Why Diversify?

Did our diversification produce a mediocre result?  That depends on how you define “mediocre.”  The average of the wealth results of the two separate investments is ($137,737 + $139,613) / 2 = $138,675.  Our diversified portfolio, consisting of the combined investments, beat this average by $372. If “mediocre” means average, then our diversified portfolio produced better than mediocre results. This, too, is a general truth: if the average returns are the same, a diversified portfolio (with equal amounts invested in each holding, and the holdings have different patterns of returns) will produce more wealth than the average of the amounts of wealth generated by putting the same total amount of money into each of the individual investments.

It gets better.

Consider the following set of hypothetical returns:

In this table, the hypothetical pattern of returns to the U.S. stock market is the same as before, but the Chinese stock market has a different pattern of returns, still averaging 7%.  And now, the diversified portfolio, with $50,000 invested in each of the two markets, produces⁵ final wealth of $139,241. This beats both of the individual markets! (It also, of course, therefore beats the average of the wealth they would produce.)

This certainly isn’t a mediocre result; it’s better than investing in either of the two individual stock markets, and it was the result of diversification.

I caution again that I’m using completely made-up numbers, and I’m not arguing that you should invest half your money in China.  But you can see why it might make sense to diversify your holdings internationally, even if you expect all of the world’s stock markets to produce about the same average return in the long run. You can expect the results to be better than mediocre.

How can this be?  Unfortunately, it requires a bit of mathematics to show how this works, and I try not to be tedious.  But the key to the underlying mathematics is correlation, the correlation among the patterns of returns produced by the different assets.

In everyday language, when we say that two things are “correlated,” we often mean that they move in lock step, like soldiers marching in parade, and if they’re “uncorrelated,” they’re completely unrelated, like dancers in two different halls, one group dancing to disco music and the other group waltzing to a ballroom dance orchestra.  But in precise statistical usage, there are degrees of correlation, whose values range on a scale from 1 through 0 to -1. The soldiers’ steps have a correlation of 1, the dancers in the separate halls have a correlation of 0.  But the dancers in one hall or the other have intermediate correlations among each other, that is, correlations between 0 and 1.  They may not be taking the same steps, but they’re dancing to the same music and rhythms, and to some degree, they tend to move together. A correlation of -1 is like a pair of ballroom dancers waltzing together, where each dancer’s moves are the exact mirror images of the other’s, so that one dancer extends her left foot at the same moment that her partner extends his right.

It turns out that, all else being equal, the lower the correlations among investment returns, the better the financial result of their combination.⁶  And negative correlations (which are nearly impossible to find in the real world of investing), are better than positive ones.  In my first example, above, the correlation of the returns between the U.S. and China was 0.66.  In my second example, the correlation was only 0.08.  Hence, the greater wealth at the end of the second example.

Another way of looking at this is that the lower the correlations, the lower the volatility of the combination of the investments, and we already know that, all else being equal, lower volatility produces greater final wealth.

Diversification and Modern Portfolio Theory

So there are a lot of moving parts in the machinery that generates investment results. The final wealth produced by diversified investments depends upon (besides the amount of money with which you start):

1. The initial proportional weighting of the investments (10% of this, 15% of that, and so on)
2. The average return of each investment;
3. The volatility of each investment⁷
4. The correlation between each pair of investments.⁸

The more individual investments you consider, the more average returns there are, the more volatilities there are, and the more correlations there are.  And there are tradeoffs to be made among all of them.

This way of looking at diversification, by considering all these four elements, is the foundation of Modern Portfolio Theory (MPT), which was first put forth by Harry Markowitz in his University of Chicago doctoral dissertation in 1955, and for which he shared in the award of the Nobel Prize in economics in 1990.  What I have laid out before you, though, it is not so much the theory, as the indisputable consequence of the mathematics of return and risk.  Markowitz saw this, added simplifying economic assumptions, and, applying the miraculous electronic computing power available to him in the early 1950s, built out the theory, which we’ll see again when we consider how best to combine investments into a portfolio. (Some traditional investors refer disapprovingly to virtually all mathematical and economic reasoning about investing as “modern portfolio theory,” but this is so vague as to rob the term of useful meaning, and is basically incorrect.)

In order to follow Markowitz in computing the results of diversification, you have to forecast a lot of statistics in order to predict your portfolio’s investment results. Of the four categories of inputs that I listed, only the initial proportional weights on the investments are known. It’s not sufficient to forecast average returns of the individual investments, which is hard enough; how can you forecast volatilities and correlations?  In practice, investment professionals can’t make all these forecasts precisely for the long term, but there are rules of thumb and intuitions that help. (For the short term, many do, indeed, use precise forecasts; as I mentioned in my previous essay, I once worked for a company that produces risk models that incorporate forecasts of volatility and correlations--but not forecasts of average returns, which we felt were the responsibility of our clients, companies that offered their own investment expertise to individuals and pension funds.⁹)

Diversification lessons

Even without working through the math and computations, however, we can explore some of the consequences of investment diversification and draw some practical inferences for investing.

First, and most important, especially if the forecasts of return are fairly similar, you want to diversify among assets whose patterns of return over time have low correlations.  Take, for a counterexample, the many mutual funds that are designed to produce the same returns as an index of the stock market as a whole.  These are called “index funds.”  Many of these track the Standard and Poor’s 500 index of the largest companies in the U.S. stock market.  There is no diversification benefit to be gained by spreading your money among different S&P 500 index funds, because they will have the same return (whatever it turns out to be), and they are all perfectly correlated, that is, every pair has a correlation of 1. The combination of these funds will have exactly the same variability, and risk, as each individual fund.¹⁰

Few investors would make this mistake, but there is a similar one that is all too common.  Repeated studies have shown that many employees, when given a choice of mutual funds in their 401(k) plans, will allocate money in equal amounts among all the options, regardless of what the options are. So, for example, a person given a choice of ten investment options, consisting of nine stock funds and one bond fund, will spread his money equally among all of them.  The same person, given a different choice of ten funds, this time consisting of nine bonds funds and one stock fund, would also spread his money equally among these.   But stocks funds of all sorts tend to have high risk with an expectation of relatively high return, and bond funds tend to have low risk with the expectation of relatively low return. (Of course, things haven’t worked out this way for stocks and bonds over the last twenty years, but I’m talking of expectations, not results.)  So it can’t possibly be right that the same person would be equally well served by a portfolio consisting of 9/10 stock and another portfolio consisting of 9/10 bonds.  In short, spreading money equally among available investment options without regard to the characteristics of these options will likely produce inappropriate investment results. This is called “naïve diversification.”

Not only the financially innocent have been led astray by misunderstanding diversification. In the last few years, salesmen of sophisticated investment products were able to practice upon the credulous simplicity of supposedly sophisticated investors who had absorbed the basic lesson that low correlations are good, but hadn’t thought through this concept much further.  Known correlations are measured with historical data, and they can therefore change over time as more data accumulate.  So the salesmen would select a measurement period (say, the trailing five years) during which their products had low correlations with both the stock market and the bond market to pitch these investment vehicles as the very thing to lower the risk of their clients’ portfolios.  This technique was a boon to the sales of some hedge funds and “structured products.”  But, unsurprisingly, longer runs of historical data would have shown that the correlations were nowhere near as low as the sales pitch said they were. In the recent great financial crisis, we have seen the results of the authentic correlations.

Correlation and independence

Even without promotion by clever Wall Street salesmanship, this lack of clarity over the meaning of correlation can lead to overestimation of the virtues of diversification.  Statisticians apply the concept of independence.  Remember those two groups of dancers, in two different halls, whose motions are uncorrelated?  I didn’t tell you that they’re all from one extended family, celebrating the patriarchs’ 60th wedding anniversary. One group was the children’s party, the other was the adults’. At 9:00 PM, they all converged in a single conga line.  The two groups were not independent, and suddenly, they’ve changed from a correlation of 0 to a correlation of 1.

We’ve encountered the same phenomenon in the financial markets. I’ve mentioned my preference for diversifying among the U.S. stock market and the foreign stock markets.  But these markets are not independent of each other (indeed, the correlations of their returns have historically been high).  And when a calamity befalls, the correlations among their returns approaches 1.  That is, when we in the United States most want the benefit of diversification, the correlations with foreign markets are most likely to leave us bereft. We’ve known this for a long time. We saw this in the market crash of October 1987.  Anyone who was surprised that diversification across international stock markets provided little protection during our recent market turmoil knew little history and wasn’t paying attention.

But these periods of convergence are always brief.  It’s as if the partiers danced the conga for five minutes, then went back to their separate dance halls.

When I address the problem of constructing a portfolio in a future essay, I will say more about the right ways to go about diversifying. But it is worth considering here the extremes of diversification.  We already know one extreme: No diversification, meaning that you hold a single investment.  The other extreme is to hold everything.

How diversification makes risk comprehensible

Let’s consider just the U.S. stock market (and ignore the bond markets, foreign stock markets, and privately-held stocks).  Holding everything means holding some amount of stock in every U.S. company that is traded on an American stock exchange, in the same proportion, by value, that each stock contributes to the overall stock market. If, for example, the stock of General Electric is 1.99%, by value, of all the stocks traded in the U.S., then it will be 1.99% of our extremely diversified portfolio. Such extreme diversification is what stock market index funds offer.

At this point, I will reveal a sleight of hand that I used in my essay on the nature of investment risk,¹¹ at the end of which I wrote, “There is an important distinction to be made between the risk of individual investment securities and the risk of amalgams or aggregations of securities, which is what mutual funds and markets are.” And then I ignored the distinction, in the noble cause of pedagogy. In that essay, I used the familiar so-called “bell curve” (also known as the normal distribution) to illustrate the risks of investment returns:

This is a graph of the (supposed) frequency of the returns, not their size. The height of the curve represents frequency.  Accordingly, the middle of the hump represents the middling returns, which occur most frequently, and the tails represent the extreme returns, both very low or negative returns (on the left) and very high returns (on the right), which occur very infrequently.

As I wrote then, quantitative investment analysts usually assume, for the purpose of analyzing investment risk, that the possible outcomes of an investment resembles the bell curve.  And I explained that this is owing to wishful thinking, because the bell curve is mathematically tractable, unlike other possible frequency distributions. There are problems with the assumption of the bell curve, in part because, in actual fact, disastrous results from the stock market are more frequent than the slender left-hand tail of the bell curve would suggest.  All the same, the bell curve beats the alternatives.

But if you think of individual stocks, as distinct from the entire stock market, no amount of wishful thinking is going to conjure a match of their periodic returns (say, daily, or monthly) to the bell curve.  A company that is consistently successful month after month, year after year—and I’m sure you can name one or two of these—will likely have very good returns that are more numerous and larger than their very bad returns, thereby stretching the (bumpy and far-from-smooth) frequency graph to the right. Correspondingly, the returns to the stock of a company like Enron that is snuffed out, if you look at the frequency of their monthly values during the last couple of years of the company’s existence, are also nothing like a bell curve.¹²

It turns out, though, that as you gather first two, then three, then, say, ten or twenty disparate stocks into a portfolio, the frequency of the various returns to that portfolio as a whole begins to fit the shape of a bell curve, more or less. And if these stocks are randomly chosen, then the average return of the portfolio (average over time, that is) will be close to the average return of the market, and the shape of the more-or-less bell curve of the portfolio’s returns will resemble the more-or-less bell curve of the stock market’s returns.

Of course, no one, not even someone reproducing the entire market, as an index fund does, chooses portfolios randomly. But while the index fund is designed to resemble the market, other portfolios may be designed with other ends in mind, and their more-or-less bell curves won’t be the same as that of the market, but will still submit to a mathematical analysis that quantifies the risks of poor outcomes.

In short, when considering a distinct individual stock or other investment, we may be able to develop some notion of what return we should expect from it, but we cannot, in a statistical sense, easily conceive its concomitant risks, in the sense of probabilities of dire outcomes. Once we combine individual investments into a portfolio, in contrast, well-understood laws of statistics begin to apply, and the probabilities of all kinds of outcomes will begin to fit a pattern that bears at least a passing resemblance to the bell curve, from which we can deduce approximate risks of dire outcomes.¹³

A fundamental analyst, that is, the kind of analyst who pores over the financial statements of the company that issues a stock or of the company or government that issues a bond, would probably object to my first proposition, in the sense that she might claim an ability to evaluate the risks to the specific company or government entity.  That may be true—but I did qualify my proposition by saying, “in a statistical sense.”  Once, however, we consider the individual investments together as a diversified collective, that is, as a portfolio, the bell curve, or something like it, rears its back, and the investments yield, even if grudgingly and imperfectly, to statistical analysis, with its implied numerical probabilities of bad results. And this suits nearly every investor’s portfolio. Hardly any investor holds only one stock or only one bond. And this is how I excuse myself for eliding the distinction between an individual stock or bond and a portfolio when I introduced the concept of investment risk.

More diversification lessons

A proper appreciation of the nature of diversification can overturn some commonly held beliefs.  So far is diversification from ensuring mediocrity, that it can actually justify holding very risky investments in a portfolio.  It may be that, despite a high level of risk, a particular investment holds out the prospect of a good return, which will be directly averaged into the return of the entire portfolio, while the risk (depending upon the correlations with the other investments in the portfolio) may be largely diversified away.

Understanding this has led to very real changes in how portfolios are managed. The management of a trust or an endowment, for example, is legally governed by what is known as the prudent person rule, which not so long ago was known as the “prudent man rule.” As enunciated by Judge Samuel Putnam in his 1830 decision in the case of Harvard College v. Amory, this stated that a trustee or fiduciary “is to observe how men of prudence, discretion, and intelligence manage their own affairs, not in regard of speculation, but in regard to permanent disposition of their funds, considering the probable income, as well as the probable safety of the capital to be invested.” This was for many years interpreted to rule out the holding of individual highly risky investments within a portfolio. If, however, you were to sift diligently through all 500 companies that constitute the S&P 500 index, you would doubtless find a few companies that, considered by themselves, were issuers of stock that a prudent fiduciary would not purchase. And, as an index fund based on the S&P 500 index would by mandate hold those stocks, such an index fund would be considered an imprudent investment for a fiduciary to purchase for a client.  The change took many years, but now, in light of modern portfolio theory, the prudent person rule has been reinterpreted, and index funds are considered among the least adventurous kinds of investments for a trust fund or endowment.

This notion of the nullification of risk can, however, be taken too far.  Mathematics alone will show that, from a set of very risky investments with the right correlations, a portfolio manager could construct a portfolio with low risk. That is, the dramatic moves in price hither and yon by the individual investments could cancel each other in the portfolio.  In practice, this is almost certainly not economically feasible, or if it appears to be, that is only because the numerical values of the correlations being used to calculate the risk of the portfolio are themselves being estimated incorrectly.  This is, however, an effective excuse for a Wall Street firm that needs to justify the unloading of risk from its own inventory onto unsuspecting customers.

Diversification not only affects portfolios; it also adds a layer of sophistication to the relationship between the risk and return of individual investments. As I have written before,¹⁴ there isn’t a direct relationship between the risk of an investment and the return you may expect from it; not all risks are compensated.  Many economists, especially the proponents of the hypothesis of market efficiency, have argued that the market, in its haggling over prices, does not provide compensatory return for risks that are easily diversified away. Whether there is no diversifiable risk that is compensated by return is a matter for continuing research by those who analyze the actual historical numbers.  What is clear, however, is that at least a very large proportion of diversifiable investment risk is not compensated by return. Consequently, you shouldn’t expect to receive a large return just because you hold very risky stocks. A very risky stock (or other investment) may provide a good return in the short run—because, after all, volatility causes prices to go up as well as down—but in the long run, if much of the risk could be sloughed off through diversification, it may very well provide no better return than a much less risky stock.

Nearly all common investors are diversified to some extent—as I’ve said, if you own more than a single investment, you’re diversified—but most are broadly diversified, because they hold mutual funds.  A case can be made (and has been by some commentators) that one of the greatest benefits of mutual funds is not that they offer professional management skill, but that they provide broad and, in many cases, inexpensive diversification. This may not be, however, and it usually isn’t, the kind of diversification that a portfolio manager or investment advisor would create to suit the needs of a particular investor.

Conclusion

What investment diversification is, and how it works, is one matter; how best to diversify and to benefit from diversification is a further matter.

Over the last year or so, some investment advisors have made a great show by noisily proclaiming that “diversification is dead.” Stated like that, the claim is, of course, utter nonsense. I’d be astonished if these managers held only a single investment each in their portfolios.  What they are almost certainly trying to say is that they believe in concentrating their portfolios in one asset class or another; that is, that they believe that they ought not to be diversifying across asset classes. They may be right or wrong—my judgment is that they’re wrong—but we will find out how to decide how much weight to give each investment or each asset class in a portfolio, and when to change this, when we reach our ultimate destination, in a pair of essays on portfolio construction.

Adam Jared Apt, CFA, is a financial advisor and the owner of Peabody River Asset Management, based in Cambridge, MA.

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