Stability and predictability of the comovement structure of returns in the american depository receipts market

STABILITY AND PREDICTABILITY OF THE COMOVEMENT STRUCTURE OF RETURNS IN THE AMERICAN DEPOSITORY RECEIPTS MARKET Mahmoud Wahab and Malek Lashgari I. INTRODUCTION Potential benefits from international portfolio diversification have long been recognized in a number of earlier studies dating back to the early work of [l, 10, 11,19, 20,21,27,29,30] among others. These benefits stem from the fact that the variancecovariance structure of returns is characterized by much higher covariances among securities within national markets than among securities in different markets so that national factors have a stronger impact on security returns relative to any assumed common world factor International diversification can be achieved through a variety of ways. Direct purchase of foreign securities is but one method, whereas incorporating American Depository Receipts (ADRs) into a domestic portfolio may be another. However, it is fairly well established that with the direct acquisition of foreign equity shares several difficulties do arise primarily due to existence of a few sources of uncertainty such as taxation, capital flow restrictions, and perhaps, more importantly, exchange rate risk. On the other hand, ADRs may escape many of those problems as they are denominated in U.S. dollars and are traded in the U.S. The portfolio choice problem inherently involves both in-sample estimation of portfolio parameters as well as out-of-sample forecasting of those parameters to construct and select ex ante mean variance efficient portfolios. Many of the earlier studies that examined international diversification benefits have suffered from an important limitation arising from an implicit assumption of intertemporal stationarity of the joint multivariate distribution of security returns. Essentially, a timeinvariance assumption about the joint distribution of security returns, variances, and covariances has been typically made for the sake of convenience and exposition of international diversification results. If the mean-return vector and the varianceMahmoud Wahab Assistant Professor of Finance and Malek Lashgari Associate Professor of Finance, University of Hartford, West Hartford, CT 06117. Global Finance Journal, 4(2): 141-169 ISSN: 1044-0283

Copynght 0 1993 by JAI Press, Inc. All rqhts of reproduction in any form reserved.

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covariance matrix of returns were time varying, portfolio selection based on simple extrapolation of ex post values of portfolio parameters into the future is likely to produce sub-optimal results at best. Indeed, evidence presented in [9,14,15, 18, 221, and, more recently, Cumby et al. [4] suggests that historical values are rather unreliable and noisy estimates of expected values. Therefore, historical estimates suffer from estimation risk or sampling error because they are not the true values of the forecasted parameters but only estimates thereof, thereby leading to poor out-of-sample performance. Accordingly, several attempts have been made to improve the quality of forecasted values of the inputs required for mean-variance optimization. Alternative models (beyond the simple historical model) have been proposed and evaluated to forecast expected returns (Jorion, [14]), as well as the expected comovement measures of security returns (for example, [4,6,8,15]). Typical samples used in the above studies included domestic and international equity shares and indexes. However, the performance of alternative models to predict the comovement structure of returns on a considered set of assets that includes ADRs is yet to be assessed. Therefore, the purpose of this study is to fill this gap. Two sets of considerations have motivated this paper. First, ADRs represent an interesting subset of equity securities in that they combine the features of both domestic shares (as they are traded in the U.S.) and foreign shares (as they are claims on cash flows generated by their underlying shares). Furthermore, the ADR market has witnessed dramatic growth over the last two decades so that it can no longer be regarded as a marginal component of U.S. equity markets? Second, ADRs have been shown to potentially provide tangible risk reduction benefits (Officer and Hoffmeister [25]), without any commensurate reduction in average portfolio returns. If indeed ADRs can provide risk diversification benefits to U.S. investors, then expected values of the comovement measures of returns determined from the variance-covariance matrix or from the correlation matrix are needed before ex ante mean variance efficient portfolios can be generated. Therefore, empirical tests of the intertemporal stationarity of the comovement measures of returns are important, particularly in the context of dynamic mean-variance portfolio optimization. In applying mean-variance analysis, knowledge of the expected variance-covariance matrix as the “broader” comovement measure should prove sufficient to maximize return at any risk level or, alternatively, to minimize risk for any desired level of return. However, additional insights into the nature of security return comovement may be also garnered from a decomposition of covariances into their component statistics: the correlations and the variances. This effort may take on added importance especially if (1) the covariance structure is intertemporally unstable so that use of ex post covariances to generate ex ante estimates can no longer be justified, and (2) if the quality of the forecasted covariances can be enhanced by separately modelling each component statistic. For example, the time-variation properties of return volatility and correlations could be modelled by the autoregressive conditionally heteroskedastic (ARCH) family of statistical techniques (or its generalized multivariate version) originally introduced by Engle [7], extended by Bollerslev [2], and Nelson [24], and specialized to the portfolio choice problem by Cumby et al. [4]?

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This study is divided into two parts. In the first part, empirical tests of the intertemporal stationarity of the variance-covariance matrix and the correlation matrix of returns are conducted, which are important prerequisite tests in order to determine whether or not, and how, time series of asset returns can be used in meanvariance portfolio optimization. These tests use multivariate test-statistics proposed by Box [3] and Jennrich [13] and have been used previously by Gibbons [9] and Kryzanowski and To [18] on a sample of purely domestic stocks and by Kaplanis [15] on a sample of international stock market price indexes. This paper finds that the covariance matrix of returns on a portfolio that includes the S&P500 index and 34 ADRs from nine different countries is non-stationary with almost certainty for alternative pairs of contiguous sub-periods. Since covariance matrix non-stationarity may be due to non-stationarity of the correlation matrix and/or timevariation in assets return variances, Jennrich’s [13] asymptotic x2 test procedure was also employed to test for stationarity of the correlation matrix of security returns. Simulations by Jenrich indicate that this test has good convergence properties for moderate size samples. Stationarity of the correlation matrix is accepted with almost complete certainty across all alternative pairs of sub-periods in our sample. Last but not least, the autocorrelation coefficients for the squared daily returns were computed to provide crude evidence of time-varying volatility, which may explain the non-stationary character of the covariance matrix of returns. The results show that, indeed, return volatility is time varying. Given these conclusions, the second part of the paper considers alternative models for forecasting the comovement measures and compares their relative performance. Forecast accuracy of the alternative models is assessed from both statistical and economic standpoints. Statistical accuracy is evaluated using two forecast error measures, notably Root Mean Square Error (RMSE) and Mean Absolute Error (MAE) criteria, with the latter used as a hedge against possible non-normality of the distribution of returns. On the other hand, economic significance is assessed using the forecasted covariance matrix elements as inputs to a quadratic optimization model and examining the performance of ex mfe portfolios constructed using alternative forecasting methods in relation to the “actual” mean-variance efficient portfolio that is constructed assuming perfect foresight. The remainder of this paper is organized as follows. Section II provides a brief description of the ADR market. Section III describes the data, methodology, characteristics of the alternative forecasting models, and measures of forecast evaluation. Section IV contains the empirical results. Finally, the last section contains the main conclusions of the paper.

II. THE ADR MARKET American resenting They are securities Securities

Depository Receipts are dollar-denominated negotiable certificates repclaims on cash flows generated by their respective underlying shares. created by large U.S. (depository) banks to facilitate trading of foreign in the U.S., particularly those which have not been registered with the and Exchange Commission. They are either “sponsored” or “unspon-

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sored.” In the former, they are created by a U.S. bank at the request of the company whose shares will underlie the ADR issue; while in the latter, they are created when sufficient demand exists for foreign shares with no active involvement of the underlying company beyond a simple solicitation of its approval before an unsponsored ADR can be issued. ADRs are regarded as convenient means to achieve international diversification without the administrative problems associated with direct ownership of foreign securities. This is so since ADRs offer U.S. investors some advantages over investing in the underlying shares in foreign markets. 1.

2. 3. 4.

5. 6.

7.

8.

Using ADRs, administrative problems associated with having to convert U.S. dollars into local currency to purchase foreign shares can be avoided because ADRs are priced in U.S. dollars. Custody charges on an ongoing basis are lower than for foreign shares. Voting rights are available on sponsored (but not on unsponsored) ADRs. The settlement period for ADRs is shorter than it is for some foreign shares. In the U.S., the ADR settlement period is five days, just as it is for U.S. shares. Usually there is no requirement for a transfer of stock certificates. Dividends declared on the underlying shares are automatically converted into U.S. dollars at the preferential wholesale exchange rate and are then distributed to ADR holders by the U.S. depository bank responsible for that ADR. ADRs can be converted into their underlying ordinary shares at any time although conversion will usually involve a cancellation (conversion) fee. Compared to sponsored ADRs, unsponsored ADRs are usually traded on the OTC market. Last but not least, ADRs can provide risk reduction benefits for any desired level of returns without having to purchase the foreign equity shares directly.

Officer and Hoffmeister [25], using a random sample of U.S. stocks and ADRs, showed that U.S. investors can reduce risk by as much as 20-25s when including ADRs in a domestic portfolio. This is possible without any accompanying reduction in average monthly returns. Wahab and Khandwala [32] compared the relative contribution of ADRs and their respective underlying shares to enhancing the investment performance of purely domestic portfolios. Once more, risk-reduction by as much as 44% (using weekly returns) is possible when including ADRs, as opposed to foreign shares in a domestic portfolio, without sacrificing expected returns. However, a major disadvantage of ADRs is their limited liquidity This may well change in the future as ADRs become increasingly accepted media for international diversification. ADR pricing is conceivably impacted in at least two interesting ways, with respect to the foreign currency share price and with respect to an exchange rate effect. Whenever ADR prices deviate from the dollar equivalent value of their respective underlying shares, efficient arbitrage should force realignment of the

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two prices. For example, suppose that ADR prices exceed the cost of buying the underlying shares in U.S. dollars. Abstracting from transaction costs and other market frictions, an arbitrageur would buy the underlying shares, which would then be deposited at a non-US. custodian bank. ADRs will be issued in the U.S. by a depository bank (typically a large U.S. bank), resulting in an arbitrage profit equal to the differential U.S. dollar price. On the other hand, if ADR prices fall below the dollar cost of buying the underlying shares (which are denominated in foreign currency), an arbitrageur would buy the less expensive ADRs, which would then be converted to their respective underlying shares (subject to a cancellation or conversion fee). The local shares would subsequently be sold at the higher foreign currency price with the proceeds converted back to U.S. dollars at the spot exchange rate prevailing at that time. This simultaneous transaction would yield a profit (in the absence of any associated costs) equal to the differential of dollar denominated prices. Thus, absence of profitable riskless arbitrage opportunities requires the U.S. dollar prices of ADRs to closely track the dollar numeraire value of their respective underlying shares’ prices.

III. DATA AND METHODOLOGY The data base consists of daily closing prices for 34 ADR issues that are traded in the U.S., representing nine different countries (namely, U.K., Japan, Sweden, Luxembourg, Australia, South Africa, Germany, Norway, and France) for the period between January 2,1988, and May 31,1991, in addition to daily values of the Financial Times (FT) World Index and the S&P500 index. The results are presented from a U.S. investor’s perspective, contemplating the addition of ADRs to a domestic portfolio that replicates the performance of the S&P500. Non-overlapping quarterly estimation and forecast periods are used to assess the ex ante performance of the different models employed to forecast the expected variance and covariance measures. The study period is divided into 13 quarters. Using the first quarter’s returns, the variance-covariance matrix of ADRs and the S&P500 returns is estimated based on each forecasting model. This essentially provides forecasted values for the next period, which are then compared to their actual values in the following quarter. Next, variances and covariances are reestimated, using the second quarter’s data, and are then compared to their actual out-of-sample values in the third quarter, and so on. Although weekly or monthly time intervals may be more appropriate than daily time intervals for an asset allocation strategy, daily returns were used in the tests for the following reasons. First, by sampling returns more frequently on a daily basis, the precision of the parameter estimates is increased without resorting to long historical samples that will be contaminated by obsolete data from the distant past. Second, in order to detect possible short-lived time dependent volatility, which may help explain the potentially-non-stationary character of the variancecovariance matrix of returns, it seems reasonable to use more frequently-sampled data. Admittedly, using daily data introduces few econometric problems that may produce downward-biased estimates of the variances and pairwise covariance sta-

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tistics (discussed later). However, recalling that interest lies in examining the relative, as opposed to the absolute performance of alternative forecasting models, any bias should not seriously jeopardize the results since all models will be estimated employing daily data. Statistical Tests of Intertemporal Stationarity If the considered assets’ risk and return parameters are constant over time, the efficient frontier will also be time invariant, and so will the composition of the optimal portfolios for any given level of expected return and risk. On the other hand, time variation of portfolio performance parameters complicates the task of constructing and selecting ex ante mean-variance efficient portfolios. Therefore, it is important to investigate the extent to which the comovement structure of ADRs and S&P500 returns can be considered intertemporally stationary. Covariance (and correlation) matrix equality tests will be conducted over contiguous sub-periods of about three months each. This, in turn, assumes that the optimal rebalancing frequency is quarterly, which coincides with the forecast evaluation period. Expected return estimates are simply the sample averages of these returns over each quarter. Stationarity tests are conducted using the forecasted and actual variance-covariante matrices based on each forecasting model. Due to the large dimensionality of the variance-covariance matrix for our sample of ADRs and S&P500 (n = 35), Box’s x2 statistic for testing equality of two sample covariance matrices is not deemed appropriate (Morrison [23], p. 252). Therefore, Box’s [3] F-approximation to the likelihood ratio statistic is used. (See Appendix A.) On the other hand, Jennrich’s 1131 asymptotic x2 test procedure (See Appendix A.) was used to test for stationarity of the correlation matrices of security returns. This test is sensitive, asymptotically, to departures from the null hypothesis of equality and has the form of a standard asymptotic x2 statistic for testing equality of two covariance matrices with a correction term added when testing equality of correlation matrices. Alternative Covariance Matrix Forecasting Models This section describes the various forecasting models used to estimate the covariance structure of ADR and S&P500 returns: two historical models (adjusted and unadjusted for estimation risk), two mean models, and five index models. Historical Covariance

Model

This model sets every variance and covariance element in the (35 x 35) variancecovariance matrix equal to its last quarter’s value. It is perhaps the simplest forecasting method since it uses historical values of the pairwise variance-covariance elements as unbiased estimators of next-period values. No assumption is made about how or why any pair of securities would move together. This model was

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used in a number of earlier studies [6,8,15], historical model.” This model is given by,

and is referred to by the “unadjusted

(1)

covij,q = COV,J,q-z + EiJq

where COV,I,~ and COV~,~_~ are variances (Vi =i) and pairwise covariances (‘v’i ;ti) in quarters q and 9-1, respectively, and E~,~is a zero mean random variable, with cov (E~,~,~kl,~)= 0; and COV(E~,~, E+_~) = 0 (V i #i); and COV(E~~,~, ~kl,~_~)= 0 (Vi +i) and (M), where (i,j) and (k,Z)are elements in the variance-covariance matrix of returns. The future performance of this model would depend on the extent to which history repeats itself from one quarter to the other. Adjusted Historical Covariance

Model

This model recognizes and adjusts for estimation risk, using, for example, Klein and Bawa’s [17] simple Bayesian adjustment technique, which involves elementwise multiplication of the historical variance-covariance matrix at 9 - 1 by a constant proportional amount contingent upon sample size (I) and the number of assets (?z).~Hence,

z; =c lz,,

(2)

where xi denotes the forecasted variance-covariance matrix, using last period’s Bayesian-adjusted historical matrix zq_l, and c is a constant equal to c = (T + l)(T - l)/T(T - IZ- 2). The historical covariance matrix may then be used to estimate next quarter’s covariance matrix. Overall Mean Model This model uses the grand mean of all variances and pairwise covariances to predict next period’s variances and covariances. It rests on the assumption that the historical variance-covariance matrix contains information about future values of the variance-covariance elements (which are estimated as simple averages of historical values) but not information about individual deviations from the average of all elements. Therefore, all individual differences between forecasted and actual variances and covariances are then estimated to be zero in an ex ante context.4 Thus, covjjq = cov~,~-l + eq,q

(3)

where Gij,q_l is the expected value formed at 9 - 1 for each element, estimated from a simple average of historic values, and eiiq are random disturbances with standard distributional assumptions. The forecasting performance of this model will be governed by the importance of deviations from the overall average; which might proxy for country and/or industry effects, that are ignored by this model.

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National Mean Model The national mean model represents a finer averaging scheme that recognizes nationalistic differences within sub-groupings of ADRs according to their nationality as well as between ADRs and the S&P500. This model uses two levels of aggregation. In the first, ADRs are classified into groups based on their country of origin so that intra-group variance and covariance elements are set equal to the average of the respective elements for that group, ignoring for the moment pairwise covariances between ADRs belonging to different groups as well as between ADRs and S&P500. A second level of aggregation explicitly considers inter-group comovement characteristics between ADRs and between ADRs and the S&P500. This involves setting every inter-group covariance element equal to the grand mean of all inter-group covariances5 Therefore, this model explicitly captures the geographical (country) characteristics in the sample through a two-level aggregation scheme.’ Thus,

and

covqjsq

= covz~q-l,ppI +

%q,jsq

(44

where COD llq,r is the pairwise covariance element for assets i and j Vi # j)or the- individual variance elements Vi #j) and where assets i and j are from country u; cOvij,q_ I,~ is the expected value at 9 - 1 of each element from country r, estimated by the simple average of all intra-country variances Vi = j), and covariances (for i’s # j); and nqq,y is a random forecast error with the usual assumptions. On the other is the pairwise covariance element of asset i from country r and asset hand, ~~~~~~~~~~ j from country s, measured at 9, while COU,,~ _Ids4 _1 represents the expected value at 9 - 1, taken as the simple average of all inter-country covariances between assets i (from country Y) and j (from country s), and E,,~,jsqis the forecast error, which has the usual assumptions if the national mean model is correctly specified. In sum, equation (4) represents the intra-country aggregation level whereas equation (4a) identifies the inter-country level of aggregation. Single World-Index Model Assuming that securities move together only because of a common comovement with an aggregate world index, this model can be presented as follows:7 R,t = o,, + BiRwt + e,t, i = l,... n

(5)

with the key assumption that cov(e,t,el,) = 0, for all i,j = 1 ...?I and i # j, and where R,, is the daily return on asset i and R,t is the daily return on the world equity index as proxied by the Financial Times (FT) world index with both series being measured over the quarterly estimation period (9 - 1). Once p’s are estimated using equation (3, pairwise covariance statistics for assets i and j are computed as

American Depository Receipts Market

covijq = &,4-&q _p$,,_ I i,j = I,...12

149

(54

where o2w,4_1represents the variance of returns on the world index estimated from the previous quarter’s returns. The systematic risk coefficients of assets i and j are estimated using data also from last quarter, which together with the variance of daily returns on the world index provide an estimate of next period’s pairwise covariances. Additionally, the variance of returns on asset i computed from last quarter’s returns as p2,,q_ 1 02w,g_ 1 + cr2E14 _ 1 is used as an estimate of next period’s forecasted value. Since forecasts are updated quarterly, no assumptions about stationarity of the component statistics. This model is referred “unadjusted” single world index model.

are made to by the

Adjusted Single World Index Model Because this study uses daily returns, and due to the familiar non-synchronous trading problem which is potentially more severe with high frequency data, the parameters of equation (5) are likely to be measured with error, especially that ADRs may trade relatively infrequently. This in turn implies that daily closing prices (used in return calculations) may represent the outcome of a transaction that occurred earlier in the day. Therefore, the covariance of asset i with the world index and thus, its beta coefficient will likely suffer from a downward bias due to serial correlation induced in “observed” returns even if “true” returns were serially independent. In turn, each pairwise covariance statistic may also be underestimated. Therefore, beta coefficients were adjusted using the Scholes-Williams [28] adjustment technique, which potentially provides unbiased and consistent estimates as a combination of least squares estimates. Specifically, the sum of betas estimated by regressing returns on the ith ADR against returns on the world market index from previous, current, and subsequent periods is divided by one plus twice the estimated first order autocorrelation coefficient for the world market index. This adjustment is analogous to an instrumental variable estimator, which uses as an instrument the moving sum of measured daily rates of return on the market for previous, current, and subsequent periods.8 To compute the adjusted betas, a modified version of eqution (5) is estimated with up to two leads and two lags (in addition to contemporaneous values) of the returns on the world index. Once adjusted estimates of /3sare obtained, variance and covariance statistics were then computed along the same lines as the unadjusted estimates. Both models are consistent with a single integrated world equity market in which the only priced risk is that of the asset’s international systematic (or world covariance) risk. Since this study does not explicitly test whether ADRs are priced as if traded in a segmented or integrated world equity market, two alternative index model specifications are employed, notably a single domestic index model that recognizes compensation for only domestic covariance risk and a two-index model that captures both domestic and international covariance risks. Single Domestic Index Model (Unadjusted) This model is specified along the same lines of equation (5) after substituting the appropriate market index so that the world market proxy is replaced by the

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S&P500 index. Pairwise covariance measures are then obtained as in equation (5a), after substituting o& _ z for c$& _ I, and where the d subscript identifies the domestic index. Single Domestic Index Model (Adjusted) The Scholes-Williams adjusted version of the Single Domestic Index model, as explained earlier, is also used to capture the effects of nonsynchroneity on the quality of the parameter estimates. Once more, only two leads and two lags of the S&P500 index values were found to be statistically significant and are thus included to obtain the extended version of the model. This is followed by estimating Scholes-Williams adjusted betas, individual variances, and pairwise-covariance measures. Multi-Index

Model

This model assumes that ADRs comove together because of their common response to movements in two market indexes. Thus, the model is R,t = oi + P;I &it + P’l2 J&t

f

%t

(6)

where Rdt and XL, are, respectively, daily returns on the S&P500 index and daily returns on the orthogonalized world index (Stehle, [31]).9 This model assumes that security returns are affected by a domestic index influence and a “pure” world factor (after removing domestic index effects from the world index).” This assumption is consistent with a partially segmented view of security pricing. Once the ps are estimated from equation (6), covariance statistics were obtained as cavij,q = P>l,q- 2 Pjl,q-I o&-I Individual

+ Fi2,q-1 s;2,, - 1 &,,,.I

(64

asset variances were also obtained by

(6b) where o$ is the estimate of asset is return variance, o2d,q- 1 and oi,, _ 1 are variances of returns on the domestic (S&P500) and orthogonal world market proxies, is the residual variance of asset i. respectively, and O$ Lq- 1 Forecast Error Evaluation With 13 quarters making up the study period, 12 generated forecasts are possible based on each model. Forecast accuracy is assessed using the Root Mean Square Error (RMSE) and the Mean Absolute Error (MAE) criteria. Given a-pairs of forecasted and corresponding actual variances and l/2(12’ - n) unique covariances, forecast errors for forecast-method i are averaged over time and cross-sectionally to compute the forecast error metrics. Thus, letting Fl~,rl_l and Alk,q denote fore-

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casted and actual values of the unique kth element of the variance-covariance matrix estimated using forecast-model i for the qth quarter, we computed the root mean square error for the kth element as follows:

RMSE,

=

By cross-sectionally averaging the root mean square errors for all k elements (b’k = { 1,2,...1/2(n2 - n)}), we obtained the Average Root Mean Square Error (ARMSE) for forecast-model i, as follows: ARMSE,

= : (rz-n)

i

RMSE,

k=l

This potentially provides us with a summary measure of the forecast accuracy of each model over the 12 forecast intervals. Similarly, computation of the time series and cross-sectional average of the Mean Absolute Error for forecast-model i followed the same logic and is given by AMAE,

=

+2-‘z)

kcl (;@

IV EMPIRICAL

(lFzk,q-l-Alk.q

RESULTS

Table (1) presents a description of the 12 pairs of contiguous sub-periods over which stationarity tests and forecast evaluation are conducted. Box’s likelihood ratio test results for testing equality of two covariance matrices are presented in Table (2). The evidence clearly suggests that the null hypothesis of equality (stationarity) of the covariance matrices is reliably and invariably rejected for all models and pairs of sub-periods with almost complete certainty. Since covariance stationarity was rejected, it is of interest to know the source of rejection, i.e., instability in the correlation matrix and/or time variation in return volatility. In order to test for intertemporal stationarity of the correlation matrix, the quantity Q2 (Jennrich x2 statistic) was estimated. The results are presented in Panel A of Table (3). None of the Qz statistics rejects the null hypothesis of intertemporally stationarity of the correlation matrix at the 20% level. These findings are in a sense expected since there is not much in the way of structural changes that could happen over relatively short horizons to cause a uniform structural change in the correlation matrix. Furthermore, these results may also be regarded as evidence about the forecasting performance of the Historical model in providing estimates of next period’s correlation matrix. Since covariance matrix equality test

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Table 1 DESCRIPTION OF THE 12 PAIRS OF CONTIGUOUS SUB-PERIODS OVER WHICH FORECAST EVALUATION IS CONDUCTED First sub-period Length of each subperiod+

Identifier

3 months 3 months 3 months 3 months 3 months 3 months 3 months 3 months 3 months 3 months 3 months 3 months

Pair

Second sub-period

Starts

Ends

1

Ol/Oz/s8

03/31/88

2 3 4

04/01/88 07/01/88 1O/01/88

06130188 09/30/88 12’31/88

5 6 7

01/02’89

03/31/89

04/01/89 07/01/89 10/01/89 01/02/90

06/30/89 09/30/89 12/31/89 03/31/90

01/02/90 04/01/90

04/01/90 07/01/90 10/01/90

06/30/90 09/30/90 12/31/90

07/01/90 10/01/90 01/02/91

8 9 10 11 12

Sfarfs

04/01/88 07/01/88 10/01/88 01/02/88 04/01/89 07/01/89 10/01/89

Ends 06/30/88 09/30/88 12/31/88 03/31/89 06/30/89 09/30/89 12/31/89 03/31/90 06/30/90 09/30/90 12/31/90 03/31/91

results indicate (judging by the magnitude of the test statistics) that the National Mean model, the Overall Mean model, the Historical model, and the Adjusted Historical model ranked as relatively best forecasters of the covariance matrix when compared to the remaining five models, it would be of interest to provide evidence on their relative performance in forecasting the subsequent quarter’s correlation matrix. Essentially then, the following null hypothesis was tested for each of the twelve forecasting quarters: Ho:

RNM,q-l = R MM+1

= RHH,q-l

= %H,q-1

= RA,q

where subscripts denote, respectively, correlation matrix forecasts provided at 9 1 by the National Mean model (NM), the Overall Mean model (MM), the Historical model (HH), the Adjusted Historical model(AH), and the Actual (A) out-of-sample correlation matrix at 9. This hypothesis is tested against the alternate hypothesis that at least one of the matrices is not equal to the remaining matrices. To this aim, Jennrich’s [13] modified x2 statistic for testing the homogeneity of a set of correlation matrices is used. (See appendix A.) Sub-period pair-by-pair results are presented in the Panel B of Table (3). These results are consistent with the proposition that the correlation matrix is intertemporally stationary regardless of the model used to generate the expected value. Therefore, the Historical model, for example, appears to be an adequately specified model of the evolving correlation patterns over quarterly forecast horizons. It appears that time-variation in individual security return variances may be the primary reason behind covariance instability and that ex ante asset allocation strategies involving ADRs should account for this characteristic in the time series of individual security returns. To pursue this point further, the average values of the autocorrelation coefficients of squared returns on the

11 12

3.512* 6.098**

8.329**

5.095** 3.006*

9.474""

5.604"*

12.158"* 4.102"

4.815" 12.494""

23.267""

AH

sqqufsmce at the 5% level ** Denotes sqpfrance at the 1% level HH = Hlstoncal (unadpsted) Model; MM = Overall Mean Model, SIM(W) = Unadpsted single world Index Model, SIM(lJ.S.) = Unadjusted smgle domestlc index Model MIM = Multi-Index Model

Notrs. *Denotes

5.479""

3.743" 5.820""

10

4.370*

12.880*+

4.586%

6

4.253** 2.500"

8.270**

4 5

7

12.473** 11.190*+

2 3

8 9

19.644""

HH

1

Pair Identifier

Sub-period

1.510" 3.265"

2.908"

2.190" 1.759"

3.862*

3.054*

2.651* 2.207*

3.543" 4.102"

k3.799**

NM

11.529"* 12.251**

8.329%"

39.427** 12.884*"

12.880"*

12.027**

7.027** 29.453*"

40.027** 11.762**

31.039**

SIM(W)

22.371"" 24.966""

6.148**

26.188," 8.577**

15.737""

25.282"" 19.620"" 14.776""

12.206** 20.160**

16.588**

SIM(US)

AH = Adpted Histoncal, NM = National Mean, ASIM(W) = Adjusted smgle world Index Model; ASIM(U.S.) = Adjusted smgle domestic index Model,

2.037% 4.514**

3.983""

3.179" 2.784"

8.387""

3.236"

3.224" 2.445"

10.196"* 5.282"

11.023**

MM

Forecasting Models

15.115**

13.814"" 8.329"" 16.445""

10.788** 16.445**

20.085**

15.109**

17.256"" 12.884"" 8.634*"

13.108**

10.424"* 8.759"* 9.806""

25.101**

11.743** 7.763"*

14.611**

12.027** 32.245**

17.330**

MIM

8.238'" 60.110"" 22.158**

19.695""

ASIM(US)

10.774**

18.865** 14.800** 6.738**

7.312**

10.343**

17.330** 18.256*" 52.070**

19.522""

ASTM(W)

Table 2 COVARIANCE MATRIX STATIONARITY TEST RESULTS USING BOX’S APPROXIMATE LIKELIHOOD RATIO TEST STATISTIC

5 CT -+

2. ii! u a” 2 z w2 Fu c: 3. ‘ci ET

GLOBAL FINANCE JOURNAL

154

CORRELATION

4(Z), 1993

Table 3 MATRIX EQUALITY TEST RESULTS USING JENNRICH x2 TEST STATISTICS (Ol/Oz/SS- 03/31/91) Sub-Period Pair Identifiers

Ted Statistic

1

2

3

4

5

6

7

8

9

10

11

12

Panel A Q2

18786 19.71220.01719.21519.32518.75919.69418.97819.00320.42922.36821.784

Q?

12.45012.82612.76412.09012.34713.01012.25812.95713.92713.40812.91111.852

Notes.

Panel B

-All chi square stahstlcs were mslgmhcant at the 20% level -Panel A contains test results of the null hypothesis of equality of two correlatnm matrices computed over two quarters 9-1 and 9 Panel B reports test results of the null hypothesis of homogeneity of five correlatmn matnces, eshmated usmg the best four cwanance forecasting models, notably the National Mean, the Overall Mean, the Historical, and the Adjusted Hlstoncal models, and the actual out-of-sample correlation matrix Results are reported for 12 pars of in-sample and out-of-sample permds

34 ADRs and the S&P500 were computed and are presented in Table (4). This may be interpreted as crude evidence that at least some of the variances may be timevarying. The sample autocorrelations series are mostly positive and decay slowly to zero, which is indicative of higher order dependence in the return series, possibly as a result of changing conditional volatility over time. Indeed, volatility appears to be time-varying for 26 assets in the sample at the 5% level or better. The next section examines whether one can improve the estimate of the expected variance-covariance matrix by employing alternative forecasting models beyond the simple historical model, which in a sense conditions the variance expectation on its past value. These results are presented in Table (5) on a periodby-period basis whereas, Table (6) provides the average (over time and cross-sectional) values of the forecast errors, summarizing the information in Table (5). The most significant finding is that the National Mean (NM) model dominates all other models in forecasting next periods values of the variance-covariance matrix. The Overall Mean model (MM), which ignores national factors, lagged behind the National Mean model. This finding somehow corroborates the importance of the country factor in determining the comovement structure of ADR returns so that excessive averaging via the Overall Mean (MM) model may have destroyed worthwhile information in the interim. Eun and Resnick [S] also found the National Mean model to be the best forecaster of the comovement structure of international security returns. The second best performing models were the Overall Mean and the Historical (HH) models, with the former outperforming the latter in seven out of the 12 quarterly samples when using the RMSE criterion (although the difference was marginal) and in four out of the twelve samples when using the Mean Absolute Error criterion. This finding contrasts with Eun and Resnick’s results; they found that the Historical model outperformed the Overall Mean model. The Overall Mean model assumes that historical data contains only information regarding mean variances and covariances so that observed pairwise differences

155

American Depository Receipts Market

%ble 4 AVERAGE SAMPLE AUTO-CORRELATION COEFFICIENTS OF SQUARED DAILY RETURNS ON ADRS AND THE S&P500 (Ol/O@S- 03/31/91) Autocorrelation Securit.y

r2

y3

Coefficients __Lofi Sauared Returns y4

f.6

r5

Ljung-Box

Barclays

0.113

0.090

0.069

0.049

0.034

0.028

BAT md. Cadbury

0.037 0.181

0.022

-0.006

-0.004

-0.003

-0.002

0.090

0.087

0.071

0.067

0.051

25.36** 1.62 51.24**

Cannon CSK

0.081 0.269 0.166

0.063 0.170

0.032 0.142 0.114

0.030 0.141

0.019 0.123

0.000 0.061

10.81 135.21*”

0.107 0.013

0.091 0.013

0.062

68.70**

0.038 0.056

0.023

0.001 0.019

8.04 15.55**

0.031

0.012

29.59**

0.018

0.016 0.031

0.011

3o.s1**

0.011

0.016 0.002

26.90*+ 15.23**

0.124 0.114

0.103 0.089

0.076 0.062

117.62** 79.10**

Daiei

0.130 0.028

De Beers Dresdner Fuji

0.088 0.188 0.189

0.066 0.086

Gambro Hitachi

0.148

0.128

0.143 0.167

0.105 0.069

0.208 0.182

0.181

0.035 0.177

0.135 0.079

0.131 0.074

0.071

0.019

-0.001

28.60**

0.080

0.079

0.181

0.092 0.110

0.029

0.024

0.016 0.021

0.004 -0.005

37.16** 26.50”’

0.059 0.251

0.051 0.219

0.021

0.021

0.011

-0.008

0.150 0.085

0.136 0.069 0.109

0.127 0.066

0.058 0.042

7.94 125.19** 37.12**

0.098

79.56**

Honda Ito Yokado Japan Air Kirin Kyocera LVMH Minorco Mitsui NEC News

0.128 0.150

0.195 0.155

Norsk Hydro Norsk Data

0.083

0.092 0.149 0.054

0.025 0.063 0.066 0.021 0.067

0.112

0.045 0.020

0.023

0.099 -0.023

10.50

0.003

-0.017 0.001

0.022

0.018

12.29

0.064

Pacific Dunlop

0.069 0.095

0.034 0.019

0.054

0.030

Pioneer

0.082

0.045

0.042

0.003

9.49

0.162

0.149 0.165

0.100

0.029 0.091

0.017

Reuters Saatchi

0.080

0.072

0.139

0.095

0.061

0.119 0.076

0.112 0.014

0.085 0.072

66.09 89.54**

0.065 0.076

0.051

-0.006 0.042

0.073 0.042

0.025

0.009

0.036

0.000 0.037

Sony TDK

0.129

Tokio Toyota Unilever National Westminister WPP S&P500 Notes

0.198 0.160 0.165 0.184 0.124 0.308 0.216 0.191

0.147 0.063 0.066 0.155 0.100 0.106

0.058

0.008 0.024

0.063 0.151

0.058

0.050

0.178

0.109

0.173

0.142

0.113

0.089 0.091

0.062 -0.016 0.025

0.081 0.070

7.98

69.98** 16.71* 27.17”” 58.86** 26.91** 99.06”’ 89.76** 83.46**

*Denotes sigmficance at the 5% level **Denotes slgmficance at the 1% level. -L~ung Box Statistic testing the Iomt null hypothesis that all serial cnrrelatmns (up to SIXlags) taken together are jointly zero The test statistic IS distnbuted as a chl square variate wth degrees of freedom equal to the number of lags Asymptohc standard error for the mdwidual autocorrelahon

coeffiaents can be approxunated

as the square root of the reciprocal of the number of observations under the null hypothesis of zero autocorrelatmn.

156

GLOBAL FINANCE JOURNAL

4(2), 1993

from the mean are random with an expected value of zero. Alternatively, the Historical model implies that setting estimated deviations (from the mean) at their historical level is better than assigning to them a value of zero. The fact that both models performed very similarly, yet each underperformed the National Mean model, implies that inter-country mean-covariances are not unbiasedly estimated by setting their values either to zero (Overall Mean) or at their historical levels (Historical model). Once more, these results simply attest to the changing character of the covariances (variances) and signifies that more elaborate modelling efforts are required before temporal shifts can be reasonably captured. In a sense, whereas the National Mean model can be regarded as being consistent with a partially segmented (partially integrated) view of the structure of international security returns, the Overall-Mean model can be regarded as indicative of a fullyintegrated view. The evidence presented so far is more in line with the former, as opposed to the latter, view. Unexpectedly, the Adjusted Historical model has performed worse than the Unadjusted Historical model in spite of random noise contained in historical data. However, in almost all of the sampling quarters, the Adjusted Historical model has outperformed the various Index models, using both the RMSE and MAE criteria. The closest competing model to the Adjusted Historical model appears to be the Unadjusted Single Domestic Index model, which renders support to the importance of the U.S. domestic factor in explaining the stochastic comovement structure of ADR returns. Another index model that appears to be a strong competitor is the Multi-Index model, which tends to outperform other index models such as the Single-World Index model. Adding an orthogonal World Index to the Domestic Index has (as expected) improved R2 although the improvement was marginal (not reported). Furthermore, variance-covariance values obtained using both models were not materially different. These results are broadly consistent with results obtained from the National Mean model and are once more supportive of a partially segmented (or partially integrated) view of ADR pricing. To sum up our empirical findings, Table (6) presents the mean forecast error values for the two forecast error criteria, computed over the entire study period, for the different forecasting models. These results simply serve to confirm our earlier findings concerning the superiority of the National Mean model. Indeed, the two error measures rank the top three performing models similarly so that determination of performance results is not dependent upon the particular evaluation criterion. Last but not least, both error criteria rank the Single World Index model as the worst performer so that ADRs are not priced as if they were traded in a single integrated world equity market. The above results provide “statistical” evidence regarding the relative performance of the different forecasting models in an ex ante setting. To assess the “economic” significance of these findings, we provide two “representative” sample plots of ex ante efficient frontiers generated by the four best performing models, notably the National Mean model, the Overall Mean model, the Historical model, and the Adjusted Historical model, which are then compared to actual out-of-sample efficient frontiers. Economic significance is assessed by examining how close the forecasted efficient frontiers are to the subsequent quarter actual frontier.

18.669

Mean Abs. Error

16.954

0.1490

0.0611

t-stat

02450

0.1069

Pair (4)

RMSE

Sub-Period

19.449

Mean Abs. Error

21.733

0.2238

0.0728

t-stat

0.3588

0.1078

Pair (3)

RMSE

Sub-Period

18.284

Mean Abs. Error

19.963

0.1399

0.0865

f-stat

0.2332

0.1365

Pair (2)

12.814

RMSE

Sub-Period

12.598

0.1966

0.0651

Mean Abs. Error

t-stat

0.4225

AH

0.1419

Pair (1)

HH

15.645

0.0513

0.0949

25.344

0.0748

0.1048

20.050

0.0919

0.1446

12.640

0.0641

0.1391

MM

8.936

0.0461

0.0616

11.096

0.0347

0.0833

20.001

0.0909

0.1031

10.367

0.0516

0.1156

NM

7.585

0.7847

0.9952

8.873

0.8277

0.9949

10.005

1.0271

1.1932

9.783

0.7328

0.8563

SIM(W)

7.598

0.3376

0.4279

8.024

0.2848

0.3539

7.618

0.4555

0.5767

6.780

0.2369

0.3145

SlM(US)

Forecasting Models

Table 5 OF ALTERNATIVE MODELS BY SUB-PERIOD

RMSE

Sub-Period

Performance Measures

PERFORMANCE

7.973

0.4094

0.5098

5.762

0.5838

0.8367

7.239

0.4149

0.5358

6.432

0.3667

0.4982

ASIM(W)

PAIRS

6.739

0.2496

0.3321

7.910

0.2719

0.3396

7.471

0.3691

0.4708

5.500

0.3447

0.5062

ASlM( US) -

(continued)

9.763

0.3007

0.3960

10.401

0.3362

0.4315

10.044

0.4147

0.5397

0.4062 9.432

0.5430

MIM

G

$ E

%:’ z

G % (D w

t-stat

Mean Abs. Error

RMSE

Sub-Period Pair (8)

t-stat

Mean Abs. Error

RMSE

Sub-Period Pair (7)

f-stat

Mean Abs. Error

RMSE

Sub-Period Pair (6)

t-stat

Mean Abs. Error

RMSE

Sub-Period Pair (5)

Performance Measures

18.521

15.523

0.0740

0.1334

0.0718

15.160

0.1377

0.2205

17.378

0.1318

19.547

0.0541

0.1593

0.0578

18.454

0.0932

0.2545

19.520

0.0958

20.878

0.0614

0.1067

0.0544

18.529

0.0982

16.463

0.1640

12.895

0.0899

18.075

0.0473

0.1063

0.0518

MM 0.0845

-

0.2273

AH

0.0871

HH

0.0667

0.0717

9.319

0.0291

0.0714

NM

13.116

0.0816

0.1120

12.556

0.0216

0.0619

10.268

0.6816

0.7867

9.983

0.7441

0.8649

8.454

0.7001

0.8544

10.696

0.942s

1.077

SIM(W

4.737

0.2018

0.3228

5.412

0.2440

0.3615

3.676

0.1814

0.3437

5.827

0.2858

0.4073

SIM(US)

Forecasting Model

(Continued)

20.113

Table 5

-

6.183

0.3300

0.4280 7.190

0.4567

7.850

0.4241

0.5310

3.815

0.1736

0.3204

8.396

0.4113

0.5031

ASlM(US)

0.5543

7.760

0.4169

0.5242

6.333

0.4417

0.6044

7.308

0.4819

0.6200

ASIM(W)

0.2917 8.516

0.4089

8.236

0.2314

0.3299

7.189

0.2008

0.3081

7.892

0.3382

0.4784

MIM

?z F

2

5 L!

!z

g $

3

Measures

9.547

t-stat

Note

14.283

+ Forecastmg Models are defined as before m Table (2)

14.796

0.5684

0.1406

Mean Abs. Error

t-stat

1.124

0.2711

RMSE

Sub-Period Pair (12)

0.3458

0.1613

Mean Abs. Error 18.165

0.5787

0.4422

23.345

RMSE

Sub-Period Pair (11)

21.515

0.2278

0.1067

Mean Abs. Error

t-stat

0.3294

0.1613

12.694

RMSE

Sub-Period Pair (10)

18.584

0.1646

0.0955

Mean Abs. Error

t-stat

0.3564

AH

0.1574

Hi3

RMSE

Sub-Period Pair (9)

Performance

24.716

0.1685

0.2367

21.932

0.1775

0.2654

24.023

0.1270

0.1809

17.541

0.0989

0.1809

MM

0.0813

0.1569

NM

20.319

0.1353

0.2551

16.650

0.1449

0.2133

17.721

0.0939

0.1717

5.429

1.349

1.9951

4.722

1.133

1.8163

8.193

0.6946

0.8567

8.300

0.9657

1.1859

SlM(W)

2.342

0.6824

1.8542

2.784

0.7705

1.809

3.110

0.2326

0.4998

5.807

0.3478

0.4966

SIM(US)

Forecasting Model

(Continued)

12.313

Table 5

3.081

0.9554

2.0681

3.669

1.0236

1.942

5.830

0.6208

0.8845

0.6671 7.820

0.8365

ASIM(W)

2.791

0.8266

1.9369

3.446

1.0309

2.048

2.997

0.2247

0.4972

0.4071 5.778

0.5827

ASIM(US)

3.422

0.5885

1.5544

4.164

0.5808

1.3034

5.233

0.2040

0.3847

0.3221 10.149

0.4175

MIM

; & 2

-G’

ct9 %I 6

R

w

B

2 u

Ti‘

4(2), 1993

GLOBAL FINANCE JOURNAL

160

Bble 6 AVERAGE VALUES OF THE RSME AND MAE FOR THE DIFFERENT FORECASTING MODELS (01/02/g&03/31/91) lUvlSE

Rank

Historical

0.16122

3

Adj.Historical(AH)

0.37624

Overall Mean

0.14918

National Mean

0.13906

1

0.06542

1

Single World Index(SIMW)

0.85637

9

0.73281

9

Single Domestic Index(SIMUS)

0.31445

4

0.23694

5

Adj.Single Domestic Index(ASIMUS)

0.50618

8

0.34468

7

Adj.Single World Index(ASIMW)

0.49824

7

0.36670

8

Multi-Index Market Model(MIM)

0.44301

6

0.30623

6

Model

MAE -__

Rank

0.08540

3

5

0.21019

4

2

0.07408

2

.t....rr--~.-----.r-....-.-...~.~~~~~~~.~......~.~~*~..~.........~~...... 0037 I

.

I

3

l

I

I

1 I o.co6 !

Return

I 0.005! I I 0.m + I I

I t

0.003 !

oan

I

I

I ;

.

I

I o.cm ;

l

I

I

I

I I

I .

l

0.033 ..~.1~~.....~~~~..~~..~....~-l..~..~...~....~..-..-.......--.-0.020 0.025 0.03 O.wO O.LKtS 0.010 0.015

0.03s

Std-dev Note:

‘1’ Natmnal Mean Model ‘4’ Histoncal Model

Figure 1.

Mean-Variance

Covariance

‘2’. Overall Mean Model ‘5’ Adjusted Historical Model

Efficient

Frontiers

‘3’. Actual (Subsequent Quarter)

Generated

Using

Forecasting Models Plus the Actual Efficient (April 3,1988-June 30,1988)

Four VarianceFrontier

American Depository Receipts Market

0.047

161

*’ ,

i 1 a.006! Return

o.ws

I I

l

I 0.002

l

0.001

;

I

I

o.ooo

l ---*----------*~-~~~~~l-~~.~~~-~~~~r-r_~~~

0.00

0.01

0.02

0.03

0.04

0.0.5

0.06

Std-dev Note.

‘1’: Natmnal Mean Model ‘4’: Historical Model

‘2’: Overall Mean Model

‘3’: Actual (Subsequent Quarter)

‘5’: Adpsted Hlstoncal Model

Mean-Variance Efficient Frontiers Generated Using Four VarianceCovariance Forecasting Models Plus the Actual Efficient Frontier (January 2,1991-March 31,199l)

Figure 2.

Figures 1 and 2 represent overlaid plots for the forecasting quarters covering the periods April 1,1988-June 30,1988, and January 2,1991-March 31,1991, respectively. As shown in Figure 1, for risk levels corresponding to a daily standard deviation of about 1.4% (or below), the NationalMean model consistently dominates other models in generating an efficient frontier that approximates the actual, followed in order by the Overall Mean model, the Historical model, and the Adjusted Historical model. Taking the 12th forecasting period as another representative sample, we still observed the superior performance of the National Mean model for risk levels corresponding to about 1% (and below) of daily standard deviation. The National Mean model is followed by the Historical model, the Overall Mean model, and the Adjusted Historical model for risk levels corresponding to a daily standard deviation range between 1% and 1.5%. Above the 1.5% risk level, the National Mean model is followed by the Overall Mean, then the Historical, with the Adjusted His-

162

GLOBAL FINANCE JOURNAL

4(Z), 1993

torical always being the worst performer within that sub-group of the four best performing models. Figure 2 reveals, substantially, a similar picture. These two plots simply serve to confirm the statistical results found earlier and indicate that relative superiority among the best performing models may be also dependent on the acceptable risk-return habitat or preferred locus on the efficient frontier. An interesting observation, however, is that all efficient frontiers plot under the actual outof-sample frontier. There appears to be a tendency for all four models to overestimate variances and pairwise covariance elements (or alternatively, to underestimate the benefits from international diversification with ADRs), with the National Mean model exhibiting the least tendency to do so.

U CONCLUSIONS Previous work with American Depository Receipts (ADRs) has indicated that indeed, ADRs can provide risk diversification benefits when combined with U.S. stocks. Therefore, this paper investigates the intertemporal stability of the comovement measures of returns on ADRs and U.S. stocks to determine whether or not, and how, the time series of these assets returns can be used in mean-variance portfolio optimization. Based on the empirical findings presented above, the following conclusions can be drawn. 1.

2.

3.

The covariance structure of ADR returns is intertemporally non-stationary on a quarterly basis, while the correlation structure is highly stationary with almost certainty for quarterly forecast horizons. And perhaps the most significant result, is that time-varying volatility of individual asset returns plays the primary role of destabilizing the covariante structure of short-horizon returns. This is an interesting result which is observed in the ADR market in that it is consistent with the mounting evidence that financial time series often possess time-varying volatilities, a phenomenon that is detected more easily when using frequently sampled returns. Given the non-stationary character of the covariance structure of returns, the out-of-sample performance of nine variance-covariance forecasting models is examined to determine the model with the relative best out-ofsample fit. Judging by the Root Mean Square Error and the Mean Absolute Error Criteria, the National Mean model strictly dominates all other models in terms of relative prediction accuracy. The Overall Mean model ranks closely in second place followed by the Historical and “Adjusted” Historical models. Of the several index models, the Single Domestic Index Model performed the best, followed by a Multi (two) Index model. The worst performing model was the Single World Index model. These results provide support to the importance of a U.S. domestic factor that seems to dominate the joint ADR-S&P500 return-generating process. However, a world factor appears to be important as well. These results are consistent with a partially segmented (partially integrated) view of ADR pricing.

163

American Depository Receipts Market

APPENDIX

A

This appendix presents the formulas for the test statistics and correlation matrices. Box’s F-approximation to the likelibood ratio statistic for testing for equality of two or more covariance matrices is given by the following Q1 quantity: Q1 = (T1 + Tz-2)lnIS(

-(T1-l)lnlS1\

-(Tz-l)lnISZ(

where S1 and S2 are the n x n matrices of sums-of-squares and cross-product moments of the sample of 35 assets, (34 ADRs and the S&P500), computed for contiguous sub-periods, and T, and T, are the lengths of the first and second contiguous sub-periods of observations, respectively, and

s=

I_

1

(T,+T,-2)

I

The quantity Q1 follows an F-distribution

dfl =

YL(n+l)

2

and

df2 =

(Tz-Wzl

[(T,-l)S,+

with degrees of freedom: [12T;(11+1)2]

[7(7~+1)~(n-l)

[iZ(n+l)

(n+2)

+4]

-6(2n2+3n-l)]

for the case where there are two covariance matrices and (n) being the number of assets. The quantity Q1 is an independent test of the homogeneity of the covariante matrices as it assumes unknown (and potentially unequal) mean vectors. The null hypothesis that two covariance matrices are equal (drawn from the same population) is rejected if Qr > Fsdfr,df2. Jennrich’s x2 statistic for testing stability (equality) of two correlation matrices is given by the following Qz quantity: The following test statistic Q2 is asymptotically distributed as a x2 variate with 1/2n(n - 1) degrees of freedom under the null hypothesis of equality and is given by Q2 = l/2 tr (Z’) - &‘(Z)U-’

dg(Z)

where Z = CZIZR-l(R1 - X2) c = TITp,/(TI + 2-2) R= (T,R1 + T2R2)/(T1 + T,) = {q> where RI and X2 are n x n matrices of sample pairwise correlations, computed for the first and second contiguous sub-periods, respectively; T1 and T2 are the lengths of the first and second contiguous sub-periods, dg(.) denotes the diagonal of a square matrix (Z) written as a column vector; tr(.) denotes trace of a matrix, U = ($ + sgrq), where 6q is the Kronecker delta and where 8 is the 9th element of the inverse correlation matrix R-l ; and all other variables are as defined above.

164

GLOBAL FINANCE JOURNAL

4(Z), 1993

Jennrich’s modified x2 statistic for testing equality of more than two correlation matrices is given by the following QS quantity:

Q3 =

5 (;tr(Z;,

-dg’(zi)S-‘dg(zl))

i=l

where Z, = T~12X-l(Xi - R) ii= (T,R, + T2R2 + . . . + T,R,)/T T = T1 + T2 + . ..+ T5 S = (~ij + rij+j).

= {YJ

Xl . . . X5 are the five n x n (n = 35) matrices of sample correlations between the n securities computed over alternative pairs of contiguous sub-periods, and all other variables are as defined earlier. The quantity Q3 has an asymptotic x2 distribution with (k - l)n(n - 1)/2 degrees of freedom where k denotes the number of matrices being tested for equality. The null hypothesis is rejected if Q3 2 &,df

APPENDIX Country Australia France Germany

Japan

Luxembourg Norway

B:

List Of ADRs

Security NEWS CORE PACIFIC DUNLOP LTD. LVMH DRESDNER BK PLC. CANNON INC. CSK CORI? DA1 EI INC. FUJI PHOTO FILM LTD. HITACHI LTD. HONDA MOTOR LTD. IT0 YOKADO LTD. JAPAN AIRLINES LTD. KIRIN BREWERY LTD. KYOCERA LTD. MITSUI & CO LTD. NEC CORI? PIONEER ELECTRONIC CORE SONY CORP TDK CORI? TOYOTA MOTOR CORP TOKIO MARINE & FIRE INSURANCE MINORCO NORSK DATA A S

165

American Depository Receipts Market

South Africa Sweden United Kingdom

NORSK HYDRO A S DE BEERS CONS MINES GAMBRO A B BARCLAYS PLC. BAT INDUSTRIES PLC. CADBURY SCHWEPPES PLC. NATIONAL WESTMINISTER BANK PLC. REUTERS HOLDING PLC. SAATCHI & SAATCHI PLC. UNILEVER PLC. WPP PLC.

ACKNOWLEDGEMENTS We are indebted to two anonymous referees for their very helpful comments. Responsibility for any remaining errors is, naturally, our own.

NOTES

1.

2.

3.

4.

5.

It has been estimated that over 850 ADRs are currently traded in the U.S., as of year-end 1989, as opposed to only 150 ADRs in 1961,390 ADRs in 1978, and 550 ADR issues by year-end 1984. Results reported in Cumby et al. [4] confirm the presence of substantial time variation in portfolio performance parameters. However, although fitting ARCH, GARCH, or exponential GARCH models to the covariance matrix seems to have captured some of the time variation in portfolio performance parameters, improvement in the quality of portfolio parameter forecasts was rather limited. This adjustment assumes that investors have a diffuse prior so that very little or no information exists about means, variances, and covariances before taking a sample. Prior information is, therefore, dominated by in-sample information. This model was first proposed and tested in Elton and Gruber [6] and reexamined in Eun and Resnick [8] by specializing it to an international setting. Elton and Gruber found that the Overall Mean model performed better than the Historical model and was intertemporally robust; whereas, Eun and Resnick documented a reverse result in an international context, despite the random noise contained in historical data. An alternative averaging procedure that uses industry classification as the disaggregation medium could have also been employed to accommodate alternative comovement structures. Since national factors would be expected to dominate industry factors (Lessard [19]) it was felt that not much information is lost by not adopting an industry-averaging model. In addition, country industry indices were unavailable to us.

166

6.

7.

8.

9.

GLOBAL FINANCE JOURNAL

4(2), 1993

Since Japanese ADRs constitute about 50% of our sample, average (overtime and cross-sectional) performance results of the National Mean model may have been driven by a systematic bias reflecting the dominance of Japanese ADRs, especially that ADRs from most other countries have a smaller representation in numbers. Thus, if the National Mean model was found to perform in a superior manner compared to other variance-covariance models, this performance may be simply an artifact of the structure of our sample and may not reflect the true performance of the National Mean model as a forecaster of the future variance-covariance values for a better diversified sample of ADRs. Alternatively, the performance of the National Mean model may relate predominantly to forecasting the variance-covariance structure of only Japanese ADRs (with the S&P500). To examine this possibility, the National Mean model was reestimated excluding Japanese ADRs from our sample (which effectively reduced our sample size to 17 ADRs) and recomputed the average RMSE and MAE over 12 quarters. We owe this idea to a referee. The results were not materially different. First, the National Mean model still dominated all other forecasting models. Second, the average RMSE showed a very slight improvement (from 0.1390 to 0.1288), whereas the MAE did not practically change (from 0.06542 to 0.06539). No systematic biases appear to have been induced by using the full sample. Furthermore, the results assert the familiar idea that inter-country mean-covariances tend to be much smaller than intra-country mean covariances so that including the Japanese subsample when averaging on an inter-country level did not alter the results. This model represents the international version of the Sharpe Single Index model, which was employed by Solnik [29] in testing the International CAPM. The Scholes-Williams [28] beta adjustment technique is not the only method to adjust for measurement errors due to non-synchroneity. Another popular technique is Dimson’s [5] aggregated coefficient method. The specification given in equaiton (6) follows from the following: first, since the S&P500 index is expected to be cross-correlated with the FT-World market proxy, we regressed the FT index on the S&P500 index to separate the impact of both factors on ADR returns, as follows: h_ut =

PO +

(1)

hXdt+ht

Given a two-index model specification (non-orthogonalized), assumed return generating process is given as follows:

Substituting

equation (1) into equation (2), we get Rit = ai + &1Rdt +

or

the alternative

P’t2[h+khRdl

+

kutl

+ eit

(3)

167

American Depository Receipts Market

Rit = ai + FilRdt + P’ixR*wt + eit where 10.

a, = a, + P’&I,

p’il = PiI + P’&,

(3a)

and jl,t = R*,t.

We also tried including a third “national” index in equation (6), representing the equity market of the country of origin of the particular ADR in question along with the S&P500 and the orthogonal world factor. This specification would be also consistent with a partially segmented view of ADR pricing. Since ADRs are traded in the U.S. but represent claims on cash flows generated by their underlying securities, this specification, which reflects the hybrid nature of ADRs may not be unreasonable. In no case were the coefficients on the national index statistically significant at conventional levels once the domestic S&P500 index and the orthogonal world factor were accounted for. We also dropped the domestic S&P500 index and ran a twoindex model with only the national index and an orthogonal world factor; however, the regression R2 measures were invariably lower if the national index was substituted for the domestic S&P500 index. It appears that the medium of trading effect dominates the country-of-origin effect. Therefore, only results based on a two-index specification that excludes the national index are reported here. REFERENCES

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