Investors’ ability to capitalize on the benefits of the diversification effect seems threatened:

• Economic globalization has reduced international asset return independence [Dunis and Shannon, 2005].
• Economic shocks spill over international borders at precisely the times investors most need the diversification benefits of asset allocation [Longin and Solnik, 1995; Calvo and Mendoza, 1999; Kodres and Pritsker, 2002].
• Correlations between asset classes and subclasses vary across time [Krein, 2007].

Why Asset Allocation Alone Is Not Sufficient

Investors apply asset allocation mainly to help reduce portfolio risk. Part of the asset allocation rationale rests on the common sense underlying the familiar idiom, “Don’t put all your eggs in one basket.” This process allocates resources to at least nominally different asset classes. Unfortunately, asset allocation’s idiomatic rationale is overrated: Nominal asset allocation alone fails to confer a meaningful effect on one important form of risk—portfolio volatility.

That’s not to say that asset allocation is useless or harmful. Asset allocation’s greatest benefit is its capacity to mitigate concentration risk. Also, if the investor gets lucky, maybe one or more of the selected asset classes will outperform a benchmark. This potential return benefit of asset allocation is not a risk management strategy at all; it’s a seat-of-the-pants attempt at increasing the rate of return.

Concentration risk is easy to manage because asset allocation is its simple solution. The greatest risk threat comes from portfolio volatility. Volatility is rapid price change. It creates havoc with investors whether prices go up or down. We can improve the effectiveness of asset allocation as a risk-reducing process by measuring the diversification effect in the process of planning asset allocation and evaluating portfolio performance.

A Review Of Diversification Effect Metrics And Their Shortcomings For Practical Applications

Diversification effect measurement is not new. Research articles (e.g., Conover et al., 2002; Abraham et al., 2001, among many others) and some commercial investment analytics programs publish correlation coefficient matrices to help gauge the diversification effect. These matrices list the correlation coefficients of all assets in a given portfolio, and they can be onerous. For example, a modest 10-asset portfolio yields 45 different correlation coefficients. How are we to draw confident conclusions about a portfolio’s diversification effect by poring over a table with so many coefficients?

Correlation coefficient matrices and correlations in general present another problem when we use them to gauge portfolio risk in an applied setting: Correlation is not a direct measure of the diversification effect. It measures covariance. Measuring covariance as a proxy for the diversification effect is somewhat like measuring absolute price change as a proxy for rate of return. Certainly, price change is a critical input for rate of return. But a direct rate-of-return measure is better. Likewise, a direct measure of diversification effect is better than its proxies.

Researchers often apply more-complicated measures of the diversification effect that surely serve their specific purposes well. Sharpe [1992], among others, applied factor models and the coefficient of determination to quantify the diversification effect. Mills [1996] used co-integration to measure the tendency for two stationary time series to move together in a long-term equilibrium state. From an applied perspective, these are indirect metrics because they only permit inference about risk reduction as a function of imperfect correlations.

Another approach to diversification measurement quantifies the gain in expected returns by allocating from a benchmark portfolio to a portfolio located at the same risk level on the efficient frontier [Li, et al., 2003; Kandel et al., 1995]. This tactic depends on the efficient frontier component of Modern Portfolio Theory. Yet, for practical applications, we need not appeal to theory if we can directly measure the diversification effect.

The Diversification Effect Metric

None of the diversification effect metrics above, although surely suitable for research purposes, seems as parsimonious for practical application as the direct measure of diversification effect used by Cheng and Roulac (2007) and De Wit (1997). The calculation directly taps the diversification effect without the need for inference or theoretical assumptions and without onerous matrices. In this calculation, the diversification effect lies in the relationship between two forms of the portfolio standard deviation equation. The first form is the generally accepted form, which weights each asset’s standard deviation by its allocation and its correlation coefficient with every other asset in the portfolio. The second form assigns only allocation weights, effectively assuming a perfect positive correlation between all portfolio assets. This form is the allocation-weighted mean standard deviation of each asset in the portfolio. The diversification effect resides in the difference between the results of these two equation forms, because the equations partition the effect of imperfect correlations. The generally accepted form incorporates the effect of correlation coefficients, and the allocation-weighted mean standard deviation does not. Cheng and Roulac expressed the relationship between the two standard deviation forms in an equally weighted portfolio as a ratio of portfolio standard deviation to the allocation-weighted mean standard deviation of individual assets:



Where:

B = Diversification effect

σ_p = Portfolio standard deviation weighted by allocations and the correlations between each asset in the portfolio

σ_j = Allocation-weighted mean portfolio standard deviation

This ratio divides the generally accepted form of the standard deviation, which incorporates both allocation and correlation weights, by the allocation-weighted mean standard deviation, which only weights asset standard deviations with allocations. It indicates the percent of risk remaining after quantifying the effect of correlation coefficients on portfolio risk. For the present article, we prefer:



Where:

DE = Diversification effect

By subtracting the Cheng and Roulac ratio from one, DE describes the percent of risk reduction that occurs as a function of correlation coefficients. A larger result indicates greater diversification effect.

Incremental diversification effect can also be useful, particularly in the process of evaluating expected effects of adding or exchanging portfolio assets. Incremental diversification effect is the change in diversification effect that occurs after assets are added or subtracted from a portfolio. We can express incremental diversification as:



Where:

IDE = Incremental diversification effect

DE_j-1 = Diversification effect of the original portfolio (the portfolio before portfolio assets are changed)

We can simplify the equation as follows:



Where:

σ_p-1 = The original portfolio standard deviation weighted by allocations and the correlations between each asset in the portfolio

This ratio indicates the change in the percent of risk reduction that occurs as a function of correlation coefficients after adding or exchanging portfolio assets. Incremental diversification uses the relationship between the allocation-weighted mean standard deviation of the original portfolio (σj-1) and the generally accepted form of the standard deviation of the new portfolio (σp), effectively using the first standard deviation as the basis for the second and all subsequent comparisons.

Data

The following illustrations use daily market data for the seven-year period January 2, 2002, through December 31, 2008. The source for long-term bond data is daily yields for Moody’s Seasoned Baa Corporate Bond Yields reported at stlouisfed.org by the Federal Reserve Bank of St. Louis. The SPDR S&P 500 ETF (SPY) serves as the U.S. large-capitalization proxy; the iShares Russell 2000 Index Fund (IWM) serves as the small-capitalization proxy; and the iShares MSCI EAFE Index Fund (EFA) serves as the non-U.S. large-capitalization proxy. The commodities data source is the Dow Jones-AIG Commodity Total Return index reported at djindexes.com by Dow Jones.

When credit, equity or commodities markets recognized different holidays or closed for other reasons, we use linear interpolation to fill in missing data. Daily return calculations apply dividend and coupon reinvestment, and the process assumes assets are held for the entire seven-year period without rebalancing. The process of calculating returns for the Moody’s Seasoned Baa Corporate Bond Yields assumes bonds were purchased at par on January 2, 2002, and coupons were paid semiannually on the first trading day on or after July 1 and January 1, each year. For all assets, daily return and standard deviation calculations are based on daily value changes that include cumulative change in the value of reinvested coupons and dividends. We omit the effects of taxes and transaction fees, although exchange-traded fund values are net of expense ratios and other costs incurred by the fund. Betas are calculated using the standard deviation of the asset or portfolio, standard deviation of the S&P 500 Index and the corresponding correlation coefficient. Sharpe ratios are calculated using the daily five-year geometric mean yield of three-month Treasury bills, which serves as the riskless rate of return.

The three illustrations that follow use historical data and make changes in portfolio components the way a portfolio manager might evaluate prospective portfolio changes in a real investment portfolio. However, investors cannot directly invest in the indexes used in these illustrations. The reader should bear in mind that all data used in portfolio analysis and projections are historical. Extrapolations from historical data, although practically necessary, must be done with caution, particularly when the extrapolations are for relatively short periods beyond the historical database and when the historical database is a relatively short period. One of the implications of these facts is that investors should regularly reevaluate their data and assumptions. And, of course, past performance cannot guarantee future results.

Diversification Effect Illustration

The first illustration applies the diversification effect metric to a theoretical, equally allocated three-asset portfolio consisting of proxies for U.S. large-capitalization stocks, long-term bonds and non-U.S large-capitalization stocks. Later we add the proxy for the Russell 2000 Small-Capitalization Index and add the Dow Jones-AIG Commodity Total Return Index. Our interest rests primarily with diversification effect metrics, though we calculate other useful portfolio statistics.

Figure 1 reports performance data for assets used in the present and subsequent illustrations and Figure 2 displays correlation coefficients for the present and subsequent illustrations. Figure 3 illustrates the effects of equally allocating the SPY, 30-year bond and EFA into a three-asset portfolio. This allocation yields a diversification effect (DE) of 19.19%, which means that correlation coefficients between all three portfolio assets decrease portfolio risk by 19.19% compared to the allocation-weighted mean standard deviation that assumes +1.00 correlations. The portfolio beta falls below the arithmetic mean beta (see Figure 1) of the three assets, also because of the influence of these correlations. The portfolio Sharpe ratio exceeds the arithmetic mean Sharpe ratio because the portfolio standard deviation has been reduced by imperfect correlations between the assets.

This illustration explains the effect of correlation coefficients on at least three commonly used portfolio metrics, and the DE statistic explains the other two. Clearly, the statistics are related by the effect correlations have on them. Next we change portfolio composition and monitor the results.

Incremental Diversification Effect Illustration: Adding An Asset With Higher Correlations

In this illustration, we add a fourth asset class with historically high returns and higher correlation coefficients with other assets in the portfolio. The rationale for adding this asset might be to increase expected returns, reduce beta, increase the Sharpe ratio or increase DE. We add incremental diversification effect (IDE) in the analysis to evaluate the diversification benefit of adding a fourth asset. We select IWM as a proxy for the Russell 2000 small-capitalization index. Again, we impose the equal allocation constraint. Figure 4 displays results.

Comparing results in Figures 3 and 4, the new four-asset portfolio return practically adds no additional return or improvements in the Sharpe ratio, and beta rises. The new asset’s correlations with the other two equity assets (Figure 2) are high. Because of the higher correlation coefficients, the new DE is smaller than the three-asset DE, which means that the new four-asset portfolio delivers less diversification effect than the original three-asset portfolio. Because the new asset adds so much risk without a corresponding decline in the correlation coefficients, the four-asset portfolio standard deviation is higher than the standard deviation of the three-asset portfolio. Consequently, IDE is negative. We conclude that adding this new asset compromises DE.

Incremental Diversification Effect Illustration: Adding An Asset With Lower Correlations

If the investor decides the risk of adding IWM exceeds its benefits, the decision might be to replace IWM with an asset that co-varies less with existing portfolio assets. The Dow Jones – AIG Commodity TR Index (DJAIGTR) correlation coefficients range from +0.2526 to -0.0221. This near-zero range could yield a favorable IDE if we use it to replace IWM. Figure 5 illustrates results of replacing IWM with DJAIGTR.

In this scenario, compared to the original three-asset portfolio, the portfolio standard deviation falls, and because of the larger DE attributable to DJAIGTR’s near-zero correlations, IDE is positive. Beta drops and the Sharpe ratio increases. This portfolio is clearly superior to the three-asset portfolio and the IWM four-asset portfolio according to these measures.

The purposes of this illustration and the one before it are to show how DE and IDE can help evaluate portfolio risk in particular and, in general, how DE and IDE can be used with other portfolio statistics to make better-informed investment decisions.

These illustrations are not intended to advocate use of DJAIGTR-linked investments. In scenarios with different details, we would not necessarily realize favorable changes in returns, betas and the Sharpe ratios when we also realize an increase in DE or a positive IDE. Results for different time periods using these assets could be very different than the results obtained in the present illustrations.

Conclusions

Diversification effect is a potential result of asset allocation and a function of imperfect correlation coefficients between the returns of portfolio assets. Meaningful diversification effect may not result from asset allocation alone, so it follows that active risk management requires measurement. This article illustrates how a direct diversification effect metric provides useful information about important portfolio properties.

The first illustration started with a three-asset portfolio. When we added the fourth asset, less imperfect correlation coefficients yielded a negative incremental diversification effect. The fourth asset’s relatively high correlations with existing assets diminished any risk-attenuating benefit that might have been gained by adding the asset. A secondary lesson in this second illustration reminds us that adding nominally different assets does not necessarily increase diversification effect.

Replacing the original fourth asset with a low-correlated asset enabled a positive incremental diversification effect. In this example, the values of other important portfolio properties improved.

DE metrics presented here directly measure diversification effect and do not require any inference or dependence upon theoretical assumptions. They also do not duplicate the information value of other metrics.

Diversification effect measurement should not be confused with total risk measurement. If an investor wants to change a portfolio’s total risk level, the standard deviation is the correct metric. However, if an investor wants to capitalize on diversification effect or evaluate the extent to which a certain allocation of certain assets reduces risk as a function of imperfect correlation coefficients, DE provides necessary and sufficient information because it directly measures diversification effect. Likewise, if an investor wants to evaluate the impact of changing portfolio assets on diversification effect, IDE provides necessary and sufficient information.

One perspective might hold that the variability of correlation coefficients diminishes the information value of DE and IDE. Research on the relationship between real estate investment trusts (REITs) and the S&P 500 Index suggests that correlations increase after REIT down months and decrease after REIT up months [Chandrashekaran, 1999]. Correlations among stocks increase during market downturns [Campbell, Koedijk and Kofman, 2002]. Also during stock market downturns, stock-bond correlations turn negative, but they approach unity during stock market upturns [Gulko, 2002]. Speidell and Sappenfield (1992) report rising correlations among developed markets due to economic convergence and interdependence. We would expect similar variability in other modern portfolio statistics such as returns, standard deviation of returns, beta and Sharpe ratios.

Yet all of these caveat examples provide evidence that favors actively measuring and managing diversification effect. None of these facts concerning variability of markets compromises the value of measuring diversification effect any more than uncertainty about the future compromises the value of planning. These facts are all the more reason to directly measure diversification effect.

References

Abraham, A, Fazal, J, and Seyyed, A A (2001) “Analysis of diversification benefits of investing in the emerging gulf equity markets,” Managerial Finance, 27, (10/11), 47-57.

Adair, A, McGreal, S and Webb, J R (2006) “Diversification effects of direct versus indirect real estate investments in the U.K.” Journal of Real Estate Portfolio Management, 12 (2), 85-90.

Calvo, G A and Mendoza, E G (1999) “Regional contagion and the globalization of securities markets,” NBER Working Papers 7153, National Bureau of Economic Research, Inc.

Campbell, R A C G, Koedijk, K C G and Kofman, P II (2002) “Covariance and correlation in international equity returns,” Financial Analysts Journal, 58 (1), 87-94.

Chandrashekaran, V (1999) “Time-series properties and diversification benefits of REIT returns,” The Journal of Real Estate Research, 17 (1), 91-112.

Cheng, P and Roulac, S E (2007) “Measuring the effectiveness of geographical diversification,” Journal of Real Estate Management, 13 (1), 29-44.

Conover, C M, Friday, H S and Sirmans, G S (2002) “Diversification benefits from foreign real estate investments,” Journal of Real Estate Portfolio Management, 8 (1), 17-25.

De Wit, D P M (1997) “Real estate diversification benefits,” The Journal of Real Estate Research, 14 (2), 117-136.

Dunis, C L and Shannon, G (2005) “Emerging markets of south-east and central Asia: Do they still offer a diversification benefit?” Journal of Asset Management, 6 (3), 168-190.

Gulko, L (2002) “Decoupling,” Journal of Portfolio Management, Spring, 59-66.

Israelsen, Craig (2007) “The benefits of low correlation,” Journal of Indexes, 10 (6), 18-26.

Kandel, S, McCulloch, R and Stambaugh, R (1995) “Bayesian inference and portfolio efficiency,” Review of Financial Studies, 8, 1-53.

Kodres, L E and Pritsker, M (2002) “A rational expectations model of financial contagion,” The Journal of Finance, 57 (2), 769-799.

Krein, D (2007) “Mapping a market for correlation,” Journal of Indexes, 10 (6), 33-35.

Li, K, Sarkar, A. and Wang, Z. (2003) “Diversification benefits of emerging markets subject to portfolio constraints,” Journal of Empirical Finance, 10 (1), 57-80.

Longin, F and Solnik, B (1995) “Is the correlation in international equity returns constant: 1960–1990?” Journal of International Money and Finance, 14 (1), 3-26.

Markowitz, H. (1991) “Portfolio Selection,” Blackwell Publishing.

Mills, T C (1996) “The Econometrics of the ‘Market Model’: Cointegration, Error Correction and Exogeneity,” International Journal of Finance & Economics, 1(4), 275-286

Sharpe, W F (1992) “Asset allocation: Management style and performance measurement,” Journal of Portfolio Management, Winter, 7-19.

Speidell, L S and Sappenfield, R (1992) “Global diversification in a shrinking world,” The Journal of Portfolio Management, 19 (1), 57-67.