Diversifying Stock Portfolios
Long before my children’s fascination with Super Mario Brothers, Tetris, and more recent addictive games, I spent interminable hours as a kid playing antediluvian, low-tech Monopoly with my two brothers. The game requires the players to roll dice and move around the board buying, selling, and trading real estate properties. Although I paid attention to the probabilities and expected values associated with various moves (but not to what have come to be called the game’s Markov chain properties), my strategy was simple: Play aggressively, buy every property whether it made sense or not, and then bargain to get a monopoly. I always traded away railroads and utilities if I could, much preferring to build hotels on the real estate I owned instead.
A Reminiscence and a Parable
Although the game’s get-out-of-jail-free card was one of the few ties to the present-day stock market, I’ve recently had a tiny epiphany. On some atavistic level I’ve likened hotel building to stock buying and the railroads and utilities to bonds. Railroads and utilities seemed safe in the short run, but the ostensibly risky course of putting most of one’s money into
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building hotels was ultimately more likely to make one a winner (especially since we occasionally altered the rules to allow unlimited hotel building on a property).
Was my excessive investment in WorldCom a result of a bad generalization from playing Monopoly? I strongly doubt it, but such just-so stories come naturally to mind. Aside from the jail card, a board game called WorldCom would have few features in common with Monopoly (but might more closely resemble Grand Theft Auto). Different squares along players’ paths would call for SEC investigations, Eliot Spitzer prosecutions, IPO giveaways, or favorable analyst ratings. If you attained CEO status, you would be allowed to borrow up to $400 million ($1 billion in later versions of the game), whereas if you were reduced to the rank of employee, you would have to pay a coffee fee after each move and invest a certain portion of your savings in company stock. If you were unfortunate enough to become a stockholder, you would be required to remove your shirt while playing, while if you became CFO, you would receive stock options and get to keep the stockholders’ shirts. The object of the game would be to make as much money and collect as many of your fellow players’ shirts as possible before the company went bankrupt.
The game might be fun with play money; it wasn’t with the real thing.
Here’s a better analogue for the market. People are milling around a huge labyrinthine bazaar. Occasionally some of the booths in the bazaar attract a swarm of people jostling to buy their wares. Likewise, some booths are occasionally devoid of any prospective customers. At any given time most booths have a few customers. At the intersections of the bazaar’s alleys are sales people from some of the bigger booths as well as well-traveled seers. They know the various sections of the bazaar intimately and claim to be able to foretell the fortunes of various booths and collections of booths. Some of these
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sales people and some of the prognosticators have very large bullhorns and can be heard throughout the bazaar, while others make do by shouting.
In this rather primitive setting, many aspects of the stock market can already be discerned. The forebears of technical traders might be those who buy from booths where crowds are developing, while the forebears of fundamental traders might be those who coolly weigh the worth of the goods on display. The seers are the progenitors of analysts, the sales people progenitors of brokers. The bullhorns are a rudimentary form of business media, and, of course, the goods on sale are companies’ stocks. Crooks and swindlers have their ancestors as well with some of the booths hiding their shoddy merchandise under the better goods.
If everyone, not just the booth owners, could sell as well as buy, this would be a better elemental model of an equities market. (I don’t intend this as an historical account, but merely as an idealized narrative.) Nevertheless, I think it’s clear that stock exchanges are natural economic phenomena. It’s not hard to imagine early analogues of options trading, corporate bonds, or diversified holdings developing out of such a bazaar.
Maybe there’d even be some arithmeticians around too, analyzing booths’ sales and devising purchasing strategies. In acting on their theories, some might even lose their togas and protractors.
Are Stocks Less Risky Than Bonds?
Perhaps because of Monopoly, certainly because of WorldCom, and for many other reasons, the focus of this book has been the stock market, not the bond market (or real estate, commodities, and other worthy investments). Stocks are, of course,
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shares of ownership in a company, whereas bonds are loans to a company or government, and “everybody knows” that bonds are generally safer and less volatile than stocks, although the latter have a higher rate of return. In fact, as Jeremy Siegel reports in Stocks for the Long Run, the average annual rate of return for stocks between 1802 and 1997 was 8.4 percent; the rate on treasury bills over the same period was between 4 percent and 5 percent. (The rates that follow are before inflation. What’s needless to say, I hope, is that an 8 percent rate of return in a year of 15 percent inflation is much worse than a 4 percent return in a year of 3 percent inflation.)
Despite what “everybody knows,” Siegel argues in his book that, as with Monopoly’s hotels and railroads, stocks are actually less risky than bonds because, over the long run, they have performed so much better than bonds or treasury bills. In fact, the longer the run, the more likely this has been the case. (Comments like “everybody knows” or “they’re all doing this” or “everyone’s buying that” usually make me itch. My background in mathematical logic has made it difficult for me to interpret “all” as signifying something other than all.) “Everybody” does have a point, however. How can we believe Siegel’s claims, given that the standard deviation for stocks’ annual rate of return has been 17.5 percent?
If we assume a normal distribution and allow ourselves to get numerical for a couple of paragraphs, we can see how stomach-churning this volatility is. It means that about two- thirds of the time, the rate of return will be between -9.1 percent and 25.9 percent (that is, 8.4 percent plus or minus 17.5 percent), and about 95 percent of the time the rate will be between -26.6 percent and 43.4 percent (that is, 8.4 percent plus or minus two times 17.5 percent). Although the precision of these figures is absurd, one consequence of the last assertion is that the returns will be worse than -26.6 percent about 2.5 percent of the time (and better than 43.4 percent with the
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same frequency). So about once every forty years (1/40 is 2.5 percent), you will lose more than a quarter of the value of your stock investments and much more frequently than that do considerably worse than treasury bills.
These numbers certainly don’t seem to indicate that stocks are less risky than bonds over the long term. The statistical warrant for Siegel’s contention, however, is that over time, the returns even out and the deviations shrink. Specifically, the annualized standard deviation for rates of return over a number N of years is the standard deviation divided by the square root of N. The larger N is, the smaller is the standard deviation. (The cumulative standard deviation is, however, greater.) Thus over any given four-year period the annualized standard deviation for stock returns is 17.5%/2, or 8.75%. Likewise, since the square root of 30 is about 5.5, the annualized standard deviation of stock returns over any given thirty-year period is only 17.5%/5.5, or 3.2%. (Note that this annualized thirty- year standard deviation is the same as the annual standard deviation for the conservative stock mentioned in the example at the end of chapter 6.)
Despite the impressive historical evidence, there is no guarantee that stocks will continue to outperform bonds. If you look at the period from 1982 to 1997, the average annual rate of return for stocks was 16.7 percent with a standard deviation of 13.1 percent, while the returns for bonds were between 8 percent and 9 percent. But from 1966 to 1981, the average annual rate of return for stocks was 6.6 percent with a standard deviation of 19.5 percent, while the returns for bonds were about 7 percent.
So is it really the case that, despite the debacles, deadbeats, and doomsday equities like WCOM and Enron, the less risky long-term investment is in stocks? Not surprisingly, there is a counterargument. Despite their volatility, stocks as a whole have proven less risky than bonds over the long run because
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their average rates of return have been considerably higher. Their rates of return have been higher because their prices have been relatively low. And their prices have been relatively low because they’ve been viewed as risky and people need some inducement to make risky investments.
But what happens if investors believe Siegel and others, and no longer view stocks as risky? Then their prices will rise because risk-averse investors will need less inducement to buy them; the “equity-risk premium,” the amount by which stock returns must exceed bond returns to attract investors, will decline. And the rates of return will fall because prices will be higher. And stocks will therefore be riskier because of their lower returns.
Viewed as less risky, stocks become risky; viewed as risky, they become less risky. This is yet another instance of the skittish, self-reflective, self-corrective dynamic of the market. Interestingly, Robert Shiller, a personal friend of Siegel, looks at the data and sees considerably lower stock returns for the next ten years.
Market practitioners as well as academics disagree. In early October 2002, I attended a debate between Larry Kudlow, a CNBC commentator and Wall Street fixture, and Bob Prech- ter, a technical analyst and Elliot wave proponent. The audience at the CUNY graduate center in New York seemed affluent and well-educated, and the speakers both seemed very sure of themselves and their predictions. Neither seemed at all affected by the other’s diametrically opposed expectations. Prechter anticipated very steep declines in the market, while Kudlow was quite bullish. Unlike Siegel and Shiller, they didn’t engage on any particulars and generally talked past each other.
What I find odd about such encounters is how typical they are of market discussions. People with impressive credentials regularly expatiate upon stocks and bonds and come to conclusions contrary to those of other people with equally imA
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pressive credentials. An article in the New York Times in November 2002 is another case in point. It described three plausible prognoses for the market—bad, so-so, and good—put forth by economic analysts Steven H. East, Charles Pradilla, and Abby Joseph Cohen, respectively. Such stark disagreement happens very rarely in physics or mathematics. (I’m not counting crackpots who sometimes receive a lot of publicity but aren’t taken seriously by anybody knowledgeable.)
The market’s future course may lie beyond what, in chapter 9, I term the “complexity horizon.” Nevertheless, aside from some real estate, I remain fully vested in stocks, which may or may not result in my remaining fully shirted.
The St. Petersburg Paradox and Utility
Reality, like the perfectly ordinary woman in Virginia Woolf’s famous essay “Mr. Bennett and Mrs. Brown,” is endlessly complex and impossible to capture completely in any model. Expected value and standard deviation seem to reflect the ordinary meanings of average and variability most of the time, but it’s not hard to find important situations where they don’t.
One such case is illustrated by the so-called St. Petersburg paradox. It takes the form of a game that requires that you flip a coin repeatedly until a tail first appears. If a tail appears on the first flip, you win $2. If the first tail appears on the second flip, you win $4. If the first tail appears on the third flip, you win $8, and, in general, if the first tail appears on the Nth flip, you win 2N dollars. How much would you be willing to pay to play this game? One could argue that you should be willing to pay any amount to play this game.
To see why this is so, recall that the probability of a sequence of independent events such as coin flips is obtained by multiplying the probabilities of each of the events. Thus the
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probability of getting the first tail, T, on the first flip is 1/2; of getting a head and then the first tail on the second flip, HT, is (1/2)2 or 1/4; of getting the first tail on the third flip, HHT, is (1/2)3 or 1/8; and so on. Putting these probabilities and the possible winnings associated with them into the formula for expected value, we see that the expected value of the game is ($2 x 1/2) + ($4 x 1/4) + ($8 x 1/8) + ($16 x 1/16) + . . . (2N x (1/2))N + ... . All of these products are 1, there are infinitely many of them, and so their sum is infinite. The failure of expected value to capture our intuitions becomes clear when you ask yourself why you’d be reluctant to pay even a measly $1,000 for the privilege of playing this game.
The most common resolution is roughly that provided by the eighteenth century mathematician Daniel Bernoulli, who wrote that people’s enjoyment of any increase in wealth (or regret at any decrease) is “inversely proportionate to the quantity of goods previously possessed.” The fewer dollars you have, the more you appreciate gaining one and the more you fear losing one, and so, for almost everyone, the likely prospect of losing $1,000 more than cancels the remote possibility that you’ll win, say, a billion dollars.
What’s 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.
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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 we’re back to Virginia Woolf’s essay.
Portfolios: Benefiting from the Hatfields and McCoys
John 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 generation’s Nobel prize-winning finance professor.
To get a taste of what a couple more of these Nobelists have written, assume you’re 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 Jake’s 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
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often become portfolios lacking in dollars. How can you more judiciously choose stocks to maximize a portfolio’s returns and minimize its risks?
Let’s first envision a simple portfolio consisting of only three stocks, Abbey Roads, Barkley Hoops, and Consolidated Fragments. Let’s 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 one’s proverbial eggs in one basket. (The result, as was the case with my WorldCom misadventure, may well be egg on one’s face and the transformation of one’s 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 can’t 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 Produc-
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tions. 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, let’s say that half the time Hatfield’s 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. McCoy’s 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 .10. 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 company’s returns.
But what if we don’t choose one or the other to invest in, but split our investment funds and buy half as much of each stock? Then we’re 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 don’t have to worry about who’s winning the battle between the Hatfields and the McCoys.
Of course, it’s difficult to find stocks that are perfectly negatively correlated, but that is not required. As long as they aren’t 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,
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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 doesn’t 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 - px) x (Y - pY)], where px and pY 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
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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 + qpY)]2, which, upon rewriting, is the expected value of [(pX - ppx) + (qY - qpY)]2, which, using the algebra rule cited above, is the expected value of [(pX - ppx)2 + (qY - qpY)2 + 2 x the expected value of [(pX - ppx) x (qY - qpY)].
Minding (that is, factoring out) our p’s and q’s, 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 homeowner’s 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 don’t, 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.
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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 portfolio’s 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 portfolio’s 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 isn’t. 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
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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 organization’s goals or products. Taken to its extreme, diversification is something we do naturally in dealing with the inevitable trade-offs in our daily lives.
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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 isn’t to insure that your money will do some good, but to maximize the good it will do. There are other situations too where bulleting one’s 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.”
Beta—Is It Better?
Returning to more quantitative matters, we choose stocks so that when some are down, others are up (or at least not as
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down), giving us a healthy rate of return with as little risk as possible. More precisely, given any portfolio of stocks, we grind the numbers describing their past performances and come up with estimates for their expected returns, volatilities, and covariances, and then use these to determine the expected returns and volatilities of the portfolio as a whole. We could, if we had the time, the price data, and fast computers, do this for a variety of different portfolios. The Nobel prize-winning economist Harry Markowitz, one of the originators of this approach, developed mathematical techniques for carrying out these calculations in the early 1950s, graphed his results for a few portfolios (computers weren’t fast enough to do much more then), and defined what he called the “efficient frontier” of portfolios.
If we were to use these techniques and construct comparable graphs for a wide variety of contemporary portfolios, what would we find? Arraying the (degree of) volatility of these portfolios along the graph’s horizontal axis and their expected rates of return along its vertical axis, we would see a swarm of points. Each point would represent a portfolio whose coordinates would be its volatility and expected return, respectively. We’d also notice that among all the portfolios having a given level of risk (that is, volatility, standard deviation), there would be one with the highest expected rate of return. If we single out the portfolio with the highest expected rate of return for each level of risk, we would obtain a curve, Markowitz’s efficient frontier of optimal portfolios.
The more risky a portfolio on the efficient frontier curve is, the higher is its expected return. In part, this is because most investors are risk-averse, making risky stocks cheaper. The idea is that investors decide upon a risk level with which they’re comfortable and then choose the portfolio with this risk level that has the highest possible return. Call this Variation One of the theory of portfolio selection.
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Don’t let this mathematical formulation blind you to the generality of the psychological phenomenon. Automobile engineers have noted, for example, that safety advances in automobile design (say anti-lock brakes) often result in people driving faster and turning more sharply. Their driving performance is enhanced rather than their safety. Apparently, people choose a risk level with which they’re comfortable and then seek the highest possible return (performance) for it.
Inspired by this trade-off between risk and return, William Sharpe proposed in the 1960s what is now a common measure of the performance of a portfolio. It is defined as the ratio of the excess return of a portfolio (the difference between its expected return and the return on a risk-free treasury bill) to the portfolio’s volatility (standard deviation). A portfolio might have a hefty rate of return, but if the volatility the investor must endure to achieve this return is roller coasterish, the portfolio’s Sharpe measure won’t be very high. By contrast, a portfolio with a moderate rate of return but a less anxiety-inducing volatility will have a higher Sharpe measure.
There are many complications to portfolio selection theory. As the Sharpe measure suggests, an important one is the existence of risk-free investments, such as U.S. treasury bills. These pay a fixed rate of return and have essentially zero volatility. Investors can always invest in such risk-free assets and can borrow at the risk-free rates as well. Moreover, they can combine risk-free investment in treasury bills with a risky stock portfolio.
Variation Two of portfolio theory claims that there is one and only one optimal stock portfolio on the efficient frontier with the property that some combination of it and a risk-free investment (ignoring inflation) constitute a set of investments having the highest rates of return for any given level of risk. If you wish to incur no risk, you put all your money into treasury bills. If you’re comfortable with risk, you put all your
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money into this optimal stock portfolio. Alternatively, if you want to divide your money between the two, you put p% into the risk-free treasury bills and (100 - p)% into the optimal risky stock portfolio for an expected rate of return of [p x (risk-free return) + (1 - p) x (stock portfolio)]. An investor can also invest more money than he has by borrowing at the risk-free rate and putting this borrowed money into the risky portfolio.
In this refinement of portfolio selection, all investors choose the same optimal stock portfolio and then adjust how much risk they’re willing to take by increasing or decreasing the percentage, p, of their holdings that they put into risk-free treasury bills.
This is easier said than done. In both variations the required mathematical procedures put enormous pressure on one’s computing facilities, since countless calculations must be performed regularly on new data. The expected returns, variances, and covariances are, after all, derived from their values in the recent past. If there are twenty stocks in a portfolio, we would need to compute the covariance of every possible pair of stocks, and there are (20 x 19)/2, or 190, such covariances. If there were fifty stocks, we’d need to compute (50 x 49)12, or 1,225 covariances. Doing this for each of a wide class of portfolios is not possible without massive computational power.
As a way to avoid much of the computational burden of updating and computing all these covariances, efficient frontiers, and optimal risky portfolios, Sharpe, yet another Nobel Prize winner in economics, developed (with others) what’s called the “single index model.” This Variation Three relates a portfolio’s rate of return not to that of all possible pairs of stocks in the portfolio, but simply to the change in some index representing the market as a whole. If your portfolio or stock is statistically determined to be relatively more volatile
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than the market as a whole, then changes in the market will bring about exaggerated changes in the stock or portfolio. If it is relatively less volatile than the market as a whole, then changes in the market will bring about attenuated changes in the stock or portfolio.
This brings us to the so-called Capital Asset Pricing Model, which maintains that the expected excess return on one’s stock or portfolio (the difference between the expected return on the portfolio, Rp, and the return on risk-free treasury bills, Rf) is equal to the notorious beta, symbolized by 3, multiplied by the expected excess return of the general market (the difference between the market’s expected return, Rm, and the return on risk-free treasury bills, Rf). In algebraic terms: (Rp-Rf) = P(Rm - Rf). Thus, if you can get a sure 4 percent on treasury bills and if the expected return on a broad market index fund is 10 percent and if the relative volatility, beta, of your portfolio is 1.5, then the portfolio’s expected return is obtained by solving (Rp - 4%) = 1.5(10% - 4%), which yields 13 percent for Rp. A beta of 1.5 means that your stock or portfolio gains (or loses) an average of 1.5 percent for every 1 percent gain (or loss) in the market as a whole.
Betas for the stocks of high-tech companies like WorldCom are often considerably more than 1, meaning that changes in the market, both up and down, are magnified. These stocks are more volatile and thus riskier. Betas for utility company stocks, by contrast, are often less than 1, which means that changes in the market are muted. If a company has a beta of .5, then its expected return is obtained by solving (Rp - 4%) = .5(10% - 4%), which yields 7% for Rp, the expected return on the portfolio. Note that for short-term treasury bills, whose returns don’t vary at all, beta is 0. To reiterate: Beta quantifies the degree to which a stock or a portfolio fluctuates in relation to market fluctuations. It is not the same as volatility.
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This all sounds neat and clean, but you beta watch your step with all of these portfolio selection models. Specifically with regard to Variation Three, we might wonder where the number beta comes from. Who says your stock or portfolio will be 40 percent more volatile or 25 percent less volatile than the market as a whole? Here’s the rough technique for finding beta. You check the change in the broad market for the last three months—say it’s 3 percent—and check the change in the price of your stock or portfolio for the same period—say it’s 4.1 percent. You do the same thing for the three months before that—say the numbers this time are 2 percent and 2.5 percent, respectively—and for the three months before that—say -1.2 percent and -3 percent, respectively. You continue doing this for a number of such periods and then on a graph you plot the points (3%, 4.1%), (2%, 2.5%), (-1.2%, -3%), and so on. Most of the time if you squint hard enough, you’ll see a sort of linear relationship between changes in the market and changes in your stock or portfolio, and you then use standard mathematical methods for determining the line of closest fit through these points. The slope or steepness of this line is beta.
One problem with beta is that companies change over time, sometimes rather quickly. AT&T, for example, or IBM is not the same company it was twenty years ago or even two years ago. Why should we expect a company’s relative volatility, beta, to remain the same? In the opposite direction is a related difficulty. Beta is often of very limited value in the short term and varies with the index chosen for comparison and the time period used in its definition. Still another problem is that beta depends on market returns, and market returns depend on a narrow definition of the market, namely just the stock market rather than stocks, bonds, real estate, and so forth. For all its limitations, however, beta can be a useful notion if it’s not turned into a fetish.
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You might compare beta to different people’s emotional reactivity and expressiveness. Some respond to the slightest good news with outbursts of joy and to the tiniest hardship with wails of despair. At the other end of the emotional spectrum are those who say “ouch” when they accidentally touch a scalding iron and allow themselves an “oh, good” when they win the lottery. The former have high emotional beta, the latter low emotional beta. A zero beta person would have to be unconscious, perhaps from ingesting too many beta- blockers. Unfortunate for the prospect of predicting the behavior of people, however, is the commonplace that people’s emotional betas vary depending on the type of stimulus a person faces. I’ll leave out the examples, but this may be beta’s biggest limitation as a measure of the relative volatility of a portfolio or stock. Betas may vary with the type of stimulus a company faces.
Whatever refinements of portfolio theory are developed, one salient point remains: Portfolios, although often less risky than individual stocks, are still risky (as millions of 401 (k) returns attest). Some mathematical manipulation of the notions of variance and covariance and a few reasonable assumptions are sufficient to show that this risk can be partitioned into two parts. There is a systematic part that is related to general movements in the market, and there is a non-systematic part that is idiosyncratic to the stocks in the portfolio. The latter, nonsystematic risk, specific to the individual stocks in the portfolio, can be eliminated or “diversified away” by an appropriate choice of thirty or so stocks. An irreducible core, however, remains inherent in the market and cannot be avoided. This systematic risk depends on the beta of one’s portfolio.
Or so the story goes. To the criticisms of beta above should be added the problems associated with forcing a non-linear world into a linear mold.
A Reminiscence and a Parable
Although the game’s get-out-of-jail-free card was one of the few ties to the present-day stock market, I’ve recently had a tiny epiphany. On some atavistic level I’ve likened hotel building to stock buying and the railroads and utilities to bonds. Railroads and utilities seemed safe in the short run, but the ostensibly risky course of putting most of one’s money into
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building hotels was ultimately more likely to make one a winner (especially since we occasionally altered the rules to allow unlimited hotel building on a property).
Was my excessive investment in WorldCom a result of a bad generalization from playing Monopoly? I strongly doubt it, but such just-so stories come naturally to mind. Aside from the jail card, a board game called WorldCom would have few features in common with Monopoly (but might more closely resemble Grand Theft Auto). Different squares along players’ paths would call for SEC investigations, Eliot Spitzer prosecutions, IPO giveaways, or favorable analyst ratings. If you attained CEO status, you would be allowed to borrow up to $400 million ($1 billion in later versions of the game), whereas if you were reduced to the rank of employee, you would have to pay a coffee fee after each move and invest a certain portion of your savings in company stock. If you were unfortunate enough to become a stockholder, you would be required to remove your shirt while playing, while if you became CFO, you would receive stock options and get to keep the stockholders’ shirts. The object of the game would be to make as much money and collect as many of your fellow players’ shirts as possible before the company went bankrupt.
The game might be fun with play money; it wasn’t with the real thing.
Here’s a better analogue for the market. People are milling around a huge labyrinthine bazaar. Occasionally some of the booths in the bazaar attract a swarm of people jostling to buy their wares. Likewise, some booths are occasionally devoid of any prospective customers. At any given time most booths have a few customers. At the intersections of the bazaar’s alleys are sales people from some of the bigger booths as well as well-traveled seers. They know the various sections of the bazaar intimately and claim to be able to foretell the fortunes of various booths and collections of booths. Some of these
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sales people and some of the prognosticators have very large bullhorns and can be heard throughout the bazaar, while others make do by shouting.
In this rather primitive setting, many aspects of the stock market can already be discerned. The forebears of technical traders might be those who buy from booths where crowds are developing, while the forebears of fundamental traders might be those who coolly weigh the worth of the goods on display. The seers are the progenitors of analysts, the sales people progenitors of brokers. The bullhorns are a rudimentary form of business media, and, of course, the goods on sale are companies’ stocks. Crooks and swindlers have their ancestors as well with some of the booths hiding their shoddy merchandise under the better goods.
If everyone, not just the booth owners, could sell as well as buy, this would be a better elemental model of an equities market. (I don’t intend this as an historical account, but merely as an idealized narrative.) Nevertheless, I think it’s clear that stock exchanges are natural economic phenomena. It’s not hard to imagine early analogues of options trading, corporate bonds, or diversified holdings developing out of such a bazaar.
Maybe there’d even be some arithmeticians around too, analyzing booths’ sales and devising purchasing strategies. In acting on their theories, some might even lose their togas and protractors.
Are Stocks Less Risky Than Bonds?
Perhaps because of Monopoly, certainly because of WorldCom, and for many other reasons, the focus of this book has been the stock market, not the bond market (or real estate, commodities, and other worthy investments). Stocks are, of course,
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shares of ownership in a company, whereas bonds are loans to a company or government, and “everybody knows” that bonds are generally safer and less volatile than stocks, although the latter have a higher rate of return. In fact, as Jeremy Siegel reports in Stocks for the Long Run, the average annual rate of return for stocks between 1802 and 1997 was 8.4 percent; the rate on treasury bills over the same period was between 4 percent and 5 percent. (The rates that follow are before inflation. What’s needless to say, I hope, is that an 8 percent rate of return in a year of 15 percent inflation is much worse than a 4 percent return in a year of 3 percent inflation.)
Despite what “everybody knows,” Siegel argues in his book that, as with Monopoly’s hotels and railroads, stocks are actually less risky than bonds because, over the long run, they have performed so much better than bonds or treasury bills. In fact, the longer the run, the more likely this has been the case. (Comments like “everybody knows” or “they’re all doing this” or “everyone’s buying that” usually make me itch. My background in mathematical logic has made it difficult for me to interpret “all” as signifying something other than all.) “Everybody” does have a point, however. How can we believe Siegel’s claims, given that the standard deviation for stocks’ annual rate of return has been 17.5 percent?
If we assume a normal distribution and allow ourselves to get numerical for a couple of paragraphs, we can see how stomach-churning this volatility is. It means that about two- thirds of the time, the rate of return will be between -9.1 percent and 25.9 percent (that is, 8.4 percent plus or minus 17.5 percent), and about 95 percent of the time the rate will be between -26.6 percent and 43.4 percent (that is, 8.4 percent plus or minus two times 17.5 percent). Although the precision of these figures is absurd, one consequence of the last assertion is that the returns will be worse than -26.6 percent about 2.5 percent of the time (and better than 43.4 percent with the
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same frequency). So about once every forty years (1/40 is 2.5 percent), you will lose more than a quarter of the value of your stock investments and much more frequently than that do considerably worse than treasury bills.
These numbers certainly don’t seem to indicate that stocks are less risky than bonds over the long term. The statistical warrant for Siegel’s contention, however, is that over time, the returns even out and the deviations shrink. Specifically, the annualized standard deviation for rates of return over a number N of years is the standard deviation divided by the square root of N. The larger N is, the smaller is the standard deviation. (The cumulative standard deviation is, however, greater.) Thus over any given four-year period the annualized standard deviation for stock returns is 17.5%/2, or 8.75%. Likewise, since the square root of 30 is about 5.5, the annualized standard deviation of stock returns over any given thirty-year period is only 17.5%/5.5, or 3.2%. (Note that this annualized thirty- year standard deviation is the same as the annual standard deviation for the conservative stock mentioned in the example at the end of chapter 6.)
Despite the impressive historical evidence, there is no guarantee that stocks will continue to outperform bonds. If you look at the period from 1982 to 1997, the average annual rate of return for stocks was 16.7 percent with a standard deviation of 13.1 percent, while the returns for bonds were between 8 percent and 9 percent. But from 1966 to 1981, the average annual rate of return for stocks was 6.6 percent with a standard deviation of 19.5 percent, while the returns for bonds were about 7 percent.
So is it really the case that, despite the debacles, deadbeats, and doomsday equities like WCOM and Enron, the less risky long-term investment is in stocks? Not surprisingly, there is a counterargument. Despite their volatility, stocks as a whole have proven less risky than bonds over the long run because
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their average rates of return have been considerably higher. Their rates of return have been higher because their prices have been relatively low. And their prices have been relatively low because they’ve been viewed as risky and people need some inducement to make risky investments.
But what happens if investors believe Siegel and others, and no longer view stocks as risky? Then their prices will rise because risk-averse investors will need less inducement to buy them; the “equity-risk premium,” the amount by which stock returns must exceed bond returns to attract investors, will decline. And the rates of return will fall because prices will be higher. And stocks will therefore be riskier because of their lower returns.
Viewed as less risky, stocks become risky; viewed as risky, they become less risky. This is yet another instance of the skittish, self-reflective, self-corrective dynamic of the market. Interestingly, Robert Shiller, a personal friend of Siegel, looks at the data and sees considerably lower stock returns for the next ten years.
Market practitioners as well as academics disagree. In early October 2002, I attended a debate between Larry Kudlow, a CNBC commentator and Wall Street fixture, and Bob Prech- ter, a technical analyst and Elliot wave proponent. The audience at the CUNY graduate center in New York seemed affluent and well-educated, and the speakers both seemed very sure of themselves and their predictions. Neither seemed at all affected by the other’s diametrically opposed expectations. Prechter anticipated very steep declines in the market, while Kudlow was quite bullish. Unlike Siegel and Shiller, they didn’t engage on any particulars and generally talked past each other.
What I find odd about such encounters is how typical they are of market discussions. People with impressive credentials regularly expatiate upon stocks and bonds and come to conclusions contrary to those of other people with equally imA
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pressive credentials. An article in the New York Times in November 2002 is another case in point. It described three plausible prognoses for the market—bad, so-so, and good—put forth by economic analysts Steven H. East, Charles Pradilla, and Abby Joseph Cohen, respectively. Such stark disagreement happens very rarely in physics or mathematics. (I’m not counting crackpots who sometimes receive a lot of publicity but aren’t taken seriously by anybody knowledgeable.)
The market’s future course may lie beyond what, in chapter 9, I term the “complexity horizon.” Nevertheless, aside from some real estate, I remain fully vested in stocks, which may or may not result in my remaining fully shirted.
The St. Petersburg Paradox and Utility
Reality, like the perfectly ordinary woman in Virginia Woolf’s famous essay “Mr. Bennett and Mrs. Brown,” is endlessly complex and impossible to capture completely in any model. Expected value and standard deviation seem to reflect the ordinary meanings of average and variability most of the time, but it’s not hard to find important situations where they don’t.
One such case is illustrated by the so-called St. Petersburg paradox. It takes the form of a game that requires that you flip a coin repeatedly until a tail first appears. If a tail appears on the first flip, you win $2. If the first tail appears on the second flip, you win $4. If the first tail appears on the third flip, you win $8, and, in general, if the first tail appears on the Nth flip, you win 2N dollars. How much would you be willing to pay to play this game? One could argue that you should be willing to pay any amount to play this game.
To see why this is so, recall that the probability of a sequence of independent events such as coin flips is obtained by multiplying the probabilities of each of the events. Thus the
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probability of getting the first tail, T, on the first flip is 1/2; of getting a head and then the first tail on the second flip, HT, is (1/2)2 or 1/4; of getting the first tail on the third flip, HHT, is (1/2)3 or 1/8; and so on. Putting these probabilities and the possible winnings associated with them into the formula for expected value, we see that the expected value of the game is ($2 x 1/2) + ($4 x 1/4) + ($8 x 1/8) + ($16 x 1/16) + . . . (2N x (1/2))N + ... . All of these products are 1, there are infinitely many of them, and so their sum is infinite. The failure of expected value to capture our intuitions becomes clear when you ask yourself why you’d be reluctant to pay even a measly $1,000 for the privilege of playing this game.
The most common resolution is roughly that provided by the eighteenth century mathematician Daniel Bernoulli, who wrote that people’s enjoyment of any increase in wealth (or regret at any decrease) is “inversely proportionate to the quantity of goods previously possessed.” The fewer dollars you have, the more you appreciate gaining one and the more you fear losing one, and so, for almost everyone, the likely prospect of losing $1,000 more than cancels the remote possibility that you’ll win, say, a billion dollars.
What’s 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.
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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 we’re back to Virginia Woolf’s essay.
Portfolios: Benefiting from the Hatfields and McCoys
John 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 generation’s Nobel prize-winning finance professor.
To get a taste of what a couple more of these Nobelists have written, assume you’re 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 Jake’s 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
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often become portfolios lacking in dollars. How can you more judiciously choose stocks to maximize a portfolio’s returns and minimize its risks?
Let’s first envision a simple portfolio consisting of only three stocks, Abbey Roads, Barkley Hoops, and Consolidated Fragments. Let’s 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 one’s proverbial eggs in one basket. (The result, as was the case with my WorldCom misadventure, may well be egg on one’s face and the transformation of one’s 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 can’t 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 Produc-
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tions. 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, let’s say that half the time Hatfield’s 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. McCoy’s 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 .10. 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 company’s returns.
But what if we don’t choose one or the other to invest in, but split our investment funds and buy half as much of each stock? Then we’re 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 don’t have to worry about who’s winning the battle between the Hatfields and the McCoys.
Of course, it’s difficult to find stocks that are perfectly negatively correlated, but that is not required. As long as they aren’t 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,
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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 doesn’t 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 - px) x (Y - pY)], where px and pY 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
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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 + qpY)]2, which, upon rewriting, is the expected value of [(pX - ppx) + (qY - qpY)]2, which, using the algebra rule cited above, is the expected value of [(pX - ppx)2 + (qY - qpY)2 + 2 x the expected value of [(pX - ppx) x (qY - qpY)].
Minding (that is, factoring out) our p’s and q’s, 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 homeowner’s 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 don’t, 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.
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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 portfolio’s 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 portfolio’s 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 isn’t. 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
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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 organization’s goals or products. Taken to its extreme, diversification is something we do naturally in dealing with the inevitable trade-offs in our daily lives.
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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 isn’t to insure that your money will do some good, but to maximize the good it will do. There are other situations too where bulleting one’s 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.”
Beta—Is It Better?
Returning to more quantitative matters, we choose stocks so that when some are down, others are up (or at least not as
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down), giving us a healthy rate of return with as little risk as possible. More precisely, given any portfolio of stocks, we grind the numbers describing their past performances and come up with estimates for their expected returns, volatilities, and covariances, and then use these to determine the expected returns and volatilities of the portfolio as a whole. We could, if we had the time, the price data, and fast computers, do this for a variety of different portfolios. The Nobel prize-winning economist Harry Markowitz, one of the originators of this approach, developed mathematical techniques for carrying out these calculations in the early 1950s, graphed his results for a few portfolios (computers weren’t fast enough to do much more then), and defined what he called the “efficient frontier” of portfolios.
If we were to use these techniques and construct comparable graphs for a wide variety of contemporary portfolios, what would we find? Arraying the (degree of) volatility of these portfolios along the graph’s horizontal axis and their expected rates of return along its vertical axis, we would see a swarm of points. Each point would represent a portfolio whose coordinates would be its volatility and expected return, respectively. We’d also notice that among all the portfolios having a given level of risk (that is, volatility, standard deviation), there would be one with the highest expected rate of return. If we single out the portfolio with the highest expected rate of return for each level of risk, we would obtain a curve, Markowitz’s efficient frontier of optimal portfolios.
The more risky a portfolio on the efficient frontier curve is, the higher is its expected return. In part, this is because most investors are risk-averse, making risky stocks cheaper. The idea is that investors decide upon a risk level with which they’re comfortable and then choose the portfolio with this risk level that has the highest possible return. Call this Variation One of the theory of portfolio selection.
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Don’t let this mathematical formulation blind you to the generality of the psychological phenomenon. Automobile engineers have noted, for example, that safety advances in automobile design (say anti-lock brakes) often result in people driving faster and turning more sharply. Their driving performance is enhanced rather than their safety. Apparently, people choose a risk level with which they’re comfortable and then seek the highest possible return (performance) for it.
Inspired by this trade-off between risk and return, William Sharpe proposed in the 1960s what is now a common measure of the performance of a portfolio. It is defined as the ratio of the excess return of a portfolio (the difference between its expected return and the return on a risk-free treasury bill) to the portfolio’s volatility (standard deviation). A portfolio might have a hefty rate of return, but if the volatility the investor must endure to achieve this return is roller coasterish, the portfolio’s Sharpe measure won’t be very high. By contrast, a portfolio with a moderate rate of return but a less anxiety-inducing volatility will have a higher Sharpe measure.
There are many complications to portfolio selection theory. As the Sharpe measure suggests, an important one is the existence of risk-free investments, such as U.S. treasury bills. These pay a fixed rate of return and have essentially zero volatility. Investors can always invest in such risk-free assets and can borrow at the risk-free rates as well. Moreover, they can combine risk-free investment in treasury bills with a risky stock portfolio.
Variation Two of portfolio theory claims that there is one and only one optimal stock portfolio on the efficient frontier with the property that some combination of it and a risk-free investment (ignoring inflation) constitute a set of investments having the highest rates of return for any given level of risk. If you wish to incur no risk, you put all your money into treasury bills. If you’re comfortable with risk, you put all your
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money into this optimal stock portfolio. Alternatively, if you want to divide your money between the two, you put p% into the risk-free treasury bills and (100 - p)% into the optimal risky stock portfolio for an expected rate of return of [p x (risk-free return) + (1 - p) x (stock portfolio)]. An investor can also invest more money than he has by borrowing at the risk-free rate and putting this borrowed money into the risky portfolio.
In this refinement of portfolio selection, all investors choose the same optimal stock portfolio and then adjust how much risk they’re willing to take by increasing or decreasing the percentage, p, of their holdings that they put into risk-free treasury bills.
This is easier said than done. In both variations the required mathematical procedures put enormous pressure on one’s computing facilities, since countless calculations must be performed regularly on new data. The expected returns, variances, and covariances are, after all, derived from their values in the recent past. If there are twenty stocks in a portfolio, we would need to compute the covariance of every possible pair of stocks, and there are (20 x 19)/2, or 190, such covariances. If there were fifty stocks, we’d need to compute (50 x 49)12, or 1,225 covariances. Doing this for each of a wide class of portfolios is not possible without massive computational power.
As a way to avoid much of the computational burden of updating and computing all these covariances, efficient frontiers, and optimal risky portfolios, Sharpe, yet another Nobel Prize winner in economics, developed (with others) what’s called the “single index model.” This Variation Three relates a portfolio’s rate of return not to that of all possible pairs of stocks in the portfolio, but simply to the change in some index representing the market as a whole. If your portfolio or stock is statistically determined to be relatively more volatile
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than the market as a whole, then changes in the market will bring about exaggerated changes in the stock or portfolio. If it is relatively less volatile than the market as a whole, then changes in the market will bring about attenuated changes in the stock or portfolio.
This brings us to the so-called Capital Asset Pricing Model, which maintains that the expected excess return on one’s stock or portfolio (the difference between the expected return on the portfolio, Rp, and the return on risk-free treasury bills, Rf) is equal to the notorious beta, symbolized by 3, multiplied by the expected excess return of the general market (the difference between the market’s expected return, Rm, and the return on risk-free treasury bills, Rf). In algebraic terms: (Rp-Rf) = P(Rm - Rf). Thus, if you can get a sure 4 percent on treasury bills and if the expected return on a broad market index fund is 10 percent and if the relative volatility, beta, of your portfolio is 1.5, then the portfolio’s expected return is obtained by solving (Rp - 4%) = 1.5(10% - 4%), which yields 13 percent for Rp. A beta of 1.5 means that your stock or portfolio gains (or loses) an average of 1.5 percent for every 1 percent gain (or loss) in the market as a whole.
Betas for the stocks of high-tech companies like WorldCom are often considerably more than 1, meaning that changes in the market, both up and down, are magnified. These stocks are more volatile and thus riskier. Betas for utility company stocks, by contrast, are often less than 1, which means that changes in the market are muted. If a company has a beta of .5, then its expected return is obtained by solving (Rp - 4%) = .5(10% - 4%), which yields 7% for Rp, the expected return on the portfolio. Note that for short-term treasury bills, whose returns don’t vary at all, beta is 0. To reiterate: Beta quantifies the degree to which a stock or a portfolio fluctuates in relation to market fluctuations. It is not the same as volatility.
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This all sounds neat and clean, but you beta watch your step with all of these portfolio selection models. Specifically with regard to Variation Three, we might wonder where the number beta comes from. Who says your stock or portfolio will be 40 percent more volatile or 25 percent less volatile than the market as a whole? Here’s the rough technique for finding beta. You check the change in the broad market for the last three months—say it’s 3 percent—and check the change in the price of your stock or portfolio for the same period—say it’s 4.1 percent. You do the same thing for the three months before that—say the numbers this time are 2 percent and 2.5 percent, respectively—and for the three months before that—say -1.2 percent and -3 percent, respectively. You continue doing this for a number of such periods and then on a graph you plot the points (3%, 4.1%), (2%, 2.5%), (-1.2%, -3%), and so on. Most of the time if you squint hard enough, you’ll see a sort of linear relationship between changes in the market and changes in your stock or portfolio, and you then use standard mathematical methods for determining the line of closest fit through these points. The slope or steepness of this line is beta.
One problem with beta is that companies change over time, sometimes rather quickly. AT&T, for example, or IBM is not the same company it was twenty years ago or even two years ago. Why should we expect a company’s relative volatility, beta, to remain the same? In the opposite direction is a related difficulty. Beta is often of very limited value in the short term and varies with the index chosen for comparison and the time period used in its definition. Still another problem is that beta depends on market returns, and market returns depend on a narrow definition of the market, namely just the stock market rather than stocks, bonds, real estate, and so forth. For all its limitations, however, beta can be a useful notion if it’s not turned into a fetish.
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You might compare beta to different people’s emotional reactivity and expressiveness. Some respond to the slightest good news with outbursts of joy and to the tiniest hardship with wails of despair. At the other end of the emotional spectrum are those who say “ouch” when they accidentally touch a scalding iron and allow themselves an “oh, good” when they win the lottery. The former have high emotional beta, the latter low emotional beta. A zero beta person would have to be unconscious, perhaps from ingesting too many beta- blockers. Unfortunate for the prospect of predicting the behavior of people, however, is the commonplace that people’s emotional betas vary depending on the type of stimulus a person faces. I’ll leave out the examples, but this may be beta’s biggest limitation as a measure of the relative volatility of a portfolio or stock. Betas may vary with the type of stimulus a company faces.
Whatever refinements of portfolio theory are developed, one salient point remains: Portfolios, although often less risky than individual stocks, are still risky (as millions of 401 (k) returns attest). Some mathematical manipulation of the notions of variance and covariance and a few reasonable assumptions are sufficient to show that this risk can be partitioned into two parts. There is a systematic part that is related to general movements in the market, and there is a non-systematic part that is idiosyncratic to the stocks in the portfolio. The latter, nonsystematic risk, specific to the individual stocks in the portfolio, can be eliminated or “diversified away” by an appropriate choice of thirty or so stocks. An irreducible core, however, remains inherent in the market and cannot be avoided. This systematic risk depends on the beta of one’s portfolio.
Or so the story goes. To the criticisms of beta above should be added the problems associated with forcing a non-linear world into a linear mold.
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