The “Most Diversified Portfolio”

CAPM / Alpha Theory 27 Mar 2011

It’s an axiom that the “most diversified” portfolio, and therefore the portfolio used as the basic benchmark for active management, is the so-called “market portfolio” – a value-weighted combination of the complete opportunity set facing investors.  For investors in public equities, the S&P 500 is often a proxy for this portfolio.  But as regular readers are aware, debate rages over the appropriateness of such a portfolio.  Commentators such as Rob Arnott say that this so-called “passive” portfolio inherently under-weights undervalued stocks and over-weights overvalued ones.  They suggest a “fundamental index” where weights are determined by fundamental economic drivers.  Today’s guest contributor takes a different tack when addressing this issue.  Steve Sapra of asset manager TOBAM (see previous AAA posts by and about Sapra here, here and here) points out that true diversification requires that holdings have a low correlation to each other.  Investors in equity markets that are dominated by one or two sectors (say, mining or oil) will find Sapra’s views particularly helpful.

Special to AllAboutAlpha.com by: Steve Sapra, CFA, Ph.D., Managing Director, TOBAM North America

Intuitively, we all know that the magic of diversification happens because a portfolio is greater than the sum of its parts. Or more precisely, the risk of a portfolio is less than the weighted-average risk of its component holdings. Consider the metric known as the Diversification Ratio developed by Choueifaty [2006]:

where wi is the portfolio weight in asset i, σi is the risk of asset i, and σp is the total risk of the portfolio. The numerator of this equation is the weighted-average volatility of the individual securities. It is also equal to the risk of a portfolio if every asset were perfectly correlated with all other assets. The denominator, on the other hand, represents the actual risk of the portfolio. Since in reality assets are not perfectly correlated, portfolios with more than two assets will always be characterized by a DR greater than one (it will equal one only for single assets). But how does one interpret the DR?

Consider a portfolio which holds equal weights in two uncorrelated assets with equal volatility. Such a portfolio will have DR = number of independent factor exposures in the portfolio. The term effective is important because market factors are never truly independent as sectors and styles often move in the same direction, albeit imperfectly. For example, in a two-factor world with a correlation between the factors of 0.3, the effective number of factors available in the market is only 1.5.

Importantly, what makes a portfolio’s DR large is not necessarily holding a large number of assets. Rather, in order for a portfolio to be characterized by a high DR it must be exposed to a . Holding 500 oil stocks is not diversifying; holding the stocks of 50 companies with relatively low correlations between their cash flows is. The latter portfolio will have a higher DR than the former, despite the fact that it holds 90% fewer names.

With the DR, we now have the ability to precisely measure the degree of diversification in any portfolio. Below, we plot the squares of the DR (DR2) over time for the MSCI U.S., MSCI Europe, and MSCI Developed World indices.

Several observations are clear from this chart. First, we see that the U.S. and European indices are less diversified than the World index. As we increase the number of countries and industries in a portfolio, we are essentially expanding the opportunity set in terms of effective risk factors, naturally resulting in greater diversification potential. Secondly, we see that the number of effective factors in the market changes through time. For example, using the MSCI World index, we see the range is roughly between 3 and 10 factors globally. As new sources of common variation evolve, such as the ‘new economy’ factors in the late 1990s, the diversification opportunities increase. In some cases effective factors ‘disappear’, as globalization of the economy reduces the opportunity set in some respects. Internet retail stocks like Amazon, for example, at one time behaved quite distinctly from other stocks. Ultimately, these companies simply became ‘Consumer Discretionary’ stocks, resulting in greater commonality with the rest of the market. Today we could be seeing a similar, albeit smaller, effect with respect to the development of ‘green’ technology.

We have shown that the DR can be used to measure diversification by determining the number of effectively independent factor exposures contained in a portfolio of risky assets. But what if we seek to build a portfolio which actually maximizes the DR? We call this portfolio the Most Diversified Portfolio (MDP). What would be some of the important characteristics of a portfolio which was designed to be the most diversified portfolio one could hold?

Before we entertain such an idea, let us explore a particular interpretation of this exercise. Assume that investors possess no a priori information on the expected returns of assets other than the general opinion that high (low) risk assets command a high (low) expected returns. Under this assumption, we can express the expected excess return of asset i as E (ri) – f = kσi where r f is the risk-free rate and k is a constant. Hence, the expected return on a portfolio, P, comprised of N assets can be expressed as E (rp) – f = k ΣNi=1 wiσi. Dividing both sides of the return equation by σp, we see that a portfolio’s expected Sharpe Ratio is linear in diversification:

Thus, in this case, maximizing the DR can be interpreted as maximizing the portfolio’s Sharpe Ratio.

In the presence of a long-only constraint, the MDP will generally hold only a subset of the assets in the investment universe. Assets which are non-diversifying – in the sense that they have relatively high correlation with to the MDP – are those securities which are excluded [Choueifaty et al., 2011]. Conversely, the assets contained in the MDP will be those securities which have the lowest (and equal) correlation with the MDP. Thus, we can think of the DR maximization problem in following way: The selection of a portfolio whose holdings have the lowest correlation to itself, while simultaneously excluding the assets with which it is most highly correlated. This statement illustrates that the MDP is so diversified, that all assets in the universe considered are effectively represented in the portfolio, even if the MDP does not physically hold them.

In order to test the benefit of diversification maximization, we used portfolio optimization to build long-only, unleveraged portfolios which maximized the ex-ante DR in each period over the MSCI World universe. The universe was pre-screened for liquidity in order to avoid any bias toward illiquid securities. We then compared the actual performance of this strategy to the MSCI World index itself. The table below compares the performance of the two portfolios:

We see that the MDP significantly outperforms the benchmark on an absolute basis, but more importantly, on a risk-adjusted basis as well. It is important to remember that maximum-diversification is a Sharpe-centric exercise; the Information Ratio of the portfolio (the ratio of active return to active risk) is not a consideration in the portfolio construction process. We see that the Sharpe Ratio of the MDP is 0.42 vs. 0 for the index. This results from the higher absolute return of the MDP as well as the materially lower volatility. Finally, recall that the square of the DR of the MDP measures the number of independent factor exposures available in the market. We see that the MDP has on average been exposed to 13.7 (3.72) independent factors while the benchmark has been exposed to only 5.3 (2.32). In other words, the MDP is exposed to 2.5 times the number of risk factors as the MSCI World, as the market index fails to fully diversify across all of the relevant dimensions of risk. Ultimately, the materially higher diversification of the MDP results in a notable improvement in risk-adjusted performance, not just ex-ante, but ex-post as well.

What does it mean to hold an efficient portfolio? The CAPM [Sharpe, 1964] tells us that under its stringent assumptions of perfect market efficiency and homogeneity of beliefs (amongst several others), that the capitalization-weighted market index is maximally efficient. This implies that we can do no better than the market in terms of risk-adjusted returns as measured by the Sharpe Ratio. However, the CAPM result, while elegant, is based on many simplifying assumptions unlikely to hold true in the real world. Sharpe himself, in his seminal paper, states that the CAPM assumptions of free borrowing and identical beliefs are “highly restrictive and unrealistic assumptions.”

What if we relax those assumptions even slightly? Treynor [2005] has shown that under fairly innocuous assumptions about the structure of market pricing, the cap-weighted market portfolio will be sub-optimal. If it is possible for asset prices to temporarily vary (in a multi-period timeframe) from their fundamental value, then by definition a cap-weighted index will have placed too much weight on over-priced securities and too little weight on underpriced securities. An implication is that which is unrelated to price will have a higher expected Sharpe ratio than the benchmark and hence be closer to the efficient frontier.

As shown in the previous section, the market portfolio is characterized by relatively concentrated exposure to the effective sources of common risk in the marketplace. A portfolio which maximizes the DR on the other hand has, by definition, a more diverse exposure to the relevant dimensions of common variation. While it is difficult to say ex-ante whether or not a portfolio will be on the efficient frontier, it is clear from the previous section that a portfolio which maximizes the DR has yielded a material improvement in the ex-post Sharpe Ratio relative to the cap-weighted index.

The Diversification Ratio is a new metric which extracts the number of effective factor exposures contained in a portfolio. Highly concentrated portfolios, not in number of positions, but in a particular sector or perhaps a certain style, will be characterized by a low DR. Conversely, portfolios exposed to a broad distribution of diverse risk factors will have a relatively high DR, regardless of the physical number of positions. We showed that when we set an objective of maximizing the DR, the resultant portfolios are characterized by materially higher ex-ante and ex-post Sharpe Ratios than the cap-weighted benchmark. The improvement in portfolio efficiency is a direct result of the fact that such portfolios are characterized by a broad exposure to a diversified set of global risk factors. Portfolios which are exposed to diverse sources of common risks, have shown to be characterized by improved performance on a risk-adjusted basis.

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6 Comments

  1. Lazy eye
    March 28, 2011 at 5:12 am

    “The first and most important feature of the S&P 500 is that it does not employ a passive selection strategy…”
    http://www.bylo.org/leggmasn.html

    “…the S&P 500 is an actively selected index. Its stocks are chosen from the nearly 9000 publicly traded securities on the New York, American and NASDAQ markets by a committee using specific investment criteria. The returns of the S&P 500 are the best evidence of the long-term advantage of active portfolio management. It has consistently beaten broader, passively constructed indices such as the Wilshire 5000, the Russell 2000, and the NYSE Composite. That it has also beaten other active money managers is not an argument against active management, it is an argument against the methods employed by most active managers.

    The S&P 500 is a long-term oriented, low turnover index employing a buy and hold strategy, which is by nature tax efficient. It lets its winners run, and selectively eliminates its losers. It never reduces a successful investment no matter how far up the stock has run, and it does not arbitrarily impose size or position limits on holdings, either by company or industry. Size is fixed at 500 names, and new names usually come into the portfolio because of the merger or acquisition of existing companies. Periodically, new names are added and others eliminated in an attempt to replace companies that are marginal with those whose position in the economy or in an industry are deemed more important.

    The overall index is positioned to represent the broad sweep of the US economy. The stock selection committee at S&P consists of 9 analysts, and new names are meant to be seasoned companies with a history of profitability, financially sound, leaders in their industry or market, with a probability of being in business over the next 10 to 20 years. Portfolio turnover averaged 25 to 30 names in the l980’s but has picked up to 40 or so in the past few years. “


  2. John Hall
    March 28, 2011 at 2:04 pm

    I thought this was a very good post.

    It would be interesting to see how these portfolios compare to other methodologies, such as minimum variance portfolios, risk parity, Sharpe ratio maximization, or mean-variance.


  3. bob
    April 1, 2011 at 7:10 pm

    I’m a bit confused about why the numerator and the denominator in the first equation represent different values.
    https://allaboutalpha.com/blog/wp-content/uploads/2011/03/understandin_formula2.jpg
    Put a different way, what is the difference between the sum of weighted individual risks (the numerator) and total portfolio risk (the denominator)? Could someone please elaborate?
    Thanks


  4. Naer
    April 3, 2011 at 5:29 pm

    total portfolio risk = square root of sum w^i X variance i ( if correlation is zero)
    which is different than sum of wieght individual asset


  5. Peter Urbani
    April 3, 2011 at 11:24 pm

    Although Interesting this metric is once again assuming that all volatility is bad and failing to differentiate between upside and downside volatility or incorporate asymmetry in its views. Whilst more diversification is in general good there are times when it definitely is not. As a portfolio manager your job is to take the minimum amount of downside risk necessary to maximise the probability of achieving the clients long run required rate of (geometric or CAGR) return as espoused in their strategic asset allocation or policy benchmark and taking into account any shorter-term tactical considerations. The converse of this implies that you should take the maximum amount of upside risk. If you fail to differentiate between upside and downside risk, overdiversify or take too little risk to have any chance of achieving your client’s objectives you are just as likely to be fired. All of these mitigate against the maximally diversified portfolio which is only really attractive in total market meltdown periods. Where this metric might be useful though is in comparing two similar funds covering similar investment universes (e.g. two Equity Sector funds ) where one fund having both a higher DR and higher rate of return than the other would imply superior stock picking skills. It is also worth noting that the measurement period (1990 – 2011) includes the 2nd and 3rd largest Equity market drawdowns ever (-52%,-46%) and is therefore quite likely biased in favour of the DR portfolios. If anything I think this is once again illustrating the asymmetric effect of downside risks and the importance of differentiating these upside and downside risks.


  6. Michael
    August 1, 2011 at 5:52 pm

    For a more basic overview of the topic covered you should read up on portfolio theory by Harry Markowitz


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