Practical analysis for investment professionals
11 April 2014

# Don’t Forget Diversification

Diversification is one of the most fundamental yet misunderstood concepts in investment management. I will begin with a basic and hopefully intuitive discussion of the principles of diversification and the mechanics by which it works. Then, I will provide a more technical presentation for those interested. Throughout, I will use numeric examples to illustrate the principles involved. Finally, I will examine some typical historical data regarding diversification across asset classes (e.g., stocks, bonds, and cash).

Simply stated, diversification results when you spread your investable funds across different assets. By chance, some will do well when others do poorly, thus reducing the overall variability (i.e., risk) of your portfolio. There are numerous methods of achieving diversification. Naive diversification is done by simply randomly choosing a set of assets for your portfolio. The law of large numbers makes this work. If you have 30 randomly chosen stocks in your portfolio, some will go up when others go down, thus reducing overall portfolio risk. But we can do better. To maximize the benefits of diversification, we should look for assets that tend not to move together. We look for some assets that “zig” when others “zag.” We can reduce risk more efficiently this way (i.e., achieve the same level of risk reduction with fewer separate investments).

The key to efficient diversification involves the statistical concept of correlation. Correlation measures the degree to which two assets move together. The maximum correlation is 1.0, or 100%; in this case, the two assets always move up and down together (though possibly by different amounts) and no diversification is achieved. The minimum correlation is −1.0, or −100%; in this case, the two assets always move in the opposite direction and perfect diversification is achieved. For correlations between these extremes, the lower the correlation, the more diversification we achieve.

An Illustration

Consider the stocks in Table 1. Each stock has its typical ups and downs. Note that Stocks 2 and 3 have the same returns but in different years. Also note that Stocks 1 and 2 move together (positive correlation) whereas Stocks 1 and 3 move opposite one another (negative correlation).

Table 1. Stock Returns

 Year 1 Year 2 Year 3 Year 4 Stock 1 10% −5% 15% −8% Stock 2 12% −7% 18% −10% Stock 3 −7% 12% −10% 18%

Table 2 shows the values of two portfolios. Portfolio 1 assumes initial equal \$100,000 investments in Stocks 1 and 2, whereas Portfolio 2 assumes initial equal \$100,000 investments in Stocks 1 and 3. Both portfolios are rebalanced to 50% weights in each stock each year. I will comment on the significance of this below.

Table 2. Portfolio Values: \$200,000 Initial Investment

 Year 1 Year 2 Year 3 Year 4 Portfolio 1: Stocks 1 & 2 \$222,000 \$208,680 \$243,112 \$221,232 Portfolio 2: Stocks 1 & 3 \$205,000 \$210,125 \$218,530 \$227,271

First, note that Portfolio 1 is much more volatile than Portfolio 2. It fluctuates from a low of \$208,680 to a high of \$243,112. In contrast, Portfolio 2 climbs steadily in value. You would sleep much better at night with Portfolio 2. The lower volatility of Portfolio 2 is a direct result of the lower correlation between Stocks 1 and 3 relative to that of Stocks 1 and 2. Second, note that Portfolio 2 has a higher ending value than Portfolio 1 despite the fact that the stocks in each portfolio have the same period-by-period returns. This result is due to the annual rebalancing of the portfolios. In Portfolio 2, the negative correlation between the stocks, combined with rebalancing, causes more weight to be put into a stock after it has fallen and less weight after it has risen. Over time, this results in a higher ending portfolio value.

Typical Historical Correlations among Asset Classes

To get an idea of the correlations that can be expected for various asset classes, let’s examine historical correlations. Although correlations among assets can change over time, it is reasonable to consider recent history as representative of potential future results. Table 3 presents correlations for eight asset classes that might be considered in building a diversified portfolio. The asset class correlations were calculated by Morningstar for the period 1 January 2002 through 31 December 2006 (60 months). The following indices are represented by number in Table 3:

1. Three-month CD — cash equivalent
2. Lehman Brothers Global Aggregate* — global bonds
3. Lehman Brothers Aggregate Bond* — US bonds
4. MSCI EAFE Growth — Europe, Australasia, and Far East growth stocks
5. MSCI World ex US — global stocks excluding the United States
6. Russell 1000 Growth — US large-capitalization growth stocks
7. Russell 2000 Growth — US small-capitalization growth stocks
8. Russell Midcap Growth — US medium-capitalization growth stocks

Table 3. Historical Correlations among Asset Classes

(represented by indices)

 1 2 3 4 5 6 7 8 1 1.0 2 −0.15 1.0 3 −0.04 0.72 1.0 4 0.10 0.13 −0.17 1.0 5 0.15 −0.35 −0.37 0.84 1.0 6 0.05 −0.23 −0.32 0.72 0.82 1.0 7 0.00 −0.13 −0.31 0.77 0.79 0.84 1.0 8 0.00 −0.17 −0.31 0.76 0.81 0.93 0.95 1.0

First, note that all correlations along the diagonal are 1.0, simply because each asset class is perfectly correlated with itself. Suppose that an investor buys two index funds, each tracking the Russell 1000 Growth Index, from two different mutual fund companies. The investor has achieved zero diversification by buying the second fund! Hopefully, few investors will make such a mistake. However, many investors have portfolios that contain overlapping asset classes with very high correlations. Such portfolios provide little more diversification than our two-fund example.

Next, consider the correlations below the diagonal in Table 3. I will not discuss all the possible combinations of assets, but let’s consider examples of asset class combinations that would provide good diversification versus those that would not. Low or — better yet — negative correlations lead to good diversification. Here, combining cash, bonds, and stocks provides good diversification. Cash is negatively correlated with bonds  (column 1, rows 2 and 3) and has a near-zero correlation with stocks (column 1, rows 4–8). Also, bonds are negatively correlated with stocks (columns 2 and 3, rows 4–8). Importantly, the correlations among non-US and US stocks and across different market capitalizations for US stocks are quite high (columns 4–7, rows 5–8). For example, adding small- or mid-cap US growth stocks to a portfolio of large-cap US growth stocks would provide relatively little diversification.

A diet analogy might apply here. Suppose a patient is quite fond of steak. His doctor realizes that too much steak is unhealthy and recommends that the patient “diversify” his diet. If the patient adds hamburgers and rib roasts, instead of fruits and vegetables, it is easy to see that the desired result will not be achieved.

The Math

The volatility of an asset or portfolio is measured by the statistical concept of variance or standard deviation. Standard deviation is simply the square root of variance. The variance of returns on a portfolio of assets depends on three things: (1) the variances of returns of the assets that make up the portfolio, (2) the correlations between the returns of the assets in the portfolio, and (3) the amounts invested in each asset (the portfolio weights). I will present the case of two assets. Similar formulas exist for more than two assets, but no further insights come from them and the notation is more cumbersome.

Portfolio return variance = $sigma^2_p = w^2_1 times sigma_1^2 + w_2^2 times sigma^2_2 + 2w_1w_2sigma_1sigma_2$,

where $sigma^2_p$ = Portfolio return variance $w_1, w_2$ = Portfolio weights invested in Assets 1 and 2 $sigma_1,sigma_2$ = Standard deviations of returns for Assets 1 and 2 $rho_12$ = Correlation between the returns on Assets 1 and 2

Consider two stocks with standard deviations of 20% and 25%, equal weights (0.50) in each stock, and a correlation between the stocks’ returns of 0.60. The portfolio’s variance would be $sigma^2_p = 0.5^2 times 0.2^2 + 0.5^2 times 0.25^2 + 2 times 0.5 times 0.5 times 0.2 times 0.25 times 0.6 = 0.040625$.

The portfolio’s standard deviation is the square root of the variance, or 0.202 = 20.2%. Note that because the two stocks are less than perfectly correlated (0.60), the portfolio’s standard deviation is lower than the weighted average of the two stocks’ standard deviations (0.5 × 0.2 + 0.5 × 0.25 = 0.225 = 22.5%). If you are interested, you can verify that the lower the assumed correlation between the two stocks, the lower the portfolio’s standard deviation. For example, if we assume a correlation of 0.3 instead of 0.6, the portfolio’s standard deviation would fall from 20.2% to 18.2%.

The Bottom Line

The bottom line is that a well-diversified portfolio reduces risk without sacrificing returns. The key to efficient diversification is combining asset classes that have low correlations. Finally, adding asset classes that are highly correlated with those already in the portfolio is redundant, achieving little benefit and adding to costs.

*Editor’s Note: Lehman Brothers indices have been rebranded after their purchase by Barclays Capital. The updated names for these indices can be found here.

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