DIVERSIFICATION

Though literally everyone talks about diversification for an investment portfolio, very few understand the true statistical data underlying the definition. As a result, few portfolios are properly diversified and an extended risk is being taken- unquestionably unwittingly, but nonetheless evident- by most consumers.

In order to properly explain the problem, certain other definitions are required:

Systematic Risk- this is risk due to the movement of the market itself. The benchmark could be the Dow (normally used by consumers and brokers alike), the S&P 500, Wilshire index, etc., but the index may also be that of select or special uses such as gold, real estate, biotech and other sector funds. If one had one or a few investments in a given area, they could compare its return to that of the benchmark index to determine how well it is doing.

Unsystematic Risk- This is the risk of ONE company causing a significant move- either up or down- in a portfolio. This is usually the risk that most consumers would want to eliminate- unless they are TRUE risk takers (few are). It may be tempered- and in fact virtually eliminated- by the purchase of a certain number of stocks or bonds. The graph below shows the statistical reference.

NO GRAPH- see video

As is easily identified, the number of stocks necessary to insulate a given portfolio from unsystematic (firm- specific) risk is around 10 to 15. Slight variations are possible depending on the volatility (beta is one of these elements) of the portfolio and correlation of the individual stock and some texts have suggested 20 stocks should be purchased. Overall however, the use of 13 stocks may be used an "acceptable" average/compromise for proper diversification- ASSUMING ADEQUATE CORRELATION described more fully below.

Below is a table from Investments by Bodie, Kane and Marcus (standard text for CFA's)
Number of stock Standard Deviation

1 50.00

2 41.83

5 36.06

10 33.91

20 32.79

100 31.86

The minimum, or nondiversifiable risk for this scenario would be 31.62 so you can see that once you get past 10 stocks, each additional stock does not reduce the volatility that much. And that's the problem if the stocks tend to move in the same direction as the above chart reflects. In such a scenario, the use of just the 13 or so stocks would not decrease the risk appreciably. Hence the next important issue is called correlation. This means that all the stocks selected all move in the same direction when something happens in the marketplace- either all up or all down. Most portfolios would prefer some negative correlation- that is some stocks/investments would go up when other investments in the portfolio were going down. If we take the same number of stocks and gave them a complete correlation, look at how the volatility (standard deviation) is reduced.

Number of Stock Standard Deviation

1 50.00

2 35.36

5 22.36

10 15.81

20 11.81

100 5.00

As you can see in the top chart, 20 stocks have a standard deviation of 32.79 versus the much lower 11.81 directly above. The first portfolio- which, I submit, would might be used by unknowledgeable and/or untrained brokers and investors- is 64% more volatile (linear). But all this is a rather moot point since investors aren't diversified to begin with. A study by the New York stock exchange shows that 49% of stockholders held only one common stock, 20% held two, 22% held between three and seven issues and only 9% held 9 or more. Thus 91% of all individual investors had portfolios of eight or less issues making the portfolios substantially more risky that acknowledged by their attitudes. That report was from the early 90's and is unquestionably tempered by the significant use of mutual funds from about 1990 onward. However, the risk is still apparent since a significant number of 401(k) plans show a large use of employer stock as identified next.

An excellent example of the problem of unsystematic risk that is endemic in society is through company incentives to employees to purchase company stock through discounts, stock options and profit sharing plans. Employees invariably end up with highly non diversified portfolios since most people do not buy other securities to offset such risk. Digital stock defines such a problem. It reached a high of 200 but subsequently dropped to about $16 before rebounding somewhat. Employees did maintain large undiversified holdings and will probably never see their investment return to even half the value in their remaining lifetime. As a question therefore, are such companies liable for not making employees aware of such risk? The answer is clearly yes under the rule 404(c) which requires that employers give employees education on risk. (In such cases, one would use a weighted portfolio to ascertain the overall risk.) It would appear that the continued use of employer stock will be cause for significant employer liability in the future.

One type of guide to diversification, though one not normally monitored except by certain portfolio managers and sophisticated investors who statistically monitor various funds, is called R squared. It's an indication of a fund's overall diversification and measures the percentage of the funds performance as compared to the overall market. A R² squared of 75 means that the fund is 75% as diversified as the S&P 500- though other indexes may be used. This means that 75% of the fund's performance is due to the overall market movement while the other 25% is unique to the fund's characteristics.

CAUTION: Though diversification has been properly defined with around 13 stocks in a SINGULAR portfolio, that is NOT to say that the ENTIRE portfolio for the investor, in and of itself, is diversified. For example, a diversified mutual fund portfolio in totally small capitalization stocks or junk bonds (under BBB rating) will unquestionably have 20 or more issues in the portfolio, but the underlying risk of owning ONLY this ONE fund is unacceptable for almost all purposes. The entire investors portfolio should contain other "diversified" investments of, say, growth stocks, bonds, municipals, etc. in order to maintain proper diversification of all the holdings. This is better known as asset allocation.

TIME DIVERSIFICATION?: It has been usually held in the industry that the longer one holds an investment, the less the overall risk. It's based on standard deviation (volatility around a norm) and it means that if you have a risky fund, risk would lower the longer the fund was held. For example, if the standard deviation for one year was 20% (about equal to the S&P overall) and you held the fund for 5 years, you would divide 20% by the square root of five for a standard deviation of 8.94%- an apparent decrease over 55%. But the standard deviation of the dollar RETURN (that's your money folks) is INCREASED by the formula (1- .0894)5th or .626. This means that the final five year wealth may be only 63% of projected. That represents a 37% loss- far greater than a one year 20% swing. Per Investments by Bodie, Kahn and Marcus, Richard Irwin Inc. 1989, page 224, ".....time diversification does not reduce risk. Although it is true that per year average rate of return has a smaller standard deviation for a longer time horizon, it is also true that the uncertainty compounds over a greater period of years. Unfortunately, this latter effect dominates in the sense that the total return becomes more UNCERTAIN the longer the investment horizon". "Investing for more than one holding period means that the amount at risk is growing. This is analogous to an insurer taking on more insurance policies. The fact that these policies are independent does not offset the effect of placing more funds at risk."

That is the major reason why I feel it is necessary to monitor investments. Maybe nothing would wrong, but are you willing to trust your entire life's savings to MAYBE'S????? The validity of that point is (allow some latitude) by the significant risks absorbed by the property casualty insurers during the last few years. They use the same type risk scenario- over time the odds of risks can actually decrease. But in just a short time, there were the hurricanes in Florida and Hawaii, flooding over two years in the mid west, an earthquake in California and the numerous major fires as well. The point is that these major insurers used all the tables to develop risk analyses and profit return. But they obviously did not properly account for what could and did happen- major catastrophes all at one time. Investors must weigh this risk and stay vigilant.

EFFICIENT FRONTIER: The chart shows the Markowitz Efficient Frontier Graph. For each level of risk and return, you can plot different curves. The essence of the graph is that point M represents the highest expected return relative to all other portfolios of comparable risk.

My point to this discussion however is to quote on aspects of diversification: (Stalla CFA Manual) "In the illustration, single asset portfolios will be well located within (below) the efficient frontier (solid line) because these portfolios have high levels of market and specific risk. Multi- asset portfolios lie closer to the efficient frontier because diversification causes their specific risk to be reduced by the law of large numbers. Ultimately, portfolios lying on the efficient frontier will be those whose specific risks have been eliminated by diversification; they are efficiently diversified portfolios."

Unfortunately, the above definitions have been lost on the brokerage community (and attorneys, expert witnesses and consumers as well). This is also evident in the initial license preparation for literally all new security licensees. As an instructor in securities licensing training for the two largest firms in the United States, I have been involved with literally all the major licenses/exams- Series 7 for stockbrokers; Series 6 for mutual funds; Series 24 and 8 for supervision and a host of others. In perhaps 10 to 15 minutes over a period of two to four days, there might be some superficial commentary regarding systematic and unsystematic risk, but in NO cases is the statistical information given- since it is not required for passing the exam.

Further, and most importantly, there has been NO continuing educational training OF ANY TYPE required for brokers, supervisors, compliance representatives, etc. once they attained a securities license- until 1995. And a good portion of this new training is still sophomoric and redundant. As a result, a goodly number of brokers and firms are designing portfolios exhibiting substantially greater risk than consumers want or should accept. Confirmation of that point is offered by a study from the New York Stock Exchange which found that 49% of all stockholders held only one common stock, 20% held two, 22% held between three and seven issues and only 9% held nine or more. Thus 91% of all individual investors had portfolios of eight or less issues making the portfolios substantially more risky than that undoubtedly acknowledged or desired by their attitudes.

And as stated, securities attorneys are ignorant of this issue as well- though not necessarily surprising since there is nothing inherent in a law degree that covers such technicalities. Additionally, none of the SEC or NASD texts address the issue either. As a result, in literally all the cases I have reviewed and in all cases where I have acted as arbitrator, there has never been one argument on improper diversification. Yet it is has been quite apparent that it should have been raised in at least 85% of such cases. A key aspect is not even addressed. Risk is the key investment factor that must be developed for all clients.

Expert witnesses have also not addressed the issue. But since most witnesses come from retired members of the brokerage community, it is not surprising that the knowledge is not there. Unless one has advanced education in investments or is a CFA (Chartered Financial Analyst), it is rarely noted. As far as consumers are concerned, unless they have attended some obscure seminar by the likes of AAII (American Association of Individual Investors), none of the investing public has any comprehension of the numbers as well. It is also apparent by the non diversified portfolios they hold/are sold.

And since the members of the arbitration panel all come from the ranks of the above, the arbitrators are therefore woefully ignorant of a material issue as well.

NYSE Rule 405 demands that brokers "know thy client". The NASD rules demand suitability. I submit it is literally impossible to discuss the singular merits of a bond, stock or the like if the portfolio consists of only two, three, four or only six stock. Regardless of a low beta and/or limited volatility of one (or just a few) purchase(s), the unsystematic risk suggests an unsuitable portfolio in and of itself.

This lack of education regarding proper diversification is the foremost issue in literally all cases- especially where individual stocks are purchased. It is IMPOSSIBLE to properly judge the correctness of an account if the material benchmark is missing- that of RISK of the entire portfolio. And even if the broker is addressing only a small part of an investors holdings, it is mandatory that they "know thy client" and therefore must require detailed knowledge about the rest of the investment assets to determine proper diversification and risk throughout.

Diversification: (2000) If you have read my article on the 2000 Investor's  Guide to Asset Allocation, you will note the new comments about the lowered correlation and the fact that you need more stocks to accomplish what (supposedly) worked with just 15 stocks. William Bernstein suggests you need up to 200+. If you say you really are an investor- you've got to read this

"if you think that you can do an adequate job of minimizing portfolio risk with 15 or 30 stocks, then you are imperiling your financial future and the future of those who depend on you. The reason is simple: There are critically important dimensions of portfolio risk beyond standard deviation. The most important is so-called Terminal Wealth Dispersion (TWD). In other words, it is quite possible to put together a 15-stock or 30-stock portfolio with a very low SD, but whose lousy returns will put you in the poorhouse."

Diversification 2001 by John Y. Campbell , Martin Lettau, Burton G. Malkiel and Yexiao Xu:  This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period from 1962 to 1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. A portfolio once required 12 stocks (I used 13 in my teachings) to properly diversify and track the S&P 500, but the reducing correlations pushed that to 50. A WSJ article noted that a University of Nevada article says that investors need up to 100 stocks to stay within 5% of average portfolio risk. Research by CooperNeff show investors need 350.

Diversification: (Clemens Sialm, Marcin Kacperczyk and Lu Zheng 2003) according to a recent study from the University of Michigan Business School. In fact, skilled fund managers actually diversify less, according to , the authors of the study.

Sialm and his colleagues studied the performance of over 2,000 actively managed diversified equity funds from 1984-1999, and found that mutual funds with above-median "concentration" yielded an average return, adjusted for risk, of 2.18% per year before expenses and 0.82% per year after expenses. By comparison, more diversified mutual funds yielded an average return, adjusted for risk, of 0.39% per year before expenses and -0.73% per year after expenses. Concentration was measured by the number of industries held by a fund, and the weightings given to the various different industries.

"I think [the study] has important implications for perceptions of mutual fund managers," said Sialm in an interview. "Are they skilled and do they add value? Our study supports that some of them are skilled, and those mutual fund managers tend to choose highly concentrated portfolios. Why do they do that? One manager might have a background in biotech stocks, maybe he is a medical doctor, or has an advanced degree in that area. And of course, he will primarily pick from this sector."

The study also found that managers who run concentrated funds tend to focus on small-cap and growth stocks. "We believe that mispricing in the asset market is most severe for small and growth stocks, so it's easier for mutual funds to find good stock picks in these areas,"

But the study says nothing about volatility. These are forms of sector funds. Remember how well technology funds did just a few years ago. Of course they sometimes can do better, but the risk is apt to be very high.

Changes in diversification. (John Campbell 2004) This is also identified extensively in my book: Thirty or forty years ago in the US market, a single randomly selected stock had a standard deviation 35 percentage points higher than a portfolio invested in an equally weighted index of all available stocks; a portfolio of 20 individual stocks could reduce this excess risk to a modest level of about 5 percentage points. This was the basis for the well-known rule of thumb that a 20-stock portfolio is adequately if not perfectly diversified. During the last decade, however, a single randomly selected stock has had a standard deviation 50 percentage points higher than an equally weighted index; and it takes a portfolio of 50 stocks to reduce excess risk to 5 percent. The old rule of thumb is no longer adequate because a 20-stock portfolio has excess risk of 10 percent, twice its former level.

What has been happening in the stock market to cause these changes? A typical individual stock is now more volatile than before, but it also has a lower correlation with other stocks. More of the individual-stock volatility is idiosyncratic, less is shared with the market as a whole, and so the volatility of the overall market has not increased. In monthly data from the early 1960’s a typical US stock had a correlation between 0.25 and 0.30 with other stocks; by the late 1990’s this correlation had fallen below 0.10. These trends reflect numerous changes in the market, including the trend away from conglomerates towards companies focused on one or two core competencies, and the tendency to list companies earlier in their life cycle, when their futures are still very uncertain.

Diversification also has important benefits for international investors. Even the US market is imperfectly diversified relative to a world portfolio; other national stock markets are generally much smaller, with poorer diversification. Figure 2 illustrates recent movements in the volatility of excess country returns over the Morgan Stanley Capital International World Index (the blue line in the figure). Volatilities are calculated separately for each country using daily data in a six-month moving window, and a valueweighted average across countries is shown in the figure. Diversifiable country volatility averaged about 9% per year in the mid-1990’s, but international crises in the late 1990’s (notably the Asian crisis in 1997 and the Russian crisis in 1998) have driven the average up around 12% in the last few years.

How Many Stocks Do You Need to Be Diversified?, (2004) By Daniel J. Burnside, Donald R. Chambers and John S. Zdanowicz . This subject has been identified here and in my new book. Essentially the old statistical reference of 10 to 20 stocks as proper diversification has been outdated for some time. I referred in the book to a study by by John Y. Campbell, Martin Lettau, Burton G. Malkiel and Yexiao Xu:  This paper uses a disaggregated approach to study the volatility of common stocks at the market, industry, and firm levels. Over the period from 1962 to 1997 there has been a noticeable increase in firm-level volatility relative to market volatility. Accordingly, correlations among individual stocks and the explanatory power of the market model for a typical stock have declined, whereas the number of stocks needed to achieve a given level of diversification has increased. A portfolio once required 12 stocks (I used 13 in my teachings) to properly diversify and track the S&P 500, but the reducing correlations pushed that to 50. A WSJ article noted that a University of Nevada article says that investors need up to 100 stocks to stay within 5% of average portfolio risk. Research by CooperNeff show investors need 350.

This more recent study validates the fact that far more stocks are needed to reduce the unsystematic risk of a portfolio. In essence, it reinforces the fact that it is foolish to attempt to buy stocks to fit a portfolio. Far too many are needed and the research beyond almost all, myself included.

"In early studies, researchers formed different-sized portfolios through random selection and observed the risks, as measured by the average portfolio standard deviation. Standard deviation measures volatility—the amount by which a portfolio’s actual returns vary around the average return; the more volatile a portfolio, the greater the risk.

The study found that as the portfolio size was increased to roughly 10 stocks, average volatility dropped dramatically. In addition, as the number of stocks increased beyond 10, portfolio volatility continued to drop, but not nearly as dramatically. More recent research has documented that there has been an increasing amount of diversifiable risk contained in stocks. What that means is that the number of stocks necessary to achieve a given level of diversification has actually increased."

The authors conducted a survey that covered the 504 months from January 1960 to December 2001.

"In the study, we examined tracking error, which measures the average amount by which the returns of the portfolios deviated from the returns of a target index. We measured the portfolios relative to two target indexes, one a value-weighted market index and the other an equal-weighted market index.

In a value-weighted market index, the weighting of each stock is based on its market capitalization; these indexes, such as the familiar S&P 500, are dominated by large stocks since the total capitalization of large stocks greatly exceeds the total capitalization of small stocks. When comparing the randomly selected portfolios against the value-weighted market index, the random portfolios were constructed to take into consideration the domination of larger-cap stocks. In an equal-weighted market index, equal dollar amounts of each stock are held.

Average tracking error for different sized portfolios (1960- 2001)

                                                  Monthly tracking error of portfolio relative to:

Number of Stocks in portfolio	Value weighted index (%)	Equal weighted index (%)
1	5.49	9.23
3	3+	5.84
5	2+	4.66
7	2+	4.01
10	1+	3.4
15	1+	2.8
20	1+	2.44
30	1+	2.02
45		1.65
65		1.39
100	0.6	1.13

Most importantly, Table 1 indicates that even a portfolio of 100 stocks will deviate from its target index by an average of 1.13% per month for the equal-weighted approach and 0.60% per month for the value-weighted approach.

Doesn’t seem like much? A monthly average deviation of 1.13% would correspond to an annualized deviation of approximately 3.9%, and a monthly average deviation of 0.60% would correspond to an annualized deviation of approximately 2.1%. Thus, even a portfolio consisting of as many as 100 stocks deviates substantially from the overall market average.

Translation: Investors with portfolios containing 100 stocks are bearing substantial diversifiable risk which, on average, is not rewarded with higher return.

The next table shows the monthly average tracking error for each portfolio size under the value-weighted strategy for the six sub-periods. The largest portfolio size, 100, has average absolute monthly deviations that increase from a low of 0.42% in the first period to a high of 0.85% in the last period.

The last table shows the corresponding results for the equal-weighted approach. For the most recent seven-year period, 1995–2001, the average absolute deviation of a single stock from the equal-weighted average of all stocks was 12.15%—which annualizes to 42%!

These last two tables indicate that diversifiable risk has been generally increasing substantially over the last 42 years. Thus, investors maintaining a fixed level of stocks in a portfolio will find that they are “less diversified” than they were in the past. This clearly corresponds to  

Portfolio size: Tracking errors relative to value weighted index over various periods

                                                   Time periods- 84 months each

Number of stocks in portfolios	1960- 1966	1967- 1973	1974- 1980	1981- 1987	1988- 1994	1995- 2001
1		4.98	5.48	5.77	5.09	7.72
3		3.01	3.34	3.46	3.16	4.76
5		2.37	2.60	2.74	2.48	3.64
7		2.02	2.21	2.34	2.12	3.11
10		1.7	1.87	1.98	1.81	2.64
15		1.39	1.52	1.62	1.48	2.14
20		1.20	1.31	1.4	1.28	1.9
30		1.0	1.08	1.15	1.05	1.55
45		0.81	0.89	0.94	0.87	1.26
65	0.52	0.68	0.75	0.77	0.72	1.07
100	0.42	0.55	0.59	0.64	0.58	0.85

Portfolio size: Tracking errors relative to equal weighted index over various periods

                               Monthly tracking error relative to equal weighted index (%)

Number of stocks in portfolio	1960- 1966	1967- 1973	1974- 1980	1981- 1987	1988- 1994	1995- 2001
1		7.95	9.26	9.86	10.09	12.15
3		4.84	5.83	6.26	6.6	7.76
5		3.82	4.62	4.99	5.39	6.2
7		3.31	3.85	4.39	4.64	5.3
10		2.75	3.31	3.66	3.93	4.61
15		2.23	2.7	3.1	3.28	3.74
20		1.94	2.36	2.64	2.89	2.7
30		1.6	1.99	2.18	2.39	2.7
45		1.32	1.6	1.79	1.96	2.23
60	0.85	1.08	1.37	1.51	1.66	1.89
100	0.69	0.88	1.09	1.19	1.36	1.54

Holding a single stock rather than a perfectly diversified portfolio increases annual volatility by roughly 30%. Thus, instead of bearing the market’s volatility of, say, 15%, the single-stock investor bears portfolio volatility of perhaps 45% [15% + 30%]. Roughly, investors may view volatility as the typical amount by which a portfolio’s return will deviate from long-term averages. Thus, the single-stock investor will experience annual returns that average a whopping 35% above or below the market—with some years closer to the market and some years further from the market.

As a rule of thumb, diversifiable risk will be reduced by the following amounts:

Holding 25 stocks reduces diversifiable risk by about 80%,

Holding 100 stocks reduces diversifiable risk by about 90%, and

Holding 400 stocks reduces diversifiable risk by about 95%.

All of these reductions are compared to the risk of holding a single stock. Thus, a single stock’s tracking error of about 40% would be reduced to about 8% if you hold 25 stocks, 4% if you hold 100 stocks and 2% if you hold 400 stocks.

In order to achieve the best diversification, stocks should be weighted by capitalization size. That means that a stock with a $300 billion total market value (currently, GE) should be held in a size about 20 times the size of a company with a market capitalization of about $15 billion (currently, Corning). Further, the larger a firm is, the more important that it be included in the portfolio. The largest 50 or so stocks should almost all be held in any portfolio of individual stocks attempting to match the performance of the overall market.

A portfolio with nearly equal weights (the same dollar amount of holdings in each stock) will not be well-diversified relative to the overall market regardless of how many stocks are selected. In fact, an equal-weighted portfolio of every U.S. stock will behave primarily like a small-cap fund, since approximately 3% of the holdings would be large cap, about 12% would be mid cap and about 85% would be small cap (and the small cap would be dominated by micro caps). Although our study reports the performance of equal portfolio weightings, this is not a recommended strategy for an investor.

The stocks should be selected at random (with selection probabilities directly proportional to capitalization size) or with careful and purposeful diversification, such as by selecting stocks from a variety of industries and balancing with respect to effects such as style (e.g., value or growth) and size. Our experience is that many investors select poorly diversified portfolios because they deliberately select a particular type of stock (e.g., low price-earnings ratios) or because they subconsciously are drawn to particular types of stocks (e.g., firms with consumer products that have familiar sounding names).

In short, investors - naive or (supposedly) sophisticated- are simply are out to lunch. To spend countless hours trying to find a portfolio of stocks is an exercise in futility. Just buy a cheap index fund. Of course to know which one to buy requires a review of FED material. Are you doing that?

Diversification: (T Rowe Price 2004) an investor who put all his money into the Standard & Poor's 500-stock index from 1955- 2003 would have earned 10.5 percent a year, on average. Had that investor moved just 20 percent of that stock portfolio into bonds, he would have enjoyed nearly as much in gains: 10 percent a year. And he would have earned those returns with 19 percent less volatility.

The problem is that you are looking at about 50 years of history. It is not the same 50 years of your retirement nor does it reflect the terrible 60's and 70's nor the debacle of 2000- 2002. Statistics are necessary to analyze- but you are not a statistic.

Nondiversified portfolios: (WSJ 2005) New research finds that stock trades made by households holding only one or two stocks outperformed those with more diversified portfolios, according to finance professors at the University of Michigan and the University of Illinois at Urbana-Champaign.

Wealthier investors -- those with brokerage accounts of $100,000 or more -- who held concentrated portfolios also outperformed comparable households with diversified accounts by at least four percentage points a year. In fact, wealthy households that invested in many stocks underperformed the markets by about 1% a year.

The study also found that the outperformance of households with concentrated portfolios is particularly strong if they hold "local" stocks -- defined as companies located within a 50-mile radius of their residences. That is because investors who live near the company they are investing in are likely to know more about whether that business is worth investing in.

On average, investors who held one or two stocks outperformed those who held three or more by about one percentage point during the year following a stock purchase. For households with portfolios of $25,000 to $100,000, the difference in returns for concentrated versus diversified investors is three percentage points. Households with stock portfolios valued at less than $25,000 didn't perform materially better.

The study analyzed 1,156,000 stock trades that individual investors made through a discount broker from 1991 to 1996.

Take specific note of the dates. Just about everything went up during that time. Everybody thought they were a winner. But the risk was unquestionably higher. Further, the study is inherently flawed unless you look at what these same people did with the stocks during 2000- 2003. Bet it wasn't pretty.

Wanna another example- take Merck. Great company, great payout, great whatever. It made Vioxx. Gee, it's price is now down about 40%. Lots of lawsuits waiting in the wings. Yet such a drop had a minuscule impact on the S&P.

Diversification Over Time (Goetzman, Kumar 2005)

During the 1991-96 sample period, the average number of stocks in investor portfolios has increased almost monotonically from 4.28 in 1991 to 6.51 in 1996 – an increase of almost 48% (see Table III, Panel A). Furthermore, the normalized portfolio variance has steadily decreased from 0.47 in 1991 to 0.31 in 1996 – a decrease of more than 34%. At a first glance, these two observations seem to imply that portfolio composition skills of investors have improved over time. However, when we compare investor portfolios to a benchmark of randomly constructed matching portfolios, we find that the average risk exposures of investor portfolios are significantly higher than that of matching benchmark portfolios. In fact, during our sample period the mean excess normalized variance (relative to benchmark portfolios) has increased from 44.14% in 1991 to 67.80% in 1996. This suggests that, to a large extent, improvements in the diversification characteristics of investor portfolios result from changes in the correlation structure of the U.S. equity market and not necessarily from investors’ improved abilities to construct better diversified portfolios.

(Malkiel and Xu (1997) report a similar finding by tracking the variation in correlations among industry portfolios during the 1970-95 time-period. They find that the mean correlation among portfolios decreases over time thereby suggesting that the risk reduction benefits of holding a diversified portfolio has increased over time.)

We find that the average correlations for both sets of portfolios decrease during the 1991-96 time period but the average correlation among stocks in actual investor portfolios is significantly higher than the average correlation among stocks in randomly constructed portfolios. For instance, the excess average correlation is 87.24% in 1991 and 61.69% in 1996. Again, these results suggest that investors’ portfolio composition skills have not improved over time.

We find that at an aggregate level reduction in portfolio variance over time is driven primarily by the changing correlation structure of the market.

DIVERSIFICATION LINK: Why Do Individual Investors Hold Under-Diversified Portfolios?,: WILLIAM N. GOETZMANN , ALOK KUMAR  The definitive article on diversification in the real world.

Investors’ personal characteristics, their stock preferences, and their behavioral biases jointly influence their diversification choices. Younger, lower-income (less wealthy), and relatively less sophisticated investors and those who follow price trends, prefer local (familiar) stocks, and exhibit over-confidence hold relatively less diversified portfolios.

Under-diversified investors exhibit strong style and industry preferences and they also prefer more volatile and positively skewed stocks. Furthermore, we find some evidence to support the asymmetric information hypothesis for under-diversification. In contrast, we find that factors such as small portfolio size, transaction costs, and search costs are unlikely determinants of investors’ diversification choices. The unexpectedly high idiosyncratic risk in investors’ portfolios results in a welfare loss.

Our results indicate that the majority of individual investors in our sample are underdiversified and the extent of under-diversification is more severe for investors who hold only retirement accounts (IRA or Keogh). Consistent with results from previous studies, we find that more than 25% of investor portfolios in our sample contain only one stock, more than 50% of them contain fewer than three stocks, and in any given month, only 5-10% of the portfolios contain more than ten stocks. Consequently, investor portfolios have extremely high volatility (more than 75% of investor portfolios have higher volatility than the market portfolio) and they have worse risk-return properties than randomly constructed matching portfolios.

In 1998, roughly 75% of households held five or fewer stocks.

Our results indicate that the majority of individual investors in our sample are underdiversified and the extent of under-diversification is more severe for investors who hold only retirement accounts (IRA or Keogh). Consistent with results from previous studies (e.g., Kelly (1995), Polkovnichenko (2003)), we find that more than 25% of investor portfolios in our sample contain only one stock, more than 50% of them contain fewer than three stocks, and in any given month, only 5-10% of the portfolios contain more than ten stocks. Consequently, investor portfolios have extremely high volatility (more than 75% of investor portfolios have higher volatility than the market portfolio) and they have worse risk-return properties than randomly constructed matching portfolios.

To gain further insights into the diversification choices of investors, we examine the time variation in the average diversification level of investor portfolios. We find that during the 1991-96 sample period, the average number of stocks in investor portfolios increases from four to seven. This results in a decrease in the average normalized portfolio variance. However, the improved diversification over time does not necessarily imply that investors’ portfolio composition skills have improved. We do not find any perceptible evidence of diversification improvement by active means. Over time, there is no decrease in either the excess average correlation (relative to benchmark portfolios) or the excess normalized variance of investor portfolios. This suggests that a significant part of diversification improvement results from passive means where investors increase the number of stocks in their portfolios without giving proper consideration to stock correlations.

Diversification level increases with income (wealth) as well as age which reflects an increasing degree of risk aversion with age and income (wealth).7 We also find that investor sophistication, in particular, their financial sophistication influences their portfolio choices. Relatively more sophisticated investors – investors who hold mutual funds, trade in options and foreign equities, and engage in short-selling – hold relatively more diversified portfolios. The degree of diversification also varies across occupation categories in a manner which further supports the view that investors’ diversification decisions depend upon age, income (wealth), and their level of financial sophistication. Investors who belong to the non-professional job category (bluecollar workers, clerical workers, sales and service workers, house-makers, and students) hold the least diversified portfolios in our sample while retired investors are on the other end of the diversification spectrum where they hold the most diversified portfolios. The diversification decisions of investors are also influenced by their preferences for certain stock characteristics and by their behavioral biases. We find that investors with strong style and industry preferences are less diversified. In addition, investors with a preference for volatile and positively skewed stocks hold relatively less diversified portfolios. Examining the influence of investors’ behavioral biases on their diversification decisions, we find that investors who are over-confident (i.e., investors who trade excessively and earn lower returns), investors who invest disproportionately in local (familiar) stocks, and those who are sensitive to price trends (i.e., trend-chasers and contrarian investors) exhibit a greater degree of underdiversification. The unexpectedly high idiosyncratic risk in investor portfolios results in a welfare loss

* Lack of diversification may also be induced by a variety of psychological factors. For instance, investors may ignore correlations due to their behavioral biases. Unlike stock prices (or returns) and trading volume, stock correlations are not directly available to investors and they must be computed. If investors formulate their investment decisions using information that is easily available, i.e., if they employ the availability heuristic (Tversky and Kahneman (1973)), they are more likely to ignore correlations and hold under-diversified portfolios. In addition, investors who rely on past price trends to formulate their trading decisions (e.g., Odean (1999), Dhar and Kumar (2001)) may ignore stock correlations (e.g., Kroll, Levy, and Rapoport (1988a, 1988b), Kroll and Levy (1992)). Furthermore, people who frame their investment decisions narrowly (e.g., Kahneman and Lovallo (1993), Barberis, Huang, and Thaler (2003), Kumar and Lim (2004)) may ignore the interactions among their individual stock selection decisions and they may ignore correlations among the stocks they hold in their portfolios.1 It is possible that investors take correlations into account but they mis-estimate the strength of correlations. Experimental research suggests that people misestimate correlations at an intuitive level – they under-estimate small correlations and they are bad at judging correlation differences when the correlations are low (Jennings, Amabile, and Ross (1982)).

Goetzman, Kumar

Diversification (2005) Investors who exhibit stronger biases hold less diversified portfolios. Our results are consistent with the findings in Odean (1999) who show that over-confident investors trade more actively and earn lower net returns. The lower level of diversification among active investors appears to be another manifestation of their over-confidence. Furthermore, as expected, the Adjusted Disposition Effect variable has a positive and significant coefficient estimate which indicates that a certain degree of diversification is an unintended consequence of investors’ reluctance to realize losses

Diversification. (2006) The Truth about Diversification by the Numbers (Ron Surz, CIMA, and Mitchell Price, Roxbury Capital Management)

"Ninety-five percent of the benefits of diversification are captured with a 30-stock portfolio." This is a common belief held by virtually all investment professionals. It’s based on research conducted by professors Lawrence Fisher and James H. Lorie (F&L) on NYSE-traded stocks during 1926-1965.1 In this article we clarify F&L’s work, offer a couple of alternative approaches, and update the analysis to incorporate January 1, 1986, through June 30, 1999.

F&L measured the percent of all possible reduction in dispersion achieved by portfolios of various sizes on average. The reductions were measured relative to the dispersion of a one-stock portfolio. In this context, “all possible reduction” is the denominator of the ratio and is the difference between the dispersion of a one-stock portfolio and the dispersion of a portfolio comprising all NYSE stocks. The numerator of the ratio is the difference in dispersion between a one-stock portfolio and that of an “N”-stock portfolio where “N” is the number of stocks. Their results are summarized in Table 1.

Hence it is commonly said that 90% of diversification is achieved with a 16-stock portfolio, and 95% is achieved with a 32-stock portfolio. We believe this is a misinterpretation of the F&L results, for reasons presented in the following.

F&L measure the reduction in total volatility, which includes both diversifiable, or specific, and non-diversifiable, or market, risk. Modern Portfolio Theory (MPT) postulates that only market risk is rewarded in the aggregate, so specific risk is to be avoided.

The reduction in specific risk is the benefit of diversification, and the basis for measures of diversification such as R-squared and tracking error. R-squared measures the percent of variance that is explained by the market, and is hence undiversifiable risk. Tracking error measures specific, or diversifiable, risk as the standard deviation of returns away from the market.

R-squared and tracking error are the measures that should be used to determine improvement in diversification since they are measures of diversification. Improvement in overall risk is interesting, but does not support statements about diversification. Some real examples will illustrate this point. We have repeated the F&L analysis using Portfolio Opportunity Distributions (PODs)2 for the period 1/1/86 to 6/30/99. PODs create all possible portfolios of a given size that could be held from stocks in the Compustat database, so the market in this analysis is broader than just the NYSE and includes NASD-traded securities. Table 2.a shows an F&L dispersion measure (standard deviation) and the two diversification measures. Table 2.b shows the percent of possible reduction derived from Table 2.a.

As can be seen from Table 2.b, we get results that are very similar to F&L for reductions in dispersion, or standard deviation, but the results for improvements in diversification are much less than previously thought. We now see that a 15-stock portfolio gets 76% of available diversification versus the F&L 93%; this improves somewhat to 82% if tracking error is used as the diversification measure, but note that the tracking error is still a formidable 8.1% per annum. Even at 60 stocks we still have less than 90% of the available diversification whereas F&L would suggest virtually full diversification at this level.

In addition to correcting the misunderstandings about the F&L work, we want to extend our analyses beyond just the average fund to encompass the full range of results. Table 3 shows the ranges of dispersion and diversification for various size portfolios. As can be seen in the table, some 15-stock portfolios have less dispersion than the market, but none comes close to the diversification of the market. Seen this way it seems so obvious that reductions in dispersion do not equate to improvements in diversification, but the old interpretation of the F&L work will probably live on.

We also want to acknowledge that most managers attempt to diversify beyond a randomly chosen portfolio. Table 4 shows how tracking error can be reduced with two different techniques—optimization and holding the largest names. As can be seen, computer optimization can significantly reduce diversifiable risk, but a less sophisticated approach of just holding the largest names can go a long way toward controlling tracking error.

Summary

The relationship between number of stocks held in a portfolio and diversification has been clarified. Fifteen-stock portfolios, on average, achieve only 75-80% of available diversification, not the 90%-plus previously believed. Even 60-stock portfolios achieve less than 90% of full diversification.

Conscious efforts to diversify can improve these figures, but even optimizations won’t achieve the diversification levels that were previously believed to be reached with simple random portfolios.

The implications of these findings for both the portfolio manager and the investor are significant.

The portfolio manager can no longer rely on a simple rule of thumb to decide on the number of stocks to include in the portfolio. Diversification is more complex than the old “30-stock” saw had suggested. Similarly, investors should be less sanguine in the achievement of their diversification objectives if this confidence has been based on a count of the names held in their portfolio.

Table 1

Percent of possible reduction in dispersion achieved through increasing the number of stocks in the portfolio.

All

1 2 8 16 32 128 Market

0 41 82 90 95 99 100

Table 2.a

Risk and Diversification Measures from 1/1/86 to 6/30/99 for portfolios of various sizes

All

                              1 15 30 60 Market

Standard Deviation 45% 16.5% 15.4% 15.2% 14.5%

R2                         0 0.76 0.86 0.88 1.0

Tracking Error      45 8.1 6.2 5.3 0

Table 2.b

Percent of possible reduction

All

1 15 30 60 Market

Standard Deviation 0% 93% 97% 98% 100%

R2 0 76 86 88 100

Tracking Error 0 82 86 88 100

Table 3 : Ranges of Dispersion and Diversification

A. Standard Deviation

15 Stocks 30 Stocks 60 Stocks Market

5 19.2 17.5 17.2

25 17.5 16.1 16.0

50 16.5 15.4 15.2 14.5

75 15.5 14.7 14.5

95 14.3 13.9 13.6

R-Squared

5 0.86 0.91 0.94

25 0.84 0.89 0.91

50 0.76 0.86 0.88 1.0

75 0.71 0.84 0.86

95 0.63 0.76 0.79

Table 4 : Reducing Tracking Error

Tracking Error

                     15 Stocks 30 Stocks 60 Stocks

Random        8.1            6.2             5.3

Largest          7.5           5.2              4.1

Optimized      5.4           4.2             3.5

Diversification

TIME DIVERSIFICATION LINK: (John Norstad 2006) While the basic argument that the standard deviations of the annualized returns decrease as the time horizon increases is true, it is also misleading, and it fatally misses the point, because for an investor concerned with the value of his portfolio at the end of a period of time, it is the total return that matters, not the annualized return. Because of the effects of compounding, the standard deviation of the total return actually increases with time horizon. Thus, if we use the traditional measure of uncertainty as the standard deviation of return over the time period in question, uncertainty increases with time.

Common variants of this time diversification argument can be found in many popular books and articles on investing, including those by highly respected professionals and even academics. For example, John Bogle used this argument in his otherwise totally excellent February, 1999 speech The Clash of the Cultures in Investing: Complexity vs. Simplicity (see his chart titled "Risk: The Moderation of Compounding 1802-1997," which if it had been properly drawn might well have been titled "Risk: The Exacerbation of Compounding 1802-1997"). Burton Malkiel uses a similar argument and chart in his classic book A Random Walk Down Wall Street (see chapter 14 of the sixth edition). (I deliberately chose two of my all-time favorite authors here to emphasize just how pervasive this fallacy is in the literature.)

The fact that some highly respected, justly admired and otherwise totally worthy professionals use this argument does not make it correct. The argument is in fact just plain wrong - it's a fallacy, pure and simple. When you see it you should dismiss it in the same way that you dismiss urban legends about alligators in sewers and hot stock tips you find on the Internet (you do dismiss those, don't you? :-). It's difficult to do this because the argument is so ubiquitous that it has become an unquestioned assumption in the investment world.

Time and Money (Norstad 2007) In the random walk model of the S&P 500 stock market index , the probability that a stock investment will earn less than a bank account earning 6% interest is 42% after 1 year. After 40 years this probability decreases to only 10%. Doesn't this prove that risk decreases with time?

The problem with this argument is that it treats all shortfalls equally. A loss of $1000 is treated the same as a loss of $1! This is clearly not fair. For example, if I invest $5000, a loss of $1000, while less likely, is certainly a more devastating loss to me than is a loss of $1, and it should be weighted more heavily in the argument. Similarly, the argument treats all gains equally, which is not fair by the same reasoning.

John Norstad has provided one of the best argument for the fallacy of time diversification . I approach the problem slightly differently since I actively manage clients' monies- but the essence is the same. Time does not reduce the risk of losing money. It INCREASES the risk. That is why literally all financial and investment plans nationally are incompetent at best and fraudulent at worst. But if one is acting as a fiduciary, the point is the same- the consumer has been provided information that is detrimental to their financial future.