An Optimal Asset Allocation Strategy with 2-3 Asset Classes

Since the advent of Mean-Variance Optimization, the approach to building an efficient strategic asset allocation portfolio was to add as many “distinct” asset class inputs to the optimizer. Several asset classes meant there was a greater potential for more efficient and more diversified strategies. What this paper does is focus on Markowitz’s intention and end with testing a more pure version of a 60% stocks, 40% bonds asset allocation strategy. 


It’s been numerous decades since the Markowitz paper was first disseminated. Numerous investment professionals starting in the early 1990s started citing the Markowitz paper and the diversification benefits of an optimized strategic asset allocation portfolio. Yet, the established practice and use of long-run expectations based on 85-year risk premiums isn’t what Markowitz proposed in his paper nor did he suggest the use of a set it and forget it strategic asset allocation approach for managing assets. He envisioned a more active approach based on forecasts.

The foundation of the paper is centered on the following section:

“The process of selecting a portfolio may be divided into two stages. The first stage starts with observation and experience and ends with beliefs about the future performances of available securities. The second stage starts with the relevant beliefs about future performances and ends with the choice of the portfolio. This paper is concerned with the second stage.” 

Harry’s intent with the first stage was not that returns be based on a historical equity risk premium taken from the Ibbotson yearbook.

Yet, the industry created strategic asset allocation and drew investment professionals to the sophisticated appeal of using desktop mean-variance optimization to create strategic asset allocation portfolios based on the Ibbotson risk premiums. This approach was spread across both advice and product models for doing business. 

Instead of a plethora of asset classes including large, mid, small, growth, value, and blend, Markowitz explicitly said “It is necessary to avoid investing in securities with high covariances among themselves.”

So unless one has “beliefs about the future performances of available securities” such as growth and value, small and mid-cap especially since they are highly correlated then the question falls to how many asset classes and representative securities are truly needed to develop a fully diversified strategy.

What we’ll learn is the use of multiple asset classes is both unnecessary and riskier than two-three broad asset classes such as US Stocks, International Stocks, and US Bonds. The major aggregate asset classes that are represented by firms that develop indices and asset managers that make available index investment vehicles.  

Efficient Capital Markets

In 1970 Eugene F. Fama, wrote a review of the work related to the efficient markets hypothesis (EMH) including the work of Benoit Mandelbrot and Paul Samuelson. EMH suggested that stock prices were impossible to predict because “all information” was already built into the prices.

Fama states “the assumption that the conditions of market equilibrium can be stated in terms of expected returns elevates the purely mathematical concept of expected value to a status not necessarily implied by the general notion of market efficiency. The expected value is just one of many possible summary measures of a distribution of returns, and market efficiency per se (i.e., the general notion that prices “fully reflect” available information) does not imbue it with any special importance.”

The implication for the asset management industry was obvious and eventually it became evident in the returns data when researchers started to look for evidence of EMH. The message: it is impossible for professional money managers to outperform the market.

The Performance Of Mutual Funds – In The Period 1945-1964

In this 1967 study using Jensen’s Alpha, the researcher Michael C. Jensen looked at net and gross fund performance of fund managers. This was his conclusion on the ability of fund managers to predict stock prices:

“The evidence on mutual fund performance indicates not only that these 115 mutual funds were on average not able to predict security prices well enough to outperform a buy-the-market-and-hold policy, but also that there is very little evidence that any individual fund was able to do significantly better than that which we expected from mere random chance. It is also important to note that these conclusions hold even when we measure the fund returns gross of management expenses (that is assume their bookkeeping, research, and other expenses except brokerage commissions were obtained free). Thus on average the funds apparently were not quite successful enough in their trading activities to recoup even their brokerage expenses.”

The Losers Game

Before Roger Ibbotson started Ibboston Associates to give the financial services industry all the tools necessary to run an advice business, Yale Professor Charles D, Ellis wrote a landmark advocacy paper for individual investors and indexation. “The Loser’s Game” contended:

“Disagreeable data are streaming out of the computers of Becker Securities and Merrill Lynch and all the other performance measurement firms. Over and over and over again, these facts and figures inform us that investment managers are failing to perform. Not only are the nation’s leading portfolio managers failing to produce positive absolute rates of return (after all, it’s been a long, long bear market) but they are also failing to produce positive relative rates of return. Contrary to their oft articulated goal of outperforming the market averages, investment managers are not beating the market: The market is beating them.”

Ellis pointed out that “Professionals win points, amateurs lose points.” He used tennis as an analogy to get across his point in the paper.

“In expert tennis, about 80 percent of the points are won; in amateur tennis, about 80 percent of the points are lost. In other words, professional tennis is a Winner’s Game – the final outcome is determined by the activities of the winner – and amateur tennis is a Loser’s Game – the final outcome is determined by the activities of the loser. The two games are, in their fundamental characteristic, not at all the same. They are opposites.”

On performance Ellis points out:

“The disagreeable numbers from the performance measurement firms say there are no managers whose past performance promises that they will outperform the market in the future. Looking backward, the evidence is deeply disturbing: 85 percent of professionally managed funds underperformed the S&P 500 during the past 10 years. And the median fund’s rate of return was only 5.4 percent – about 10 percent below the S&P 500.”

Ellis went on to found Greenwich Associates, become part of Yale’s investment committee, and join the board of his friend’s John Bogle’s, The Vanguard Group.

Graham’s Later View of Active Management

Benjamin Graham, the dean of active value investing and security selection, was also well aware that the superior rewards he had reaped using his valuation principles would be difficult to achieve in the future. In a 1976 interview, he made this remarkable concession, “I am no longer an advocate of elaborate techniques of security analysis in order to find superior value opportunities. This was a rewarding activity, say, 40 years ago, but the situation has changed a great deal since then. In the old days, any well-trained security analyst could do a good professional job of selecting undervalued issues through detailed studies; but in the light of the enormous amount of research now being carried on, I doubt whether in most cases such extensive efforts will generate sufficiently superior selections to justify their cost.”

Over time, more academics started to look at portfolio management models and performance.

On the Impossibility of Informationally Efficient Markets

As Sanford Grossman and Joseph Stiglitz point out in their 1980 paper, On the Impossibility of Informationally Efficient Markets, there must be “sufficient profit opportunities, i.e., inefficiencies, to compensate investors for the cost of trading and information-gathering.” While they argue that there are some returns for investors, they suggest that the rewards investors gather are commensurate with the costs they bear. Investors clearly seek and exploit obvious profit opportunities (which is why they are so rare).

On Persistence in Mutual Fund Performance

In 1997, Mark M. Carhart, released his paper “On Persistence in Mutual Fund Performance” in the Journal of Finance. The research “finds that “performance does not reflect superior stock-picking skill. Rather, common factors in stock returns and persistent differences in mutual fund expenses and transaction costs explain almost all of the predict-ability in mutual fund returns. The results do not support the existence of skilled or informed mutual fund portfolio managers.”

The Intelligent Asset Allocator

In 2000, William J Bernstein wrote The Intelligent Asset Allocator which laid the foundation for individual self-directed investors to utilize modern portfolio theory to build multi-asset strategies.

His book looks at multiple asset classes including T-bills, treasuries, stocks, REITS, small and value stocks, international, emerging markets stocks, and precious metals. It uses standard deviation and return for comparisons.

It also focuses on correlation and the development of efficient strategies by blending asset classes to improve risk-adjusted returns.

The book expounds on model portfolios, utilizing index funds. He emphasizes that:

“Asset allocation is the only factor affecting your investments you can actually influence.”

He lays out several multi-asset class model variations for the individual investor and makes it clear:

“The book is aimed at the investor who wishes to squeeze every bit os return possible out of a given degree of risk. As we have seen, the essence of this involves splitting your portfolio into many small imperfectly correlated parts.” Bernstein points out that if you go a more simple route of just two “all-U.S.” funds, one stock and one bond fund, “you are probably sacrificing 1$ to 2% of long-term return for a given degree of risk.”

The emphasis here is that more asset classes is better.

Unconventional Success: A Fundamental Approach to Personal Investment

By the time David F. Swenson wrote this investment book for individuals he was widely regarded for his institutional investment book Pioneering Portfolio Management: An Unconventional Approach to Institutional Investment (2000) and for the Yale Endowment’s use of alternative asset classes to reduce risk. Later came the ridicule for the dismal portfolio performance during the ’08-’09 market drop. Apparently the Yale model’s alternative asset classes turned out to be more correlated than expected and the Yale portfolio performed worse than expected.

As for his recommendation to asset allocation for individuals, here is how he viewed the process and the number of asset classes to use in his 2005 book, Unconventional Success: A Fundamental Approach to Personal Investment.

“In the portfolio construction process, diversification requires that individual asset-class allocations rise to a level sufficient to have an impact on the portfolio, with each asset-class accounting for at least 5 to 10 percent of assets. Diversification further requires that no individual asset class dominate the portfolio, with each asset class amounting to no more than 25 to 30 percent of assets.”

When Diversification Fails

In their 2018 paper, When Diversification Fails, authors and T. Rowe Price portfolio managers, Sébastien Page, CFA, and Robert A. Panariello, CFA conclude:

“One of the most vexing problems in investment management is that diversification seems to disappear when investors need it the most. We surmise that many investors still do not fully appreciate the impact of extreme correlations on portfolio efficiency—in particular, on exposure to loss.”

They recommend:

“Prudent investors should not use them (correlations) in risk models, at least not without adding other tools, such as downside risk measures and scenario analyses. To enhance risk management beyond naive diversification, investors should re-optimize portfolios with a focus on downside risk, consider dynamic strategies, and depending on aversion to losses, evaluate the value of downside protection as an alternative to asset class diversification.”

The paper proceeds to ask: Is the Stock–Bond Correlation the Only True Source of Diversification?

“When market sentiment suddenly turns negative and fear grips markets, government bonds almost always rally because of the flight-to-safety effect (Gulko 2002). In a sense, duration risk may be the only true source of diversification in multi-asset portfolios. Therefore, the expected stock-bond correlation is one of the most important inputs to the asset allocation decision.”

Common Sense on Mutual Funds

In 2009, John Bogle released the 10th Anniversary Edition of Common Sense on Mutual Funds.

Bogle makes the point that consistently on a risk-adjusted basis low-cost funds provide better returns (small-cap blend IS the exception).

In the mutual fund world, future fund expense ratios, unlike future fund relative returns, are highly predictable. We now know — as a certainty — that cost matters. It matters for equity funds in the aggregate; it matters far more for bond funds.

Some of Bogle’s notable colleagues also had positive views on indexing.

Paul Samuelson, Nobel Laureate: “The most efficient way to diversify a stock portfolio is with a low-fee index fund. Statistically, a broadly-based stock index fund will outperform most actively managed equity portfolios.”
Professor Burton Malkiel, author of A Random Walk Down Wall Street: “I recommend a total market index fund—one that follows the entire U.S. stock market. And I recommend the same approach for the U.S. bond market and international stocks.”

Bogle on Asset Allocation

“Stick to simplicity. Don’t complicate the process. Basic investing is simple—a sensible asset allocation to stocks, bonds, and cash reserves; a selection of middle-of-the-road funds that emphasize high-grade securities; a careful balancing of risk, return, and (lest we forget) cost.”

As far as implementation:

“When you select your portfolio’s long-term allocation to stocks and bonds, you must make a decision about the real returns you can expect to earn and the risks to which your portfolio will be exposed. You must also consider the costs of investing that you will incur. Costs will tend to reduce your return and/or increase the risks you must take. Think of return, risk, and cost as the three spatial dimensions—the length, breadth, and width—of a cube.

“With the stage thus set, however roughly, for future market returns, what does Reversion To The Mean (RTM) suggest about equity investment strategy? Since RTM prevails among all market sectors such as growth stocks and value stocks, large-cap stocks and small-cap stocks, and U.S. stocks and international stocks, most investors should own equity funds that represent a broad cross-section of the U.S. stock market, in which large-cap stocks are the predominant component. Investors who believe they can garner a performance edge by selecting (or even overweighting) funds with different investment styles and strategies should be aware of the risks involved in doing so. For those who believe that the clear lessons of history are pointing us in the wrong direction (always a risky bet), an equally risky bet remains: determining which of these countervailing segments will in fact prove to be superior in the years to come. If, for example, large-cap and small-cap stocks do not revert to the market mean over the next 10 to 20 years, an investor has to guess which of the two is more likely to provide superior returns. It is for this reason that I prefer, on both theoretical and practical grounds, index funds that track the total U.S. stock market. With their extraordinarily broad diversification, over a wide-ranging spectrum of large-, mid-, and small-cap stocks alike, these funds are the ultimate response to the power of RTM in the stock market.

A decision to own an all-stock market index fund also solves the problem of fund selection. Why fly in the face of historical evidence by trying to select individual mutual funds in the hope of picking a  big winner? Given the power of mean reversion in the returns of individual mutual funds, an index fund provides the most reliable participation in the future returns of equities as a group. Surely it has proved its worth in the past. Notwithstanding my preference for the total market fund, a Standard & Poor’s 500 Index fund is by no means an unacceptable choice. This large-cap index fund carries a 75 percent weight in the U.S. stock market, and cannot diverge widely from the total market, even in short-term periods. RTM suggests that its long-run returns will closely parallel those of the total market. Given low costs, either index fund should provide investors with the best possible opportunity to earn returns approaching 100 percent of the market return.”

Bogle on International Allocations

“I would add that  I am not persuaded that international funds are a necessary component of an investor’s portfolio. Foreign funds may reduce a portfolio’s volatility, but their economic and currency risks may reduce returns by a still larger amount. The idea that a theoretically optimal portfolio must hold each geographical component at its market weight simply pushes me further than I would dream of being pushed. My best judgment is that international holdings should comprise 20 percent of equities at a maximum and that a zero weight is fully acceptable in most portfolios.”

“Large additional exposure to foreign stocks to invest in foreign nations is not essential. In terms of risk and return, the record of the past — whether prologue to the future or not — does not provide compelling reasons to abandon the acres of diamonds that can be unearthed at home in order to seek unknown diamond lodes abroad.”

“I want to reemphasize my reluctance to embrace the idea of holding a true global portfolio, in which a U.S. investor’s market weighting would be based on the weights of the markets of each major nation, resulting, in mid-2009, in 44 percent U.S. stocks and 56 percent international stocks. But I have no reluctance whatsoever to emphasize a truly global strategy, focused largely on U.S. stocks. After all, the major U.S. corporations include some of the largest firms in the world, doing business all over the globe. In 2008 foreign sales represented 48 percent of all sales for the firms in the Standard & Poor’s 500 Index, up from 42 percent in 2003. So I continue to believe it is not necessary to stray too far from home.”

Bogle on Implementation

Bogle saw that three funds were better. He thought, “The beauty of owning the market is that you eliminate individual stock risk, you eliminate market sector risk, and you eliminate manager risk; There may be better investment strategies than owning just three broad-based index funds but the number of strategies that are worse is infinite.”

“A recent study by Morningstar Mutual Funds—to its credit, one of the few publications that systematically tackles issues like this one— concluded essentially that owning more than four randomly chosen equity funds didn’t reduce risk appreciably. Around that number, risk remains fairly constant, all the way out to 30 funds (an unbelievable number!), at which point Morningstar apparently stopped counting. Figure 4.6 shows the extent to which the standard deviation of the various fund portfolios declined as more funds were added.”

A Case for Index Fund Portfolios

In their paper, A Case for Index Fund Portfolios, authors Richard A. Ferri and Alex C. Benke, state that “investors holding only index funds have a better chance for success.” To avoid the survivorship bias that exists commercially available mutual fund databases, in their research, Ferri and Benke used the “CRSP Survivor-Bias-Free US Mutual Fund Database” which includes funds that have failed or merged over time.

They “looked at the performance of the Vanguard Total Stock Market Index Fund Investor Shares relative to actively managed large-cap fund portfolios, the performance of the Vanguard Total International Stock Index Fund Investor Shares relative to foreign stock funds, etc. We then combined these probabilities together to estimate the probability that a portfolio of index funds will outperform, and compared this to the actual results of the scenario.

Table 2 lists individual category results for the index fund portfolio funds used in Scenario 1. The “Index Portfolio Win %” column is the probability of the fund to outperform a randomly selected actively managed fund in its category over the 16-year period. “Median Performance Loss” is the relative median performance of the active funds that Line 4 in Table 2 shows the weighted average of the three individual index funds based on 40% US equity, 20% international equity, and 40% US investment-grade bond fund portfolio. Weighting the three index funds using 40%/20%/40% suggests an expected index portfolio win rate for the portfolio to be 79.9%. In reality, the actual win rate was 82.9% from Scenario 1. Our estimate was 3.0% below. This was an unexpected outcome.

In addition, the “Median Performance Win” of 0.74% based on the 40%/20%/40% asset allocation method was higher than the actual outcome. In the actual 5,000 simulated trials, the median outperforming actively managed portfolio won by only 0.52% annually, one-third less than the expected outperformance.

The paper concludes:

Mutual fund portfolios holding only index funds have performance advantages over comparable portfolios that hold only actively managed funds. These advantages were quantified by running several scenarios that measured and compared strategy performance over time, both nominally and risk-adjusted.

How many index funds are optimal?

To get to the heart of this question The Vanguard Group of funds admiral shares classes were utilized because of their representation at the major and then the sub-asset class level. The following breaks out two sets of asset class types Set A without international and Set B with international.

Correlations – Set A

The following correlation table clearly shows that for Set A, the two sets of funds with the lowest correlation and diversification benefits are the Total Stock & Total Bond and Small Cap & Total Bond. Total Bond has a clear advantage over the bond sub-asset classes. Total Stock is not a clear winner over the other stock sub-asset classes.

Correlations – Set B

The following correlation table clearly shows that for Set B, the two sets of funds with the lowest correlation and diversification benefits are the Total Stock & Total Bond and Small Cap & Total Bond. The international funds had negligible correlations and diversification benefits.

Summary Statistics – Set A

The following summary statistics for the larger asset class strategy set show no benefits advantage across any measurement. 

Summary Statistics – Set B

The following summary statistics show higher risk-adjusted return measurements for the three asset class strategy.


The performance during drawdown periods, reflect an advantage for a 2-3 asset class strategy.

Disclosure: The use of Vanguard funds as proxies for asset classes is not a recommendation nor an endorsement of the funds. Individual circumstances should play a role when it comes to investment advice and recommendations. Additionally, the author is not a registered investment advisor.


Bernstein, William J., The Intelligent Asset Allocator

Swenson, David F., Unconventional Success: A Fundamental Approach to Personal Investment

Bogle, John, Common Sense on Mutual Funds


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