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Optimization, Cointegration and Diversification Gains from International Portfolios: An Out-of-Sample Analysis Chanwit Phengpis Department of Finance California State University, Long Beach Long Beach, CA 90840 Phone: (562) 985-1581 Fax: (562) 985-1754 E-mail: [email protected] and Peggy E. Swanson* Department of Finance and Real Estate UTA Box 19449 The University of Texas at Arlington Arlington, TX 76019 Phone: (817) 272-3841 Fax: (817) 272-2252 E-mail: [email protected]

* Corresponding author

Optimization, Cointegration and Diversification Gains from International Portfolios: An Out-of-Sample Analysis Abstract This study investigates comparative performance of iShares and their underlying market indices in a portfolio context from the perspective of U.S. investors. Two aspects are important. First, portfolios based on standard optimization procedures and a portfolio based on cointegration procedures are created and out-of-sample performance is compared. The portfolio utilizing cointegration inputs shows superior out-of-sample performance. Second, portfolio performance measurement is extended to different holding periods. The findings do not differ.

1

Optimization, Cointegration and Diversification Gains from International Portfolios: An Out-of-Sample Analysis 1. Introduction Numerous studies have identified international diversification gains when foreign assets are added to a domestic portfolio. Most works such as Levy and Sarnat (1970), Errunza (1977), Bailey and Stultz (1990), Harvey (1991) and Li, Sarkar and Wang (2003) use foreign stock market indices to represent foreign investments from the perspective of U.S. investors. These stock indices however are not tradable assets and hence are not easily replicable by investors. Many countries, especially lesser developed ones, also have investment restrictions making their markets partially or entirely unavailable to foreign investors. Therefore, realistically obtainable benefits from international diversification may deviate from the empirical magnitudes suggested by these studies. Realizing the obtainability issue, several other studies investigate the benefits from foreign investment opportunities based on assets that are readily available for purchase in the U.S. market. These include, for example, open-end international mutual funds, closed-end country funds and American Depository Receipts (e.g., Cumby and Glen, 1990; Chang, Eun and Kolodny, 1995; Jiang, 1998; Errunza, Hogan and Hung, 1999). Despite U.S. availability, these assets are not necessarily well diversified and hence may not appropriately reflect the true diversification potential of each country market. That is, the first two, open-end international mutual funds and closed-end country funds, are designed to generate profits and not to be representative of the individual country markets while the third, American Depository Receipts, represent only individual companies and not countries. The introduction of iShares (originally known as WEBS) into the U.S. market has

2 resolved both the obtainability and representativeness issues. iShares are exchange traded funds that are designed to track the MSCI broad market indices for individual countries. Thus, diversification benefits that are representative of the entire foreign country markets can be realistically obtained from domestic purchases of iShares. However, iShares are a recent phenomenon causing analyses of international diversification gains over long-term horizons to require the use of historical data for MSCI indices together with the iShares data. Conceptually, this approach is valid because, by design, each iShare fund should behave as its underlying market index behaves. Whether or not this suggested behavior holds true empirically should be investigated when the gains based on the underlying market indices are measured. The next relevant issue is the portfolio selection process. An obvious goal is to select the markets and construct an international portfolio with significant diversification potential relative to the domestic market. Typical selection in prior related studies involves the optimization procedure in a mean-variance framework (Markowitz, 1952). For example, using the U.S. market index and closed-end country funds (Chang et al., 1995) or the U.S. market index and ADRs (Jiang, 1998) as an opportunity set, the authors identify international diversification gains by comparing the excess return per unit of risk of the tangent or optimized portfolio with that of the U.S. market. However, whether or not the magnitude of suggested gains actually can be realized after portfolio formation is questionable because the historically optimized portfolio tends to perform poorly out-of-sample due to the estimation error, especially of average returns (e.g., Best and Grauer, 1991). To address this problem, Jorion (1985) employs the Bayes-Stein or shrinkage estimators to adjust historical average returns of the U.S. and six developed-country market indices as inputs for the optimization procedure. He finds that the resulting optimized portfolio shows an

3 improvement in out-of-sample performance. Whether or not this approach works well with a considerably greater number of markets, however, has not been addressed. Further, when the shrinkage estimators are employed, the estimation error in average returns is reduced but not entirely eliminated since the “adjusted” average returns are essentially historical average returns with adjustments. In this instance, an equal weight portfolio of all available assets may perform better than the historically optimized portfolio because the equal allocation helps reduce the overweighting/underweighting possibility (e.g., Jobson and Korkie, 1980; Michaud 1998). Nonetheless, if the number of assets n in an opportunity set is large, the 1/n allocation to each asset in the equal-weight portfolio can become complex and incur considerable transaction costs to initiate and maintain through time. Another possible portfolio selection process is based on cointegration. Rather than relying on individual asset performance, the process relies on the knowledge of possible long-run equilibrium relationships or cointegrating relationships among assets (e.g., Engel and Granger, 1987). These relationships emerge because the assets share, and hence their prices are driven by, common stochastic trends (e.g., Gonzalo and Granger, 1995). In an international context, the common trends among stock markets of different countries may result from similar economic fundamentals (e.g., Chen et al., 2002) and capital market integration (e.g., Masih and Masih, 2001) within or across regions. The implication is that cointegrated assets exhibit significant long-term comovements and their diversification potential is lessened (e.g., Taylor and Tonks, 1989; Kasa, 1992; Chen, Firth and Rui, 2002; Zhong and Yang, 2005). To maximize diversification benefits, identification of assets that are independent of these significant longterm comovements becomes the crux of the cointegrated-based portfolio selection process. Previous studies typically provide implications of cointegration in portfolio selection only in a

4 qualitative fashion. That is, they do not show how to practically use the finding of cointegrating relationships (or lack thereof) in portfolio selection. Thus, the extent of international diversification gains from utilizing this long-term information has not been quantitatively verified, especially during the out-of sample period following the cointegration-based portfolio formation. Therefore, the primary objectives of this study are (1) to investigate whether the performance characteristics of iShares differ from those of the underlying market indices, in both individual and portfolio contexts and (2) to measure out-of-sample performance and diversification gains of an international portfolio derived from cointegration analysis. Comparisons of this cointegration-based portfolio will be made to the portfolios that have been examined or suggested by prior studies including the historically optimized portfolios with and without shrinkage estimators and the equal-weight all-available-market portfolio. The remainder of the paper is organized as follows. Section 2 describes the data, the analytical approach and econometric methodology employed, with results presented in Section 3. Section 4 concludes the paper.

2. Data, Analytical Approach and Methodology 2.1 Data The data set is obtained from the Datastream International databank and divided into the estimation period and the out-of-sample period. The estimation period (during which optimized portfolio weights are estimated, cointegration analysis is conducted and the results are used as prior information in portfolio construction) is from January 1988 through July 2000. The out-ofsample period (during which performance of the constructed portfolios is evaluated) is from

5 August 2000, the earliest month in which monthly returns on iShares can be obtained, through December 2006. End-of-month observations on total return indices for Russell 3000 and for 20 MSCI foreign country indices are collected for the estimation period. The Russell 3000 index represents the domestic market. In contrast, the 20 MSCI foreign country indices (comprising Australia, Austria, Belgium, Brazil, Canada, France, Germany, Hong Kong, Italy, Japan, Malaysia, Mexico, Netherlands, Singapore, South Korea, Spain, Sweden, Switzerland, Taiwan and U.K.) that have corresponding iShares traded in the U.S. market during the out-of-sample period reflect foreign stock market representations. End-of-month observations of 1) total return indices for Russell 3000 and for 20 MSCI foreign country indices and 2) total return indices for iShares Russell 3000 and 20 corresponding foreign country iShares are collected for the out-of-sample period. These paralleling data allow performance comparisons between portfolios of the underlying market indices and portfolios of iShares during the out-of-sample period. 2.2 Analytical Approach and Methodology Average returns, standard deviation of returns, and pair-wise return correlations are computed for Russell 3000 and 20 foreign market indices during the estimation period. Similar to Chang, Eun and Kolodny (1995) and Jiang (1998), among others, the optimization procedure is performed to obtain portfolio weights of various markets in the optimized portfolio. Subject to non-negative weights and the 100% weight sum (i.e., Wi ≥ 0 and

n

Σ Wi = 100%), the weights are i=1

chosen such that the portfolio total-risk-adjusted performance or Sharpe ratio (Sharpe, 1966) in (1) is maximized.

6

Sharpe Ratioi =

R i − rf Si

(1)

where R i is an average return on the portfolio, rf is the average monthly return on 3-month U.S.

Treasury bills used as a proxy for the risk-free rate and Si is standard deviation of returns of the portfolio. The optimization procedure, however, has a well-known drawback resulting from the estimation error (e.g., Michaud, 1989; Best and Grauer, 1991; Britten-Jones, 1999). The inputs for the optimization procedure are average returns and the variance-covariance matrix for the assets being considered. While the estimate of the variance-covariance matrix tends to be robust, the estimates of average returns are considerably noisy (e.g., Broadie, 1993). As a result, the optimized portfolio based on historical data is often found to perform poorly out-of-sample. This is primarily because assets with historically high average returns are overweighted while assets with historically low average returns are underweighted (i.e., have positive but minimal weights or even have zero weights). To alleviate the estimation error, Jorion (1985, 1986) uses the Bayes-Stein or shrinkage estimators to adjust historical average returns and finds that out-of-sample performance of the optimized portfolio derived from adjusted average returns is improved. Specifically, average returns for various assets are adjusted or “shrunk” towards a global average as in (2). Adjusted R i = α R 0 + (1- α) R i

(2)

where α is the shrinkage factor; R 0 is the average return on the global minimum variance portfolio which is chosen because the weights for this portfolio mathematically depend on the variance-covariance matrix, not on average returns of individual assets; and R i is the unadjusted

7 average return on asset i. Then, the shrinkage factor α which ranges between 0 and 1 can be estimated as in (3). α =

N+2 N + 2 + M(R − R 0 )T ∑ −1 (R − R 0 )

(3)

where N is the number of assets, M is the sample size, R is a vector of unadjusted average returns, R 0 is a vector with each element equal to R 0 , and ∑ is the variance-covariance matrix. With shrinkage estimators, the relatively high (low) average returns are adjusted downward (upward) prior to the optimization procedure. Thus, the number of assets with non-zero and positive weights in the optimized portfolio will typically increase as the adjustments help balance the extremely high and low average returns on various assets. This reduces the estimation error and contributes to the improvement in out-of-sample performance of the historically optimized portfolio (e.g., Chopra and Ziemba, 1993). Despite the shrinkage estimators, the adjusted average returns are essentially “adjusted” historical information, implying that the ensuing optimization procedure continues to rely on historical performance of individual assets as inputs. A different approach where the portfolio selection process is based on cointegration analysis may become beneficial. Rather than relying on historical individual asset performance that is particularly sensitive to estimation errors, the selection process relies on the knowledge of cointegrating relationships as a result of shared common trends among assets. Thus, due to their long-term characteristic, the relationships and their implications on portfolio performance should remain relevant as time horizons are extended beyond the estimation period. Because cointegrating relationships imply significant long-run comovements, the assets that are independent of the relationships should be identified and included in a portfolio to enhance its diversification potential.

8 The first step is to identify cointegrated and non-cointegrated markets. To achieve this, Johansen cointegration methodology (e.g., Johansen, 1988, 1991), which is based on the vector autoregressive (VAR) process, is performed on foreign market index prices. The test for the presence of cointegration is the rank test for r non-zero eigenvalues (λi). The test statistic for the null hypothesis of at most r long-run relationships or cointegrating vectors (CIVs) against the alternative of p CIVs where p is the number of assets being considered is the λtrace statistic given in (4). p

λtrace = - T Σ ln(1-λi)

(4)

i = r +1

The distributions of the λtrace statistics are non-standard and partially dependent on the specification for deterministic components in the VAR. Hence, the appropriate deterministic specification must be identified. Since market index prices show upward trends over the estimation period, the VAR accounts for these trends but the remaining question is whether or not the trends should also be present in the cointegration space. Therefore, the null hypothesis of no linear trends in the CIVs is tested against the alternative of linear trends the CIVs. This is achieved by computing the G(r) statistic in (5) which is distributed as χ2(r) (Johansen, 1994). r ⎡ (1 − λ * ) ⎤ i G(r ) = T ∑ ln ⎢ ⎥ (1 ) − λ ⎢ i =1 ⎣ i ⎥⎦

(5)

where λi* and λi are the eigenvalues from the VARs under the alternative and null hypotheses, respectively. The finding of cointegration does not necessarily indicate that all market indices subject to prior cointegration rank tests are cointegrated. Some markets may not be part of long-run relationships in that do not share common trends with cointegrated markets (as evidenced by the

9 rank tests) and are excludable from the cointegrated group. Thus, the excludable markets should be included in the portfolio due to the absence of significant long-term comovements among themselves and with the cointegrated markets. The null hypothesis that the foreign market i can be excluded from cointegration relationships can be tested using the likelihood ratio test statistic derived in Johansen and Juselius (1990, pages 193-194). Further, within the non-excludable or cointegrated group, some markets may be weakly exogenous while some others are endogenous. Weakly exogenous markets are the source of common trends that drive the prices of endogenous markets in the cointegrated group. Specifically, weakly exogenous markets independently follow their own time paths and do not respond to deviations from cointegrating relationships. However, they serve as avenues through which endogenous markets adjust to deviations to retain the relationships and to share the common trends. Therefore, weakly exogenous markets are part of the cointegrated group only to the extent that endogenous markets behave as connectors to maintain the cointegrating relationships. Without endogenous markets, the weakly exogenous markets are not cointegrated among themselves (nor with the excludable markets already selected earlier) and should be included in the portfolio due to their long-run independence. Selection of weakly exogenous markets from the cointegrated group is consistent with suggestions in prior studies such as Chen et al. (2002) that there are essentially fewer independent assets available to investors than a simple count of number of assets in the cointegrated system. The null hypothesis that the market index i is weakly exogenous can be tested using the likelihood ratio test statistic shown in Johansen (1992, page 323). Identification of markets with non-zero and positive weights in the cointegration-based portfolio is now complete. The markets are 1) the U.S. market (by construction to represent the

10 domestic market exposure); and 2) the non-cointegrated excludable foreign markets and the weakly exogenous foreign markets which are free from significant long-term comovements. Unlike the optimization procedure, cointegration analysis identifies the markets to be included in the portfolio but does not explicitly provide specific portfolio weights for the selected markets. The equal-weight allocation is a reasonable alternative. This allocation simplifies the allocation decision by assuming that the selected markets must be present in the portfolio and are equally important diversification gain contributors. The above procedures provide all prior information needed for portfolio construction. Four portfolios of market indices are formed, and their risk-adjusted performance over the outof-sample period (i.e., August 2000 through December 2006) is evaluated based on the Sharpe ratio. Portfolio 1 is an equal-weight portfolio of all 21 market indices. Because no prior information is needed to constitute this portfolio, data from the estimation period are not necessary. Given the error in asset returns estimation and despite the use of shrinkage estimators as a partial remedy, the historically optimized portfolio may continue to overstate diversification benefits actually received. Therefore, during the out-of-sample period, an equal-weight portfolio which includes all assets and implicitly assumes their equal importance may in fact be closer to the true mean-variance optimality than the optimized portfolio based on historical data. The equal-weight scheme is advocated by Jobson and Korkie (1980), Michaud (1998) and Windcliff and Boyd (2004). Portfolios 2 and 3 are historically optimized portfolios in which the weights for various

markets are obtained from the optimization procedure over the estimation period (1988:01 2000:07) without and with average returns adjusted by shrinkage estimators, respectively. Portfolio 4 is an equal-weight cointegration-based portfolio for which the market selection is

11 done based on cointegration analysis over the estimation period. It is expected that this portfolio performs well during the out-of-sample period if cointegration analysis is in fact useful in the portfolio selection process. Once the composition of the various portfolios has been established, performance comparisons are made. To measure if the each of the four portfolios provides significant international diversification to U.S. investors, the portfolio Sharpe ratio is compared against the Sharpe ratio of the Russell 3000 index (US) based on the Jobson-Korkie Z-statistic (Jobson and Korkie, 1981). This Z-statistic is under the null hypothesis that the two Sharpe ratios are equal and is computed as in (6). Z=

where θ =

R i SUS − R US Si θ

(6)

⎤ 1⎡ 2 2 1 2 2 1 2 2 R i R US 2 2 (SiUS + Si2SUS ) ⎥ ; R i and R US are ⎢ 2Si SUS − 2SiSUS SiUS + R i SUS + R US Si − T⎣ 2 2 2SiSUS ⎦

average excess returns on an equal-weight portfolio i and US, respectively; Si and SUS are the estimated return standard deviation of i and US, respectively; and SiUS is the estimated return covariance between i and US. Finally, the above portfolio construction and performance tests are repeated using the iShares data during the out-of-sample period. This procedure serves two purposes. First, whether or not performance rankings based on portfolios of underlying market indices are consistent with those based on portfolios of iShares, the truly obtainable portfolios, can be investigated. Second, whether or not the diversification gains provided by underlying market index portfolios and iShares portfolios significantly differ can be determined. This is done by computing the JobsonKorkie Z-statistic as in (6) but replacing the parameters for U.S. and the market index portfolio with those for the underlying market index portfolio and the corresponding iShares portfolio.

12 3. Results

Table 1 presents relevant descriptive statistics of iShares and their underlying market indices during the out-of-sample period (2000:08 - 2006:12) for which the two data sets are concurrently available. As expected, the return correlations between iShares and the market indices are very close to one. The minimum is 0.954 for the Austria market (AL) while the maximum is 0.998 for the U.S. market (US). This finding implies that iShares managers are able to closely adjust their holdings in securities to replicate the corresponding market indices. Further, the average returns of iShares are slightly lower than those of the underlying market indices for 15 out of 21 markets. However, the Z-statistic on the difference between average returns on each pair of iShares and the market index is not statistically significant at any conventional level. The insignificance result also holds true for the F-statistic on the difference between return variances (and hence return standard deviations) on each pair. Therefore, the riskreturn characteristics of individual iShares and their underlying market indices do not significantly differ. This finding validates the use of market index data for a prior period (i.e., the estimation period during which the iShares do not exist) to form iShares portfolios for investments in a later period (i.e., the out-of-sample period during which iShares do exist). Table 2 shows results from estimating the optimized portfolio weights using market index data during the estimation period (1988:01 - 2000:07) without and with shrinkage estimators. Without the shrinkage estimators, the optimized portfolio yields the maximized Sharpe ratio of 0.313. It comprises 7 markets with non-zero portfolio weights including US, Brazil (BZ), Mexico (MX), Netherlands (NT), Sweden (SW), Switzerland (SZ) and Taiwan (TW). With the shrinkage estimators, the average returns on the markets with relatively high average returns are adjusted downwards including, for example, BZ (from 3.382% to 2.269%)

13 and MX (from 2.623% to 1.913%). In contrast, the average returns on the markets with relatively low average returns are adjusted upwards including, for example, AT (from 0.639% to 0.982%) and BG (from 1.234% to 1.261%). The optimized portfolio which is derived using adjusted average returns as inputs yields the Sharpe ratio of 0.302. The adjustments, however, do not drastically alter the number nor identification of markets with non-zero portfolio weights. The portfolio comprises 7 markets (which have been previously chosen based on the optimization procedure without shrinkage estimators) plus BG (for which its portfolio weight becomes nonzero after its average return is adjusted upwards). The similarity between composition of the two portfolios is due to the fact that the shrinkage factor estimate is 0.531 rather than being close to one (which would make the adjustments more significant and possibly make the compositions discernibly different - see Equation (2)). This implies that the two portfolios may perform similarly out-of-sample, thereby negating the suggested advantage of applying shrinkage estimators to historical data. An alternative to the optimization procedure is to select the markets based on cointegration. Table 3 shows results from multivariate cointegration tests of foreign market indices during the estimation period. In the first column, the G(r) statistics indicate rejection of the null hypothesis of no linear trends in the cointegration space for all potential r cointegrating relationships except where r = 1. Thus, the VAR for cointegration rank tests is constructed such that linear trends are incorporated into the CIVs. Also, the VAR has one lag which is the minimum sufficient in eliminating serial correlation in residuals based on the LM test statistic (Dennis, Hansen, Johansen and Juselius, 2005; Page 177). The λtrace statistics suggest that the null hypothesis of r ≤ 4 cannot be rejected (second and third columns). Therefore, there are 4 cointegrating relationships among foreign markets. Conditional on the cointegration finding, the

14 exclusion test statistics reveal that the French (FR), Malaysian (ML) and Taiwanese (TW)

markets are excludable from the relationships and thus from the cointegrated group. These three markets should be included in the portfolio since they do not exhibit significant long-term comovements with one another nor with the non-excludable/cointegrated markets as implied by the cointegrating relationships. The next step is the identification of weakly exogenous markets within the cointegrated group that should be further included in the portfolio. Therefore, the VAR comprising only the non-excludable/cointegrated markets (all markets except FR, ML and TW) is estimated and the weak exogeneity tests are performed, with results shown in the last column of Table 3. The test statistics indicate non-rejection of the null hypothesis of weak exogeneity for the Austrian (AT), Italian (IT), Japanese (JP), Mexican (MX), South Korean (SK), Spain (SP), Swiss (SZ) and British (UK) markets. Therefore, these markets are weakly exogenous in that they follow their own time paths without responding to deviations from cointegrating relationships over time. They are part of the cointegrated group only to the extent that other markets (i.e., endogenous markets) adjust to maintain cointegrating relationships once deviations from the relationships occur. Without endogenous markets, the weakly exogenous markets become an additional source of diversification gains and should also be included in the portfolio due to the absence of longrun comovements with one another and with the excludable markets already selected. Now the composition of the markets in the equal-weight cointegration-based portfolio can be identified. They are 1) the U.S. market (US) to represent domestic exposure; and 2) the excludable foreign markets (FR, ML and TW) and the weakly exogenous foreign markets (AT, IT, JP, MX, SK, SP, SZ and UK), totaling 12 markets in all.

15 Table 4 presents the performance test results for various portfolios during the out-ofsample period (2000:08 - 2006:12) based on the selection process with the market index data during the estimation period (1988:01 - 2000:07). Results are shown separately for market index portfolios and iShares portfolios. Focusing on market index portfolio results (upper portion of the table), Portfolio 1, which is an equal-weight portfolio of all 21 markets and thus assumes no asset selection, performs very well against the domestic market benchmark. Its Sharpe ratio of 0.169 is obviously higher than the Sharpe ratio of 0.014 for the Wilshire 3000 index (US). Its Jobson-Korkie Z-statistic (henceforth, J-K statistic) against US of 2.698 is statistically significant at the 1% level. This implies significant diversification gains relative to the domestic market. Portfolio 2, which is the historically optimized portfolio without shrinkage estimators, also

performs quite well against US. However, its Sharpe ratio is lower at 0.100 and is statistically significant at the weaker 5% level based on the J-K statistic against US of 2.214. Contrary to the suggestions and findings by prior studies, Portfolio 3, which is the historically optimized portfolio with shrinkage estimators, performs even more poorly than Portfolio 2 out-of-sample. Its Sharpe ratio of 0.075 is lower than the Sharpe ratios of Portfolios 1 and 2. Also, its J-K statistic against the US of 1.728 is statistically significant at only the 10% level. Portfolio 4, which is an equal-weight cointegration-based portfolio, performs better than Portfolios 1 through 3. Its Sharpe ratio of 0.181 is the highest and its J-K statistic against US of

2.693 is statistically significant at the 1% level. Relative to Portfolio 1, an investor who constructs Portfolio 4 can obviously benefit from reduced complexity and transaction costs of initiating and maintaining the portfolio as a result of a reduction in the number of invested markets (i.e., from 21 to 12). The above performance rankings also hold true for the corresponding iShares portfolios

16 (lower portion of Table 4). Portfolio 4 yields the highest and statistically significant Sharpe ratio of 0.160 and hence is the best performing, followed by Portfolios 1, 2 and 3. Further important results are that the Sharpe ratio of each iShares portfolio is lower than the Sharpe ratio of the underlying market index portfolio. However, the J-K statistics of iShares portfolio against underlying market index portfolio for the two better performing portfolios -- Portfolio 4 (-1.408) and Portfolio 1 (- 1.450) -- are not statistically significant. This indicates insignificant performance deviation of iShares portfolios from the underlying market index portfolios. The same comparison for lesser performing portfolios shows different results. The relevant J-K statistics for Portfolio 2 (-1.647) and Portfolio 3 (-1.705) are statistically significant. This suggests poorer performance of iShares portfolios relative to the underlying market index portfolios, albeit at the relatively weak 10% significance level. The paralleling performance of the iShares portfolio and the underlying market index portfolio for Portfolios 1 and 4 is likely the result of an equal-weight allocation which balances portfolio weights of the included markets. In contrast, the historically optimized allocation in Portfolios 2 and 3 may assign disproportionate portfolio weights to the markets for the out-of-

sample period. This accentuates the negative effect of individual iShares with poorer performance than the underlying market indices (despite their statistical insignificance on an individual basis) on portfolio performance. As a result, the significantly poorer performance of the iShares portfolio relative to the underlying market index portfolio is observed for Portfolios 2 and 3. To verify that the performance difference between iShares and the underlying market index portfolios, if observed, is the result of allocation and not necessarily fundamental differences in risk and return, the equal weights are allocated to the markets in these two portfolios. The results shown in the Appendix indicate that the J-K Z-statistics of iShares

17 portfolio against market index portfolio for the equal-weight Portfolios 2 and 3 are statistically insignificant. In sum, with well-balanced portfolio weights of included markets such as Portfolios 1 and 4, the performance of iShares and underlying market index portfolios do not

differ. This finding validates the use of underlying market indices as criteria for constructing iShares portfolios when long spans of data are needed for analyses. Since the construction of Portfolio 4 is conditional on the absence of long-term comovements implied by the cointegrating relationships among foreign markets, it is informative to test how differing holding periods after portfolio formation affect portfolio performance. Table 5 shows the Sharpe ratios of the portfolios (identified in Table 4) during various out-ofsample horizons following the estimation period. The results for market index portfolios are strongly consistent with those for iShares portfolios. Over a 1-year horizon after portfolio formation, Portfolio 4 yields the Sharpe ratios of -0.455 based on market index data and -0.459 based on iShares data. Thus, its performance is poorer than that of the domestic market and Portfolios 2 and 3. This finding implies that the 1-year holding period may be too short for

investors to reap possible diversification benefits from the absence of long-term comovements among foreign markets. Longer horizons may be needed for such benefits to be discernible. The above possibility is indeed validated through the results for longer holding periods. The Sharpe ratio and hence performance of the cointegration-based Portfolio 4 improve as the horizon is extended. Importantly, the portfolio shows the best performance in that its Sharpe ratios are higher than those of the US and are the highest among all portfolios for all horizons beyond one year. Therefore, consistent with the cointegration concept that cointegration is a long-term property, investors can significantly enhance diversification benefits of the

18 cointegration-based portfolio derived from the absence of significant long-term comovements by holding the portfolio over longer horizons.

4. Conclusions

This study investigates comparative performance of iShares and their underlying market indices in both individual and portfolio contexts and measures out-of-sample performance and diversification gains of international portfolios from the perspective of U.S. investors. A primary motivation is that the historically optimized portfolio typically does not perform well out-ofsample due to the overweighting/underweighting problem resulting from estimation errors caused by using historical average returns on assets as inputs for the optimization procedure. Further, even when shrinkage estimators are employed, the overweighting/underweighting problem may not necessarily be resolved. The average returns used as inputs in the optimization procedure are essentially historical individual asset performance after adjustments. In addition, an equal-weight all-available-asset portfolio which may reduce the overweighting or underweighting possibility due to its balanced allocation can become complex and result in substantial transaction costs if the number of all available assets is large. Thus, we give specific attention to the construction of a portfolio based on cointegration analysis during the estimation period (1988:01 - 2000:07) and to its out-of-sample performance (2000:08 - 2006:12) relative to the U.S. market together with other portfolios based on different formation criteria employed or suggested by prior studies. Using cointegration analysis, which relies on the knowledge of significant long-term comovements implied by cointegrating relationships, is a potentially useful alternative for portfolio selection. Due to their long-term

19 nature, the relationships and their implications on portfolio diversification should remain valid as time horizons are extended and the out-of-sample period is evaluated. Unlike the findings in prior studies, the results indicate that the risk and return characteristics of individual iShares do not differ significantly from those of the underlying market indices. Further, with the allocation that balances portfolio weights of the included markets, the Sharpe ratios and performance of iShares portfolios and underlying market index portfolios do not deviate significantly from each other. These findings validate the use of the underlying market index data together with the iShares data in empirical work. The combination of the two data sets is especially relevant for an analysis that requires long time spans of data because iShares have been in existence only a relatively short period of time. Using the underlying market index data, the cointegration-based portfolio is derived. It comprises: 1) the U.S. market to represent the domestic market exposure; and 2) the excludable foreign markets and the weakly exogenous foreign markets. Specifically, the excludable markets are not part of the cointegrated group while the weakly exogenous markets, despite being part of the cointegrated group, independently follow their own stochastic trends. Therefore, both of these markets show strong diversification potential due to their independence from significant long-term comovements implied by the cointegrating relationships. Out-of-sample results indicate that the cointegration-based portfolio provides significant diversification gains relative to the U.S. market and shows the best performance relative to other portfolios. In relation to the equal-weight all-available-market portfolio, the smaller number of included markets suggests decreases in complexity and transaction costs of initiating and maintaining the portfolio. Further, the relatively poor out-of-sample performance of the historically

optimized

portfolio

emphasizes

the

estimation

errors

and

hence

the

20 overweighting/underweighting problem identified by prior studies. Unlike the findings in earlier work, however, out-of-sample performance of the historically optimized portfolio with shrinkage estimators is also poor, indicating that the weighting problem is not adequately corrected. Therefore, relying on the absence of significant long-run comovements is more beneficial than relying on individual asset performance (even with the adjustments) in the portfolio selection process using historical data. Finally, consistent with the long-term characteristic of cointegration, the out-of-sample performance of the cointegration-based portfolio improves as the investment horizons are extended and remains the best for all investment horizons beyond one year.

References

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24 Table 1: Descriptive Statistics of iShares Index Funds and Underlying Market Indices (2000:08 - 2006:12) iShares

Market Index

Market Abbreviation

Name

Average Return (%)

Standard Deviation (%)

US AL AT BG BZ CN FR GM HK IT JP ML MX NT SG SK SP SW SZ TW UK

iShares Russell 3000 iShares MSCI Australia iShares MSCI Austria iShares MSCI Belgium iShares MSCI Brazil iShares MSCI Canada iShares MSCI France iShares MSCI Germany iShares MSCI Hong Kong iShares MSCI Italy iShares MSCI Japan iShares MSCI Malaysia iShares MSCI Mexico iShares MSCI Netherlands iShares MSCI Singapore iShares MSCI South Korea iShares MSCI Spain iShares MSCI Sweden iShares MSCI Switzerland iShares MSCI Taiwan iShares MSCI United Kingdom

0.273 1.541 2.255 1.217 1.986 0.886 0.551 0.638 0.591 0.798 0.258 0.909 1.896 0.432 0.951 1.727 1.250 0.787 0.728 0.139 0.618

4.261 5.333 5.315 5.490 11.115 5.687 5.538 7.125 5.527 4.989 5.146 5.933 6.827 5.536 5.668 9.009 5.457 8.136 4.393 8.255 3.880

Z-Statistic F-Statistic on on Average Standard Return Difference Difference Name Return Deviation Correlation Between Between Average (%) (%) Variances Returns Russell 3000 0.290 4.204 0.998 -0.025 1.028 MSCI Australia 1.574 4.747 0.970 -0.041 1.262 MSCI Austria 2.247 4.884 0.954 0.009 1.185 MSCI Belgium 1.223 5.439 0.968 -0.007 1.019 MSCI Brazil 2.249 11.082 0.983 -0.147 1.006 MSCI Canada 0.826 5.373 0.978 0.068 1.120 MSCI France 0.609 5.416 0.991 -0.065 1.046 MSCI Germany 0.680 6.984 0.992 -0.037 1.041 MSCI Hong Kong 0.670 5.373 0.968 -0.090 1.058 MSCI Italy 0.875 5.121 0.989 -0.094 0.949 MSCI Japan 0.304 4.977 0.976 -0.056 1.069 MSCI Malaysia 0.898 5.206 0.956 0.012 1.299 MSCI Mexico 1.914 6.387 0.983 -0.016 1.142 MSCI Netherlands 0.581 5.531 0.982 -0.167 1.002 MSCI Singapore 0.917 5.466 0.962 0.037 1.075 MSCI South Korea 1.814 8.918 0.980 -0.060 1.020 MSCI Spain 1.235 5.727 0.984 0.017 0.908 MSCI Sweden 0.759 7.961 0.982 0.021 1.044 MSCI Switzerland 0.831 4.176 0.963 -0.149 1.107 MSCI Taiwan 0.325 8.450 0.968 -0.138 0.954 MSCI United Kingdom 0.693 3.898 0.956 -0.119 0.991

*, ** and *** would indicate statistical significance of the Z- and F-statistics at the 10%, 5% and 1% levels, respectively.

25 Table 2: Optimized Portfolio Weight Estimation on Russell 3000 and All Foreign Market Indices (1988:01 - 2000:07) Without Shrinkage With Shrinkage Estimators Estimators Optimized Portfolio Average Return (%) 1.719 1.528 Standard Deviation (%) 4.098 3.620 Sharpe Ratio 0.313 0.302 Number of Indices with Non-Zero Portfolio Weight 7 8 Average Portfolio Adjusted Portfolio Index Return (%) Weight (%) Average Return (%) Weight (%) US 1.432 38.375 1.354 51.439 AL 0.945 0.000 1.126 0.000 AT 0.639 0.000 0.982 0.000 BG 1.234 0.000 1.261 6.687 BZ 3.382 5.412 2.269 2.324 CN 1.157 0.000 1.225 0.000 FR 1.454 0.000 1.364 0.000 GM 1.309 0.000 1.297 0.000 HK 1.769 0.000 1.512 0.000 IT 0.987 0.000 1.145 0.000 JP 0.308 0.000 0.827 0.000 ML 1.107 0.000 1.201 0.000 MX 2.623 11.103 1.913 3.284 NT 1.468 30.415 1.371 24.054 SG 1.229 0.000 1.259 0.000 SK 0.994 0.000 1.149 0.000 SP 1.180 0.000 1.236 0.000 SW 1.874 8.055 1.562 2.765 SZ 1.410 4.886 1.344 6.664 TW 1.615 1.755 1.440 2.784 UK 1.087 0.000 1.192 0.000 Note: All abbreviations are referred to market indices. Given Russell 3000 and all 20 foreign market indices as an opportunity set and subject to the non-negative weight and the 100% weight sum, the portfolio weights are such that the portfolio Sharpe ratio is maximized. The adjusted average returns are computed based on Equations (2) and (3) with the average return on the global minimum variance portfolio ( R 0 ) of 1.285% and the shrinkage factor (α) of 0.531.

26 Table 3: Multivariate Cointegration Tests of Foreign Market Indices (1988:01 - 2000:07) Specification and Cointegration Rank Testsa G(r) Statistic H0: λtrace Statistic N/A r=0 1064.848*** 0.001 r≤1 925.572*** 14.605*** r≤2 803.851*** r≤3 697.697* 32.454*** 41.902*** r≤4 609.769 r≤5 536.681 43.105*** 45.362*** r≤6 464.147 r≤7 399.302 48.384*** 52.416*** r≤8 339.014 r≤9 282.592 62.051*** 62.483*** r ≤ 10 235.751 r ≤ 11 193.618 63.540*** 67.390*** r ≤ 12 157.294 r ≤ 13 126.775 67.454*** 67.633*** r ≤ 14 96.583 r ≤ 15 67.431 77.537*** 77.634*** r ≤ 16 47.853 r ≤ 17 30.522 80.676*** 85.089*** r ≤ 18 16.58 87.391*** r ≤ 19 7.058

Foreign Market Index AL AT BG BZ CN FR GM HK IT JP ML MX NT SG SK SP SW SZ TW UK

Exclusion Test Statisticb 21.948*** 33.569*** 27.177*** 23.582*** 31.003*** 5.834 22.076*** 15.561*** 15.643*** 14.313*** 3.097 38.315*** 33.943*** 26.555*** 19.667*** 38.111*** 40.239*** 9.787** 6.656 32.433***

Weak Exogeneity Test Statisticc 11.639** 6.442 33.053* 12.445** 11.311** N/A 9.963* 12.225** 6.703 4.867 N/A 4.799 18.600*** 25.129*** 6.824 4.008 16.881*** 6.54 N/A 8.074

Note: a The G(r) statistic is under the null hypothesis of no linear trend in the cointegration space and distributed as χ2(r) where r is the number of cointegrating relationships. Because the null hypothesis can be rejected for all potential r’s (except where r = 1), the VAR accounts for linear trends in the data and incorporates time trends into cointegrating relationships. The λtrace statistic is adjusted for the sample size using the Barrett correction derived in Johansen (2002). b Conditional on the cointegration finding, the exclusion test statistic is under the null hypothesis that the market index i is excludable from the cointegrating relationships. c Because FR, ML and TW are excludable from the cointegrating relationships (i.e., from the cointegrated group), the VAR is re-estimated without these three market indices. The weak exogeneity test statistic is under the null hypothesis the market index i is weakly exogenous and hence does not respond to deviations from the cointegrating relationships. *, ** and *** indicate rejection of the null hypothesis at the 10%, 5% and 1% levels, respectively.

27 Table 4: Performance Tests of Market Index Portfolios and iShares Portfolios (2000:08 2006:12) US Market Index Portfolioa Average Return (%) Standard Deviation (%) Sharpe Ratio Jobson-Korkie Z-Statistic against US Number of indices with Non-Zero Portfolio Weight iShares Portfoliob Average Return (%) Standard Deviation (%) Sharpe Ratio Jobson-Korkie Z-Statistic against US Number of iShares with Non-Zero Portfolio Weight Jobson-Korkie Z-statistic of iShares Portfolio against Market Index Portfolio Portfolio Weightc (%) US AL AT BG BZ CN FR GM HK IT JP ML MX NT SG SK SP SW SZ TW UK

1

2

Portfolio 3

4

0.290 4.204 0.014

1.024 0.730 4.700 4.976 0.169 0.100 2.698*** 2.214** 21 7

0.571 4.510 0.075 1.728* 8

1.003 4.257 0.181 2.693*** 12

0.273 4.261 0.010

0.973 4.914 0.151 2.530** 21

0.655 5.054 0.084 1.949* 7

0.508 4.578 0.060 1.449 8

0.950 4.487 0.160 2.491** 12

-1.450

-1.647*

-1.705*

-1.408

4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762 4.762

38.375 0.000 0.000 0.000 5.412 0.000 0.000 0.000 0.000 0.000 0.000 0.000 11.103 30.415 0.000 0.000 0.000 8.055 4.886 1.755 0.000

51.439 0.000 0.000 6.687 2.324 0.000 0.000 0.000 0.000 0.000 0.000 0.000 3.284 24.054 0.000 0.000 0.000 2.765 6.664 2.784 0.000

8.333 0.000 8.333 0.000 0.000 0.000 8.333 0.000 0.000 8.333 8.333 8.333 8.333 0.000 0.000 8.333 8.333 0.000 8.333 8.333 8.333

Note: a Market index portfolio performance is based on market index data over the period 2008:08 - 2006:12. Portfolio 1 is an equal-weight portfolio of all 21market indices. Portfolios 2 and 3 are constructed based on optimized portfolio weights of market indices with and without shrinkage estimators, respectively, over the period 1998:01 - 2000:07 (See Table 2). Portfolio 4 is an equal-weight cointegration-based portfolio comprising Russell 3000 and the foreign market indices based on cointegration analysis over the period 1998:01 - 2000:07 (See Table 3). b iShares Portfolio performance is based iShares data over the period 2008:08 - 2006:12. Identifications and weights of iShares in the portfolios correspond to those of the underlying market indices in the market index portfolios. c Abbreviations are referred to market indices (if market index portfolios) or iShares (if iShares portfolios). *, ** and *** indicate statistical significance at the 10%, 5% and 1% levels, respectively.

28 Table 5: Portfolio Sharpe Ratios during Various Out-of-Sample Horizons US Market Index Portfolioa 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year Entire Out-of-Sample Period iShares Portfoliob 1 Year 2 Year 3 Year 4 Year 5 Year 6 Year Entire Out-of-Sample Period

Portfolio 1

2

3

4

-0.265 -0.341 -0.162 -0.084 -0.022 -0.016 0.014

-0.456 -0.290 -0.104 0.005 0.096 0.132 0.169

-0.343 -0.331 -0.135 -0.049 0.030 0.066 0.100

-0.365 -0.351 -0.151 -0.063 0.013 0.042 0.075

-0.455 -0.245 -0.086 0.019 0.109 0.145 0.181

-0.256 -0.344 -0.164 -0.087 -0.026 -0.020 0.010

-0.451 -0.281 -0.105 -0.004 0.081 0.116 0.151

-0.362 -0.348 -0.154 -0.066 0.013 0.050 0.084

-0.373 -0.367 -0.167 -0.078 -0.003 0.027 0.060

-0.459 -0.240 -0.087 0.008 0.090 0.125 0.160

Note: The numbers shown above are the Sharpe ratios of the portfolios during various out-of-sample horizons after portfolio formation at the end of the estimation period. For instance, “1 year” corresponds to the horizon from 2000:08 through 2001:07 and “2-year” corresponds to the horizon from 2000:08 through 2002:07. Hence, the longest horizon is for the entire out-of-sample period from 2000:08 through 2006:12. a Market index portfolios are based on market index data and have the same compositions as the market index portfolios in Table 4. b iShares portfolios are based on iShares data and have the same compositions as the iShares portfolios in Table 4.

29 Appendix: Performance Tests of Portfolios 2 and 3 with Equal-Weight Allocation (2000:08 - 2006:12) US Market Index Portfolioa Average Return (%) Standard Deviation (%) Sharpe Ratio Jobson-Korkie Z-Statistic against US Number of indices with Non-Zero Portfolio Weight iShares Portfoliob Average Return (%) Standard Deviation (%) Sharpe Ratio Jobson-Korkie Z-Statistic against US Number of iShares with Non-Zero Portfolio Weight Jobson-Korkie Z-statistic of iShares Portfolio against Market Index Portfolio Portfolio Weightc (%) US AL AT BG BZ CN FR GM HK IT JP ML MX NT SG SK SP SW SZ TW UK

Portfolio 2

3

0.290 4.204 0.014

0.993 5.667 0.134 2.230** 7

1.021 5.481 0.144 2.318** 8

0.273 4.261 0.010

0.892 5.802 0.114 1.999** 7

0.932 5.609 0.125 2.126** 8

-1.544

-1.578

0.143 0.000 0.000 0.000 0.143 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.143 0.143 0.000 0.000 0.000 0.143 0.143 0.143 0.000

0.125 0.000 0.000 0.125 0.125 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.125 0.125 0.000 0.000 0.000 0.125 0.125 0.125 0.000

Note: a Market index portfolio performance is based on market index data over the period 2008:08 - 2006:12. Portfolios 2 and 3 comprise equal weights of Russell 3000 and foreign markets that have non-zero and positive weights in their corresponding Portfolios 2 and 3 in Table 4. b iShares Portfolio performance is based iShares data over the period 2008:08 - 2006:12. Identifications and weights of iShares in the portfolios correspond to those of the underlying market indices in the market index portfolios. c Abbreviations are referred to market indices (if market index portfolios) or iShares (if iShares portfolios). ** indicates statistical significance at the 5% level.