stocks in the world, there would be 220 or approximately 1 million possible subsets of these stocks-1 million possible mutual funds. Of course, most of these subsets would not have a compelling reason for existence. Something more is needed, and that is the financial balancing act that ensures diversification and low volatility. We can increase the number of possibilities even further by extending the notion of diversification. Instead of searching for individual stocks or whole sectors that are negatively correlated, we can search for concerns of ours that are negatively correlated. Say, for example, financial and social ones. A number of portfolios purport to be socially progressive and politically correct, but in general their performance is not stellar. Less appealing to many are funds that are socially regressive and politically incorrect but that do perform well. In this latter category many people would place tobacco, alcohol, defense contractors, fast food, or any of several others. The existence of these politically incorrect funds suggests, for those passionately committed to various causes, a nonstandard strategy that exploits the negative correlation that sometimes exists between financial and social interests. Invest heavily in funds holding shares in companies that you find distasteful. If these funds do well, you make money, money that you could, if you wished, contribute to the political causes you favor. If these funds cool off, you can rejoice that the companies are no longer thriving, and your psychic returns will soar. Such diversification has many applications. People often work for organizations, for example, whose goals or products they find unappealing and use part of their salary to counter the organizations goals or products. Taken to its extreme, diversification is something we do naturally in dealing with the inevitable trade-offs in our daily lives. Of course, extending the notion of diversification to these other realms is difficult for several reasons. One is that quantifying contributions and payoffs is problematic. How do you place a numerical value on your efforts and their various consequences? The number of possible funds, subsets of all your possible concerns, also grows exponentially. Another problem derives from the logic of the notion of diversification. It often makes sense in life, where some combination of work, play, family, personal experiences, study, friends, money, and so forth, seems more likely to lead to satisfaction than, say, all toil or pure hedonism. Nevertheless, diversification may not be appropriate when you are trying to have a personal impact. Take charity, for example. As the economist Steven Landsburg has argued, you diversify when investing to protect yourself, but when contributing to large charities in which your contributions are a small fraction of the total, your goal is presumably to help as much as possible. Since you incur no personal risk, if you truly think that Mothers Against Drunk Driving is more worthy than the American Cancer Society or the American Heart Association, why would you split your charitable dollars among them? The point isnt to insure that your money will do some good, but to maximize the good it will do. There are other situations too where bulleting ones efforts is preferable to a bland diversification. Metaphorical extensions of the notion of diversification can be useful, but uncritical use of them can lead you to, in the words of W. H. Auden, commit a social science. |
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