## Abstract

In a multi-factor world, diversification benefits do not generally depend on correlation. Investors can restructure portfolios to align factor sensitivities. This implies that diversification benefits depend only on the idiosyncratic volatility that remains after restructuring. Similarly, the risk reduction that follows adding an asset to an existing portfolio does not depend on the asset’s correlation with the portfolio. These implications evince the fundamental importance of measuring the underlying factors and estimating factor sensitivities for every asset. Other researchers have investigated several methods for measuring factors. An easy-to-implement general method involves specifying a group of heterogeneous indexes or traded portfolios. Exchange-traded funds (ETFs) could be well suited to this purpose.

**TOPICS:** Portfolio construction, volatility measures, statistical methods

There are two striking facts about portfolios of assets. First, even really well-diversified portfolios are quite volatile. The volatility of a large positively-weighted portfolio is often around half as large as the average volatility of its constituents.^{1} Second, although well-diversified portfolios are highly correlated within the same asset class, they are much less correlated across asset classes; i.e., across bonds vs. equities vs. commodities, or across countries or industry sectors.^{2}

The first empirical fact is sufficient to suggest the existence of common underlying systematic influences, (or “risk drivers” or “factors”) that limit diversification within an asset class; otherwise, diversified portfolios would have much smaller volatilities. The second fact intimates the presence of multiple systematic factors; otherwise, diversified portfolios would be more correlated across asset classes, countries, and sectors.

Correlation is unlikely to reveal a multi-dimensional structure in the underlying systematic factors. To illustrate this most easily, consider the simplest possible multi-dimensional world, wherein all asset returns are driven by just two common factors that affect every asset linearly at time t according to the following return generating model:

where f_{1} and f_{2} denote the two common factors that influence the return R on asset i through its sensitivity coefficients, the ß’s. By assumption and without loss of generality, the factors have zero means, as does the idiosyncratic risk, e, while the expected return on asset i is E_{i}. Note that everything is specific to asset i (and thus carries an i subscript), except the common factors. Also, in this most elementary of all multi-dimensional models, the asset’s expected return and its sensitivities (ß’s) are assumed to be time-invariant constants.

Given this simple world, consider now the correlations of well-diversified portfolios across asset classes. For example, suppose that two asset classes, A and B, have broad, widely followed, well-diversified market indices. Let’s suppose initially that the indices are so well-diversified that both have negligible remaining idiosyncratic volatility; i.e.,

The returns on both of these indices are explained entirely by the same two underlying systematic factors. What about their correlation? It turns out that their correlation will be perfect if (and only if) for some constant of proportionality, k ? 0, both ß_{A,1} = kß_{B,1} and ß_{A,2} = kß_{B,2}. For any other set of sensitivity coefficients, the correlation will be imperfect.^{3} Conceivably, the correlation can be quite low, even though both index A and index B are driven by the same two common influences and are thus perfectly integrated in the sense used by Pukthuanthong and Roll [2009].

To illustrate the range of possibilities, let’s allow two possibly different constants of proportionality, k_{1} and k_{2}, that relate the sensitivities as follows: ß_{A,1} = k_{1}ß_{B,1} and ß_{A,2} = k_{2}ß_{B,2}. For ease of illustration, assume that the factors have the same variance and that they are uncorrelated. In this situation, the surface shown in Exhibit 1 plots the correlation between A and B for different values of k_{1} and k_{2} ranging between -1 and +1. As the figure shows, the correlation is +1 when the constants are positive and equal, while the correlation is -1 when they are negative and equal. There is a discontinuity when the constants are zero. Notice too that the correlation is exactly zero for an entire set of non-zero constants of proportionality with opposite signs.

Within an asset class, portfolios have similar sensitivities to the underlying factors, so correlations are relatively high. But across asset classes, this is evidently not the case.

One might well imagine why it’s not by using the example of equities and bonds. Suppose factor 1 is related to shocks in real output and factor 2 is related to shocks in expected inflation. Then a positive shock in factor 1 would increase equity returns, but not affect bonds all that much. Conversely, a reduction (a positive shock) in expected inflation would drive up nominal bond prices, but have a more attenuated impact on equities.

The result over many periods, when there are shocks in both real output and expected inflation, is a relatively low correlation between stocks and bonds. Of course, this is just an example for discussion and is not meant to imply that equities and bonds really are so divergently sensitive to the true underlying factors. Indeed, there could be other systematic factors, such as investor confidence, that drive them in the same direction.

Another example comes from the rather puzzling low correlation often observed between equity indices in two countries. Hong Kong and Saudi Arabia might both be driven similarly by global shocks to investor confidence, but driven differentially by global energy shocks. Saudi stocks are driven upward by energy price increases, but the opposite is true for Hong Kong, an energy importer. These two countries could be very well integrated, in the sense that they both depend on the same global factors, yet their simple correlation could be small or even negative, depending on the relative volatilities of investor confidence and energy shocks.

**IMPLICATIONS FOR PORTFOLIO MANAGEMENT**

The previous remarks, though extremely simple, have profound implications for portfolio management. Since Markowitz, diversification has been thought to be the most effective when assets or portfolios are not very correlated. But this intuition is misleading! Low correlation between bundles of assets fails to properly measure the potential benefits of diversification.

To see why, consider two diversified portfolio/indexes, A and B, perhaps in different asset classes, whose returns are driven by the same underlying systematic factors, but with diverse sensitivities. Assume their simple correlation is relatively low, for the reasons previously mentioned. Diversification into the two indexes might seem powerful, because many allocations between them (such as 50-50) appear to substantially reduce volatility. But this overstates the true benefit, because the respective index compositions are held constant when making the allocation between them.

Instead of allocating a fraction of investment funds to index A and the complementary fraction to index B, consider structuring a different investment portfolio from the individual assets in the first class (A) that matches extremely well the factor sensitivities of the original index B. This is straightforward when there is a large enough menu of available derivatives, or when short positions are feasible and inexpensive. The resulting returns would then conform to the following generating models:

Notice that the sensitivity coefficients (ß’s) from the restructured portfolio of A assets now match the original sensitivity coefficients of index B. To allow for generality, there is still some remaining idiosyncratic risk, as represented by the e’s.

What, then, is the actual diversification benefit available from combining A and B? We can gain some insight about this question by finding the minimum variance portfolio from combining index B with the ß_{B} re-structured portfolio composed of assets in class A. It is straightforward to show^{4} that this portfolio has a weighting w in index B (and 1-w in the re-structured portfolio A) equal to

In words, if the re-structured portfolio from the class A assets has no idiosyncratic component, diversifying into B brings absolutely no benefit in terms of risk reduction; w is zero. This is true even when the correlation is weak between the original indices of classes A and B. Any benefit from combining B with A would have to be in terms of enhanced return, not reduced risk.

The logic behind this result is that a mimicking portfolio for B can be engineered from assets in A to have the same ßs as B. This engineered A-asset-only portfolio was available within A all along. It could have been combined with the original A index. At an extreme, if the engineered portfolio has no idiosyncratic volatility, it strictly dominates index B, provided that B does have at least a modicum of idiosyncratic volatility. Consequently, B provides no genuine risk-reducing benefit over what could have been obtained with the A assets by themselves.

If the engineered A-asset-only portfolio retains some idiosyncratic risk, there is still some diversification benefit. But that benefit has nothing to do with the correlation between the original indices A and B. Also, if E_{B} happens to be large compared to E_{A}, there would be a benefit from combining B with A, but not because of risk reduction.

**DIVERSIFICATION BENEFITS: A BETTER MEASURE FOR PORTFOLIO COMBINATIONS**

If the ß_{B}-structured, B-mimicking portfolio composed of A assets has an R-square on the underlying factors close to 1.0, there will be negligible diversification benefits from combining B and A. (The same would be true going in the other direction; i.e., restructuring B to match the factor sensitivities of the A index.)

The initial impression of strong diversification benefits, suggested by the simple correlation between the initial indexes from classes A and B, does not account for the entire range of possibilities. Thus, the simple correlation between portfolios of assets is a bit misleading. There is nothing wrong with the Markowitz efficient set math, but it doesn’t go far enough when dealing with portfolios, as opposed to individual assets. There is virtually no benefit from diversification when factor R-squares are close to 1.0 (in either A or B), even when correlation appears to be weak between their indexes. The R-squares of the indices on the portfolios, not their correlation, are a better (inverse) measure of potential diversification benefits; high R-square, low benefits.

Of course, a full-blown mean/variance analysis of individual assets in A and B would yield a correct measure, but this is rarely considered for good reasons. It is impossible to estimate the covariance matrix for a large number of individual assets, except with a very long time series sample, and then non-stationarity becomes a serious problem.

**THE DIVERSIFICATION BENEFIT OF ADDING INDIVIDUAL ASSETS TO DIVERSIFIED PORTFOLIOS**

In the well-known Treynor/Black [1973] analysis, the impact of adding an individual asset to an existing portfolio depends on their relative expected returns and their correlation of individual assets with the portfolio.^{5}

But suppose the individual asset i has the same form of return-generating equation (where, for illustration, we again assume that there are only two factors): and suppose a perfect i-mimicking portfolio (P) with exactly matching sensitivities can be engineered from among the portfolio of assets already held; i.e.,

There can be no genuine reduction in diversified risk from adding asset i, regardless of its correlation with the original portfolio. (Adding i to the portfolio would be beneficial only if E_{i} > E_{P}.)

But what are the factors?

Given the importance of assessing the true diversification benefit when combining portfolios or when adding a single asset to an existing portfolio, it is clearly essential to develop estimates of the ßs in every situation. But what exactly are the factors, the underlying risk drivers?

They cannot be the infrequently published official numbers about macro-economic variables, because market prices move around much too rapidly. Instead, they must be high-frequency changes in market perceptions of pervasive macro-economic conditions. Perceptions could include (a) rational anticipations of change in macro conditions that are truly pervasive, such as real output growth, real interest rates, inflation, energy, and so on, and (b) behavior-driven, pervasive shocks in confidence or risk perceptions such as panics, liquidity crises, and so on.

To do a really good job of optimal diversification, we must be able to identify and measure the pervasive factor perceptions and then to estimate factor sensitivities for any asset or portfolio of interest. The first job is to identify and measure the factors. Existing literature has studied several alternative approaches.

One approach relies on an entirely statistical method, such as principle components or factor analysis (e.g., Roll and Ross [1980], Connor and Korajczyk [1988]).

A second approach pre-specifies macro-economic variables that seem likely to be pervasive, and then pre-whitens the official numbers pertaining to such low-frequency constructs as industrial production, inflation, and so on (e.g., Chen, Roll, and Ross [1986] for equities, Litterman and Scheinkman [1991] for bonds). Then there is the approach of relying on firm characteristics (e.g., size, book/market, momentum) to develop proxies that are empirically related to average returns (e.g., Fama/French [1992], Carhart [1997]). Finally, a lesser-known but simpler approach is to employ a handful of rather heterogeneous indices or tradable portfolios.

Each of these approaches has particular limitations. Purely statistical methods are theoretically sound, but everything has to be stationary. Pre-specified macro-economic variables are the most theoretically solid, but are observed with excruciatingly low frequency. Factor proxies suggested by characteristics are weak theoretically and are not necessarily even related to risk. A group of heterogeneous diversified portfolios can have non-stationary compositions and be observed at high frequency—but heterogeneity must be sufficient to span all relevant and pervasive underlying risk drivers.

Heterogeneous portfolios work well for spanning global factors. Pukthuanthong and Roll [2009] went to a lot of trouble to extract 10 global principal components. They employed the extracted global principal components as factor proxies and demonstrated a substantial increase in global market integration for many countries. Then, as a robustness check for their purely statistical procedures, they replaced the principal components with broad indices from 10 large countries and found virtually identical results. Country indices are evidently sufficiently heterogeneous to span the same underlying macro perceptions as principal components.

Using a set group of portfolios is arguably the easiest and best approach to factor estimation if heterogeneity can be assured, which suggests that a well-chosen set of exchange-traded funds (ETFs) might serve the purpose quite well. ETFs are often diversified portfolios or derivatives-based equivalents. As such, their returns must be driven mainly by underlying factors, i.e., by high-frequency changes in market perceptions of macro-economic conditions. Their idiosyncratic volatility should be relatively small. Moreover, they are generally liquid, transparent, and cheap to trade. Their variety across several asset classes suggests a healthy degree of heterogeneity.

**THE ETF MARKETPLACE**

According to the NYSE web site, there are now more than 20,000 ETPs (exchange-traded products) listed on exchanges around the world. The NYSE ARCA, the electronic network where many ETFs are traded domestically, has almost 2,000 ETF listings. NYSE Euronext has additional listings, and so do other exchanges.

ETF trading volume has been accelerating recently, particularly in relation to trading in related cash assets and in other derivatives, such as options and futures. For example, Roll, Schwartz, and Subrahmanyam [2011] document an increase in trading volume for the S&P 500 ETF “Spider” of roughly 10,000% from 1996 through 2009. This compares with increased volume of around 100% in the underlying S&P 500 cash assets and a similar increase in options on the S&P 500 index. Meanwhile trading in the original futures contract on the index has actually declined (though this has been offset by trading in the newer “E-mini” futures.)

Just on the NYSE, there are ETFs in many asset classes, including equities, bonds, commodities, and currencies. But as might have been anticipated, heterogeneity is not that impressive within each class. Domestic equity ETFs in particular are highly correlated with each other and with the broad market indices.

For example, Exhibit 2 shows the correlation between 25 prominent equity-class ETFs and the S&P 500 Index. As can be seen there, all of these ETFs are very highly correlated with the index (and, of course, with each other.) There is probably not enough heterogeneity across many equity ETFs to provide a sufficient spanning of the underlying factors.

Across asset classes, however, ETF heterogeneity might be acceptable. Within the class of bond ETFs, there is more heterogeneity than among equity ETFs, while commodity and currency ETFs provide still more. (See Roll [2010] for details.) Heterogeneity is also enhanced by bear market ETFs. As the name suggests, these funds are designed to move inversely to their underlying cash assets. Hence, they possess a relatively large degree of diversification potential.

Further enhancements are provided by the recent appearance of spread ETFs, funds constructed to be long in one asset class and short in another. Spread ETFs also offer a unique opportunity to measure the market’s consensus beliefs about the future correlations between asset classes. (See the appendix.)

**CONCLUSIONS**

When evaluating the diversification benefit from combining two portfolios or when combining a single asset with an existing portfolio, the simple correlation is misleading in a multi-factor world. This is because a mimicking portfolio can be constructed from one portfolio to match the other portfolio or the single asset. A mimicking portfolio has the same risk profile as the portfolio or asset being mimicked, the same sensitivities to the true underlying high-frequency macro perception shocks that comprise the factor risk drivers. When a mimicking risk profile can be engineered, the only thing that matters for diversification is the residual volatility that is not explained by the factors. If, in the limit, there is no residual volatility, there can be no benefit from diversifying.

This altered way of thinking about diversification makes it apparent that a fundamental investment concern involves measuring the underlying factors and then using those measurements to estimate risk profiles. Several different methods for measuring factors have been developed in the literature. Each has its own peculiar difficulties.

However, one of the seemingly most attractive methods is simply to rely on a set of heterogeneous existing indices or liquid tradable portfolios. With sufficient heterogeneity and liquidity, the true underlying factor space is likely to be spanned and the observations are of sufficiently high frequency.

Exchange traded funds (ETFs) exist in a large variety of flavors, are highly liquid for the most part and are generally well diversified. ETFs from divergent asset classes might very well be some of the best proxies for the unobservable macro perceptions that are the true underlying risk drivers.

**APPENDIX**

**OPTIONS ON SPREAD ETFs AND IMPLIED CORRELATIONS**

Spread ETFs are long one asset class, perhaps equities, and short another asset class, such as bonds. They may have substantial leverage. The implied volatility from an option on a spread ETF depends on the correlation between the long and short constituent asset classes. This implies that market consensus anticipations about future correlations can potentially be directly observed from prices.

To make the required calculations, we need three simultaneous option prices for the long (L) and short (S) components of the ETF and for the ETF itself.

To work out this calculation algebraically, it is convenient to use the following notation:

R

_{L}: return on the long asset held in the spread ETF, (subscript “L”)R

_{S}: return on the short asset held in the spread ETF, (subscript “S”)w

_{j}: weighting (or leverage ratio) for spread ETF position j, (j = L,S)s

_{j}: implied volatility, standard deviation, of asset j (from an option written on j)?: implied correlation between the returns on assets held long and short in the ETF

Assuming that there is no volatility in the borrowing undertaken to finance the leveraged spread ETF, the risky part of the ETF’s daily return is

For example, with 2:1 leverage rebalanced at the end of each trading day, the weightings at the beginning of each day are w_{L} = 2.0 and w_{S} = -2.0.

The variance of a spread ETF’s return is where ? is the correlation whose value is the objective of our calculations.

The implied (i.e., market consensus) value of the spread ETF’s volatility can be obtained from its option, while the implied volatilities of the ETF’s constituents can be obtained from separate options on those assets. Consequently, provided that all three options are actively traded, everything in the above formula is known or can be obtained from market prices, except for the correlation.

To solve for the correlation, we invert the formula and collect term to obtain

This is the general formula for the implied correlation between the two asset classes held in a spread ETF.

For example, given the values of w_{L} = 2.0 and w_{S} = -2.0, the correlation above is

As a numerical example, suppose that option prices on the long asset, the short asset, and the spread ETF imply, respectively, annual percentage volatilities of 20%, 15%, and 35%. The latter seems large, but remember that the example ETF is levered 2:1. Plugging in these numbers in the expression just previous, we obtain

This particular correlation, slightly above 0.5, might strike some traders as implausibly high (or low); if so, the ETF option is too cheap (expensive), and a profitable trade would involve buying (selling) it and offsetting the trade with a dynamically hedged short (long) position in the ETF itself.

Because correlation is always between +1 and -1, it is not hard to prove that the implied volatility from options on a spread ETF must lie between 10% and 70%, given the long and short volatilities assumed in this numerical example. Anything outside that range would offer a pure arbitrage trade.

## ENDNOTES

↵

^{1}For example, during the decade from 2001 through 2010, the monthly total return on the S&P 500 had an annualized volatility (standard deviation) of 16.3%. During the same period, the average volatility for the S&P’s constituents was 36.1%.↵

^{2}From 2001 through 2010, the monthly total return correlation between the S&P 500 and Barclay’s Bond Aggregate Index was -0.0426. The return correlations between these two indices and the Goldman Sachs Commodity index were 0.266 and 0.0113, respectively.↵

^{3}The formal proof is delivered by the Cauchy inequality; it generalizes to any number of factors (greater than one). The correlation is +1 (-1) when k > (<) 0.↵

^{4}Assuming, as usual, that the idiosyncratic terms are uncorrelated with the factors and with each other.↵

^{5}For instance, see Bodie, Kane, and Marcus [2011, pp. 933-937.] The beta discussed there is proportional to the correlation between the individual asset and the existing portfolio.↵

^{6}EDM (%) is the maximum percentage reduction in volatility from combining the ETF with the S&P 500 Index without shorting either. A zero in this column signifies that the index has a smaller volatility than any long-only combination of the index and the ETF.

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