CMBF Papers - Pennsylvania State University

No. 17

byElizabeth Sheedy

Centre for Studies in Money, Banking and Finance

Macquarie UniversityJune 1997

CMBF Papers

Correlation in International Equity andCurrency Markets:

A Risk Adjusted Perspective

CMBF Papers are generally by members or affiliates of the Centrefor Studies in Money, Banking and Finance, Macquarie University.The papers are reviewed by an editorial board: Bill Norton(chairman), Sheelagh McCracken and Rob Trevor.

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No. 17

Centre for Studies in Money,Banking and Finance

CMBF Papers

Correlation in International Equity andCurrency Markets:

A Risk Adjusted Perspective

The author gratefully acknowledges support from the Australian Research Council, MacquarieInvestment Management Limited and Axiom Funds Management Corporation under CollaborativeResearch Grant C595301128.† Centre for Studies in Money, Banking and Finance, School of Economic and Financial Studies,Macquarie University, Sydney, NSW 2109, Australia. Phone: +61-2-9850-7755,Fax: +61-2-9850-7281, E-mail: [email protected]

byElizabeth Sheedy†

ii

The contents of this publication may be reproducedprovided the source is acknowledged.

Published in 1997 by

Centre for Studies in Money, Banking and FinanceMacquarie UniversityNORTH RYDE NSW 2109 AUSTRALIA

ISBN No. 1 86408 352 2

Printed in Australia by Macquarie University

1

Contents

Abstract 3

1. Introduction 5

2. Current Understanding of Return Correlationand Volatility 7

3. Multivariate Models 12

4. Data Analysis 23

5. Asset Allocation Decisions 42

2

6. Conclusions 52

Appendix 1Asset Allocation DecisionsShould we use daily or monthly data? 54

Appendix 2The Currency of Denominationfor Equity Returns 58

References 68

3

Correlation in International Equity and CurrencyMarkets: A Risk-Adjusted Perspective

Abstract

Several studies published since the 1987 stock market crash have foundevidence of structure and variability in the correlation of equity marketreturns. This evidence has undermined the argument for internationaldiversification and has spawned the development of complex multivariatemodels for explaining variance/covariance. In spite of the well-knownevidence for structure in volatility, many correlation studies have failed toadjust for volatility clustering. After rectifying this problem this paperfinds that correlation can be assumed constant with some qualifications.After standardising to allow for changing risk, correlations of daily MSCIreturns from 1982-1996 are generally stable. While correlation of currencyreturns is more variable, the changes are not economically significant in anasset allocation context.

The study finds that multivariate specifications designed to captureco-movements in volatility and changes in correlation are of dubious value.More parsimonious specifications generally perform best, provided thatthey capture volatility clustering. The study also highlights the pitfalls ofusing the traditional Fixed Window method to estimate variance/covariancefor asset allocation decisions. The implications for asset allocation decisions,hedging decisions and international diversification strategies are examined.

4

1. Introduction

Correlation estimates are essential to many areas of finance. Assetallocation decisions, index replication strategies and estimation of marketrisk, for example, all require correlation as an input. While much researcheffort has been devoted to modelling variance, understanding of correlationis relatively poor. Several multivariate modelling strategies have beenproposed, but evidence for the relative efficiency of these models is lacking.

This study explores the behaviour of correlation and examines the meritsof various multivariate modelling strategies. The research differs frommany previous studies of correlation by using daily rather than monthlydata.1 Appendix 1 shows that daily data provide superior risk estimates. Inaddition, daily data facilitate the use of more complex GARCH stylemodelling strategies that often require many observations for estimation.2

Previous studies have argued that return correlation is changing over time.Ironically, however, most of these studies have used a measure of correlationthat assumes constant correlation in each sample period. The current studyrectifies this defect by testing GARCH based measures of return correlation.These measures potentially account for variation in correlation over time.

This study further differs from others in that it considers the interactionbetween correlation in returns and volatility. Previous studies of returncorrelation have not adequately adjusted for changing risk, even thoughthe GARCH literature clearly demonstrates patterns in risk (see Bollerslev,Chou and Kroner, 1992). Specifically, the study considers whether thesewell-known patterns in volatility explain patterns in return correlation notedby other researchers.

1 The use of such higher frequency data has been recommended by Foster and Nelson (1996) andothers.

2 Nelson (1992) shows that the accuracy of conditional covariance estimates under various GARCHmodels improves as the frequency of observation increases. Even misspecified GARCH modelscan produce ‘good’ estimates of volatility provided high frequency data are employed.

5

Finally, the study contributes to the literature by examining both currencyand stock markets. To the extent that a literature on correlation exists, it isalmost entirely devoted to equities. Many financial institutions now requirecorrelation estimates for currencies and other assets in order to measureand manage market risk (see BIS, 1996). The study bridges this gap in theliterature by undertaking analysis of currency correlations. Knowledge ofcurrency returns is in turn used to analyse the impact of the currency ofdenomination on equity returns.

Seven multivariate modelling strategies are examined, thus providingempirical evidence that has previously been lacking in the literature. Themultivariate models are first evaluated in terms of their ability to explaincurrency and equity market behaviour. Secondly, the strategies are ratedaccording to their capacity to generate efficient asset allocation outcomesin the same markets. With respect to equity markets, the study finds thatthe emphasis on patterns in return correlation found elsewhere in theliterature is misplaced. Rather, the single most important issue inmultivariate modelling is to capture volatility clustering. Parsimoniousmultivariate specifications often outperform more complex, generalalternatives, provided they adequately incorporate this crucialautocorrelation in variance. For currency markets, the evidence for variationin correlation is stronger, but not economically significant. Thus the studyfinds that in currency, as well as in equity markets, a parsimoniousmultivariate specification is useful for financial decisions.

The paper is divided into six sections with two appendices. Section 2 reviewsthe “stylised facts” for both risk and correlation explored by previous studies.The links between these two strands of the literature are noted, raising thepossibility that they have a common cause. Section 3 explains sevenmultivariate specifications, drawing attention to their assumptions regardingthe structure of risk and return correlation. Section 4 tests the data forevidence of stability in return correlation and investigates the explanatorypower of the multivariate specifications in both currency and equity markets.Section 5 applies the multivariate specifications in a typical asset allocationdecision framework, allowing the economic significance of various risk

6

estimation techniques to be assessed. Section 6 concludes the discussionby linking return correlation and risk and by exploring implications forpractice. Choices regarding frequency of data and currency of denomination(for equity data) are discussed in Appendices 1 and 2 respectively.

7

2. Current Understanding of Return Correlationand Volatility

A number of studies of return correlation, volatility and volatility correlationin equity markets have been published, especially in the period since the1987 stock market crash (see Bennett and Kelleher, 1988, for example). Incontrast, the parallel literature relating to currencies is quite limited. Studiesof stock markets vary in their currency of denomination, that is, localcurrency versus a common currency such as US dollars. As shown inAppendix 2, the broad conclusions of this study are not sensitive to thecurrency of denomination. The most important ideas to emerge from theliterature on correlation and risk are as follows:

2.1. Return Correlation Related to Volatility

Correlation in stock returns tends to increase when volatility is high. Infact, October 1987 was the only month in the 1980s in which every majormarket moved in the same direction. Bertero and Mayer (1989) and Leeand Kim (1993) document an increase in return correlations at the time ofthe crash and report that the increase persisted subsequently.

The findings linking correlation and volatility are not all drawn from the1987 crash, however. Bennett and Kelleher (1988) find evidence of arelationship between return correlation and volatility prior to October 1987.Bertero and Mayer (1989) also find that the degree of intercorrelationbetween markets is related to the degree of cross-border trading. Lee andKim (1993) report that co-movements among national stock markets arestronger when the US stock market is more volatile. A recent study bySolnik, Boucrelle and Le Fur (1996) confirms a positive relationshipbetween correlation and volatility across both bond and stock markets.

8

Many investors predicate an international investment strategy ondiversification benefits. For such investors, the finding that returncorrelations increase with volatility is problematic since diversificationbenefits are reduced at the very time they are most needed.

2.2. Asymmetry in Return Correlation

This apparent “chink in the armour” of international diversification is furtherhighlighted by findings of asymmetry in correlation. Obviously, correlationbetween two markets is positive when movement, either up or down, issynchronised. Erb, Harvey and Viskanta (1994) identify an asymmetry inthis relationship for equity markets. Correlation is higher when both marketsare moving down than when both markets are moving up; a result which isconfirmed by Longin and Solnik (1995).3

A related finding concerns the linkage between stock return correlationand the business cycle. Erb, Harvey and Viskanta (1994) find thatcorrelation is higher in common recession rather than common growthphases. Return correlation is lowest when the two countries concerned areat different stages of the business cycle.

2.3 Stability in Return Correlation

Previous studies have tested the stability of return correlation in successivesample periods with mixed results. For example, Kaplanis (1988) uses atest procedure developed by Jennrich (1970) to test the stability of returncorrelation. Kaplanis (1988) is unable to reject the hypothesis that thecorrelation matrix is constant over adjacent sub-periods of 46 months each.She considers the monthly returns of ten stock markets over a fifteen yearperiod from 1967-82. Tang (1995) uses weekly stock market returns

3 Darbar and Deb (1997) provide evidence that conflicts with this hypothesis, however. Theyreport a decrease in correlation in some country pairs after the US “minicrash” in 1989. Thestudy used GARCH based correlation estimates (in contrast to the usual practice of estimatingsample correlation).

9

from January 1981 to June 1992 for 12 countries, expressed both in localcurrency terms and US dollar terms. In both cases, results support theconstant correlation hypothesis across most sub-periods.

In contrast, Longin and Solnik (1995) test monthly national excess returnsfor seven countries computed over periods of five years. Using the Jennrichprocedure, they find the unconditional correlation matrix to be unstableover time. Darbar and Deb (1997) find that conditional correlation,estimated using GARCH procedures, changes considerably over time.

2.4 Long-term Trends in Return Correlation

Longin and Solnik (1995) find that stock market return correlation increasedover the 30 year period from 1960 to 1990. Bennett and Kelleher (1988)show that return correlation was higher in the decade of the 80s than in thedecade of the 70s. Both hypothesise that this increase is related to a higherdegree of global integration due to cross-border investments and listings.According to Roll (1989) “most casual observers presume that markets arebecoming more related. The free flow of capital to locales with the mostfavourable risk/return trade-off is certainly a strong force for the alignmentof price innovations.” In contrast, a more recent study by Solnik, Boucrelleand Le Fur (1996) concludes that return correlations have remainedrelatively stable in the ten year period to 1995 following a slight long-termincrease.

Much of the evidence gathered in these studies has supported the widespreadview that return correlation varies over time. Like volatility, correlationcan change significantly, depending on the period of estimation. This raisesthe possibility that return correlation, like volatility, can be modelled usingpast price information. Several multivariate risk estimation models arebased on this premise (see Section 3).

10

2.5 Volatility Clustering and Asymmetry

None of the studies described above correct for both volatility clusteringand leverage effects4 . The GARCH literature provides strong evidencethat volatility is autocorrelated, that is, high volatility in one period is likelyto be followed by high volatility in the next. This finding applies to allmajor classes of financial assets, including equities, currencies and interestrates (see Bollerslev, Engle and Kroner, 1994).

In equity markets, these clustering effects are exaggerated at times whenthe market is falling. An increase in volatility is likely to be greater followinga large downward move than that following a large upward move. Thisrelationship was first noted by Black (1976), but has subsequently beenstudied by Nelson (1990); Glosten, Jagannathan and Runkle (1993); Engle(1993) and others. Asymmetry can be explained by the argument that theleverage of stocks increases after a large downward price move, thusincreasing conditional variance. Since the same argument cannot be appliedto other assets (such as currency), the importance of asymmetry outside ofequity markets is questionable.

This study considers whether these findings are linked to the evidence ofstructure in return correlations referred to earlier. In particular, asymmetryin the correlation of returns could be explained by a combination ofpersistence and asymmetry volatility as well as correlation in volatilityacross markets.

2.6 Volatility Correlation

Evidence of volatility correlation has been observed in a number of studiesrelating to both equity and currency markets. King and Wadhwani (1990)develop a “contagion” model of international volatility transmission

4 Longin and Solnik (1995) do, however, account for volatility clustering.

11

between stock markets. The underlying idea is that rational traders in onecountry should use price movements in another country to deduce changesin underlying economic fundamentals. This implies that a price “mistake”in one country will be transmitted to others like an infectious disease. Kingand Wadhwani claim that contagion increases with volatility.

Many studies have been undertaken on the transmission mechanism betweenstock markets. Hamao, Masulis and Ng (1990); Becker, Finnerty and Gupta(1990); Karolyi (1995); Lin, Engle and Ito (1994) and Eun and Shim (1989)all find evidence that shocks in one stock market can influence other marketsin subsequent trading sessions. The notion that volatility is correlated isconsistent with a study by Bennett and Kelleher (1988) who claim thatlarge price movements in one market are associated with large pricemovements in others. The actual level of volatility during October 1987,however, was much higher than one would normally expect based on thehistorical relationship across countries.5

The spillover of volatility between currency markets is described by Engle,Ito and Lin (1990) as a “meteor shower” effect. Najand, Rahman andYung (1992) further find evidence of volatility spillovers in currency futuresmarkets. Their study indicates that volatility spillover emanates fromstronger currencies such as the Deutschemark.

The evidence for volatility correlation may be connected with the findingsfor return correlation discussed earlier. This study considers the relationshipbetween correlations in returns and volatility. Specifically, what are thestylised facts for return correlation after adjusting for volatility clustering,asymmetry effects and volatility correlation?

5 Note that Roll (1989) criticises the Bennett and Kelleher study on the grounds that the explanatoryvariable in their model is only a sample estimate of population standard deviation. Samplingerror in the explanatory variable causes attenuation bias, meaning that the coefficient in theirmodel is biased toward zero.

12

3. Multivariate Models

Capturing all the stylised facts described above in a multivariate modelcan be extremely difficult. The difficulty compounds as the number ofassets grows, particularly since it is not always possible to ensure that agiven modelling technique will produce a positive definite variance/covariance matrix. Several attempts have been made to find parsimoniousmultivariate specifications which capture some, if not all, of these stylisedfacts. Very little evidence is available regarding the empirical support forany one specification.

In this study, seven approaches to modelling the variance/covariance ofmultivariate asset returns are considered. This section highlights the factthat measures of risk and return correlation can vary significantly dependingon the method used to estimate the variance/covariance matrix. Conclusionsabout the nature of return correlation and variance may differ as aconsequence.

3.1 Fixed Window

The Fixed Window approach implicitly assumes that return correlationand variance are constant over the sample period. This method is difficultto justify in light of the studies mentioned above, but serves as a usefulbenchmark for comparison. Further, estimating variance/covariance witha Fixed Window is still common, possibly due to ease of estimation. Mostprevious studies of correlation have been based on this method. Eachelement in the variance/covariance matrix can be represented by:

σε ε

ij t

i t a j t a

a

m

m,

, ,

+− −

=

−

= ∑10

1

(1)

where εi t i tt a

a

m

RR

m, ,= −

−

=

−

∑0

1

and m= number of observations in sample.

13

3.2 Exponential Smoothing

This method uses all available data until the point of estimation, but on anexponentially weighted basis. This method of measuring risk has enjoyedrenewed popularity in the finance industry with the distribution of J.P.Morgan’s RiskMetrics (see Longerstaey and Martin, 1996; Brown, 1990).The estimate of the variance/covariance for next period is:

( )σ λ ε ε λσij t i t j t ij t, , , ,+ = − +1 1 (2)

where εi,t is the mean-corrected return for asset i at time t.

The smoothing parameter, λ, is assumed to equal 0.94.6 Equation (2)incorporates an autoregressive structure for variance/covariance, thusreflecting the concept of volatility clustering7 . It should be noted, however,that this specification for variance is constrained relative to the more generalapproach of GARCH models (compare equations (2) and (4)). Returncorrelations can be implied from the variance/covariance estimates.

A further risk specification (Exponential Smoothing - CC, or ESCC) canbe created by using Exponential Smoothing to estimate variance, butapplying the assumption of constant correlation to obtain covariances.Under this specification variance is defined as previously shown in equation(2), where i=j. Covariances are formulated as follows:

σ ρ σ σij t ij i t j t, , ,+ + +=1 1 1 (3)

where ρij is the sample correlation of mean-corrected returns.

6 JP Morgan’s RiskMetrics (Longerstaey and Spencer, 1996) recommends the assumption thatλ=0.94 for daily data.

7 Longerstaey and Spencer (1996) provide evidence of autocorrelation in squared returns and theproduct of returns for currencies in support of this specification.

14

This measure differs from regular Exponential Smoothing only with regardto the calculation of correlation. Including this new specification thusisolates the influence of correlation in determining investment outcomes.

3.3 GARCH - Constant Correlation

GARCH - Constant Correlation (GARCHCC) is a multivariate version ofthe standard univariate GARCH risk estimation approach (see Bollerslev,1990). Variance is estimated using a simple GARCH(1,1) formulation:

σ γ γ σ γ εi t i i i t i i t, , ,+ = + +12

0 12

22 (4)

where εi,t is the mean-corrected return for asset i at time t.

Like the Exponential Smoothing approach discussed above, these equationsexpress the expected variance of next period’s return as a function of theexpected variance for the current period and the news on volatility (ε

it2)

revealed by returns in the current period. The covariances are formulatedas follows:

σ ρ σ σij t ij i t j t, , ,+ + +=1 1 1 (5)

The parameters of the model (γi0, γ

i1, γ

i2, ρ

ij) are estimated by maximum

likelihood procedures. The γ‘s are constrained to be non-negative to ensurethat all variance forecasts are positive.

The main disadvantage of the GARCHCC risk measure is that it presentsgreater technical complexity for the industry practitioner compared to FixedWindow and Exponential Smoothing. GARCHCC is, however, muchsimpler to estimate than most other multivariate models in the GARCHfamily, thus contributing to its popularity in the literature (see for exampleLongin and Solnik, 1995; Kroner and Claessens, 1991; Schwert and Seguin,1990; Bollerslev, 1990; Baillie and Bollerslev, 1989). Ease of estimationis achieved by constraining the correlation to be constant over the estimationperiod and by ignoring cross-market effects.

15

Other researchers have cautioned that the GARCHCC model does notnecessarily produce a positive definite variance/covariance matrix. Kronerand Ng (1995) note that the model is positive definite if and only if thecorrelation matrix is positive definite. In this study, however, GARCHCCalways generates positive definite variance/covariance matrices.

3.4 BEKK

BEKK is an alternative multivariate GARCH style specification. Engleand Kroner (1995) propose this model on the basis that its quadratic formguarantees that the conditional covariance matrix will be positive definite:

H C C B H B A At t t t+ = + +1 ' ' ' 'Ε Ε (6)

where Et = [ε

1t, .., ε

Nt]’ the vector of return shocks at time t;

εi,t is the mean-corrected return of asset i at time t;

C, A, and B are (estimated) NxN matrices of parameters; andN = the number of assets.

Consider, for example, a 2x2 BEKK model. The expected risk of asset 1 inthe next period is defined as:

h c c c b h b b h b h a a a at t t t t t t t11 1 112

12 21 112

11 11 21 12 212

22 112

12

11 21 1 2 222

222 2, , , , , , , ,+ = + + + + + + +ε ε ε ε

(7)

Equation (7) highlights the cross-market effects which are captured by theBEKK model. Not only is the variance of asset 1 a function of its own pastvalues and price shock, but it is also a function of the variance and pastvalues of asset 2 and the past covariance. Thus the BEKK model potentiallyreflects both volatility clustering and cross-market effects in volatility. Theconstraints of the BEKK model are evident from a closer inspection ofequation (7). Note that the coefficients of h

11,t, h

22,t, ε

1,t2 and ε

2,t2 are all

constrained to be non-negative, meaning that variance must be

16

non-negatively correlated with past price shocks and variance in all markets.Since most evidence supports such a non-negative relationship (see Kingand Wadhwani, 1990; Bennett and Kelleher, 1988), it is unlikely that thisconstraint will be binding.

The covariance in the two asset case is also a function of its own pastvalues, the twovariances, and the two price shocks, as follows:

( )( )h c c c c b b h b b b b h b b h a a

a a a a a a

t t t t t

t t t

12 1 12 11 22 12 11 12 11 11 22 12 21 12 22 21 22 11 12 12

11 22 12 21 1 2 21 22 22

, , , , ,

, , ,

+ = + + + + + + +

+ +

ε

ε ε ε

While BEKK is the most general model considered, it suffers thedisadvantage of being the most difficult to estimate, especially for largenumbers of assets. This characteristic has no doubt contributed to thepaucity of studies in the literature which apply the BEKK specification.8

Figure 1 contrasts correlation estimates for US and Japanese equity returns(expressed in local currencies) based on three of the specifications discussedso far. The BEKK and Exponential Smoothing models display considerablevariability over times. For example, under BEKK, the US/Japan equitymarket correlation reaches a peak of 0.66 in 1990 and a low in 1994 of -0.30. In contrast, the correlation estimates resulting from the ExponentialSmoothing approach range between 0.79 (in 1990) and -0.38 (in 1992).Return correlation estimated by GARCHCC is, of course, constant; theparameter estimate for the period in question is 0.147.

8 Karolyi (1995) is one of the few published studies employing the BEKK specification. Darbarand Deb (1997) use a constrained BEKK (diagonal) specification.

(8)

17

Figure 1Comparing Multivariate Specifications

US/Japan Stock Return Correlation

-1.000

-0.600

-0.200

0.200

0.600

1.000

Jan-

90

Apr

-90

Aug

-90

Dec

-90

Apr

-91

Jul-9

1

Nov

-91

Mar

-92

Jun-

92

Oct

-92

Feb

-93

Jun-

93

Sep

-93

Jan-

94

May

-94

Sep

-94

Dec

-94

Apr

-95

Aug

-95

Dec

-95

Mar

-96

Jul-9

6

Nov

-96

Date

Constant Correlation Exp. Smoothing BEKK

Return correlation estimates using daily MSCI data for US and Japan (excluding dividends andexpressed in local currency terms).Constant Correlation: A 4x4 GARCHCC model is used to estimate variances and correlations for fourequity markets (US, Japan, Germany, UK) from January 1990 until October 1996.BEKK: A 4x4 BEKK model estimates variances and covariances from January 1990 until October1996. Correlations for US/Japan are then implied from these estimates.Exponential Smoothing: Estimates of variances and covariances are obtained for the US and Japanesemarkets from January 1990 until October 1996. Return correlations are implied from variance/covariance estimates.

3.5 Factor Model/Constant Correlation

A further multivariate specification from the GARCH family is tested,namely, a Factor Model with Constant Correlation (FactorCC). Thisapproach is very similar to GARCHCC, but attempts to incorporate cross-market effects in risk. For equities this can be achieved by using the returns

18

of the MSCI World Index9 (excluding dividends) as a common factor. Therisk for each country is a function of its own past variance and price shock,and the past price shock for the factor (ε

f,t), as shown in equation (9):

σ γ γ σ γ ε γ εi t i i i t i i t i f t, , , ,+ = + + +12

0 12

22

32 (9)

where εi,t is the mean-corrected return of asset i at time t;

εf,t is the mean-corrected return of the factor at time t;

γ γ γ γi i i i0 1 2 3 0, , , ≥ .

The covariances are estimated as follows:σ ρ σ σij t ij i t j t, , ,+ + +=1 1 1 (10)

The FactorCC specification is a variation of factor models proposed byother researchers.10 Cross-market effects apply only to the estimates ofconditional variance, and they do so only through price shocks in the factor.Thus, while the basic concept is similar to BEKK, the “specification” ismore constrained.

This specification nests GARCHCC, so it is possible to test directly thespillover hypothesis. For the period 1990-1996, three of the four markets11

yield estimates of γi3 that are positive and significant at the 5% level. This

result suggests that a common market factor does help to explain volatilityin local markets (and expressed in local currency terms) during the period1990-1996.

9 Returns on the MSCI US index are also considered as a possible common factor (not reported).This common factor has less explanatory power than the MSCI World index returns for this dataset.

10 Other factor model specifications are proposed in Engle, Ng and Rothschild (1990); Lin (1992);Ng, Engle and Rothschild (1992); Engle and Susmel (1993). All of these specifications werefound to have less explanatory power for this data set than that illustrated in equations (9) and(10) (results not reported).

11 UK, Germany and Japan (but not US).

19

3.6 Asymmetry/Constant Correlation

This multivariate specification is proposed in order to account forasymmetry in volatility. A number of univariate specifications have beendeveloped to capture the asymmetry in the volatility response to positiveand negative return shocks (see Engle, 1993 and Glosten, Jagannathan andRunkle, 1993). Under the assumption of asymmetry, variance has thefollowing specification:

( )[ ]σ γ γ σ γ ε γ εi t i i i t i i t i i tMin, , , ,,+ = + + +12

0 12

22

4 0 (11)

where εi,t is the mean-corrected return of asset i at time t.

Typically, γi4 is negative. Thus, negative return shocks generate more

volatility than positive return shocks of the same size. In a multivariateframework the univariate asymmetric approach is incorporated byestimating covariances as follows:

σ ρ σ σij t ij i t j t, , ,+ + +=1 1 1 (12)

The (constant) return correlation estimate obtained under Asymmetry/CCis shown in Table 1, along with parameter estimates from other specificationsassuming constant correlation. For the US/Japanese equity markets (1990-96) the four specifications yield almost identical estimates of correlation.

20

Table 1Multivariate Specifications With Constant Correlation

US/Japan Stock Return Correlation1990-1996

Specification Correlation Standard ErrorFixed Window 0.147 na

Exponential Smoothing - CC 0.149 naGARCH/Constant Correlation 0.141 0.022

Asymmetry/Constant Correlation 0.145 0.022Factor/Constant Correlation 0.129 0.021

Return correlation estimates using daily MSCI data for US and Japan (expressed in local currencies). In each case, the sampleperiod is January 1990 - October 1996.

3.7 Summary of Multivariate Specifications

The seven multivariate specifications described above differ considerably.Table 2 summarises the key attributes of each:

Table 2Attributes of Multivariate Specifications

Characteristic FixedWindow

Exponen-tialSmoothing

Exponen-tialSmoothing - CC

GARCHCC

BEKK FactorCC

Asym-metry/CC

VolatilityClustering

No Yes Yes Yes Yes Yes Yes

Cross-MarketEffects inVolatility

No No No No Yes Yes No

Changes inCorrelation

No Yes No No Yes No No

Asymmetry inVolatility

No No No No No No Yes

Asymmetry inCorrelation

No No No No No No No

Yes - indicates that the specification attempts to incorporate the characteristic.No - indicates that the specification is not designed to incorporate the characteristic.

21

Five of the specifications (Fixed Window, Exponential Smoothing - CC,GARCHCC, FactorCC, Asymmetry/CC) assume that correlation is constantfor the sample period. The hypothesis that the five estimates of returncorrelation are identical cannot be rejected at the 95% confidence level(see Table 1). In contrast, the other two specifications (BEKK andExponential Smoothing) allow for variability in correlation. Figure 1 showsthat the these two styles of correlation track each other to some extent butcan vary significantly over time.

All specifications except Fixed Window accommodate volatility clustering.The five unique and changing estimates of risk for the US and Japaneseequity markets from 1990-96 are compared in Figure 2. A broadly similarpattern emerges in all five cases, with some minor variation depending onthe inclusion of spillover and asymmetry effects in volatility. Essentially,there are a number of episodes of high volatility but risk tends to revertback to its long-run mean over time.

Note that none of the specifications tested here is specifically designed tocapture asymmetry in return correlation. This could potentially be achievedby modifying the BEKK model (see Kroner and Ng, 1995), but was notattempted in this study. As will become apparent in later sections,asymmetry in return correlations can be captured in other ways.

22

Figure 2Estimates of Stock Market Risk - 1990-1996

Exponential Smoothing

0%

20%

40%

60%

80%

Vol

atili

ty p

a US

Japan

GARCH/Constant Correlation

0%

20%

40%

60%

80%

Vol

atili

ty p

a

BEKK

0%

20%

40%

60%

80%

Vol

atili

ty p

a

Factor/Constant Correlation

0%

20%

40%

60%

80%

Vol

atili

ty p

a

GJR/Constant Correlation

0%

20%

40%

60%

80%

Jan-

90

Jul-9

0

Jan-

91

Jul-9

1

Jan-

92

Jul-9

2

Jan-

93

Jul-9

3

Jan-

94

Jul-9

4

Jan-

95

Jul-9

5

Jan-

96

Jul-9

6

Jan-

97

Vol

atili

ty p

a

S

Jan-

90

Apr-9

0

Aug-

90

Dec

-90

Jan-

91

Apr-9

1

Aug-

91

Dec

-91

Jan-

92

Apr-9

2

Aug-

92

Dec

-92

Jan-

93

Apr-9

3

Aug-

93

Dec

-93

Jan-

94

Apr-9

4

Aug-

94

Dec

-94

Jan-

95

Apr-9

5

Aug-

95

Dec

-95

Jan-

96

Apr-9

6

Aug-

96

Oct

-96

Asymmetry/Constant Correlation

Standard deviation of MSCI returns expressed in local currency, January 1990 to October 1996

23

4. Data Analysis

In this section several testing methods are applied; each provides furtherevidence regarding the stability of return correlations and theappropriateness of the various multivariate specifications. The comparativeexplanatory power of each specification described above is first examinedusing the Schwarz criterion. The next series of tests (in Section 4.2)considers the evidence for correlation stability at a micro level: to whatextent do the various specifications account for structure found in rawreturns from one observation to the next. Finally, the stability of correlationover a longer time horizon is examined in Section 4.3. Section 4.4 discussesthe implications of test results.

For equity markets, the study initially uses daily MSCI data expressed inlocal currency12 from January 1990 to October 1996, thus avoiding themodelling complications associated with the 1987 stock market crash. Itcould be argued that previous studies of correlation, especially thosepublished in the late 1980s, have been unduly influenced by the impact ofthe crash. Returns are calculated using price index data (excludingdividends) for the US, UK, Japan, Germany and the World Index (in thecase of the FactorCC specification).

Currency data are synchronised daily observations of the US dollar (USD)relative to the Japanese Yen (JPY), British Pound (GBP), French Franc(FFR) and the US Trade-Weighted Index (TWI)13. For the FactorCCspecification, the returns of the TWI are used as the common factor. Theinitial period of analysis is January 1990 to December 1996 - a timespancomparable to that selected for equity markets.

12 Analysis is undertaken in local currency terms to reflect the perspective of a hedged, US basedasset allocator. Analogous results from a US dollar (unhedged) perspective are presented inAppendix 2. The conclusions are essentially the same, regardless of the currency of denomination.

13 Currency data are provided by the Federal Reserve Board of Chicago at web site www.frbchi.org/econinfo/finance/for-exchange/welcome.html. Daily observations are taken at 12 noon, NewYork time.

24

4.1 Tests of Explanatory Power

The explanatory power of the four specifications estimated with maximumlikelihood procedures can be examined using the Schwarz criterion (seeJudge, Hill, Griffiths, Lutkepohl and Lee, 1988). The Schwarz criteriontrades off goodness of fit and parsimony to identify the most attractivespecification.14 Table 3 shows that according to the Schwarz criterion,FactorCC has greatest explanatory power for equity data. In the case ofcurrency data, however, BEKK has the greatest explanatory power. Thissuggests that the assumption of constant correlation is harder to sustain incurrency markets.

Table 3Explanatory Power of Multivariate Specifications

Daily Observations, 1990-1996Specification Equity Data

(local currency)Currency Data

GarchCC -61,025 -70,980BEKK -61,205 -71,983FactorCC -61,365 -70,840Asymmetry/CC -61,186 -70,834

Table 3 displays the Schwarz Criterion (Judge, Hill, Griffiths, Lutkepohl and Lee 1988) for each specification. Eachspecification is tested for the period 1990-96 using daily data. The specification with the lowest Schwarz Criterion has the bestexplanatory power, after adjusting for the number of parameters.

14 Schwarz Criterion = -2(log likelihood function) + (log(T)xK), where K=number of parametersand T= number of observations.

25

4.2 Tests of Standardised Residuals

This test procedure follows that employed by Engle (1993), Kroner andNg (1995) and others. Certain “stylised facts” are commonly understoodto characterise risk and correlation (as described in Section 2). By definition,a successful multivariate modelling strategy must incorporate all theinformation available in the history of prices. The capacity of eachspecification to account for the stylised facts is tested by standardisingreturns using the conditional variance/covariance matrix. If a givenspecification is successful, the standardised returns (z

it) should show no

evidence of the characteristics described in Section 2. Success in this regardimplies that the assumptions underlying the specification are appropriatefor the data set; thus conclusions can be drawn regarding the stability ofreturn correlations and other issues.

Each test is performed first with the raw data to identify initial evidence ofstructure. The tests are then repeated using standardised returns, todetermine the success of the risk measure in accounting for this structure.

a) Stylised Facts for Risk

1. Volatility Clustering - a Lagrange Multiplier test for serial correlationin the squared returns. Looks for “clustering” or “persistence” at 12lags15 in squared, raw (standardised) residuals.

2. Asymmetry - a test of the relationship between a negative/positive returnlast period and variance this period. Regress the squared, raw(standardised) returns against x

i,t , y

i,t and w

i,t and test for joint

significance.

15 Tests (not reported) were also undertaken at 20 and 30 lags which were generally consistent withthe results at 12 lags.

26

Let I(.) be an indicator function that equals one if the argument is trueand zero otherwise.

xi,t = I(z

i,t-1 < 0) y

i,t = (1-x

i,t)z

i,t-1

wi,t = x

i,tz

i,t-1z

i,t-1 is the raw (standardised) return

3. Cross-Market Effects - a test of the relationship between volatility thisperiod and the volatility of other assets in the previous period. Regressthe squared, raw (standardised) returns of asset i against the squaredraw (standardised) returns of all other assets in the previous period, andtest for joint significance.

The first panel of Table 4 provides the results of these tests for raw equitymarket data expressed in local currency terms. The raw data provide ampleevidence for the stylised facts discussed elsewhere in the literature, that is,volatility clustering, asymmetry in volatility, and cross-market effects involatility. A p-value less than 0.05 indicates that the hypothesis cannot berejected at the 5% level of significance.

Subsequent panels show the results for standardised data for each of theseven multivariate specifications. Fixed Window shows little improvementon the raw data, highlighting the limitations of this technique. All otherspecifications have some degree of success in accounting for the stylisedfacts, but none is entirely successful. The capacity of the specifications toexplain structure in the data does not appear to vary significantly with thegenerality of the model (once having excluded Fixed Window).

27

Table 4Stylised Facts for Risk in Equity Markets

Daily Observations, Jan 1990- Oct 96 (Local Currency)(p-values)

UK US Japan GermanyRaw Data

Volatility Clustering .00 .00 .00 .00Asymmetry .00 .00 .00 .00

Cross Market Effects .00 .00 .00 .99Fixed Window


Cross Market Effects .00 .00 .00 .00Exponential Smoothing


Cross Market Effects .00 .68 .15 .13Exponential Smoothing -

CCVolatility Clustering .74 .42 .24 .99

Asymmetry .88 .50 .00 .30Cross Market Effects .00 .88 .22 .22

GARCHCCVolatility Clustering .85 .45 .57 .99


BEKKVolatility Clustering .56 .05 .28 .23


FactorCCVolatility Clustering .75 .44 .63 .78


Asymmetry/CCVolatility Clustering .79 .60 .34 .99


A series of tests are performed on raw and standardised returns as described in Section 3, with p-values as shown here. A p-value of 0.05 (highlighted in bold typeface) indicates that the results are significant at the 5% level, and therefore the hypothesisdescribed in the far-left column cannot be rejected. Tests are conducted for daily MSCI market returns, expressed in localcurrencies, for the period January 1990 to October 1996.

28

Only one specification specifically attempts to capture asymmetry(Asymmetry/CC), and it is successful in achieving this goal in all fourcases. Interestingly, all the GARCH based specifications have comparablesuccess in eliminating asymmetry. The specifications which attempt toaccount for cross-market effects are both largely unsuccessful (ie BEKKand FactorCC). In contrast, Exponential Smoothing is much more successfulin capturing cross-market effects, despite the fact that it is not specificallydesigned to do so.

b) Stylised Facts for Correlation

1. Clustering - a test for autocorrelation in the product of raw (standardised)returns at 12 lags.

2. Volatility - a test that considers the relationship between correlation/covariance this period and volatility in the last period. Regress theproduct of raw (standardised) returns for assets i and j against x

1t and x

2t

which are defined as:x

1t = z

i,t-12 and x

2t=z

j,t-12

where zi,t-1

is the raw (standardised) return.

3. Asymmetry - a joint test of sign bias in the product of returns for assetsi and j. Regress the product of raw (standardised) returns against x

3t and

x4t which are defined as:

Let I(.) be an indicator function that equals one if the argument is trueand zero otherwise.

x3t = I(z

i,t-1 < 0) and x

4t = I(z

j,t-1<0)

4. “Cross sign bias” - a further test of asymmetry explores the relationshipbetween correlation/covariance this period and the interaction of thereturns of assets i and j in the last period. Four cases are considered:both returns positive; both returns negative; asset i positive and asset jnegative; and asset i negative and asset j positive. The test is a joint test

29

covering all four cases. Regress the product of raw (standardised) returnsagainst x

5t, x

6t, x

7t, x

8t which are defined as:

x5t = I(z

i,t-1<0;z

j,t-1<0) x

6t = I(z

i,t-1<0;z

j,t-1>0)

x7t = I(z

i,t-1>0;z

j,t-1<0) x

8t = I(z

i,t-1>0;z

j,t-1>0)

5. “Cross size bias” - a further test of asymmetry that scales the signindicators by the size of the shocks. This captures the possibility thatthe effect of the sign of a shock may depend on the size of the shock.Regress the product of raw (standardised) returns against x

9t, x

10t, x

11t,

x12t

, where, x

9t = z

i,t-12I(z

i,t-1<0) x

10t = z

i,t-12I(z

j,t-1<0)

x11t

= zj,t-1

2I(zi,t-1

<0) x12t

= zj,t-1

2I(zj,t-1

<0)

The first panel of Table 5 shows the results for raw equity market dataexpressed in local currency terms. There is some evidence for clusteringin correlation and also for a link between volatility and correlation. Thesefindings are consistent with the previous studies discussed in Section 2which highlight the structure in return correlations. The evidence forasymmetry in correlation is not as compelling, however, except in the caseof cross size bias (where sign is scaled by the size of the shock).

30

Table 5 Stylised Facts for Correlation in Equity Markets

Daily Observations, Jan 1990 - Oct 96 (Local Currency)(p-values)

Pair US/UK

US/Japan

US/Germany

UK/Japan

UK/Germany

Japan/Germany

Raw DataClustering .00 .00 .55 .00 .00 .27Volatility Bias .00 .00 .00 .07 .69 .00Asymmetry .71 .64 .10 .02 .39 .50Cross Sign Bias .89 .19 .29 .08 .58 .85Cross Size Bias .32 .00 .00 .00 .05 .00

Fixed WindowClustering .08 .00 .01 .00 .00 .01Volatility Bias .18 .01 .01 .54 .16 .40Asymmetry .76 .67 .16 .08 .64 .82Cross Sign Bias .94 .72 .45 .26 .86 .10Cross Size Bias .70 .00 .00 .00 .10 .01

ExponentialSmoothingClustering .47 .06 .61 .34 .99 .78Volatility Bias .00 .46 .90 .51 .89 .81Asymmetry .52 .62 .61 .05 .71 .63Cross Sign Bias .79 .91 .88 .17 .83 .54Cross Size Bias .00 .60 .89 .01 .93 .92

ExponentialSmoothing - CCClustering .92 .40 .99 .19 .99 .99Volatility Bias .00 .97 .98 .57 .98 .78Asymmetry .56 .87 .19 .03 .91 .93Cross Sign Bias .89 .73 .50 .03 .89 .19Cross Size Bias .00 .59 .82 .08 .92 .60

GARCHCCClustering .92 .46 .99 .27 .99 .99Volatility Bias .00 .44 .99 .47 .85 .73Asymmetry .57 .81 .36 .07 .97 .97Cross Sign Bias .87 .73 .70 .11 .96 .39Cross Size Bias .02 .16 .98 .23 .96 .36

BEKKClustering .71 .10 .53 .34 .94 .47Volatility Bias .00 .07 .51 .95 .96 .54Asymmetry .48 .27 .32 .12 .91 .58Cross Sign Bias .83 .61 .65 .37 .99 .60Cross Size Bias .00 .01 .33 .27 .82 .61

FactorCCClustering .87 .51 .63 .20 .62 .99Volatility Bias .01 .56 .79 .51 .88 .59Asymmetry .64 .42 .80 .03 .84 .86Cross Sign Bias .93 .77 .65 .08 .98 .13Cross Size Bias .02 .07 .89 .14 .95 .27

Asymmetry/CCClustering .80 .38 .95 .34 .99 .98Volatility Bias .00 .50 .68 .43 .97 .69Asymmetry .73 .46 .39 .09 .93 .74Cross Sign Bias .96 .73 .74 .19 .99 .15Cross Size Bias .00 .26 .65 .38 .98 .18A series of tests are performed on raw and standardised returns (see Section 3). A p-value of 0.05 indicates that the results aresignificant at the 5% level, and therefore the hypothesis described in the far-left column cannot be rejected. Tests are for dailyMSCI market returns, expressed in local currency, for the period January 1990 to October 1996.

31

Subsequent panels show how the various multivariate specifications accountfor these stylised facts. Apart from Fixed Window, the multivariatespecifications are generally quite successful in accounting for the stylisedfacts in correlation. The specifications which incorporate volatilityclustering leave almost no evidence of structure in the correlations ofstandardised data at the 5% level. The specifications assuming constantcorrelation are most successful in eliminating structure, further supportingthe hypothesis of constant correlation, and the use of parsimoniousmultivariate specifications.

The tests show that the apparent characteristics of correlation are eliminatedwhen the returns are standardised to adjust for volatility clustering. Mostlikely, the apparent stylised facts for correlation are artefacts of the patternsin risk, such as volatility clustering, which are now well understood in theGARCH literature. Once patterns in risk have been adequately accountedfor, patterns in correlation described elsewhere in the literature cannot bediscerned.

Tables 6 and 7 present a parallel analysis for currency data for the period1990-1996. As for equity data, raw currency data generally provide evidenceof clustering in volatility, asymmetry in volatility, cross-market effects involatility, clustering in correlation, a link between volatility and correlation,and asymmetry in correlation. The evidence of asymmetry in volatility isnot easily explained in currency markets since the leverage hypothesis,used to explain the same phenomenon in equity markets, does not apply.

32

Table 6Stylised Facts for Risk in Currency Markets

Daily Observations, January 1990 to December 96(p-values)

FFR DEM GBP JPYRaw Data


Cross Market Effects .00 .00 .00 .00Fixed Window











FactorVolatility Clustering .68 .00 .00 .94




A series of tests are performed on raw and standardised returns as described in Section 3, with p-values as shown here. A p-value less than 0.05 (highlighted in bold typeface) indicates that the results are significant at the 5% level, and therefore thehypothesis described in the far-left column cannot be rejected. Tests are conducted for daily currency returns, expressed in USdollars, for the period January 1990 to December 1996.

33

Table 7Stylised Facts for Correlation in Currency MarketsDaily Observations, January 1990 to December 1996

p-valuesPair FFR/JPY DEM/JPY GBP/JPY FFR/GBP DEM/GBP FFR/DEMRaw DataClustering .00 .00 .00 .00 .00 .00Volatility Bias .00 .00 .82 .00 .00 .00Sign Bias .57 .71 .94 .67 .48 .80Cross Sign Bias .11 .37 .12 .66 .22 .96Cross Size Bias .08 .00 .76 .33 .00 .03

Fixed WindowClustering .00 .00 .00 .00 .00 .00Volatility Bias .00 .01 .00 .01 .47 .00Sign Bias .50 .05 .79 .85 .07 .82Cross Sign Bias .24 .20 .59 .73 .19 .60Cross Size Bias .01 .00 .01 .01 .00 .00

ExponentialSmoothingClustering .02 .52 .91 .61 .99 .90Volatility Bias .10 .54 .33 .11 .71 .47Sign Bias .05 .72 .57 .41 .06 .76Cross Sign Bias .21 .94 .89 .68 .24 .57Cross Size Bias .00 .64 .70 .37 .00 .12

ExponentialSmoothing - CCClustering .44 .00 .79 .03 .00 .00Volatility Bias .06 .17 .08 .09 .57 .00Sign Bias .23 .46 .05 .42 .15 .86Cross Sign Bias .54 .73 .16 .78 .43 .39Cross Size Bias .49 .13 .01 .63 .00 .00

GARCHCCClustering .00 .00 .83 .00 .06 .12Volatility Bias .17 .10 .39 .05 .20 .00Sign Bias .62 .54 .48 .59 .05 .99Cross Sign Bias .50 .40 .83 .89 .17 .95Cross Size Bias .24 .01 .65 .05 .00 .00

BEKKClustering .05 .04 .83 .00 .89 .04Volatility Bias .11 .41 .12 .05 .09 .21Sign Bias .72 .81 .39 .35 .09 .85Cross Sign Bias .44 .94 .38 .31 .29 .97Cross Size Bias .05 .25 .01 .19 .00 .25

FactorCCClustering .11 .01 .54 .02 .02 .00Volatility Bias .12 .01 .16 .03 .14 .00Sign Bias .83 .94 .33 .31 .09 .83Cross Sign Bias .29 .53 .69 .56 .26 .64Cross Size Bias .32 .00 .35 .02 .00 .00

Asymmetry/CCClustering .01 .00 .66 .01 .10 .01Volatility Bias .08 .05 .53 .05 .07 .00Sign Bias .49 .39 .34 .18 .08 .70Cross Sign Bias .51 .27 .66 .37 .24 .59Cross Size Bias .26 .00 .69 .03 .00 .00A series of tests are performed on raw and standardised returns (see Section 3). A p-value of 0.05 indicates that the results aresignificant at the 5% level, and therefore the hypothesis described in the far-left column cannot be rejected. Tests are conductedfor daily currency market returns, expressed in US dollars, for the period January 1990 to December 1996.

34

When data are standardised using multivariate specifications, many of thesestylised facts are again eradicated from the data. For this data set,Exponential Smoothing is arguably the most successful in eliminatingstructure in returns, especially when Table 7 is considered. Recall that thisspecification accommodates changing correlation; the implication is thatthe assumption of constant correlation is violated for currency markets.

In the currency case, cross-market effects in volatility are captured byFactorCC with some degree of success. The BEKK specification, however,again fails to capture the transmission of volatility between markets.

4.3 Tests of Stability in Return Correlation

Having examined the micro-structure of returns from 1990-96, the nextgroup of tests consider the stability of return correlation over longer periods.Jennrich’s (1970) approach is applied to daily data (both raw andstandardised) from 1980-96 for currencies, and 1982-96 for equities. Theperiod is broken into annual sub-periods for testing purposes as shown inTable 8.

In the case of equity return data, the raw data provide mixed evidenceregarding the stability of return correlations, consistent with the results ofprevious studies described in Section 2. It has already been argued thatprevious studies of correlation stability have failed to take account ofvolatility clustering, so this fault is rectified by also testing standardiseddata. When the equity data are standardised (using the conditional variance/covariance matrix from the Exponential Smoothing specification), returncorrelations appear far more stable. The only sub-periods in which returncorrelations change significantly are those affected by the 1987 stock marketcrash. This test further supports the case that the evidence of instability inequity correlations, noted in the literature, may be a by-product of volatilityclustering.

35

Table 8Is Return Correlation Matrix Constant Over Time?

p-valuesYears Equity Data

(Raw)Equity Data(Standard-

ised)

CurrencyData(Raw)

Currency Data(Standardised)

1980 vs 1981 .00 .111981 vs 1982 .00 .001982 vs 1983 .70 .92 .00 .001983 vs 1984 .65 .99 .00 .001984 vs 1985 .04 .99 .00 .001985 vs 1986 .00 .96 .00 .001986 vs 1987 .00 .01 .00 .471987 vs 1988 .00 .01 .00 .001988 vs 1989 .20 .34 .00 .001989 vs 1990 .01 .36 .00 .001990 vs 1991 .26 .70 .00 .001991 vs 1992 .01 .54 .00 .001992 vs 1993 .07 .99 .00 .001993 vs 1994 .32 .99 .00 .531994 vs 1995 .56 .98 .00 .001995 vs 1996 .81 .94 .00 .00

An asymptotic test for the equality of two correlation matrices (see Jennrich, 1970). Marginal significance values (p-values) are displayed. A p-value greater than 0.05 indicates that the hypotheses that the correlation matrices are equalcannot be rejected at the 5% level of significance. Unconditional correlation matrices are calculated using one yearsample periods, with daily return data. Daily data are used for both equity (expressed in local currency terms) andcurrency data (expressed in US dollar terms). Raw data are standardised using the variance/covariance matrix fromthe Exponential Smoothing model.

In contrast to equity data, the currency data consistently reject the hypothesisof stability, even after standardisation. This discrepancy between the twodata sets raises the issue of the currency of denomination for equity markets.It is possible that the equity results will differ if equity returns are reportedin common (say, US dollar) terms. Later analysis will show, however, thatthis is not the case (see Appendix 2).

36

A further test using the GARCH methodology also assesses the stability ofcorrelation. The estimation of GARCHCC models with maximumlikelihood procedures provides parameter estimates of correlation, alongwith standard errors. The GARCHCC model is estimated in successiveannual periods, so that each correlation estimate can be examined in turn,as shown in Tables 9 and 10. If the 95% confidence intervals of twosuccessive correlation estimates overlap, the hypothesis of constantcorrelation cannot be rejected at the 5% level of significance. Correlationestimates based on raw data are shown in italics for comparison.

Table 9 lends further support for the stability of return correlations in equitymarkets. Previous studies (see Section 2) find evidence that correlationsincreased at the time of the 1987 stock market crash and remained high forsome time afterwards. Table 9 compares correlation estimates for raw return(in italics), as well as those which are adjusted for volatility clustering16.In four of the six cases the raw data provide evidence of a significant increasein correlation at the time of the crash. After standardisation, the evidencefor an increase in correlation associated with the crash is weaker. Allcorrelation estimates are higher in 1987 than in 1986, but only two aresignificantly higher at the 95% level of confidence. Thus, this test suggeststhat after accounting for volatility clustering, the increase in returncorrelation around the crash is neither long-lived nor universal.

Not only is return correlation in equity markets relatively stable from oneyear to the next, there is little evidence of medium term trends in correlation.Comparing (standardised) correlation estimates in 1982 to those in 1996,none exhibit significant change. This result conflicts with the finding of(Longin and Solnik 1995) that return correlations are increasing over timeas markets become more integrated.

16 A GARCH specification with constant correlation is used to estimate the parameters in eachyear. In 1987, a dummy variable is used to account for the unusually high level of US volatilityon October 19. For further information on the use of the dummy variable, see Section 5.

37

Table 9Is Return Correlation Constant Over Time?

Equity Data 1982-1996 (Local Currency)Year US/

UKUS/Japan

US/Germany

UK/Japan

UK/Germany

Japan/Germany

1982 .257(.062).255

.128(.069).131

.067(.062).055

.226(.065).165

.267(.060).258

.335(.063).312

1983 .123(.066).152

.161(.059).152

.092(.071).085

.225(.062).234

.255(.061).295

.239(.064).263

1984 .213(.066).227

.099(.071).108

.199(.070).206

.216(.068).207

.337(.052).353

.273(.062).292

1985 .038(.061).041

-.002(.083).021

.028(.062).059

.020(.076).028

.183(.066).214

.161(.074).184

1986 .363*(.062).355*

.141(.080).154

.070(.061).067

.090(.063).090

.116(.066).115

.145(.064).112

1987 .475(.050).515

.163(.071).152

.353*(.061).391*

.254(.075).481*

.368*(.055).521*

.293(.060).411*

1988 .400(.047).432

.109(.068).131

.024*(.054).030*

.186(.064).192*

.243(.057).267*

.361(.060).388

1989 .327(.069).311

.000(.076).000

.024(.097)-.040

.165(.069).172

.286(.074).280

.201(.087).251

1990 .380(.057).385

.200(.068).204

.243(.050).278

.320(.068).318

.360(.053).387

.306(.065).322

1991 .354(.056).366

.240(.066).251

.284(.054).300

.420(.068).416

.522(.065).528

.469(.057).480

1992 .277(.070).289

.204(.069).185

.106(.069).099

.346(.057).328

.378(.065).376

.180*(.064).192*

1993 .218(.067).226

.104(.057).084

.066(.065).066

.097*(.059).086*

.262(.063).277

.105(.074).096

1994 .383(.065).357

.065(.078).053

.131(.074).108

.139(.065).133

.360(.054).357

.214(.061).234

1995 .327(.062).331

.032(.071).030

.056(.065).058

.109(.058).119

.338(.055).349

.226(.066).247

1996 .323(.067).327

.036(.079).035

.149(.075).154

.155(.073).152

.447(.060).440

.319(.069).312

In each calendar year a GARCHCC model is estimated; estimates of correlation from the GARCHCC model are shown here,with standard errors in parenthesis. Sample correlation estimates of raw returns are shown in italics. An asterisk highlights thoseestimates of return correlation more than 2 standard errors from the previous year’s estimate. In such cases, the change in returncorrelation is deemed to be statistically significant.

38

In the case of currency markets a different picture once again emerges.Statistically significant changes in correlation occur with much greaterfrequency. Return correlations in the currency markets tend to be muchgreater in magnitude, with lower standard errors. This is particularly trueof the currencies which comprise the European Exchange Rate Mechanism(ERM).17

While statistically significant changes in correlation occur regularly, thesechanges may not be economically significant in many cases. The economicimportance of these small changes in correlation is examined in Section 5.

The relative instability of currency correlations could be explained byepisodes of government intervention and also by structural changes incurrency arrangements. In 1985, then US Treasury Secretary James Bakerintroduced the Plaza Accord to systematically intervene in the exchangemarkets, followed by the Louvre Accord in February 1987. In this latteragreement, the finance ministers of the leading industrialised nations agreedto cooperate closely in fostering the stability of exchange rates aroundtheir then-current levels, seen by some as an attempt to return to a managedexchange rate system. Early in 1988, the major central banks coordinateda new round of official intervention after the USD reached new lows (seeDavies, 1988; Rowen, 1986; Almekinders and Eijffinger, 1992). Theseinitiatives could explain the high levels of currency correlation relative tothe US dollar in 1985 - 1988.

Unique structural changes have characterised European exchange rates,possibly explaining trends in correlation for European currencies. Forexample, the Great Britain Pound (GBP) entered the ERM and wassubsequently withdrawn in 1992 (Cookson, 1992). Currencies within theERM are subject to realignments from time to time which may also causediscrete shocks to correlation estimates.

17 FRF and DEM operate within the ERM, however, GBP was removed from the ERM in 1992.

39


Currency Data 1980-1996Year FFR/JPY DEM/JPY GBP/JPY FFR/GBP DEM/GBP FFR/DEM

1980 .387(.061).303

.449(.057).367

.165(.071).121

.417(.058).441

.357(.066).403

.862(.016).834

1981 .513(.058).506

.527(.056).516

.288(.060).275

.584(.028).571

.591*(.029).571

.955*(.004).954*

1982 .570(.040).568

.682(.026).684

.615*(.041).622*

.687(.038).693

.773*(.021).693*

.888*(.011).875*

1983 .657(.065).651

.808*(.030).809*

.589(.045).579

.553(.073).537

.697(.039).537*

.785*(.039).769*

1984 .608(.045).598

.678*(.034).672*

.374(.062).409

.557(.046).600

.561(.039).600

.911*(.005).886*

1985 .774*(.026).767*

.781(.026).778

.716*(.033).716*

.866*(.017).870*

.857*(.018).870*

.992*(.001).993*

1986 .719(.034).706

.737(.029).719

.547*(.048).527*

.764*(.025).767*

.772*(.023).767*

.978*(.003).971*

1987 .720(.045).744

.741(.041).764

.480(.053).501

.614*(.037).608*

.620*(.036).608*

.986(.003).988*

1988 .883*(.015).889*

.889*(.014).894*

.789*(.027).794*

.871*(.018).882*

.889*(.015).882*

.978(.003).979

1989 .818(.025).814

.816(.025).812*

.779(.028).787

.842(.021).846

.838(.020).846

.987(.002).987*

1990 .894*(.014).632*

.902*(.012).633*

.803(.023).601*

.882(.018).745*

.896(.013).745*

.979(.002).991

1991 .818(.026).570

.816*(.025).555*

.781(.028).539

.842(.021).888*

.838(.020).888*

.987(.002).990

1992 .621*.041).662

.629*(.039).655

.582*(.049).683

.728*(.030).954

.720*(.032).954

.991(.001).995

1993 .576(.062).406*

.562(.064).422*

.566(.064).325*

.887*(.015).697*

.876*(.016).697*

.989(.001).958*

1994 .663(.046).339

.658(.047).400

.680(.043).237

.949*(.010).711

.949*(.011).711

.995*(.001).932*

1995 .444*(.059).730*

.460(.066).763*

.373*(.064).541*

.741*(.034).703

.743*(.032).703

.955*(.007).932

1996 .403(.061).443*

.421(.061).468*

.245(.063).115*

.720(.047).487*

.730(.044).487*

.945(.012).904

In each calendar year a GARCHCC model is estimated; estimates of correlation from the GARCHCC model are shown here,with standard errors in parenthesis. Sample correlation estimates of raw returns are shown in italics. An asterisk highlights thoseestimates of return correlation more which are significantly different (at the 95% level) from the previous year’s estimate. Dailycurrency data are used, with all currencies expressed in US dollar terms.

40

4.4 Implications of Data Analysis

The analysis of equity data in Section 4 supports the hypothesis of constantcorrelation both at a micro level and for longer periods. Firstly, tests ofexplanatory power (Table 3) show that the specifications assuming constantcorrelation outperform those that do not. Secondly, the stylised facts forcorrelation are generally eliminated by the specifications which incorporatevolatility clustering, even when correlation is assumed constant. Thesetests imply that the apparent structure in correlation is merely a by-productof structure in risk. Once patterns in risk have been captured, correlationof the standardised returns may be assumed constant.

The final group of tests specifically assess the stability of return correlationover time, as shown in Tables 8 and 9. With few exceptions, the hypothesisof constant correlation from one year to the next is supported. Thecontention that return correlations are increasing over time is not supportedby this study. Correlations in 1996 are not significantly different fromthose in 1982. There is, however, some evidence that return correlationsincreased at the time of the 1987 crash, although this is neither universalnor long-lived.

The poor performance of the Fixed Window estimator in both equity andcurrency markets is noteworthy. Fixed Window consistently fails to capturethe stylised facts discussed in the literature for both risk and returncorrelation. The many studies of return correlation based on the FixedWindow method must therefore be called into question, along with theirconclusions regarding the deficiency of international diversificationstrategies. Previous findings of structure in return correlations for equitymarkets can be explained by weakness in the methodology. Those studiesbase their claims on the evidence for changes in return correlation, evidencethat is flawed due to the failure to account for volatility clustering.

41

For those wishing to make asset allocation decisions between internationalequity markets, the primary focus should be to model volatility clusteringeffectively. Having found a suitable specification for variance, a simpleconstant correlation assumption for the standardised returns should suffice.This approach is tested in Section 5, by comparing the performance of thevarious multivariate specifications in an asset allocation context.

Examination of currency data leads to a very different conclusion concerningthe stability of return correlations. The tests of explanatory power presentedin Table 3 show that the constant correlation assumption is inferior to theBEKK specification. The need to specifically capture changes in correlationfor currency data is confirmed in Table 7. Here, Exponential Smoothing isfound to be the most successful specification in terms of its ability toeliminate structure from return correlation. This finding is furtherconfirmed in Tables 8 and 10 which highlight the instability in currencyreturn correlations from one period to the next.

While changes in currency return correlations are statistically significant,their economic significance is, however, questionable. The choice ofspecification will depend on the economic significance of changes in returncorrelations. If they are significant, multivariate specifications whichspecifically account for changes in correlation over time, as well as volatilityclustering, are likely to be most successful. If not, the simpler constantcorrelation approach may succeed. Section 5 tests these alternatives.

42

5. Asset Allocation Decisions

The test results in Section 4 suggest certain hypotheses concerningappropriate multivariate modelling strategies. These hypotheses are nowtested in the context of an asset allocation decision. Are differences in themultivariate specifications economically important when applied toinvestment decisions? In particular, do those specifications whichspecifically model changes in correlation outperform those that do not?

The multivariate specifications are used here to estimate variance/covariance matrices for portfolio allocation decisions. Variance/covariancematrices are estimated using each of seven specifications highlighted inSection 3, and one year of past data (260 observations).18 This informationis then used to form the minimum variance portfolio, assuming short-sellingconstraints. The minimum variance portfolio is chosen in this study so asto eliminate the influence of expected returns and to focus on the importanceof variance/covariance estimation in determining asset allocation outcomes.The asset weights are stored and the ex-post standard deviation of returnssubsequently calculated. The “best” multivariate specification is that whichproduces the lowest ex-post portfolio risk.

Analysis is performed firstly for equity markets and then for currencymarkets. The longest possible time horizon is used for analysis (subject toavailability of data) to ensure that results are not period-specific.

18 The number of observations used to estimate GARCHCC and other GARCH based models isrelatively small. The study finds, however, that 260 observations is sufficient for maximumlikelihood procedures to estimate the parameters in the vast majority of cases. When largersamples are used, the efficiency of asset allocation decisions is no better and the computer timetaken for estimation increases dramatically. See Appendix 1 for further explanation of this point.

43

5.1 Analysis of Equity Data

In the case of equity markets, the perspective is that of a US-based investorallocating funds between the US, Japanese, British and German marketson a hedged basis. Analysis uses daily MSCI data from January 1982 toOctober 1996, excluding dividends and expressed in local currency terms.19

The first asset allocation decision is made in January 1983, and thenceweekly until October 1996.

The period of investigation includes the 1987 stock market crash whichcan pose a challenge for modelling risk. The MSCI index of the US marketfell by 10.4% on 19 October 1987. This sharp decline was followed befurther large movements over the following days, including -8.6% on 20October, +8.1% on 21 October, and -6.2% on 26 October. Within weeks,however, the daily price movements reverted to more usual levels. Isolatedperiods of exceptional volatility may unduly influence parameter estimatesso dummy variables are used to avoid this problem. For GARCHCC, theUS variance specification is amended as follows:

( )σ γ γ σ γ ε γ γ γ γi t i i i t i i t i t i i i tDV DV, , ,+ −= + + + − +12

0 12

22

3 3 1 2 1 (12)

where DV is set equal to unity on 19 October 1987 and zero otherwise.

A comparable adjustment is made to the Asymmetry/CC, BEKK andFactorCC specifications for US variance to account for the 1987 crash.

The results of analysis in equity markets are shown in Table 11. The firstsub-period of analysis (Jan 1990 - Oct 1996) is identical to that used fordata analysis in Section 4.

19 Analysis from a US dollar perspective is presented in Appendix 2.

44

The results for this sub-period are generally consistent with those presentedin Section 4; that is, specifications which incorporate volatility clusteringand constant correlation perform best. The best specification, Asymmetry/CC, outperforms the worst, Fixed Window, by 48 basis points, showingthat the method of risk estimation can make a significant difference toportfolio efficiency.

The specifications designed to capture cross-market effects (BEKK andFactorCC) perform relatively poorly in 1990-96. This could be explainedin one of two ways: either co-movements in volatility are not significant inthis period, or the specifications are not able to capture them adequately.Based on the evidence in Table 4, the latter interpretation seems more likely.

The analysis of the earlier sub-period, 1983-89, is complicated by the 1987stock market crash. If the crash period is included in the sample, BEKKoutperforms all other specifications. This outcome suggest that cross-marketeffects and correlation changes are particularly important in understandingthe period surrounding the 1987 crash; a point which has been emphasisedin previous studies. In contrast to the later period of analysis, BEKK is infact able to capture these effects better than other specifications.

Performance of the various specifications in 1987 is noteworthy asdifferences between the specifications are exacerbated in that year. Ex-post risk ranges from 22.40% to 25.39% and the overall level of risk isgreatest, reflecting the impact of the October crash. Asymmetry/CC alsoperforms strongly in 1987, suggesting that asymmetries in volatility arealso significant at the time of the crash.

45

Table 11MSCI Equity Data (Local Currency)

Asset Allocations 1983-96Ex-Post Standard Deviation % pa

Period

(a)

FixedWindow

(b)

Exp’-tial

Smooth-ing(c)

ESCC**

(d)

GARCH/CC

(e)

BEKK

(f)

Factor/CC

(g)

Asymmetry/CC

(h)1990-

Oct19969.39 9.04 9.00 9.06 9.11 9.18 8.91

1983-1989 11.90 12.59 12.25 11.82 11.59 11.76 11.701983-1989*

9.11 9.22 9.19 9.06 9.08 9.12 9.07

All 10.74 10.97 10.76 10.55 10.44 10.56 10.41

All -exceptcrash*

9.27 9.15 9.11 9.08 9.11 9.16 9.01

1983 7.50 7.24 7.12 7.16 7.41 7.14 7.271984 8.46 8.89 8.71 8.66 8.67 9.23 8.631985 6.64 6.60 6.62 6.51 6.37 6.23 6.281986 10.02 9.98 9.62 9.46 9.28 9.75 9.591987 23.30 25.39 24.27 23.25 22.40 23.03 22.741988 10.11 10.45 10.59 10.67 10.86 10.18 10.551989 8.18 8.40 8.78 7.59 7.47 7.61 7.971990 13.62 12.78 12.87 12.90 12.75 12.73 12.781991 11.14 11.69 11.65 11.05 11.13 10.92 10.981992 9.04 7.75 7.66 8.56 8.76 9.02 8.011993 6.92 6.69 6.74 6.80 6.67 7.07 6.461994 8.10 7.67 7.56 7.70 7.80 7.73 7.731995 6.89 6.73 6.61 6.72 7.06 7.29 6.69

Jan-Oct 96 7.46 7.48 7.26 7.32 7.42 7.46 7.26* E l d b i d i d di h O b 1987 k k h* Excludes a twenty business day period surrounding the October 1987 stock market crash.

** Exponential Smoothing - CC (ESCC) uses Exponential Smoothing for variance estimates and applies the assumptionof constant correlation to imply the covariance estimates.The variance/covariance matrix is estimated each week from January 1983-October 1996 using seven multivariatespecifications. This matrix is used to select the minimum variance portfolio (with short-selling constraints). The ex-post standard deviation of portfolio returns is then calculated using daily MSCI data (expressed in local dollars). Themost effective specification is deemed to be that which results in lowest ex-post risk.

46

When the twenty day period containing the crash is excluded from theanalysis, a different picture emerges. BEKK is less attractive in this context,suggesting that the significance of co-movements in volatility andcorrelation changes is limited to the immediate crash period (or alternativelythat BEKK has difficulty capturing these effects in other periods). Thesimple GARCHCC specification emerges as the most effective risk measurein this sub-period, closely followed by Asymmetry/CC. Asymmetry/CC isthe most effective risk measure for the entire period of analysis (excludingthe crash), supporting the asymmetry hypothesis for equity markets.20 Ifthe 1987 stock market crash is considered an aberration, the hypothesis ofconstant correlation has good support.

The success of the constant correlation hypothesis can be further assessedby comparing columns (c) and (d); the only difference between these tworisk measures is the treatment of correlation. The constant correlationmethod in column (d) is superior across the board, and in most sub-periods,providing further support for the constant correlation approach.

5.2 Analysis of Currency Data

Results of analysis using currency data are shown in Table 12. Themethodology is exactly the same as in the equity case, with weekly assetallocation decisions from January 1980 until December 1996. This timethe asset allocation decision is made between the competing currencymarkets of JPY, DEM, GBP and FFR, relative to the USD. Note that interestincome is excluded from the analysis (just as dividend income is excludedfor the earlier analysis of equity markets).

20 The importance of asymmetry has less support when returns are measured in terms of a commoncurrency - see Appendix 2. For returns in US dollars, this study finds that the preferred specificationis Exponential Smoothing - CC (see Appendix 2).

47

As anticipated in Section 4, Exponential Smoothing performs very wellwhen applied to currency data. The consistently strong performance ofExponential Smoothing may be linked to its capacity to capture the changesin correlation evident in currency data. A comparison of Columns (c) and(d) highlights the fact that imposing a constraint of constant correlation isdetrimental to portfolio efficiency. The difference is so slight, however,that one could argue that the constraint is economically insignificant. Basedon this evidence, the success of the Exponential Smoothing specificationis probably based on superior variance estimates rather than exceptionalcorrelation estimates.

The similarity between Columns (c) and (d) could be explained by the factthat correlations are generally high in the currency markets from a USperspective. This means that diversification effects are less significant forthis data set and therefore the estimation process for correlation of reducedimportance.

* Excludes a twenty business day period surrounding the October 1987 stock market crash.** Exponential Smoothing - CC (ESCC) uses Exponential Smoothing for variance estimates andapplies the assumption of constant correlation to imply the covariance estimates.The variance/covariance matrix is estimated each week from January 1983-October 1996 using sevenmultivariate specifications. This matrix is used to select the minimum variance portfolio (with short-selling constraints). The ex-post standard deviation of portfolio returns is then calculated using dailyMSCI data (expressed in local dollars). The most effective specification is deemed to be that whichresults in lowest ex-post risk.

48

Table 12Currency Data


Period

(a)

FixedWindow

(b)

Exp’-tial

Smooth-ing(c)

ESCC*

(d)

GARCH/CC

(e)

BEKK

(f)

Factor/CC

(g)

Asymmetry/CC

(h)1990-1996

8.20 7.95 7.99 8.08 8.13 8.10 8.91

1980-1989

9.27 8.96 9.00 9.05 9.20 9.30 9.53

All 8.85 8.56 8.60 8.67 8.78 8.82 9.28

1980 7.32 7.00 7.09 6.98 6.98 6.96 6.931981 11.42 11.08 10.99 11.35 11.32 11.30 11.671982 9.72 8.73 9.01 9.08 9.93 9.13 8.801983 7.22 6.87 6.89 6.81 7.03 7.08 7.431984 7.06 7.06 7.18 7.06 7.10 7.45 8.881985 9.07 9.07 9.11 9.04 9.05 9.43 11.891986 10.09 9.49 9.64 9.80 10.09 10.22 9.751987 9.41 9.02 8.99 9.17 9.08 9.25 9.121988 9.42 9.12 9.22 9.27 9.26 9.70 9.231989 10.49 10.63 10.52 10.30 10.53 10.90 10.151990 7.88 7.80 7.99 8.60 7.88 8.17 8.321991 10.11 9.50 9.44 9.76 10.04 9.81 11.441992 9.02 9.27 9.18 9.04 9.05 9.03 11.621993 9.15 8.44 8.72 8.69 8.98 8.72 9.311994 6.65 6.19 6.17 6.20 6.53 6.27 6.421995 8.52 8.38 8.36 8.20 8.41 8.65 8.221996 4.95 5.04 5.06 4.99 4.89 4.94 5.03

* Exponential Smoothing - CC (ESCC) uses Exponential Smoothing to estimate variance and then applies the assumption ofconstant correlation to imply covariance estimates.The variance/covariance matrix is estimated each week from January 1981- December 1996 using seven multivariatespecifications. This matrix is used to select the minimum variance portfolio (with short-selling constraints). The ex-poststandard deviation of portfolio returns is then calculated using daily exchange rate data (excluding interest income). The mosteffective specification is deemed to be that which results in lowest ex-post risk.

Table 13 supports this conclusion by comparing the asset weights in equityand currency markets. In equity markets, the standard deviations of assetweights are generally lower, suggesting a more stable and diverse portfolioallocation designed to capture diversification benefits. In currency markets,diversification benefits are limited by the high degree of correlation so theminimum variance portfolio is often weighted entirely (or mostly) to thesingle currency with lowest variance.

49

Note for example that the average weight in DEM is only 4%. This may beexplained by the fact that DEM is closely correlated with the FFR (seeTable 10). If the variance of DEM is higher than FFR, it will rarely beoptimal to include holdings of the DEM in the portfolio. Thus the portfolioweights are driven primarily by variance.

Table 13 suggests that the currency weights vary significantly over time,possibly to reflect changes in variance. Since correlations are generallyhigh between the currencies in this study, small changes tend not to beeconomically significant, despite their statistical significance highlightedin Table 10. In other words, whether return correlation is assumed constantor not, the asset weights will be much the same, as they are primarily drivenby changes in variance. Figure 3 supports this interpretation of the data.

Table 13Asset Weights*

Asset Average Weight Standard Deviationof weight

Equity MarketsUK 0.14 0.11US 0.53 0.15Japan 0.18 0.13Germany 0.15 0.11Currency MarketsFFR 0.21 0.28DEM 0.04 0.13GBP 0.32 0.29JPY 0.43 0.32Analyses asset weights for the period January 1983 - October 1996 for equities, and January 1981 - December 1996 forcurrencies. Market weights are determined weekly by re-estimating a 4x4 GARCHCC model using the past 260 dailyobservations for four major markets. The variance/covariance matrix thus obtained is used to determine the weights of theminimum variance portfolio. The average and standard deviation of the weight for each market over time is calculated.

50

Figure 3Currency Weights - Do Correlations Matter?

Portfolio Weighting - GBP

0%

20%

40%

60%

80%

100%

1/01

/80

9/12

/80

05/2

8/81

01/2

1/82

09/1

6/82

05/2

7/83

01/2

0/84

10/1

2/84

6/07

/85

3/03

/86

11/2

4/86

07/2

0/87

4/12

/88

1/04

/89

08/3

0/89

04/2

5/90

12/1

9/90

08/1

4/91

4/08

/92

12/0

2/92

07/2

8/93

03/2

3/94

11/1

6/94

7/12

/95

Date

Per

cent

Exp. Smoothing Exp. Smoothing/Constant Correlation

Comparison of portfolio weights for Great Britain Pound in minimum variance portfolio. Figure 3 contrasts the differencein weights depending whether correlation is assumed constant. Asset weights for the period January 1981 - December1996 are determined weekly for a portfolio of four major currencies expressed in US dollar terms. Variances aredetermined using Exponential Smoothing, covariances are determined either using Exponential Smoothing, or constantcorrelation. The variance/covariance matrix thus obtained is used to determine the weights of the minimum varianceportfolio.

Judging from the relatively poor performance of Asymmetry/CC,specifically modelling asymmetry effects cannot be justified in currencymarkets. Table 6 provides evidence of asymmetry in volatility, despite thelack of theoretical justification (see Section 2.4). The Asymmetry/CCspecification is, however, less successful that Exponential Smoothing incapturing this feature. Once again, the Fixed Window method performsvery poorly, adding to the argument discussed earlier that it fails to capturethe important stylised facts.

Note that the more general BEKK specification, which also accommodateschanges in correlation, is not as effective as Exponential Smoothing. Thisconfirms the hypothesis outlined above that variance forecasts are moreimportant to portfolio decisions than correlation estimates for this data set.The results suggest that the constraints of Exponential Smoothing are not

51

binding when applied to currency markets. Despite its lack of generality,Exponential Smoothing results in efficient investment decisions for currencymarkets.21

The relatively poor performance of BEKK and FactorCC again raisesquestions about cross-market effects. Either volatility correlation is notsignificant for this data set, or the specifications are inadequate to captureit. The results presented in Table 6 suggest that the FactorCC approach isquite successful in capturing cross-market effects for this data set, but thatBEKK is not.

Differences between the analysis for equity markets and currency marketshighlight the issue of the currency of denomination for equity market returns.The findings for equity markets may vary, depending on whether returnsare expressed in local currency terms or in terms of some common currencysuch as the US dollar. This is an important question for fund managerswho may leave currency returns unhedged, thus incorporating currencyeffects into portfolio returns. The analysis is therefore repeated from a USdollar perspective in Appendix 2. Despite the inclusion of currency effects,the findings are quite similar, at least in terms of the major conclusions.

21 An IGARCH/Constant Correlation model (see Chou, 1988) was also examined for this data set.The IGARCH specification is similar to Exponential Smoothing in that the sum of the parametersγi1 and γi2 (from equation 4) is constrained to equal unity. When this specification was used tomake asset allocation decisions, the results, as measured by ex-post portfolio risk, were inferiorto both Exponential Smoothing and GARCHCC.

52

6. Conclusions

The purpose of this study is to foster a better understanding of returncorrelation, and in turn, determine appropriate methods for modelling riskin a multivariate setting. Past studies have examined return correlations inequity markets, while largely ignoring currency markets. Most past studiesprovide evidence for changes and patterns in return correlation, but theycan be criticised for:

• using low frequency data (ie. monthly or weekly data instead of dailydata) thus increasing the sampling error;

• failing to adjust for structure in volatility including volatility clustering,asymmetry in volatility, and volatility correlation between markets; and

• estimating return correlation by assuming correlation constant for thesample period, with no empirical or theoretical justification.

Is correlation constant after all? With respect to equity markets, this studyfinds that correlation can be assumed constant with some qualifications.At a micro level, accounting for volatility clustering effectively eliminatesstructure in return correlation. Given an effective variance estimate, thecorrelation of standardised returns is generally stable over time. There is,however, some evidence of short-term instability in return correlations atthe time of the 1987 stock market crash. While previous studies havenoted a significant increase in return correlations associated with the crash,this study suggests that such an increase is neither universal nor long-lived.Apart from the crash period, return correlations have remained relativelystable over the period 1982-1996.

In light of these findings, it is imperative that a multivariate specificationfor equity returns capture volatility clustering and, to a lesser extent,asymmetry in volatility (especially if returns are measured in local currencyterms). Specifications specifically designed to capture co-movements involatility and changes in correlation (such as BEKK) are of dubious value.The analysis shows that more parsimonious specifications generally perform

53

as well as BEKK, if not better, in capturing the structure of co-movementsevident in the raw data. The study finds that a more parsimonious constantcorrelation specification generally outperforms more complex alternatives,except during the 1987 crash.

For currency data, evidence for variation in return correlation is stronger,both at a micro level and over the medium term. The study finds, however,that correlations between currencies in this data set are relatively high, sodifferences in estimation methods for correlation are not economicallysignificant. In the asset allocation context, the assumption of constantcorrelation makes very little difference to portfolio efficiency.

The study highlights the pitfalls of using Fixed Window as a measure ofrisk in finance applications. Previous studies using this approach havecontributed to a misunderstanding of return correlation in equity markets.Stylised facts for return correlation are artefacts of the patterns in risk thatare now well understood in the GARCH literature. The Fixed Windowmethod cannot be justified in a world where risk is known to be changing.Asset allocations based on this risk measure are shown to be less efficient.

Finally, it is worthwhile to consider the implications of the study for portfoliodiversification. Previous studies of equity markets highlight asymmetry inreturn correlation, a positive relationship between volatility and returncorrelation and a long-term uptrend in return correlation. All of these claimshave undermined the argument for international diversification. Thecontemporaneous fall in all major equity markets in 1987 further encouragedthe belief that diversification is powerless to prevent significant portfoliolosses. By reassessing these claims in the light of the GARCH literature,the evidence provided by this study has confirmed the strength of thediversification argument. Ten years after the events of 1987, they can beseen as an aberration in the usual pattern of market movements. This studyargues that return correlations in equity markets are generally stable, thussupporting international portfolio diversification as an effective riskminimisation tool.

54

Appendix 1Asset Allocation Decisions

Should we use daily or monthly data?

When estimating variance/covariance matrices, should daily or monthlydata be used? Practitioners involved in option pricing and estimating value-at-risk normally use daily data.22 However, an argument could be madethat asset allocation decisions are made for the medium term, and thereforelower frequency data are more appropriate.

Consider a fund manager who adjusts portfolio weightings on, say, amonthly basis. Estimates of variance/covariance for the relevant assetreturns (partly) influence the portfolio allocation decision. Does it matterwhether high frequency or low frequency data are used to form suchestimates of variance/covariance? The analysis shows that daily data (ratherthan monthly data) significantly improve portfolio efficiency, even whenportfolio adjustments are made only monthly.

Methodology

The portfolio allocation decision is based on MSCI data for the US, Japan,UK and Germany, expressed in US dollars. The minimum varianceportfolio23 under short-selling constraints is selected on a monthly basisfrom 1990-1995. Variance/covariance are estimated in several ways,consistent with the discussion is Section 3:

a) Fixed Window - Sample variance/covariance estimated using either thepast 1, 3 or 5 years of data.

22 Value-at-risk is a measure of possible losses resulting from adverse market movements assumingnormal market conditions (see Longerstaey and Spencer, 1996). New central bank requirementsnow permit banks to form their own estimates of value-at-risk for capital adequacy purposes (seeBIS, 1996).

23 Selection of the minimum variance portfolio does not rely on estimates of expected returns, thusisolating variance/covariance as the only inputs into the asset allocation decision.

55

b) Exponential Smoothing - Variance and covariance estimated using alldata to date, but with declining weights. The smoothing parameter, λ,is set equal to either 0.94 or 0.97.

Note that GARCH techniques cannot be tested in this way because thereare insufficient observations to estimate a GARCH model using monthlydata. Nelson (1992) shows, however, that the accuracy of GARCH varianceforecasts increases with frequency of observation. He finds that evenmisspecified GARCH models can produce ‘good’ estimates volatility whenhigh frequency data are employed.

Either monthly or daily data are used to form the variance/covarianceestimates. The portfolio weights are determined each month on the basisof these estimates. The ex-post monthly portfolio returns are then analysed.

Results

The most suitable risk measure is that which results in the most efficientportfolios, that is, with lowest ex-post risk. Table 14 shows that for everyrisk measure considered, portfolio efficiency is improved if variance/covariance is estimated using higher frequency data. This could beexplained by the fact that higher frequency data reduces the sampling errorassociated with risk estimation.

56

Table 14Analysis of Monthly Portfolio Returns 1990-95

Standard Deviation % pa

Basis of Risk Measure Monthly Data

DailyData

Fixed Window - 1 year 12.30 11.43Fixed Window - 3 years 13.09 11.40Fixed Window - 5 years 12.42 12.00Exponential Smoothingλ=0.97

12.24 11.09

Exponential Smoothingλ=0.94

12.21 11.08

The variance/covariance matrix is estimated each month from January 1990 - December 1995 using various multivariatespecifications. This matrix is used to select the minimum variance portfolio (with short-selling constraints). The ex-post standard deviation of portfolio returns is then calculated using monthly MSCI data expressed in US dollars. Themost effective specification is deemed to be that which results in lowest ex-post risk.

This result is consistent with Freund and Walpole (1992) who show therelationship between sample size and the degree of confidence associatedwith an estimate of variance. The (1-α)100% confidence interval forvariance is:

( ) ( )P

n s n s

n n

−< <

−

= −

− − −

1 11

2

2 12

22

1 2 12χ

σχ

αα α/ , / ,

(13)

where σ2 is the sample variance of a random sample of size n from a normalpopulation, σ2 is the population variance and data are independent andidentically distributed. The greater the sample size, n, the narrower will bethe confidence interval, and thus the greater the accuracy of the estimatedsample variance. For example, suppose that volatility is estimated from atwelve month window to be 20% pa. If monthly data are used (twelveobservations), the 95% confidence interval for the true or populationvolatility is 14.17% pa - 34.03% pa. If daily data are used (260observations), the 95% confidence interval narrows to 18.42% pa - 21.89%pa, suggesting a much higher degree of accuracy for the volatility estimate.

57

While a greater number of observations leads to greater accuracy, extendingthe sample horizon does not necessarily help when risk is changing. Theuse of a five year horizon (using daily data) actually decreases portfolioefficiency; ex-post risk rises from 11.43% pa to 12.00% pa. Foster andNelson (1996) show that when conditional heteroscedasticity is present,shorter sample horizons are most effective in order to capture changes inrisk over time.

When measuring risk under heteroscedasticity, there is a trade-off betweenthe need to reduce sampling error, and the need to capture changes in risk.Table 14 demonstrates this trade-off by showing that when observing higherfrequency data, short sampling horizons (say, 1 year or less) generallyprovide more favourable outcomes than longer sampling horizons. TheExponential Smoothing measure is particularly successful in this regard;Longerstaey and Spencer (1996) show that if λ=0.94, ExponentialSmoothing effectively uses only the past 74 observations to estimatevariance.24 The success of Exponential Smoothing may also be explainedby the exponential weighting of the observations, consistent with evidencefor volatility clustering.

Implications for Model Estimation In Current Study

When GARCH models are estimated it is generally considered desirableto have at least 200-300 observations. In the asset allocation study (Section5) a sample horizon of one year, or 260 observations, is used. Table 14shows that a sampling period beyond one year does not significantly add toportfolio efficiency, and may in fact have the reverseeffect if risk is changing.A one-year sample period is therefore used for estimating Fixed Windowand also for estimating GARCH models. This sample size is found to beadequate for maximum likelihood procedures to estimate the parametersin the vast majority of cases. When larger samples are used, the computertime taken for estimation increases dramatically, and asset allocationdecisions are no more efficient.

24 That is, all observations prior to the last 74 are deemed to be insignificant because their weightingin the volatility estimation process is less than 1%.

58

Appendix 2The Currency of Denomination for Equity Returns

This appendix considers the impact of currency on stock return correlationsand hence asset allocation decisions. Earlier sections separately examineequity returns in local currency and also currency returns from a USperspective. The interaction of currency and local equity returns can beobserved by expressing equity returns in US dollar terms. This appendixtherefore presents a parallel analysis to that conducted in the body of thisstudy, this time in US dollar terms.

Previous studies of correlation differ in their currency of denomination. Inmost studies, returns in each market are analysed in local currency terms.In others (see for example Erb, Harvey and Viskanta, 1994) analysis isundertaken in a common currency (US dollars). Becker, Finnerty and Gupta(1990) and Lee and Kim (1993) perform analysis both in local currencyterms and common currency terms. These latter studies show that findingsare not greatly sensitive to the choice of currency of denomination.

Table 15 shows how return correlations differ depending on the currencyof denomination. In every case, correlation in local currency is higher, butthe estimates are not significantly different at the 5% level.

Table 15Comparing Multivariate Specifications

Estimates of Return Correlation (US and Japan)Specification Correlation

(US dollars)Correlation

(Local Currencies)Fixed Window 0.124 (na) 0.147 (na)

GARCH/Constant Correlation 0.094 (0.021) 0.141 (0.022)Asymmetry/Constant Correlation 0.099 (0.021) 0.145 (0.022)

Factor/Constant Correlation 0.100 (0.021) 0.129 (0.021)Return correlation estimates using daily MSCI data for US and Japan. Standard errors are shown in parenthesis. In each case, thesample period is January 1990 - October 1996.

59

The similarity in the correlation estimates may be explained by the factthat currency correlations are high from a US dollar perspective (see Table10). Further, as equity markets are generally more volatile than currencymarkets, variation in currency returns tends to be swamped by movementsin the local markets. The implication for asset allocators is that the currencyhedging decision is not significant for the co-movement of assets (althoughit may have implications for the variance of international assets).

Table 16 revisits the explanatory power of multivariate specifications froma local currency perspective. The currency of denomination does have aslight impact on the rankings of the specifications; but in both cases,specifications assuming constant correlation perform best.

Table 16Explanatory Power of Multivariate Specifications

Daily Observations, 1990-1996Specification Equity Data

(US dollars)Equity Data

(Local currency)GarchCC -59,741 -61,025BEKK -59,660 -61,205FactorCC -59,486 -61,365Asymmetry/CC -59,589 -61,186

Table 16 displays the Schwarz Criterion (Judge, Hill, Griffiths, Lutkepohl and Lee 1988) for each specification. Eachspecification is tested for the period January 1990 - October 96 using daily data. The specification with the lowest SchwarzCriterion has the best explanatory power, after adjusting for the number of parameters.

Tables 17 and 18 repeat the analysis in Tables 4 and 5 from a local currencyperspective. These tests consider the capacity of multivariate specificationsto capture structure found in raw data. The conclusions are similar,regardless of the currency of denomination. As in the US dollar analysis,all specifications which capture volatility clustering have reasonable successin removing structure in both risk and return correlation.

The main difference in the US dollar based data is the success of ExponentialSmoothing - CC. Of all the specifications that take account of volatilityclustering, this is the least general. The success of Exponential Smoothingmay be explained by the fact that it is such an effective measure of variancefor currency returns (see Sections 4 and 5). Since the returns are expressed

60

in US dollar terms (from an unhedged perspective), currency effects areincorporated into the analysis, thus causing a preference for ExponentialSmoothing.

Specifications designed to capture cross-market effects and asymmetry donot fulfil their promise in this case. Parsimonious specifications (especiallyESCC) have greater success in capturing these characteristics of the rawdata.

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Table 17 Stylised Facts for Risk in Equity Markets

Daily Observations, Jan 1990- Oct 96(p-values)

UK US Japan GermanyRaw Data

Volatility Clustering 0.00 0.00 0.00 0.00Asymmetry 0.00 0.00 0.00 0.00

Cross Market Effects 0.00 0.00 0.00 0.00Fixed Window











FactorCCVolatility Clustering .51 .56 .24 .00




A series of tests are performed on raw and standardised returns as described in Section 3, with p-values as shown here. A p-value less than 0.05 (highlighted in bold typeface) indicates that the results are significant at the 5% level, and therefore thehypothesis described in the far-left column cannot be rejected. Tests are conducted for daily MSCI market returns, expressed inUS dollars, for the period January 1990 to October 1996.

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Table 18Stylised Facts for Correlation in Equity Markets

Daily Observations, Jan 1990 - Oct 96(p-values)

Pair US/UK

US/Japan

US/Germany

UK/Japan

UK/Germany

Japan/Germany

Raw DataClustering 0.00 0.00 0.00 0.01 0.00 0.00Volatility Bias 0.00 0.00 0.00 0.00 0.00 0.00Asymmetry 0.19 0.13 0.51 0.17 0.33 0.29Cross Sign Bias 0.40 0.03 0.70 0.43 0.64 0.61Cross Size Bias 0.02 0.00 0.00 0.00 0.00 0.00

Fixed WindowClustering .00 .00 .04 .00 .00 .17Volatility Bias .42 .03 .01 .00 .17 .68Asymmetry .11 .76 .07 .11 .23 .93Cross Sign Bias .35 .78 .21 .26 .17 .83Cross Size Bias .00 .00 .04 .00 .11 .16

ExponentialSmoothingClustering .25 .09 .33 .14 .95 .64Volatility Bias .03 .78 .79 .58 .30 .34Asymmetry .33 .28 .15 .14 .22 .84Cross Sign Bias .60 .47 .28 .29 .47 .89Cross Size Bias .35 .53 .18 .32 .10 .81

ExponentialSmoothing - CCClustering .92 .63 .36 .43 .97 .97Volatility Bias .19 .54 .78 .45 .29 .35Asymmetry .16 .88 .09 .12 .11 .89Cross Sign Bias .46 .76 .24 .32 .15 .54Cross Size Bias .50 .12 .73 .19 .06 .46

GARCHCCClustering .61 .71 .29 .28 .67 .97Volatility Bias .20 .48 .91 .92 .76 .08Asymmetry .12 .90 .05 .03 .18 .78Cross Sign Bias .38 .91 .18 .13 .33 .67Cross Size Bias .50 .00 .62 .11 .97 .19

BEKKClustering .59 .75 .13 .56 .95 .34Volatility Bias .03 .54 .93 .99 .98 .73Asymmetry .19 .43 .04 .14 .16 .98Cross Sign Bias .45 .56 .07 .41 .29 .99Cross Size Bias .64 .06 .20 .83 .73 .84

FactorClustering .49 .68 .40 .28 .56 .71Volatility Bias .36 .59 .27 .94 .07 .60Asymmetry .09 .79 .25 .03 .31 .70Cross Sign Bias .26 .85 .54 .13 .13 .46Cross Size Bias .26 .01 .89 .26 .37 .71

Asymmetry/CCClustering .67 .59 .20 .41 .41 .92Volatility Bias .11 .82 .21 .69 .79 .19Asymmetry .07 .79 .20 .06 .39 .55Cross Sign Bias .25 .81 .48 .24 .21 .08Cross Size Bias .27 .00 .79 .14 .65 .25A series of tests on raw and standardised returns (see Section 3). A p-value of 0.05 or less indicates that the hypothesis describedin the far-left column cannot be rejected. Tests are conducted for daily MSCI market returns, expressed in US dollars, for theperiod January 1990 to October 1996.

63

Table 19 reconsiders the stability of the return correlation matrix over timein US dollars. Using standardised data, the correlation matrix still appearsgenerally stable. Whether local currency returns or US dollar based returnsare examined, the only periods of instability are those affected by the 1987stock market crash. This conclusion is confirmed in Table 20; whencorrelation is estimated with a GARCHCC model, parameter estimatesgenerally show stability over time.

Table 19Is Return Correlation Matrix Constant Over Time?

Years Equity Data(USD)raw

Equity Data(USD)

standardised

Equity Data(Local

Currency)Raw

Equity Data(Local Currency)

Standardised

1982 vs 1983 .22 .93 .70 .921983 vs 1984 .28 .80 .65 .991984 vs 1985 .79 .68 .04 .991985 vs 1986 .00 .88 .00 .961986 vs 1987 .00 .02 .00 .011987 vs 1988 .00 .01 .00 .011988 vs 1989 .12 .99 .20 .341989 vs 1990 .01 .93 .01 .361990 vs 1991 .00 .64 .26 .701991 vs 1992 .00 .44 .01 .541992 vs 1993 .04 .99 .07 .991993 vs 1994 .01 .99 .32 .991994 vs 1995 .21 .99 .56 .981995 vs 1996 .10 .97 .81 .94

An asymptotic test for the equality of two correlation matrices (see Jennrich, 1970). Marginal significance values (p-values) aredisplayed. A p-value greater than 0.05 indicates that the hypotheses that the correlation matrices are equal cannot be rejected atthe 5% level of significance. Unconditional correlation matrices are calculated using one year sample periods, with daily returndata. Raw data are standardised using the variance/covariance matrix from the Exponential Smoothing model.

64


Equity Data 1982-1996Year US/

UKUS/Japan

US/Germany

UK/Japan

UK/Germany

Japan/Germany

1982 .259(.064).266

.164(.066).163

.119(.066).138

.417(.052).399

.534(.045).541

.615(.048).609

1983 .162(.064).177

.171(.062).194

.111(.063).131

.430(.060).452

.424(.063).448

.534(.059).550

1984 .145(.068).160

.078(.082).102

.108(.073).125

.393(.059).384

.537(.045).546

.471(.052).472

1985 .042(.061).038

.058(.071).066

.061(.064).075

.373(.058).383

.513(.047).540

.411(.058).434

1986 .345*(.060).330*

.150(.076).157

.122(.058).120

.201(.062).195

.200*(.058).204*

.179(.062).159*

1987 .370(.057).414

.034(.081).072

.285(.061).338

.271(.077).506*

.336(.058).480*

.302(.068).407

1988 .286(.061).312

-.010(.054)-.009

-.033*(.055)-.034*

.233(.057).234

.248(.050).265

.090(.059).418

1989 .261(.068).268

.015(.074).032

-.008(.090)-.019

.433(.055).444

.398(.072).371

.362(.066).371

1990 .291(.071).274

.170(.066).184

.234(.057).254

.331(.065).325

.401(.053).386

.306(.060).314

1991 .384(.055).390

.247(.059).264

.322(.057).327

.502(.062).518

.698*(.043).720*

.516(.059).534

1992 .190(.070).191

.159(.066).154

-.003(.080).000*

.405(.058).407

.467*(.053).465

.310(.065).326

1993 .107(.073).134

.108(.058).060

.049(.068).054

.150(.074).186

.447(.074).537

.154(.073).169

1994 .344(.064).345

-.002(.075)-.002

.069(.071).074

.104(.070).107

.312(.052).328

.177(.064).206

1995 .206(.172).229

-.115(.062)-.068

-.119(.058)-.066

.172(.061).151

.362(.057).365

.386(.055).362

1996 .269(.078).262

.051(.085).055

.115(.077).110

.026(.069).026

.351(.087).354

.175(.067).197

In each calendar year a GARCHCC model is estimated. The estimates of correlation obtained from the GARCHCC model areshown here, with standard errors in parenthesis. An asterisk highlights those estimates of return correlation significantlydifferent (at the 95% level) from the previous year’s estimate. The figures in italics are estimates of return correlation using rawdata. Daily MSCI data are used for the analysis, expressed in US dollars.

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Finally, Table 21 revisits the asset allocation study from a US dollarperspective. Consistent with the local currency results, the analysis supportsthe constant correlation hypothesis and the importance of volatilityclustering. The main difference is the improved performance of ExponentialSmoothing - CC relative to Asymmetry/CC which is discussed below.

The first sub-period of analysis is Jan 1990 - Oct 1996. Here, the threebest performing specifications (Asymmetry/CC, ESCC and BEKK) achievea similar outcome, despite significant differences in their underlyingassumptions regarding co-movements in returns, volatility spillovers andasymmetry in returns. In fact, volatility clustering is the sole commoncharacteristic linking the three most successful specifications in 1990-96.

For the earlier sub-period, 1983-89, the BEKK and FactorCC specificationsoutperform all others by a wide margin. Recall that while both specificationsincorporate cross-market effects in volatility, only BEKK allows correlationin returns to vary. This outcome suggests that cross-market effects involatility are more important than changes in correlation in understandingthe period surrounding the 1987 crash. Asymmetry/CC also performsstrongly in 1987, suggesting that asymmetries in volatility are significantat the time of the crash.

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Table 21MSCI Equity Data


Period

(a)

FixedWindow

(b)

Exp’-tial

Smoothing(c)

ESCC**

(d)

GARCH/CC

(e)

BEKK

(f)

Factor/CC

(g)

Asymm-etry/CC

(h)1990-

Oct19969.92 9.75 9.63 9.73 9.64 9.82 9.59

1983-1989

13.51 14.13 13.79 13.24 12.84 12.84 13.08

1983-1989*

11.68 11.43 11.36 11.50 11.60 11.67 11.40

All 11.88 12.17 11.92 11.64 11.38 11.46 11.49

All* 10.28 10.26 10.15 10.22 10.21 10.28 10.23

1983 10.61 10.52 10.47 10.45 10.61 10.47 10.371984 10.07 10.57 10.47 10.19 10.23 10.27 10.441985 8.72 8.74 8.63 8.95 8.90 8.50 8.981986 12.06 12.39 12.03 11.89 11.78 12.16 12.121987 24.90 26.98 25.87 23.90 22.33 22.45 22.901988 10.61 10.65 10.79 10.93 10.78 10.47 11.091989 10.22 10.01 10.12 9.89 10.08 10.16 10.121990 13.91 13.90 13.81 13.50 13.46 13.44 13.351991 13.14 13.21 12.83 13.43 12.84 13.16 13.031992 9.26 8.50 8.49 8.73 8.96 9.08 8.431993 7.53 7.77 7.67 7.53 7.44 7.76 7.651994 8.06 7.61 7.63 7.63 7.71 7.73 7.711995 7.01 7.05 6.84 6.92 6.98 7.29 6.89

Jan-Oct 96 7.58 6.97 6.93 7.25 7.28 7.62 7.02

* Excludes a twenty business day period surrounding the October 1987 stock market crash.** Exponential Smoothing - CC (ESCC) uses Exponential Smoothing to estimate variance, but applies the assumptionof constant correlation to imply covariance estimates.The variance/covariance matrix is estimated each week from January 1983-October 1996 using seven multivariatespecifications. This matrix is used to select the minimum variance portfolio (with short-selling constraints). The ex-post standard deviation of portfolio returns is then calculated using daily MSCI data. The most effective specificationis deemed to be that which results in lowest ex-post risk.

When the twenty day period containing the crash is excluded, the simpleESCC specification emerges as the most effective risk measure in this sub-period, and also for the entire period of analysis (excluding the crash).This result contrasts with the finding that Asymmetry/CC performs bestfor local currency data. The difference could be explained in two ways:either asymmetries are overwhelmed by currency effects in the US dollar

67

data; or the asymmetries exist, and the specification is not effective incapturing them. Table 17 suggests that the latter interpretation is correct;ESCC is actually more effective than Asymmetry/CC in explainingasymmetry in volatility. This outcome is a further reminder of a commontheme throughout this study: parsimonious specifications often outperformthose with greater theoretical justification.

Both ESCC and Asymmetry/CC share, however, the common assumptionof constant correlation, thus confirming the appropriateness of thisassumption once variance has been appropriately estimated. The successof the constant correlation hypothesis is further supported by comparingcolumns (c) and (d).

Table 15 shows that the currency of denomination has little impact on returncorrelations, yet Table 21 demonstrates that portfolio efficiency is influencedby the currency choice. Comparing Tables 11 and 21, the ex-post portfoliorisk is generally lower in the local currency case, which reflects a hedgedcurrency perspective. Although co-movements are unchanged by thehedging decision, variances are clearly reduced, thus reducing the risk ofan optimised portfolio. The hedging decision is most beneficial early inthe period of investigation, with ex-post portfolio risk reducing from 11.40%pa to 9.07% pa in 1983-89 (excluding crash).25 The saving in risk is lesssignificant later in the period, possibly because risk is lower generally. Infact, the decision to hedge in 1996 would have marginally increased portfoliorisk from a US perspective.

In summary, the currency of denomination does not significantly affect thefindings of this study. The analysis in US dollar terms confirms the necessityto capture volatility clustering when estimating risk in a multivariate setting.Subject to the requirement to model changing risk, the assumption ofconstant correlation is a reasonable one.

25 Assuming the Asymmetry/CC specification.

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ReferencesAlmekinders, G. and S. Eijffinger (1992). “Daily Bundesbank and FederalReserve Intervention and the Conditional Variance Tale in DM/$ Returns”Board of Governors of the Federal Reserve System, International FinanceDiscussion Papers Number 438, December 1992, Washington D.C.

Baillie, R. and T. Bollerslev (1989). “A multivariate generalized ARCHapproach to modeling risk premia in forward foreign exchange ratemarkets.” Journal of International Money and Finance 9: 309-324.

Becker, K., J. Finnerty and M. Gupta (1990). “The intertemporal relationbetween the US and Japanese stock markets.” The Journal of Finance XLV(4)(September 1990): 1297-1306.

Bennett, P. and J. Kelleher (1988). “The international transmission of stockprice disruption in October 1987.” The Federal Reserve Bank of New YorkQuarterly Review(Summer 1988): 17-33.

Bertero, E. and C. Mayer (1989). “Structure and performance: Globalinterdependence of stock markets around the crash of October 1987.”London, Centre for Economic Policy Research.

BIS (1996). “Amendment to the capital accord to incorporate market risks.”Basle, Basle Committee on Banking Supervision.

Black, F. (1976). Studies in stock price volatility changes. Proceedings ofthe 1976 Business Meeting of the Business and Economic Statistics Section,American Statistical Association.

Bollerslev, T. (1990). “Modelling the coherence in short-run nominalexchange rates: A multivariate Generalized ARCH model.” The Reviewof Economics and Statistics 72: 498-505.

Bollerslev, T., R. Chou and K. Kroner (1992). “ARCH modelling in finance:A review of the theory and empirical evidence.” Journal of Econometrics52: 5-59.

69

Bollerslev, T., R. Engle and D. Nelson (1994). ARCH models. Handbookof Econometrics. R. Engle and D. McFadden. Amsterdam, Elsevier ScienceB.V. 4: 2961-3038.

Brown, S. (1990). Estimating Volatility. Financial Options: From theoryto practice. S. Figlewski, W. Silber and M. Subrahmanyam. Homewood,Illinois, Business One Irwin: 516-537.

Chou, R. (1988). “Volatility persistence and stock valuations: Someempirical evidence using GARCH.” Journal of Applied Econometrics 3:279-294.

Cookson, R. (1992). “Models of imperfection” Risk 5(9)(October 1992).

Darbar, S. and P. Deb (1997). “Co-movements in International EquityMarkets” forthcoming in Journal of Financial Research

Davies, A. (1988). “Economics: Bucking the Market” Banking World6(6)(June 1988): 34-35

Engle, R. (1993). “Measuring and testing the impact of news on volatility.”The Journal of Finance(December 1993): 1749-1771.

Engle, R., T. Ito and W. Lin (1990). “Meteor showers or heat waves?Heteroskedastic intra-daily volatility in the foreign exchange market.”Econometrica 58(3)(May 1990): 525-542.

Engle, R. and K. Kroner (1995). “Multivariate simultaneous generalizedARCH.” Econometric Theory 11: 122-150.

Engle, R., V. Ng and M. Rothschild (1990). “Asset pricing with a factor-ARCH covariance structure.” Journal of Econometrics 45: 213-237.

Engle, R. and R. Susmel (1993). “Common volatility in international equitymarkets.” Journal of Business & Economic Statistics 11 (2)(April 1993):167-176.

Erb, C., C. Harvey and T. Viskanta (1994). “Forecasting international equitycorrelations.” Financial Analysts Journal(November-December 1994): 32-45.

70

Eun, C. and S. Shim (1989). “International transmission of stock marketmovements.” Journal of Financial and Quantitative Analysis 24 (2)(June1989): 241-256.

Foster, D. and D. Nelson (1996). “Continuous record asymptotics for rollingsample variance estimators.” Econometrica 64(1)(January, 1996): 139-174.

Freund, J. and R. Walpole (1992) Mathematical Statistics. Englewood Cliffs,New Jersey, Prentice-Hall.

Glosten, L., R. Jagannathan and D. Runkle (1993). “Relationship betweenthe expected value and the volatility of the nominal excess return on stocks.”The Journal of Finance 48 (5): 1779-1801.

Hamao, Y., R. Masulis and V. Ng (1990). “Correlations in price changesand volatility across international stock markets.” The Review of FinancialStudies 3 (7): 281-307.

Jennrich, R. (1970). “An asymptotic chi-squared test for the equality oftwo correlation matrices.” Journal of the American Statistical Association65(June 1970): 904-912.

Judge, G., C. Hill, W. Griffiths, H. Lutkepohl and T.C. Lee (1988).Introduction to the Theory and Practice of Econometrics, John Wiley &Sons.

Kaplanis, E. (1988). “Stability and forecasting of the comovement measuresof international stock market returns.” Journal of International Money andFinance 7(March 1988): 63-75.

Karolyi, A. (1995). “A multivariate GARCH model of internationaltransmissions of stock returns and volatility: The case of the United Statesand Canada.” Journal of Business & Economic Statistics 13 (1)(January1995): 11-25.

Kennedy, P. (1992). A Guide to Econometrics. Oxford, Blackwell.

King, M. and S. Wadhwani (1990). “Transmission of volatility betweenstock markets.” The Review of Financial Studies 3 (1): 5-33.

71

Kroner, K. and S. Claessens (1991). “Optimal dynamic hedging portfoliosand the currency composition of external debt.” Journal of InternationalMoney and Finance 10: 131-148.

Kroner, K. and V. Ng (1995). “Modeling the time varying comovement ofasset returns.” Tucson, Arizona, University of Arizona.

Lee, S. B. and K. J. Kim (1993). “Does the October 1987 crash strengthenthe co-movements among national stock markets?” Review of FinancialEconomics 3(1)(Fall 1993): 89-102.

Lin, W. L. (1992). “Alternative estimators for factor GARCH models - AMonte Carlo comparison.” Journal of Applied Econometrics 7: 259-279.

Lin, W. L., R. Engle, T. Ito (1994). “Do bulls and bears move across bor-ders? International transmission of stock returns and volatility.” The Re-view of Financial Studies 7 (3)(Fall 1994): 507-538.

Longersteay, J. and M. Spencer (1996). RiskMetrics - Technical Docu-ment, Fourth Edition, J.P. Morgan and Reuters, 17 December 1996, NewYork, http://www.jpmorgan.com/RiskManagement/Riskmetrics.

Longin, F. and B. Solnik (1995). “Is the correlation in international equityreturns constant: 1960-1990?” Journal of International Money and Finance14 (1)(February 1995): 3-26.

Najand, M., H. Rahman, K. Yung (1992). “Inter-currency transmission ofvolatility in foreign exchange futures.” The Journal of Futures Markets12(6): 609-620.

Nelson, D. (1990). “Conditional heteroskedasticity in asset returns: A newapproach.” Econometrica 59: 347-370

Nelson, D. (1992). “Filtering and forecasting with misspecified ARCHmodes I.” Journal of Econometrics 52: 61-90.

Ng, V., R. Engle and M. Rothschild (1992). "A multi-dynamic-factor modelfor stock returns." Journal of Econometrics 52: 245-266.

Roll, R. (1989). "Price volatility, international market links, and their im-plications for regulatory policies." Journal of Financial Services Research3: 211-246.

72

Rowen, H. (1986). "Jim Baker's global blueprint" Institutional Investor20(9)(September 1986): 302-310

Schwarz, G. (1978). "Estimating the dimension of a model." Annals ofStatistics 6: 461-464.

Schwert, W. and P. Seguin (1990). "Heteroskedasticity in stock returns."The Journal of Finance 45(4)(September 1990): 1129-1155.

Solnik, B., C. Boucrelle and Y. Le Fur (1996). "International Market Cor-relation and Volatility." Financial Analysts Journal(September/October1996): 17-34.

Tang, G. (1995). "Intertemporal stability in international stock market re-lationships: A revisit." The Quarterly Review of Economics and Finance35(Special 1995): 579-593.

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CMBF Papers

Paper Number

1. Conference on Inflation, Edited by Bill Norton,October 1991

2. Conference on Monetary and Financial SupervisionEdited by Bill Norton, August 1992

3. Asian Financial Markets, Paper at AustralasianFinance and Banking Conference, December 1992

4. Australian Financial Markets, Paper at AustralianInstitute of Bankers Conference, July 1993

5. Alternative Measures of Financial Development,Paper at Conference of Economists, September 1993

6. Saving, Investment and Government Saving: AsianEvidence, Paper at Conference of Economists,September 1993

7. Conference on Financial Stability, Edited by BillNorton, October/November 1993

8. Money, Budget Deficits, Economic Activity andPrices: Asian Evidence, paper at Australian Financeand Banking Conference, by Edward Nelson,December 1993

9. Derivatives: Growth, Benefits and Dangers, InvitedPaper at the Asia-Pacific Forex Assembly, Singapore, by Bill Norton, November 1994

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10. Economic Growth and Financial Sector Development,Paper at Conference of Economists, by David Lynch,1994

11. Equity Markets in Asia-Pacific Economic Develop-ment, Paper at Australian Institute of Bankers Con-ference, by David Lynch, July 1995

12. Conference on Monetary Policy, Edited by BillNorton, October 1995

13. Evaluating the Performance of Portfolios withOptions, by Elizabeth A. Sheedy & Robert G. Trevor,February 1996

14. Asia-Pacific Money Markets in Financial SectorDevelopment, by David Lynch, October 1996

15. Asset Allocation Decisions in a World With ChangingRisk, by Elizabeth Sheedy, Robert Trevor & JustinWood, October 1996

16. Asia-Pacific Bond Markets, by David Lynch,November 1996

17. Correlation in International Equity and CurrencyMarkets: A Risk Adjusted Perspective, by ElizabethSheedy, June 1997

75

Centre for Studies in Money, Banking and FinanceMacquarie UniversityNORTH RYDE NSW 2109 AUSTRALIA

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CMBF Papers - Pennsylvania State University