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 *w _{i}* is the portfolio weight in asset

*i*,

*σ*is the risk of asset

_{i}*i*, and

*σ*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

_{p}*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* (*DR*^{2}) 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 *(*r _{i}*) –

*r*=

^{f}*k*where

*σ*_{i}*r*is the risk-free rate and

^{f}*k*is a constant. Hence, the expected return on a portfolio,

*P*, comprised of

*N*assets can be expressed as

*E*(

*r*) –

_{p}*r*=

^{f}*k*

*Σ*

^{N}

_{i=1}*w*. Dividing both sides of the return equation by

_{i}σ_{i}*σ*

*, we see that a portfolio’s expected Sharpe Ratio is linear in diversification:*

_{p}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 (3.7^{2}) independent factors while the benchmark has been exposed to only 5.3 (2.3^{2}). 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.

## 6 Comments

March 28, 2011 at 5:12 am## Lazy eye

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

April 1, 2011 at 7:10 pm## bob

April 3, 2011 at 5:29 pm## Naer

April 3, 2011 at 11:24 pm## Peter Urbani

August 1, 2011 at 5:52 pm## Michael