GLOBAL DIVERSIFICATION

JOIM JOURNAL OF INVESTMENT MANAGEMENT, Vol. 3, No. 1, (005), pp. 1 11 JOIM 005 www.joim.com GLOBAL DIVERSIFICATION Meir Statman a, and Jonathan Scheid b Correlations between the returns of US stocks and international stocks were higher recently than in the past, reaching 0.86 during the 60 months ending in December 003. Today s investors note the high correlations between US and international stocks and doubt the benefits of global diversification. We argue that the benefits of global diversification remain high and that the correlation between US and international stocks is a misleading measure of the benefits of global diversification. This is for two reasons. First, the benefits of global diversification depend not only on the correlation between the returns of US and international stocks but also on the standard deviations of these returns. Second, we tend to have poor intuition about the link between correlation and the benefits of diversification. A 0.86 correlation seems high enough to eliminate the benefits of diversification, but even correlations much higher than 0.86 are associated with substantial benefits. Dispersion of returns is a better measure of the benefits of diversification because it accounts for the effects of both correlation and standard deviation and because it provides an intuitive measure of the benefits of diversification. We present the relationship between correlation, standard deviation, and dispersion. Anyone following the news cannot escape the impression that correlations among global stock returns are higher now than in the past. Rallies in US a Santa Clara University, Leavey School of Business, Santa Clara, CA 95053, USA. b Loring Ward Advisor Services, a Division of LWI Financial Inc. 3055 Olin Avenue, Suite 000 San Jose, CA 9518, USA. Corresponding author. E-mail: mstatman@scu.edu stocks are accompanied by rallies in international stocks while slumps are accompanied by slumps. Even though Europe, Asia, and Latin America are not suffering the corporate scandals or recordbreaking bankruptcies that have rattled the United States, wrote Landler (00), Markets on all three continents have been moving in lock step with Wall Street. Today s trading was a stark illustration. Shares in Tokyo, Seoul, Hong Kong, Paris, SECOND QUARTER 005 1

MEIR STATMAN AND JONATHAN SCHEID Amsterdam, and London all fell sharply in the wake of a bleak Tuesday in New York. (p. C1). Goetzmann et al. (001) studied the world equity market during the past 150 years and found that correlations were particularly high during the late 19th century and the late 0th century, periods of economic and financial integration. Correlations between the returns of US stocks and international stocks were indeed higher recently than in the past, reaching 0.86 during the 60 months ending in December 003. Today s investors note the high correlations between US and international stocks and doubt the benefits of global diversification. For example, Fuerbringer (00) wrote Americans once invested abroad to reduce portfolio risk if the United States market fell, foreign ones often rose. But this diversification has been hard to get in Europe in recent years because its markets have become very closely correlated with the performance of Wall Street. (BU6). We argue that the benefits of global diversification remain high and that the correlation between US and international stocks is a misleading measure of the benefits of global diversification. This is for two reasons. First, the benefits of global diversification depend not only on the correlation between the returns of US and international stocks but also on the standard deviations of these returns. Second, we tend to have poor intuition about the link between correlation and the benefits of diversification. A 0.86 correlation seems high enough to eliminate the benefits of diversification, but even correlations much higher than 0.86 are associated with substantial benefits. Dispersion of returns is a better measure of the benefits of diversification because it accounts for the effects of both correlation and standard deviation and because it provides an intuitive measure of the benefits of diversification. 1 Dispersion, correlation, and standard deviation Dispersion is the standard deviation of the returns of individual assets around the mean return of all assets. We know dispersion as tracking error, unsystematic risk and diversifiable risk. Solnik and Roulet (000) presented correlation as a function of dispersion and the standard deviation of the returns of a market portfolio. Statman and Scheid (005) presented dispersion as a function of correlation and the standard deviation of individual assets. Dispersion, in the case of two assets, is the deviation of the expected return of each of the two from the return of a portfolio divided equally between the two. The total risk of an asset, measured as the variance of its returns, σ, is the sum of its diversifiable risk, σ D, and undiversifiable risk, σ UD. σ = σ UD + σ D. (1) Consider two funds, one representing US stocks and one representing international stocks. Consider, for simplicity, the case where the standard deviations of the returns of the two funds are the same, σ, and consider a global portfolio that combines the two funds in equal proportions. The global portfolio is fully diversified and its risk is undiversifiable. The risk of the global portfolio, σ UD, is: σ UD = ( 1 ) σ + ( ) 1 σ + ( 1 )( ) 1 σ ρ () = 1 σ + 1 σ ρ (3) (1 + ρ) = σ. (4) The diversifiable risk is the difference between total risk and undiversifiable risk: [ ] σd = σ (1 + ρ) σ (5) JOURNAL OF INVESTMENT MANAGEMENT SECOND QUARTER 005

GLOBAL DIVERSIFICATION 3 = σ [ 1 = σ [ (1 ρ) ] (1 + ρ) (6) ]. (7) Diversifiable risk expressed as standard deviation, σ D,is (1 ρ) σ D = σ. (8) We see that diversifiable risk depends not only on the correlation between returns but also on the standard deviations of returns. Higher correlations reduce the benefits of diversification since they reduce diversifiable risk while higher standard deviations increase the benefits of diversification since they increase diversifiable risk. Campbell et al. (001) found that the benefits of diversification among US stocks increased during 196 1997 because of the combined effects of correlations and standard deviations. Correlations declined during the period while standard deviations increased, each increasing the benefits of diversification. Campbell et al. (001) referred to diversifiable risk as excess standard deviation but we also know it as dispersion. To see that, consider R US as the return of the US fund and R IN as the return of the international fund. The return of a global portfolio that combines the two in equal proportions is R G = (R US + R IN ). (9) The dispersion is [ ] [ ] R US (R US+R IN ) + R IN (R US+R IN ) σ G = (10) = R US R IN. (11) So dispersion is the deviation of the returns of the US and international funds, R US and R IN, from the return of the global portfolio, R G. Table 1 Expected dispersion of the returns of US stocks and international stocks from the returns of a global portfolio divided equally between US stocks and international stocks for various combinations of correlation and standard deviation. Standard deviation Correlation 10.00 (%) 15.00 (%) 0.00 (%) 0.99 0.71 1.06 1.41 0.9.4 3.35 4.47 0.8 3.16 4.74 6.3 0.5 5.00 7.50 10.00 0 7.07 10.61 14.14 Dispersion = standard deviation [(1 correlation)/] 1/. Table 1 shows some examples of the relationship between correlation, standard deviation, and expected dispersion. Note, for example, that expected dispersion is 1.06% when correlation is 0.99 and standard deviation is 15%. Dispersion in global markets Figure 1 and Table show that the correlation between the returns of US stocks and international stocks ranged from a low of 0.35 in the 198 1986 period to a high of 0.86 in the 1999 003 period. The mean annualized standard deviation of the returns of US and international stocks ranged from a low of 11.18% in the 199 1996 period to a high of 0.74% in the 1986 1990 period. There was a mild decrease in the correlations between US and international stocks until 1986 and a marked increase in correlations since then. There was a mild increase in standard deviations until 1990 and a mild decrease since then. The ups and downs of correlations and standard deviations correspond to ups and downs in expected dispersion. Table 1 shows that the expected dispersion between the returns of US stocks and international stocks and the returns of a global portfolio increased from SECOND QUARTER 005 JOURNAL OF INVESTMENT MANAGEMENT

4 MEIR STATMAN AND JONATHAN SCHEID Correlation 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.0 0.10 Avg. SD Correlation Expected annualized dispersion 0.00 1970 1975 1980 1985 1990 1995 000 Year a Correlations and standard deviations are based on monthly returns during 5 years. They correspond in the figure to the middle year in each 5-year period (e.g., 1985 for 1983 1987). Dispersion = standard deviation * [(1 correlation)/] 1/. 5% 0% 15% 5% 0% Avg. SD Dispersion Figure 1 Correlation, standard deviation, and expected dispersion of US and international stocks. a 7.17% in the 1969 1973 period to 11.03% in the 1986 1990 period, and declined to 4.55% in the 1999 003 period. The data show the joint effect of correlation and standard deviation on dispersion. While the correlation between the returns of US and international stocks increased from 0.36 in 199 1996 to 0.86 in 1999 003, the average annualized standard deviation of returns increased from 11.08 to 17.0%. The increase in correlation decreased expected dispersion and the benefits of diversification, but the increase in standard deviations countered some of that effect. For example, if the average standard deviation were at 11.08% in 1999 003 rather than rise to 17.0%, expected dispersion would have been 3.06% rather than 4.55%. Do not put all your eggs in one basket was the maxim of diversification long before mean-variance theory recast it in the language of correlation and standard deviation. Dispersion of returns underlies the baskets maxim. We are wise to disperse our assets among many baskets because all eggs in one basket are likely to meet similar fates while eggs in different baskets are likely to meet different fates. The expected fates of the US and international baskets were very different in 1986 1990 when expected dispersion was 11.03% so the expected benefits of diversifying between them were large. But the expected fates of US stocks and international stocks were more similar in 1999 003, when expected dispersion was 4.55% so the benefits of diversifying between them were smaller. Diversified investors eliminate the fear of lagging a diversified portfolio but they also give up hope of beating it. The first question investors ask as they consider diversification is: By how much should I expect to lag or lead a diversified portfolio if I fail to diversify? Dispersion answers this question, and the answer for any year during 1999 003 was 4.55%. The second question they ask is: Am I willing to lag a diversified portfolio by 4.55% if I have an equal (or perhaps better) chance to lead it by 4.55%? Compare an investor with information about dispersion to an investor with information about correlation. The correlation between US stocks and JOURNAL OF INVESTMENT MANAGEMENT SECOND QUARTER 005

GLOBAL DIVERSIFICATION 5 Table Correlation, standard deviation, and expected dispersion of US and international stocks. a Period Expected annualized dispersion (%) Correlation between the returns of US and international stocks 1969 1973 7.17 0.54 14.88 1970 1974 7.99 0.53 16.56 1971 1975 7.73 0.63 17.9 197 1976 7.79 0.63 18.0 1973 1977 8.03 0.60 17.95 1974 1978 8.95 0.48 17.64 1975 1979 8.18 0.45 15.57 1976 1980 8.7 0.36 14.63 1977 1981 8.47 0.36 14.9 1978 198 9.18 0.39 16.58 1979 1983 7.36 0.54 15.3 1980 1984 7.69 0.54 16.04 1981 1985 7.67 0.47 14.90 198 1986 8.99 0.35 15.75 1983 1987 8.9 0.50 17.84 1984 1988 9.41 0.47 18.6 1985 1989 9.73 0.41 17.89 1986 1990 11.03 0.43 0.74 1987 1991 9.97 0.51 0.11 1988 199 9.03 0.4 16.73 1989 1993 9.03 0.4 16.74 1990 1994 8.65 0.44 16.30 1991 1995 6.8 0.44 1.83 199 1996 6.31 0.36 11.18 1993 1997 6.4 0.46 1.07 1994 1998 5.66 0.67 14.03 1995 1999 5.64 0.68 14.1 1996 000 5.44 0.76 15.67 1997 001 5.49 0.81 17.60 1998 00 5.0 0.85 18.41 1999 003 4.55 0.86 17.0 a Dispersion = standard deviation [(1 correlation)/] 1/. Average annualized standard deviation of the returns of US and international stocks (%) SECOND QUARTER 005 JOURNAL OF INVESTMENT MANAGEMENT

6 MEIR STATMAN AND JONATHAN SCHEID international stocks in 1999 003 was 0.86. An investor asks By how much might I lag a diversified global portfolio when I hold an undiversified portfolio containing only US or only international stocks? The investor might know that a correlation of 0.86 is high and that high correlation is associated with low dispersion but he/she is left to guess the magnitude of that relationship. Does a 0.86 correlation imply an expected lag of, 5, or? The precise relationship between correlation and dispersion is not available to our intuition. Moreover, there is no one-to-one relationship between correlation and dispersion, since dispersion depends not only on correlation and also on standard deviation. It might be best to set aside correlation when we assess the benefits of diversification, and focus on dispersion. We consider the 4.55% expected dispersion for any year during the 1999 003 period as high. Moreover, expected dispersion might understate the full benefits of diversification since realized dispersion might exceed its expected value. Consider, for an extreme example, the case of 1986. International stocks gained 69.97% that year, much of it due to gains in Japanese stocks, while US stocks gained only 16.15%. The realized dispersion was 6.91%, exceeding by a wide margin the 9.41% expected dispersion. Table 3 and Figure present expected and realized dispersion. 1 smaller benefits in down markets than in up markets. Dispersion between US and international stock was lower, on average, when both groups had negative returns than when they had positive returns. But diversification provided benefits in all periods. The monthly dispersion of US and international stocks from the global portfolio during 1969 003 ranged from a low of 0.01% in July 1977 when US stocks lost 1.48% and international stocks lost 1.46%, to a high of 8.4% in October 1990, when international stocks gained 15.61% and US stocks lost 1.3%. Both US and international stocks lost during 99 of the 40 months of the 1969 003 period, both gained during 185 months, and one gained while the other lost during the remaining 136 months. The mean dispersion during the months when both groups gained was 1.31%, higher than the 1. mean dispersion during the months when both lost, so the benefits of diversification were indeed smaller during the months where stocks lost than during months where they gained. Still, diversification provided benefits in all months and it provided the greatest benefits during months where one group of stocks gained and the other lost. The mean dispersion during these months was.7% (see Figure 3). 3 Dispersion in up and down global markets Odier and Solnik (1993) found that correlations among US and international markets were higher when markets dropped than when they rose. So diversification offered smaller benefits in down markets, precisely when the benefits of diversification would have been most welcome. Observation of dispersion, like observation of correlation, indicates that diversification provided Conclusion US investors in the late 1970s and 1980s nodded their heads when advisors explained the benefits of global diversification, but they bought international stocks as sure winners rather than as contributors to diversification. As Middleton (003) wrote, Foreign investing became fashionable when US markets were relatively weak, beginning in the late 1970s. Between 1976 and 1989, the Europe, JOURNAL OF INVESTMENT MANAGEMENT SECOND QUARTER 005

GLOBAL DIVERSIFICATION 7 Table 3 Expected dispersion and realized dispersion of the returns of US stocks and international stocks. Period Expected dispersion (%) Realized return of US stocks a (%) Realized return of international stocks a (%) Realized return of a global portfolio diversified equally between US and international stocks a (%) Realized dispersion of the returns of US and international stocks (%) Difference between realized dispersion and expected dispersion (%) 1969 1973 7.17 16.16 31.19 3.68 7.5 0.35 1970 1974 7.99 16.94 37.65 7.30 10.36.36 1971 1975 7.73 18.16 14.18 16.17 1.99 5.74 197 1976 7.79 7.16.1 4.64.5 5.7 1973 1977 8.03 38.69 37.03 37.86 0.83 7.0 1974 1978 8.95 6.74 3.79 15.7 11.48.53 1975 1979 8.18 4. 19.38 7.58 11.80 3.6 1976 1980 8.7 7.49 34.3 0.91 13.4 5.15 1977 1981 8.47 3.01 6.16 14.59 8.43 0.05 1978 198 9.18 3.65 4.44 8.55 4.11 5.07 1979 1983 7.36 3.68 1.04.36 1.3 6.04 1980 1984 7.69 0.88 0.83 10.03 10.86 3.16 1981 1985 7.67 1.99 4.57 3.8 1.9 6.38 198 1986 8.99 4.48 7.88 6.18 1.70 7.9 1983 1987 8.9 3.16 56.73 44.45 1.9 3.37 1984 1988 9.41 16.15 69.97 43.06 6.91 17.50 SECOND QUARTER 005 JOURNAL OF INVESTMENT MANAGEMENT

8 MEIR STATMAN AND JONATHAN SCHEID Table 3 (Continued ) Period Expected dispersion (%) Realized return of US stocks a (%) Realized return of international stocks a (%) Realized return of a global portfolio diversified equally between US and international stocks a (%) Realized dispersion of the returns of US and international stocks (%) Difference between realized dispersion and expected dispersion (%) 1985 1989 9.73 1.7 4.93 13.33 11.61 1.88 1986 1990 11.03 18.0 8.60 3.31 5.9 5.74 1987 1991 9.97 8.95 10.79 19.87 9.08 0.89 1988 199 9.03 5.94 3.0 14.57 8.63 0.40 1989 1993 9.03 34.69 1.47 3.58 11.11.08 1990 1994 8.65 9.78 11.81 1.0 10.80.14 1991 1995 6.8 11.13 3.86.00 10.87 4.05 199 1996 6.31 0.04 8.01 3.99 4.03.9 1993 1997 6.4 36.77 11.63 4.0 1.57 6.33 1994 1998 5.66 1.31 6.3 13.77 7.54 1.88 1995 1999 5.64 31.39.0 16.71 14.69 9.04 1996 000 5.44 4.41 0.5.33.08 3.36 1997 001 5.49 5.4 7.7 6.35 0.9 4.57 1998-00 5.0 11.33 13.95 1.64 1.31 3.71 1999-003 4.55 11.13 1.1 16.17 5.04 0.49 000 004 NA 1.14 15.64 18.39.75 NA 001 005 NA 31.06 39.17 35.1 4.06 NA a Return is for the middle year in the 5-year period (e.g., 001 for the 1999 003 period). JOURNAL OF INVESTMENT MANAGEMENT SECOND QUARTER 005

GLOBAL DIVERSIFICATION 9 30% 5% Realized Dispersion 0% 15% 5% Expected 0% 1974 1979 1984 1989 1994 1999 Year Figure A comparison of expected dispersion and realized dispersion of the returns of US stocks and international stocks from the returns of global portfolios divided equally between them. Australian, Far East Index surged more than sixfold, while the S&P 500 did not even quadruple. International stocks that zoomed past US stocks in the late 1970s and 1980s lagged behind them in the 1990s and early 000s and the hindsight that brought international stocks into fashion in the 1970s and 1980s took them out of fashion in the 1990s and early 000s. The sagging interest in Asian and European funds has sounded alarm bells among many financial experts whose mantra is diversification, wrote Tam (1998). Dispersion is the deviation of the returns of undiversified portfolios from that of a diversified portfolio and we explore its relationship to correlation and standard deviation. For example, the correlation between US stocks and international stocks was 0.86 in the 60-months during 1998 003, higher than at any time during 1969 003. But the benefits of global diversification remained high. We find that the expected annual dispersion in 1999 003 was 4.55%, so investors who held undiversified portfolios containing only US stocks or only international stocks should have expected to lead or lag by 4.55% a fully diversified global portfolio divided equally between US and international stocks. Some have argued that the sagging interest of US investors in international stocks in the late 1990s and early 000s is due to the increasing correlations between the returns of US stocks and international stocks. But this cannot be true. The benefits of global diversification remained high in the 1990s and early 000s. Rather, global diversification lost its popularity among US investors in the 1990s and early 000s because of high regret among US investors who bought international stocks as sure winners only to find, in hindsight, that international stocks were losers. We can see that high correlations are not responsible for the declining interest in global diversification as we compare diversification between US stocks and international stocks to diversification between small US stocks and large US stocks. The correlation between the returns of small US stocks and large US stocks was 0.89, higher than the recent 0.86 correlation between US stocks and international stocks, yet no one employs that high correlation to advocate concentrating US portfolios in small stocks. Global diversification remains useful since dispersion between US stocks and international stocks remains high and we do not know in foresight SECOND QUARTER 005 JOURNAL OF INVESTMENT MANAGEMENT

10 MEIR STATMAN AND JONATHAN SCHEID 0% a: Periods where both returns were positive, 1969 003 18% 16% 14% Returns 1% 8% 6% 4% % 0% Jan-69 Jan-71 Jan-73 Jan-75 Jan-77 Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 b: Periods where both returns were negative, 1969 003 0% Jan-69 Jan-71 Jan-73 Jan-75 Jan-77 Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 5% Returns 15% 0% 5% 0% c: Periods where one return was positive and the other was negative, 1969 003 15% Returns 5% 0% 5% 15% Jan-69 Jan-71 Jan-73 Jan-75 Jan-77 Jan-79 Jan-81 Jan-83 Jan-85 Jan-87 Jan-89 Jan-91 Jan-93 Jan-95 Jan-97 Jan-99 Jan-01 Jan-03 a The top and bottom of each line are the returns of U.S. stocks and international stocks during a month. Figure 3 Differences in returns between US and international stocks during each month (1969 003). a JOURNAL OF INVESTMENT MANAGEMENT SECOND QUARTER 005

GLOBAL DIVERSIFICATION 11 whether US stocks or international stocks would be the future winners. Don t put all your eggs in one basket is the golden rule which, is a more sophisticated form, underpins much of modern investment theory, wrote The Economist (1997). Translated into investment theory this means investors should aim to hold a portfolio of assets whose returns are not highly correlated. Many, including academics, investment professionals, and The Economist (1997) stumble when they try to explain the seemingly simple concept of correlation. The Economist was left with egg on its face when we looked at correlation in a recent Economics Focus. This noted that, among nine big economies, stock market correlations have averaged around 0.5 since the 1960s. We translated this to mean that for every 1% rise (or fall) in, say, American share prices, share price in other markets will typically rise (fall) by 0.5%. Two weeks later, we published a missive from a reader aiming to set us straight. This provoked a barrage of correspondence from readers, all of which was perfectly correlated agreeing that both our article and the letter were off the mark. The unanimity, alas, broke down when it came to saying precisely what correlation means. We argue that dispersion, not correlation, underlies the baskets diversification maxim. Dispersion is easy to calculate directly from returns data or as a function of correlation and standard deviation, and it provides a more intuitive measure of the benefits of diversification. and tilted toward large stocks, is replaced by an index that reflects better all stocks in all international markets. This is the correlation between monthly returns of small stocks (CRSP 6 10) and large stocks (CRSP 1 5) during 196 003. References Campbell, J., Lettau, M., Malkiel B., and Xu, Y. (001). Have Individual Stocks Become More Volatile? An Empirical Exploration of Idiosyncratic Risk. The Journal of Finance 56(1), 1 43. The Economist (1997). Unscrambling Correlation. December 6, 8. Fuerbringer, J. (00). Weaker Dollar Adds to Potential of Foreign Stocks. Wall Street Journal, May 6, BU6. Goetzmann, W., Li, L., and Geert Rouwenhorst, K. (001). Long-Term Global Market Correlations. National Bureau of Economic Research, working paper. Landler, M. (00). As the Work Tracks Wall St., US Leadership is Two-Edged. The Wall Street Journal July 5, Cl&. Middleton, T. (003). Why you Don t Need Foreign Stocks. http://moneycentral.msn.com, Posted on May 6, 1 4. Odier, P. and Solnik, B. (1993). Lessons for International Asset Allocation. Financial Analysts Journal March April, 63 77. Solnik, B. and Roulet, J. (000). Dispersion as Cross- Sectional Correlation, Financial Analysts Journal, January February, 56(1) 54 61. Statman, M. and Scheid, J. (005). Correlation and the Benefits of Diversification, Santa Clara University, working paper. Tam, P. W. (1998). Sagging Interest in International Mutual Funds Alarms Advisers. Wall Street Journal May 7, C5. Keywords: Diversification; global diversification; correlation; behavioral finance Notes 1 Expected dispersion is likely to be even higher if the EAFE index, which is limited to developed international markets SECOND QUARTER 005 JOURNAL OF INVESTMENT MANAGEMENT