ELSEVIER Mathematics and Computers in Simulation 39 (1995) 417-423

~ i MATHEMATICS AND

COMPUTERS N SIMULATION

Modelling implications of the instability of the mean-variance paradigm in international finance

Jason Mitchel l

Department of Accounting and Finance Universi~ of Western Australia, Nedlands W.A. 6907, Australia

"'Only the clairvoyant could hope to predict with certainty, clairvoyant analysts have no need for the techniques of this monograph. The existence of uncertainty does not mean careful security analyses are valueless."

H.M. Markowitz

I. Introduction

The mean-variance paradigm is a cornerstone of both empirical international finance and finance in general. Commonly referred to as portfolio analysis, it provides a robust, deceptively simplistic mechanism for asset investment determination. Essentially, the allocations of wealth decision can be decomposed into two basic constructs, which are none other than the respective investment assets' mean return (change in wealth) vector and the corresponding correlation or covariance matrix to represent risk comovement. An advantage of the model is its general normative application to all identifiable asset investments. Ex ante estimates of expected future changes in wealth are required by the model. However, to the extent ex post results persist, they are indicative of future relationships. This approach assumes no identifiable structural change. The portfolio model has been applied extensively in empirical finance research and advocates the benefits of marginal risk assessment and diversification.

Research on the comovement of international stock indices has important implications regarding risk reduction through international diversification. Empirical investigations of the comovement relationship, and any deterministic patten thereof, are vital to international investment decisions. Numerous studies (including [2]) have indicated the potential gains from international portfolio diversification. Further, papers such as [6,7,9,10] examined the intertem- poral stability of the international comovement structure of equity investments. For a review of this and the associated literature, see [5]. International investment strategies and the accuracy of prediction of international asset returns depends greatly on the stability of the international comovement structure.

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418 J. Mitchell~Mathematics and Computers in Simulation 39 (1995) 417-423

Makridakis and Wheelwright [7] suggest that in order to obtain the benefits from interna- tional diversification, two conditions are necessary: the correlation coefficients must be sub- stantially less than one and, given the investor is not able to predict future relationships among countries' stock markets, the respective relationship between the price movements, must be stable across time. Early studies as noted above, in particular [6,9], are based on relatively few observations and it is difficult to generalise the results. In addition, the techniques for the identification of comovement stability and any deterministic structure are poor. They lack power at best, at worst are inappropriate and reflect the general confusion regarding the implications of stability in this area. As a result, these studies have unsuccessfully evaluated volatility and predicability of comovement for short-term horizons.

The diverse nature of empirical findings of inter-temporal comovement stability are sum- marised in [9]. [10] found strong stability for one-, two- and four-year periods. In contrast, [7] provided support for the instability of intertemporal relationships. Nonetheless, in the latter case all the subperiods examined were fairly short (less than 1 year) and no formal 'tests' were conducted. In reexamining [6]'s data and methodology, [9] concluded long-term stability exists, but short-term results are inconclusive. [6,9] employed first-, and higher-order, autocorrelation estimates of the comovement to identify the time series structure. First-order autocorrelation estimates are generally biased if /3 = 1 and conventional t-statistics tend to incorrectly reject the hypothesis H 0 :/3 = 1. Also, this approach uses a test for stationarity as a test for structural change. The flaws in the simplistic interpretation of stability and the methodology used by [6,7,9,10] are self-evident.

The aim of this paper is to examine the issue of inter-temporal stability of the comovement between several stock market indices, namely: Australia, France, Germany, Japan, UK, USA, Italy and Switzerland. Stock market indices provide an appropriate general surrogate of an asset investment in a particular country. Monthly indices for the selected countries as compiled by Morgan Stanley Capital International Perspective are obtained. Such indices are calculated from closing equity share prices using a value-weighted arithmetic average (Laspeyres) method with chain-linking. Share prices are compiled using end of daily trade prices adjusted for all capitalisation changes, rights issues and dividends. The indices assume gross dividend reinvest- ment. All indices are characterised by non-normality and are transformed by natural loga- rithms. The transformation helps alleviate this common distributional problem. Post transfor- mation, some evidence of leptokurtic non-normality remains, although the distributions are approximately symmetrical and bell shaped. Overall, the results generally suggest that there is substantial instability in the comovement measures. Intertemporal modelling of the comove- ment measures, using regression and time series analysis, reveals some deterministic structure. The explanatory power of the models is generally low.

2. Empirical results

Stationarity tests

A conventional econometric test for a (first-order) non-stationary series is the unit root test based on the Dickey-Fuller (hereafter DF) and Augmented Dickey-Fuller (hereafter ADF)

J. Mitchell/Mathematics and Computers in Simulation 39 (1995) 417-423 419

statistics. None of the A D F statistics for the raw indices is significant at convent ional significance levels. Plots of the data clearly indicate that each index is a non-stat ionary series with trend. Figures and summary tables of the empirical results, referred to th roughout the paper , are available on request. Box-P ie rce and L jung-Box statistics indicate that there are significant autocorrelat ions (prob-value < 0.01) up to lag 12. Consequent ly, first differences of the index series are obtained. A D F tests (without t rend) all indicate the first differences of the indices are stationary. Graphical and spectral plots of the first differences exhibit a stochastic non-determinis t ic series and suppor t the A D F tests. The first difference of the natural logged price series has a convenient in terpre ta t ion as it represents a cont inuous rate of re turn on the index. These cont inuous re turn series are used in the remainder of the paper . Comovemen t relat ionships derived f rom the stationary re turn series have the advantage of exhibiting desirable propert ies , namely that significance tests are reliable because of the consistency of the s tandard errors.

Intertemporal correlations

Comovemen t be tween the various countr ies ' re turns are es t imated using convent ional measures of correlat ions and covariances. Typically the correlat ion is used in preference to the covariance, a l though knowledge of one allows the calculation of the other. This is mainly due to the correlat ion 's in terpre ta t ion as a measure of association. Notwi ths tanding this, [3] has shown that the stability of the covariances may differ relative to the correlations.

The correlat ions are es t imated from the monthly indices data over the per iod 31 December 1970 to 31 Dec 1991. Full- and various sub-period correlat ions are computed . The annual (calendar year) and various sub-periods des ignated (a), (b), and (i)-(iv) are chosen on an arbitrary basis. Sub-periods (1)-(4) are selected after prel iminary data analysis which indicated a l ikelihood of structural change in the markets. The respective sub-periods (n-designates the n u m b e r of observations) are:

(a) Jan 1970-Dec 1980 (11 years; n = 132).

(b) Jan 1981-Dec 1991 (11 years; n = 132).

(i) Jan 1970-June 1975 (5½ years, n = 66).

(ii) July 1975-Dec 1980 (5½ years, n = 66).

(iii) Jan 1981-June 1986 (5½ years, n = 66).

(iv) July 1986-Dec 1991 (5½ years, n = 66).

(1) Jan 1970-Sep 1974 (n = 57).

(2) Oct 1974-Nov 1983 (n = 110).

(3) Dec 1983-Sept 1987 (n = 46).

(4) Oct 1987-Dec 1991 (n = 51).

The full t ime-per iod clearly encompasses a variety of possible influences and structural change on the comovements of the various indices. A number of 'shocks' such as the 1987 crash, oil

4 2 0 J. Mitchell~Mathematics and Computers in Simulation 39 (1995) 417-423

price shock of the 70's, and deregulation of the Australian financial market in 1983, may all cause structural change. Some structural change is inevitable and anticipated. Comovement relationships could hardly be construed as being generated by a static environment. Correla- tions relative to Australia appear to fluctuate over time but no ready pattern is noticeable.

Correlation coefficients are calculated for the above sub-periods for each of the index-pair relationships. Initially, a Fisher transformation is conducted to ensure the correlations are normal variates. Stability of the correlations for each series of sub-periods is analysed using a chi-squared test which tests the equality of multi-period correlations. This results in four multi-sub-period series tests, categorised for convenience as (A)-(D). The respective series sub-period tests are: (A) annual periods, (B) periods (a) and (b), (C) periods (i), (ii), (iii) and (iv), and (D) periods (1), (2), (3) and (4). The time (time series) and country (countries involved) effects are evaluated in each case. The chi-squared results indicate: (1) stability increases with an increase in the length of the period. Three out of 28 annual correlations (A) are not significantly different over time relative to only three being significant for the test of the equally divided sub-sample (B). (2) Stability depends on the sub-period identified. Tests of the relative instability between the various sub-samples in (C) and (D) indicate a significant difference in the changes in the correlations (direction) when matched between identified unstable periods (D) and an arbitrary determination of equal sub-samples (C). (3) Cross-sectional differences in correlations are not as apparent as the time-series differences. The significance of the cross-sectional differences does increase with the length of the sub-sample.

For modelling the instability of the correlation structure, focus is concentrated on the annual (calendar year) correlations. This reflects a fairly short-term investor focus. ANOVA and non-parametric rank sum two-way A N O V A (FRIEDMAN test) reveal the existence of a highly significant time and country effect for the correlations as a whole. Further procedures using time series analysis and time trend regression are used to identify if there is any structure to the identified instability. Only five of the individual 28 inter-country correlations have a significant time trend deterministic structure. Of primary interest here is the modelling of any determinis- tic structure of the correlations as a whole.

Three models are hypothesised and tested as being appropriate in modelling the correlations based on the results above, and the empirical literature [1,3]. Following [2], and as some time trend (T) has been established, the following model is postulated:

Ps, = aos + [3oT~ + vst (1)

where last = correlation s (between index i and j where i ~:j) for time period t. Allowing for transformation, this can be restated as:

Model 1: Time Trend Model:

Zs, -~- Oils "b / 3 1 L "+- Est (2)

where z is the conventional Fisher transformation. An alternative approach is an AR (1) model:

Model 2: A R (1) Model. Where /3 = 1, [1] uses the designation Full Historical Model:

zst = Oi2s + 132Zst-1 + est. (3)

Another possible approach to predicting the individual correlations is the use of a prior-period averaging techniques. Model 3 includes an average correlation effect (areas) for specific

J. Mitchell/Mathematics and Computers in Simulation 39 (1995) 417-423 421

countries and an average world correlation effect (avews), again specific to the indices i and j involved in the appropriate correlation s being predicted (areas is generally referred to as the National Mean Model, see [1]). A general effect, which includes the average of all the countries (aveag) and world (avewg) correlations is also included (aveag, similarly, is referred to as the Overall Mean Model [1]).

Model 3: Combined Averaging Model:

Zst = O13s + ~lTs + [~2Zs t_ l + f l 3 a v e a S s t _ l + f l 4 a v e W S s t _ l + ~saveagst_l + f l 6 a v e w g s t _ l + es t .

(4)

aveaSst = average correlations of countries i and j (of zst) relative to the other countries

at t ime t.

aveWSst = average correlations of countries i and j (of zst ) relative to the world index at

time t.

aveagst = average correlations of all countries (including i and j ) relative to all o ther

countries at time t.

avewgst = average correlations of all countries (including i and j ) relative to the world

index at time t.

est "~ niid (0, tr 2) in each model as the Fisher t ransformation induces normality. Rather than estimate the regressions separately for s = 1, 2 , . . . , 28 , the regressions are pooled and run concurrent ly to ascertain a general model. The A D F tests indicate the dependen t variables and est are both stationary for each model. The pooled regressions do not allow fur ther time series specifications of the models.

The models were recalculated using alternative month ends for the annual periods rather than the calendar year (December) period end. No major differences were found. Further , the mean returns are not significantly different across months. The results suggest that the dynamic

T a b l e 1

M o d e l M o d e l S p e c i f i c a t i o n /~2 F - l o g -

S t a t l i k e

1 z s t = - 0 . 8 6 6 * * + 0 . 0 1 6 * * T 0 . 0 6 2 4 0 . 2 * * - 2 4 9

( - 4 . 2 3 ) ( 6 . 3 4 )

2 z s t = 0 . 3 8 4 * * + 0 . 1 0 8 *z,t_ 1 0 . 0 1 0 6 . 7 * - 2 6 5

( 1 6 . 3 7 ) ( 2 . 6 0 )

3 zst = - 0 . 8 3 4 * * + 0 . 1 8 7 * * T + O.030Zst_ 1 + 0 . 5 2 2 * aveasst_ 1 0 . 0 9 2 1 0 . 9 * * - 2 3 7

( - 3 . 9 1 ) ( 7 . 0 2 ) ( 0 . 4 3 ) ( 2 . 1 2 )

- O.O00avewsst - 1 - - 0 . 2 8 4 *aveagst_ 1 - 0.417avewg~t- 1 ( - 0 . 0 1 ) ( - 1 . 9 9 ) ( - 0 . 9 3 )

4 z~t = 1 2 . 7 * * - 0 . 3 2 2 * * T + 0 . 5 9 2 * * aveasst_ 1 - 0 . 8 6 3 * * avewg~t_ l 0 . 1 1 8 2 0 . 7 * * - 2 2 9

( 4 . 1 0 ) ( - 4 . 1 8 ) ( 3 . 9 6 ) ( - 5 . 1 0 )

+ 0 . 0 0 2 * * Sqr. ( 4 . 4 2 )

* S i g n i f i c a n t a t 5 % . * * S i g n i f i c a n t a t 1 % . t - v a l u e s i n p a r e n t h e s e s , l o g - l i k e d e n o t e s t h e l o g - l i k e l i h o o d f u n c t i o n .

422 J. Mitchell~Mathematics and Computers in Simulation 39 (1995) 417-423

comovement environment is difficult to explain (/~2 between 1.0-11.8%). A time trend effect, as well as a drift component, a general effect in relation to the world and the average country specific effect are all significant. A final model, Model 4, is developed incorporating the variables from Model 3 and testing for variable deletion a n d / o r addition. A further variable to denote a additional time trend component variable, Sqr, is added. Sqr denotes the square of time and effectively surrogates for aveag. The time trend, T, is also a proxy for the first-order lag (zst). Overall, Model 4 appears the most appropriate. Diagnostic tests suggest models 3 and 4 are acceptable. Non-nested tests indicate Model 4 rejects Model 3 but Model 3 does not reject Model 4.

3. Concluding remarks

Structural fluctuations of comovement come mainly from exogenous stochastic processes although some limited dependency, which is related to time, should be incorporated. The implication for investment techniques and associated models is twofold. First, unless an at tempt is made to consider the dependence structure identified above, full utilisation of the available information set, is not made. Second, to at tempt a more accurate prediction, there is a need to divert at tent ion to a more worthwhile identification of the covariance or correlation matrix. In the words of Markowitz [8, p. 28]: "The security analyst is the meteorologist of stocks and bonds. I f he is thorough, his statements about the future of a security will be based on general conditions and prospects for the economy and the market. . . A portfolio analysis begins where security analysis leaves off."

Nothing in life is certain. Except Uncertainty! Anon.

Acknowledgements

The helpful comments of H.Y. Izan and Michael McAleer are gratefully acknowledged.

References

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