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A Trip Down Memory Lane – Revisiting Portfolio Optimization

October 26
02:15 2011

(The following is one of a special five part series meant to be shared by professionals and non-professionals alike. This particular series covers only one of the 7 Deadly Sins Every ERISA Fiduciary Must Avoid.)

Let’s get one thing straight from the start: This is about unsystematic risk, not systematic risk. Systematic risk affects all investments. Diversification does not address systematic risk. Unsystematic risk is limited to very specific investments. 329557_1162_chevy_cherry_3_stock_xchng_royalty_free_300That’s why they also call it “specific” risk. You can diversify away unsystematic risk. And that’s precisely what a fellow by the name of Harry Markowitz began toying with before Elvis Presley even graduated high school. Today we call it “portfolio optimization,” but in 1952, Markowitz titled his Journal of Finance (7,1) paper “Portfolio Selection.”

By 1959, the father of Modern Portfolio Theory had refined his description when he published his book Portfolio Selection: Efficient Diversification of Investments (Wiley, New York). Those interested might note in those seven years, the crooner from Tupelo had become pretty efficient himself, with a diversified portfolio of an even dozen (including eleven in a row) chart-topping #1 hit songs on the Billboard Hot 100. Markowitz had been just as prolific, if only in the rarified air of finance, and not yet quite as famous. More important, he had come up with the beginnings of a quantifiably based approach to reducing “risk” in a portfolio. In effect, he had figured out a way to scientifically prove what the court had intimated in Harvard College v. Amory some 150 years earlier – adding more stocks in a portfolio could diversify a substantial amount of the risk.

From that point, the race was on. Who would be the first to answer the question “What’s the fewest number of stocks a portfolio should optimally hold?” Anyone taking an advanced degree program in economics might smile at the question as the mathematical elegance of those portfolio optimization formulae foisted upon them by a force of finance professors bubbles up within their rosy memories. Let’s take a trip down memory lane and recall some of the highlights of portfolio optimization and how it influenced our thoughts on diversification.

The next notable step in the search for the optimal portfolio size occurred in 1968 with the publication of “Diversification and the Reduction of Dispersion: an Empirical Analysis” by J.L. Evans and S.H. Archer in the  Journal of Finance (23, 761-767). The two researchers looked at several randomly selected portfolios and tried to determine how much risk was reduced based on the number stocks held. They concluded 10 randomly selected stocks can, for all intents and purposes, achieve sufficient diversification. This development immediately questioned the economic justification of any portfolio consisting of more than 10 stocks. Needless to say, this rocked the small world of portfolio managers that then existed on their daily ration of the Nifty Fifty.

Two years later, L. Fisher and J.H. Lorie published “Some Studies of Variability of Returns on Investments in Common Stocks” in the Journal of Business (43 (1970), 99-134). This duo lent further support to the thesis developed by Evans and Archer. Fisher and Lorie created simulated results for a broad array of portfolio sizes using every stock listed on the New York Stock Exchange. By looking at the return distributions for the years between 1926 and 1965, they showed one could diversify away the unsystematic risk by 80% using a portfolio of just eight stocks.

By now the go-go 60’s had begun to melt down into the stagnant 70’s, and the prominence of the Blue Chips faded to black with the rise of the small caps (multiple year recessions tend to do that). A new breed of renegades left the stultifying trust departments and brokerage firms, creating a slew of registered investment advisers. They more aggressively pursued the previously neglected smaller companies as the no-longer-Nifty Fifty wallowed in infamy.

Within the walls of ivy, a different kind of newness had taken hold. A growing number of published pieces began speaking of “the market” as if it could be a portfolio. Portfolio optimization, in a sense, became a loser’s game, with the only pure diversification consisting of market-sized portfolios. Of course, the business schoolmarms would insist an “index” portfolio represented a mere theoretical construct and could not be created in the real world. But by mid-decade, Bogle had birthed his baby proving the folly of those professorial disclaimers.

But, could such a large portfolio really achieve optimal diversification, even in theory? In the year marking the death of Elvis Presley, along came E.J. Elton and M.J. Gruber with their paper entitled “Risk Reduction and Portfolio Size: an Analytical Solution” (Journal of Business, 50 (1977), 415-437). Rather than continuing on the road paved by Evans and Archer, these two went back to Markowitz. Elton and Gruber then developed a precise quantitative formula relating risk to the number of holdings in a portfolio. They then used that device to attack the work of Elton and Gruber as well as Fisher and Lorie. Implying the prevailing consensus suggesting optimal diversification would be achieved with only 10 or 20 stocks, Elton and Gruber concluded risk can be significantly reduced when one increases the size of the portfolio from 15 to 100 stocks.

But the pressure of real world dynamics could not be ignored, not even by academia. J. Mayshar, with his publication of “Transaction Cost in a Model of Capital Market Equilibrium” in the Journal of Political Economy (87 (1979), 673-700), was the first to point out the problems association with overdiversification. Mayshar, repeating the practical constraints of the business school professors, cited practical considerations would mute any perceived benefits of supposed diversification of portfolios with a large number of holdings. Specifically, he argued increasing transaction costs hurt more than any good derived from diversification. He then presented an “equilibrium model,” showing, in light of transaction costs, a portfolio is optimized when it limits the number of stocks it holds.

Mayshar’s research nipped Elton and Gruber in the bud, and later studies returned to the issue of trying to determine how many stocks the optimally diverse portfolio should hold. In 1982, T. Tole paper “You Can’t Diversify Without Diversifying” appeared in the Journal of Portfolio Management, (8, 5-11). Tole conducted his research using a method employing R2 rather than the usual averaging technique. He concluded optimal diversification occurred in a portfolio consisting of 25-40 stocks.

In 1987, the appropriately named M. Statman published the seminal paper on this subject, coincidentally entitled “How Many Stocks Make a Diversified Portfolio?” (Journal of Financial and Quantitative Analysis, 22 (1987), 353-363). He concluded a well-diversified portfolio should optimally hold between 30-40 stocks. Needless to say this became fodder for the standard MBA curriculum from the late 1980’s. But, lest you think it’s outdated, you should note well the 2010 CFA Study Guide still highlights this particular piece of research.

Statman’s work seems to have exhausted any further new discoveries in this line of research. Most of the ensuing studies appear to merely confirm Statman, albeit using foreign market data. Huck Khoon Chung’s 2000 PhD thesis for Universiti Sains Malaysia “How Many Securities Make a Well-Diversified Portfolio: KLSE Stocks.” PhD Thesis, was based on the data of Malaysian stocks from 1988-1997. It used the Statman technique and the Markowitz model in its analysis. Under Statman, the optimal portfolio contained 27 stocks. Using the Markowitz model, the optimal portfolio contained 22 stocks.

Most recently, F. S. Al Suqaier  and H. A. Al Ziyud, (“The Effect of Diversification on Achieving Optimal Portfolio,” European Journal of Economics, Finance and Administrative Sciences, 32 (2011)) likewise looked at data from the Amman Stock Exchange (ASE) over the period 2/12/2005 to 13/3/2010. They concluded “that (43%) of the total risk could be eliminated by holding (2) stocks portfolio, and holding only (3) stocks portfolio could reduce half of the total risk. As we hold (15) stocks portfolio, investors can reduce (92%) of the total risk, and to eliminate (96%) of the total risk investors need to hold (25) stocks portfolio finally, (55) stocks portfolio to eliminate (99%) of the total risk reduction obtained through diversification. In addition, holding (15-16) stocks are required to capture most of the benefits associated with diversification.”

After more than a half century of research, academia seems to have settled upon a range defining the limits of the usefulness of diversification regarding the number stocks in a portfolio. This range appears to be from 15-50 stocks with the most likely optimal amount between 30 and 40 stocks. Beyond that, the cost of diversification both in terms of out-of-pocket expenses and the risk/return tradeoff start to eat into a portfolio’s performance.

Ironically, just as this process began, The King sang us the ultimate advice. On January 4, 1956, Elvis released the single “Too Much.” Two days later he performed it on the Ed Sullivan Show. It would reach #1 on the Billboard chart. If we only knew then what we know today…

Part I: 7 Deadly Sins Every ERISA Fiduciary Must Avoid: The 4th Deadly Sin – The Overdiversification
Part II: A Trip Down Memory Lane – Revisiting Portfolio Optimization
Part III: Overdiversification and the 401k Investor – Too Many Stocks Spoil the Portfolio
Part IV: Why Overdiversification Matters to the ERISA Fiduciary
Part V: How Plan Sponsors Can Help 401k Investors Avoid Overdiversification

About Author

Christopher Carosa, CTFA

Christopher Carosa, CTFA


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