THE U-SHAPE AUTO CORRELATION PATTERN IN INTERNATIONAL STOCK MARKETS.

Kraizberg, Eli ; Kellman, Mitchell

Eli Kraizberg [*]

Mitchell Kellman [**]

Abstract

Employing monthly data from twenty stock markets, this paper tests for the international applicability of a U-Shape autocorrelation pattern of stock market returns. It is demonstrated that the U-Shape autocorrelation pattern in stock returns is typical of many stock markets, an observation that may be exploited in an attempt to generate a trading strategy which yields an abnormal return. The paper constructs several trading strategies which differ from one another along three dimensions: the sources of information, the weights set for an international portfolio and hedging strategies of foreign exchange risk exposure. The results clearly indicate that above normal returns can be obtained from past autocorrelation patterns in different markets.

I. Introduction

The issue of internationally diversified portfolios has attracted attention, especially since Solnick (1974, 1983). Additional papers by Stehle (1977), Adler and Dumas (1983), Errunza (1985), Cho (1986) and Whitley (1988) contributed to the understanding of this issue. Recent work by Chan and Stulz (1992), Atje (1993), Ferson (1993), Dumas (1994) addressed to the general issue of global markets efficiency as opposed to the local efficiency of specific markets.

Stambaugh (1986) and Fama and French (1988) reported statistically significant autocorrelations of five-year equity returns. The later authors reported that these coefficients are small for a very short horizon and large for a medium horizon, decaying in the long run, thereby producing the famous U-Shape pattern. More specifically, Fama and French directly estimated the time series autocorrelation between average returns for the NYSE, and found that while the short run (under one year) autocorrelations were not significantly different from zero, a clear U-Shaped pattern emerged when the horizon was lengthened beyond one year. For three to five years horizon, over one third of the variation of future returns was explained by past rates of return. It was argued that this finding reflected the sum of a random walk plus a systematic mean-reverting behavior. This finding has been replicated and verified subsequently by many studies of the United States stock market. [1]

Kandel and Stambaugh (1991) made an attempt to rationally explain the U-Shape pattern using Utility Theory.

Regardless of the underlying theoretical rationale and regardless of contemporaneous behavior of fundamental factors, it seems intuitively clear that the presence of a systematic mean-reverting component in stock price behavior implies substantial forecastability of intermediate-term rates of return.

In this paper we show that the general pattern demonstrated in the literature to apply to the NYSE characterizes the behavior of returns over time in markets other than that of the United States. For each of the twenty international stock markets, we calculate overlapping returns following the methodology of Fama and French (1988), for periods of one to forty-five months. The estimates were not adjusted for possible biases. [2] We directly test the proposition that this U-Shaped autocorrelation pattern provides significant opportunities to gain above-normal returns by employing a three-tier simulation design.

The issue of international barriers on capital movements is well demonstrated in the literature. For example, recent work by Stulz (1981), BosnerNeal et al. (1990), Bailey (1994) and Bekaert (1995). The two dimensions of restriction which may affect the analysis are restrictions on foreign exchange and foreign ownership, and restrictions on short sales.

Section II describes the U-Shape pattern in twenty markets. Section III describes the model and simulation methodology. Section IV describes the data and Section V describes the results of the trading rules.

II. The U-Shaped Autocorrelation Pattern

It has been generally observed that for short-horizons, the autocorrelations of monthly rates of return on the NYSE index tend to be statistically insignificant and quite small; in the range of 0.1 to 0.2. This is typically taken as a support for the efficient market hypothesis. However, when the time horizon is extended to over two years, negative autocorrelations begin to predominate. In the range of three to five years, they tend to be statistically significant and as low as -0.5. For longer time horizons, the sample autocorrelations increase back to the zero range (see Figure 1).

Figure 2 presents results for sixteen countries for which were found evidence of U-Shape patterns. Since the overall sample period is 102 observations, findings for up to 45 months only are presented. [3]

A clear U-Shape pattern is indicated in Japan, Germany, Norway, Switzerland, Sweden, Belgium, Austria, France, Netherlands, Pakistan and Venezuela. In India, Canada, Finland and Italy, the rising portion of the "U" is observed after 36 months, while it is relatively short in the United States and United Kingdom. No pattern has been observed in South Korea.

III. The Model and Simulation Methodology

This section presents simulations using three alternative procedures. The first one tests the proposition that an investor, situated in a particular country and limited to investing in his or her own country's market, can obtain abnormal returns using a set of coefficients derived solely from his own country's past observations. The second procedure considers an investor who is not constrained to investment in his own country. Here, we test the proposition that utilization of multi-market information can generate above normal returns for an investor located in any specific country who may invest abroad. The third procedure examines the case in which the investor may continuously update the weights which determine the amount to be invested in each market.

The paper accounts for various sources of risk exposure. Since all trades are made from the point of view of an investor from a particular country, the risk exposure which is used as a point of reference is that associated with his own market index and currency. An adjustment is made to equalize the overall risk to that of the investor's own market, by a specific allocation of each period's investment into risky assets and local government bonds.

The model is described as follows:

Let [p.sub.t] be the natural log of the stock price where it is the sum of the random walk components [q.sub.t] and a stationary component, [z.sub.t], then,

[p.sub.t] = [q.sub.t] + [z.sub.t] (1)

where

[q.sub.t] = [q.sub.t-1] + [micro] + [[eta].sub.t] (2)

[micro] is the drift term and [eta] is the white noise subject to the appropriate modeling of the time dependency.

Define the continuously compounded rate of return from time to to time T as:

r(t,T) = [p.sub.T] - [p.sub.t] (3)

If stock prices have both random walk and slowly decaying stationary components, the slopes of regression of r(t,T) on r(t - 1, T - 1) tend to form a U-Shaped pattern, starting around zero for short horizons, becoming more negative as T increases, and then moving back toward zero as the white-noise variance begins to dominate at long horizons. Under the assumptions of the autoregressive model, the lag coefficients were computed using the Maximum R Squared Improvement regression procedure. The best 'n' explanatory lags of the stock indices rates of return were selected, using the values of the F statistics as criteria, based only on prior data. [4]

The data were denominated in terms of nominal rates of return in each country's currency or alternatively in terms of United States dollars. Simulation results were derived using two distinct models: The 'Limited' Model utilizes only one vector j in the matrix j,T (country, period) of predicted rates of return. In other words, the 'Limited' Model uses only observations from country j. The 'General' Model utilizes the above matrix by using observations from five countries (United States, United Kingdom, Japan, Germany and country j) to determine the predicted rates in country j.

The 'Limited' Model, therefore, is:

Rt + 1; j = [[beta].sub.1]Rt - 1; j

Rt + 2; j = [[beta].sub.2]Rt - 2; j

[symbol not reproducible] (4)

Rt + T; j = [[beta].sub.T]Rt - T; j

where Rt + 1; j is the predicted rate of return in country j from period t to t + 1. Rt - l;j is the actual rate of return in country j from t - 1 till the present time t.

Since not all Betas were statistically significant, the matrix of predicted rates was not complete. (4) was transformed into a complete matrix extrapolating the implied expected rates of return as follows:

Rt + [m.sub.1];j = Rt + [m.sub.1];j = [beta][m.sub.1]Rt + [m.sub.1];j

Rt + [m.sub.2];j = exp{(Rt + [m.sub.2];j)[m.sub.2]/12 - (Rt + [m.sub.1])[m.sub.1]/12}-1

[symbol not reproducible]

Rt + [m.sub.n];j = exp{(Rt + [m.sub.n];j)[m.sub.n]/12 - (Rt + [m.sub.n-1])[m.sub.n-1]/12}-1

Where Rt + [m.sub.1];j is the predicted rate of return over the first [m.sub.1] months from the present time t, where [beta][m.sub.1] is the first significant Beta coefficient. Similarly, Rt + [m.sub.2];j is the predicted rate of return as of the t'th month with respect to the subsequent period [m.sub.2] -- [m.sub.1]. Thus, the vector R is continuous in the sense that it covers the entire horizon. Note that each predicted rate may pertain to future periods which differ in length.

The 'General' version of the Model uses information from five countries to form the predicted rates in country j. Equivalently to (4),

Rt + 1;j = [B.sub.1][RR'.sub.t-1]

Rt + 2;j = [B.sub.2][RR'.sub.t-2] (6)

[symbol not reproducible]

Rt + T;j = [B.sub.T][RR'.sub.t-T]

where Rt + 1;j is the predicted rate of return in country j from period t to t + 1. R is the scalar product of two vectors: B--Betas of five countries, and RR--the actual returns from the past to the present, t.

The implied predicted rates are extrapolated in the same fashion as in (5).

The simulations consist of three procedures:

Procedure 1

The first procedure is applied to each country separately, using its own currency. Two strategies are considered. The first is the 'Hold' strategy where an investor starts with an initial investment of 1000 units of his own currency. He buys the stock index at the initial date and sells at the end of the sample period. The transaction costs, tc are assumed to be paid at once.

The second strategy uses the Model described in (4) and (5). An investor starts with an initial wealth of 1000 of his own currency. Each month he computes the vector of predicted rates R, and invests in his country the amount of p.1000 in accordance with the following rule:

0 if Rt + [m.sub.1];j[less than or equal to]tc + [a.sub.l]

p = 1 if Rt + [m.sub.1];j[less than or equal to]tc + [a.sub.h] (7)

f otherwise

where [a.sub.l] is some arbitrary low threshold and [a.sub.h] is some arbitrary high threshold. f is the log of the absolute value of the ratio of the excess return divided by the difference [a.sub.h] - [a.sub.l]. The remaining portion (1 - p).1000 is invested in a one-month government bond over the investment period. The investment period, in order to avoid unnecessary transaction costs, is determined as follows:

If 0 [less than or equal to] Rt + [m.sub.1] [less than or equal to] tc + [a.sub.1] and there exists the first [m.sub.i] such that:

exp {(Rt + [m.sub.l])[m.sub.l]/12 + (Rt + [m.sub.2])([m.sub.2] - [m.sub.1])/12 + (Rt + [m.sub.i])([m.sub.i] - [m.sub.j-l])12} - 1 [greater than or equal to] tc + [a.sub.h] (8)

then, the investment period is [m.sub.i], transaction costs are paid once and p is computed in accordance with (7) using the left hand side of (8).

If any predicted rate or a sequence of rates in R are negative, two alternative strategies are compared:

a. p = 0, i.e., no investment in the index is made at this period.

b. p [less than] 0, i.e., the index is sold 'short', however, no use of proceeds is allowed. Hence, only net wealth is invested in government bills. This alternative may not be realistic in the case of a country which does not allow trading in index-futures.

Procedure 2

An investor from country j allocates his initial investment among all the countries. Both Models, the 'Limited' and the 'General', are applied to each country in a fashion similar to Procedure 1. The end result is that for each period we derive a subset of countries in which investment is warranted, the amount to be invested, p (as per (7)) for each country, and the length of the investment period. The 1000 initial wealth of an investor from country j can be allocated in two fashions:

a. Equal weights of 0.05, among all countries, (i.e., actual initial investment in the index in a selected country j is 0.05[p.sub.j]1000 and (0.05(1 - [p.sub.j])1000) in government bonds.

b. Constant weights over the entire sample period, where the weights are determined by an optimization procedure which approximates the solution to

[MATHEMATICAL EXPRESSION IS NOT REPRODUCIBLE IN ASCII] (9)

where q is the weight of the investment, [sigma.sub.ij] is the variance-covariance matrix of the countries' rates of return and [VA.sub.j] is a benchmark risk level set here to be the variance of the stock index in country j. Thus, the amount actually invested in the index in country j is initially 1000.[p.sub.j].[q.sub.j]

The amount of investment, denominated initially in the currency of an investor from country j is converted to the currency of country i and then converted back at the end of the investment period.

Since an investor from a specific country is exposed to currency risk, the full exposure outcome is compared with the strategy where the investor hedges his currency risk by rolling over one-month forward contracts. The price of the forward contract is set by the Interest Rates Parity derivation using the rate of interest of the investor's country and the rate of the country whose currency is used. The amount of foreign currency hedged is the actual investment in this currency multiplied by the correlation coefficient of the stock index prices and the exchange rate in terms of the investor's own currency.

Procedure 3

Similar to Procedure 2, the investor is free to allocate his investment in all countries. The innovation of this procedure is that the weights of investment in each country are updated each period based on the procedure described in (9). Additionally, all indices are denominated in terms of the investor's currency. In other words,

Rt + 1;j = [B.sub.1][RF'.sub.t-1]

Rt + 2;j = [B.sub.2][RF'.sub.t-2]

[symbol not reproducible] (10)

Rt + T;j = [B.sub.T][RF'.sub.t-T]

where RF is the vector of actual rates of return in the five countries as in (6) where the rates for four countries (except own country, j) are computed based on the index price divided by the exchange rate in j, i.e., the price of one unit of currency in country j in terms of the other four countries in the data set.

Transaction costs and entry restrictions

Information about transaction costs were not available for all the countries in the data set. The main component; the bid-ask spread varies significantly across countries due to different liquidity characteristics. We assume that there exists a liquid `market fund' in each country which represents the market index with a correlation coefficient close to unity between the two.

An entry fee of 2.5% is charged for each new transaction. We suspect that this fee is on the high side, since in each country there exists a sufficient number of local traders who incur lower transactions and would be willing to transact on behalf of a significant foreign investor for less than 0.025.

During the sample period, 18 of the 20 countries studied have had no foreign currency restrictions. Even in two countries with some restrictions, investment in foreign securities could have been feasible through entities with exporting activities. With the exception of only one country in our sample, no restriction existed on foreign investment in domestic securities.

IV. The Data

The data utilized in this study were continuously compounded monthly returns of twenty stock market indices, each expressed in terms of its own domestic currency. The data, obtained from monthly tapes of the International Monetary Fund's International Financial Statistics, include stock market indices for twenty countries, for the period 1980 through 1988. The total number of observations for each market is 102. The twenty stock markets selected were the only markets which had a full set of observations for the entire sample period.

The data set also included exchange rates, and rates of interest on government securities. The foreign exchange forward prices were computed from this data using the Interest Rate Parity Theorem.

V. Simulation Results

The results of the three procedures are summarized in the following three Tables. Table 1 presents the results in which the investor is constrained to invest in his respective market. The first two columns present the naive 'Hold' strategy, obtained by investing 1000 domestic currency units till the end of the sample period. The first column presents the actual outcome and the second presents the monthly rate of return above transaction costs.

Columns three to five examine the results associated with the 'Limited' Model, which allows a domestic investor to invest in his own market using information derived from his own respective market. The fifth column compares the result of the 'Hold' strategy and the 'Limited' Model.

The 'General' Model where an investor is constrained to his own market but uses information from past history of five markets is presented in the last three columns. As hypothesized, the behavior of the stock markets in the United States, United Kingdom, Japan and Germany as well as the domestic market added systematic predictive power.

The 'hold' strategy, demonstrating actual returns varied from 0.2% in Germany to 4.4% in Korea. These ex-post returns are, per se, uninteresting, reflecting non-systematic behavior, currency devaluation and the arbitrary choice of the sample period. What is of interest, however, is the differential or the 'marginal' gain attributed to the various Model trading rules over the 'hold' strategy.

Even in the case of the 'Limited' Model, the possibility of exploiting information from past behavior to obtain above normal returns was demonstrated. Excess returns were found for 14 of the 20 countries. The Italian market exhibited the highest excess return of 0.25% per month. Interestingly, though expected, there is a negative correlation between the excess returns and the degree which the market was relatively isolated from free capital movements (India, Venezuela and South Korea).

When comparing the 'General' Model to the 'Limited' one, there are strong indications that exploiting information from other markets leads to superior results. Again, in 14 of the 20 countries, the excess rates of return of the 'General' Model were higher than those of the 'Limited' Model. The potential gain while utilizing multi-market information is especially noticeable in markets which are less well integrated with global capital trends. Investors from the Philippines (1.8% per month), India, Norway, Italy and South Korea gained more by utilizing this information. Investors from Germany, Netherlands, Switzerland, Finland and Austria found no use of this information.

Table 2 summarizes the results when the investor is no longer constrained to invest in his or her own domestic market. Under the 'Hold' strategy the investor allocates initial wealth of 1000 at a rate of 0.05 in each of the twenty countries.

The results of the Model using weights which are selected such that the risk level is equal to that of his or her own market are uniformly superior to those of the 'Hold' strategy. These weights are initially set and remain fixed for the entire sample period. Comparison of the Model to the 'Hold' strategy should be done using the same weights (columns 3--4 to columns 5--6). Still the Model shows superior results for 19 investors ranging from 0.28% per month to a negative 0.06% in one country (Philippines). These superior results are attributed to the investor's ability to utilize valuable information and not his ability to diversify his portfolio, since diversification is obtained in both the 'Hold' and Model strategies.

Interestingly, the investors who gain the most out of the ability to invest internationally (and not due simply to diversification as noted above) are from Belgium and Germany which are not considered to be isolated markets.

When investors hedge away their exposure to currency risk, on average as is expected, the rates of return are lower. The return differentials between the hedged and non-hedged results reflect the costs of risk avoidance and naturally it is different from country to country. It is relatively high in non-integrated equity markets such as Venezuela (costs of 4.26% per month), Philippines (2.17%), Pakistan (2.11%) and as low as 0.29% in New Zealand. The average cost of hedging is 1.18% per month.

Up to this point, the simulations were performed over a period of time which followed that from which the coefficients were derived. The last four columns in Table 2 reflect simulation results obtained by information derived from the entire sample period and applied to the same period, yielding very high rates of return. These results reflect in part the fact that the data from the same sample is used to predict behavior over the same period, but they also reflect the unusually high returns obtained by the 'Hold' model for the period 1980-1988.

Table 3 extends the simulations one step further. The weights of investment in different countries are continuously updated, rather than being initially set. Note that though the weights are linked to risk adjustments, the ability to continuously recalculate them allows the investor to change the weights such that higher expected return is obtained.

The results clearly suggest that the ability to update the weights leads to superior performance. For example, when the results are denominated in terms of a single currency (the U.S. dollar), superior results are obtained for 19 of the 20 investors as compared with 14 out of 20 in the fixed weights case.

VI. Conclusion

This paper demonstrates that the U-Shaped autocorrelation pattern applies to many international stock markets when the time horizon is extended to five years. These results are in agreement with other studies in the United States stock market, such as Fama and French (1988) and Poterba and Summers (1988). The systematic mean-reverting component in international stock prices was hypothesized to imply substantial forecastability of intermediate-term returns. This was verified with a set of simulations, which were based on various ex-ante trading rules. The paper tested and demonstrated that this U-Shaped autocorrelation pattern provides significant opportunities to gain above-normal returns.

It is argued that investors aware of these international patterns may exploit this knowledge to attain a superior performance in international stock markets.

We believe that the implications of these results should be further explored; and such efforts are likely to prove fruitful. Several issues suggest themselves. Is there a relationship between the existence and timing of the U-Shaped patterns, and country fundamentals? Is the time series behavior of the U-Shaped patterns stationary over periods of several decades? What is the effect of particular currency denominations on market U-Shaped pattern, e.g., is Japan's U-Shaped pattern preserved when denominated in the terms of the United States dollar?

Finally, the empirical model can be extended for various time relationships between past behavior and predicted rates of return, e.g., the relationships between past n-periods return and predicted m-periods return.

Notes

(1.) Studies which have identified and discussed the U-Shaped autocorrelation pattern of NYSE returns are Fama and French (1988), Stambaugh (1986), Lo and MacKinlay (1988), Poterba and Summers (1989), and Kandel and Stambaugh (1989). For a good survey of the evidence De Bondt, Werner and Thaler (1989). Kandel and Stambaugh (1991) explained the U-Shaped pattern in terms of rational consumer utility theory.

(2.) This was done for two reasons. First, Kandel and Stambaugh (1989) report that the exact nature of the biases are analytically intractable. Secondly, Fama and French (1988) report simulation evidence that the bias in the autocorrelations is, in general, not severe when the true autocorrelations are similar to those calculated for the NYSE (as is suggested to be generally true in the following section).

(3.) The use of a single decade's monthly data was decided upon since we believe that the use of data for earlier periods would clearly violate stationary assumptions. The Depression of the Thirties, the World War of the Forties, and the fixed exchange rate regime of the Fifties to Seventies clearly represent external financial environments different from that of the Eighties.

(4.) i.e., the coefficients were estimated for early subsets of the data (e.g., 1980-1983); and the coefficients thus obtained were applied to simulations over the subsequent period (1984-1988).

(*.) School of Business, Bar Ilan University, Ramat Gan 52900, ISRAEL; Bitnet: kraizbe@ashur.cc.biu.ac.il.

(**.) City College and Graduate Center of CUNY, Phone: 1 212 6066203; Bitnet: ecomhk@ccny.cunyvm.cuny.edu.

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Investor Limited Model Investment General Model
from Hold Strategy in own Country in own
country A B1 A B1 B2 A B1
Belgium 1712 0.0146 1702 0.0144 -0.00016 2219 0.0217
India 1102 0.0026 1195 0.0048 0.00219 2046 0.0195
Finland 2999 0.0301 3105 0.0311 0.00097 2273 0.0224
France 1441 0.0099 1480 0.0106 0.00072 1387 0.0088
Germany 1069 0.0018 1068 0.0017 -0.00002 998 -0.001
Italy 1895 0.0174 2077 0.0199 0.00252 3249 0.0323
Japan 2228 0.0218 2238 0.0221 0.00012 2583 0.0259
Korea 4992 0.0444 5305 0.0461 0.00171 7848 0.0572
Netherlands 1086 0.0022 1118 0.003 0.00078 1048 0.0012
Norway 1218 0.0053 1155 0.0039 -0.00144 1899 0.0174
New Zealand 2135 0.0207 2279 0.0225 0.00181 3378 0.0334
Pakistan 1393 0.0091 1307 0.0072 -0.00173 1808 0.0161
Philippines 2728 0.0274 2701 0.0272 -0.00027 5138 0.0452
Sweden 2113 0.0204 2310 0.0228 0.00243 2436 0.0243
USA 1494 0.0109 1549 0.0119 0.00098 1728 0.0148
Venezuela 3644 0.0355 3889 0.0373 0.00182 4220 0.0396
Canada 1173 0.0043 1185 0.0046 0.00023 1623 0.0131
Austria 1066 0.0017 1081 0.0021 0.00037 523 -0.0173
UK 1455 0.0101 1418 0.0084 -0.00071 1624 0.0131
Switzerland 1166 0.0041 1244 0.0059 0.00175 983 -0.0015
Average 1905 0.0147 1970 0.0154 0.00070 2451 0.0193
Investor Investment
from Country
country B2
Belgium 0.0071
India 0.0169
Finland -0.0076
France -0.0011
Germany -0.0018
Italy 0.0149
Japan 0.0041
Korea 0.0128
Netherlands -0.0009
Norway 0.0121
New Zealand 0.0127
Pakistan 0.0071
Philippines 0.0177
Sweden 0.0039
USA 0.0039
Venezuela 0.0041
Canada 0.0088
Austria -0.0191
UK 0.0031
Switzerland -0.0046
Average 0.0047

Outcomes from an initial amount of 1000, and rates of return for an investor, in terms of his own currency. 'Limited' Model uses own country information. 'General' Model uses five countries. A-outcome in own currency terms. B1-excess return. B2-difference, Model-Hold.

 Second portion of data is tested based
 on model derived from first portion
Investor
from Hold Strategy Hold Strategy
country equal weights selected weights
 A B A B
Belgium 2180 2.13 2220 2.18
India 4084 3.87 4157 3.92
Finland 2427 2.42 2428 2.42
France 2326 2.32 2364 2.35
Germany 2098 2.02 2105 2.03
Italy 2441 2.44 2485 2.49
Japan 2003 1.91 2011 1.91
Korea 3124 3.07 3164 3.16
Netherlands 2086 2.01 2108 2.04
Norway 2645 2.66 2682 2.71
New Zealand 2397 2.39 2422 2.42
Pakistan 4152 3.92 4227 3.97
Philippines 4519 4.16 4709 4.28
Sweden 2513 2.52 2554 2.57
USA 3664 3.57 3781 3.66
Venezuela 7192 5.48 7311 5.52
Canada 3307 3.29 3345 3.31
Austria 2099 2.02 2131 2.07
UK 2608 2.62 2643 2.66
Switzerland 2072 1.99 2106 2.03
Average 2997 2.84 3048 2.89
Investor
from General Model Hold Strategy General Model
country selected weights FEX hedged FEX hedged
 A B A B A B
Belgium 2460 2.46 1869 1.71 2031 1.93
India 4476 4.13 2226 2.23 2504 2.51
Finland 2620 2.63 1950 1.85 2197 2.15
France 2549 2.52 1871 1.69 2091 2.01
Germany 2301 2.28 1596 1.27 1773 1.56
Italy 2677 2.69 2087 2.01 2337 2.32
Japan 2062 1.98 1579 1.24 1753 1.53
Korea 3423 3.38 1719 1.47 1913 1.77
Netherlands 2288 2.26 1659 1.38 1841 1.66
Norway 2900 2.92 2891 2.91 2023 1.92
New Zealand 2629 2.63 1709 1.46 2367 2.36
Pakistan 4555 4.18 1903 1.76 2135 2.07
Philippines 4623 4.22 1889 1.75 2118 2.05
Sweden 2756 2.78 1789 1.58 2002 1.89
USA 4077 3.87 1741 1.51 1936 1.81
Venezuela 7912 5.73 1334 1.21 1728 1.49
Canada 3365 3.49 1811 1.62 2034 1.94
Austria 2301 2.28 1602 1.28 1785 1.58
UK 2861 2.88 1854 1.68 2072 1.99
Switzerland 2273 2.24 1578 1.24 1760 1.54
Average 3255 3.08 1833 1.64 2020 1.90
 Model and test from all data
Investor
from Hold Strategy General Model
country selected weights selected weights
 A B A B
Belgium 2780 2.81 6656 5.26
India 1665 1.62 3968 3.82
Finland 2838 2.95 6797 5.31
France 2534 2.79 6070 4.96
Germany 3568 3.95 8548 5.97
Italy 2390 2.83 5723 4.59
Japan 4745 4.23 9367 6.23
Korea 2557 2.83 8124 5.83
Netherlands 3520 3.63 8427 5.93
Norway 2430 2.71 5821 4.88
New Zealand 2172 2.14 5202 4.56
Pakistan 1517 1.53 3630 3.55
Philippines 1023 0.05 2449 2.45
Sweden 2198 2.15 5265 4.59
USA 2734 2.76 6549 5.21
Venezuela 1211 0.53 3281 3.26
Canada 2691 2.71 6450 5.18
Austria 3578 3.51 8592 5.99
UK 2330 2.31 5582 4.75
Switzerland 3862 3.72 9252 6.21
Average 2617 2.59 6288 4.93

Selected weights are fixed and determined in the beginning of the period. A-each outcome in each country's currency from initial 1000. B-monthly rates of return. Note that last four columns, both the derivation of the model and the test are done on the same data.

Investor
from Hold Strategy
country equal weights General Model, nominal, variable weight
 A B A B1 B2
Belgium 2180 2.13 2472 2.48 0.35
India 4084 3.87 5404 4.67 1.21
Finland 2427 2.42 2687 2.71 0.29
France 2326 2.32 2593 2.61 0.31
Germany 2098 2.02 2468 2.47 0.45
Italy 2441 2.44 2785 2.81 0.37
Japan 2003 1.91 2847 2.87 0.97
Korea 3124 3.07 4079 3.87 0.81
Netherlands 2086 2.01 2340 2.32 0.31
Norway 2645 2.66 3043 3.05 0.39
New Zealand 2397 2.39 3176 3.17 0.78
Pakistan 4152 3.92 5651 4.79 0.87
Philippines 4519 4.16 5919 4.92 0.76
Sweden 2513 2.52 2890 2.91 0.39
USA 3664 3.57 5027 4.46 0.89
Venezuela 7192 5.48 11570 6.84 1.36
Canada 3307 3.29 4602 4.21 0.92
Austria 2099 2.02 2669 2.69 0.67
UK 2608 2.62 2888 2.91 0.29
Switzerland 2072 1.99 2295 2.27 0.28
Average 2997 2.84 3870 3.45 0.63
Investor
from
country General Model, FEX, variable weights
 B3 A B1 B2 B3
Belgium 0.02 2418 3.38 1.25 0.92
India 0.54 6162 5.04 1.17 0.91
Finland 0.08 3795 3.67 1.25 1.04
France 0.33 3696 3.59 1.29 1.32
Germany -- 3249 3.24 1.22 0.54
Italy 0.83 3901 3.74 1.31 1.76
Japan 0.53 2964 2.98 1.02 1.00
Korea 1.61 3825 3.69 0.62 1.43
Netherlands 0.04 3215 3.21 1.19 0.29
Norway 0.41 4283 4.01 1.35 1.36
New Zealand 0.52 4518 4.19 1.76 0.01
Pakistan 0.57 7400 5.56 1.64 1.34
Philippines 2.14 7665 5.66 1.50 2.88
Sweden 0.13 4051 3.89 1.33 0.02
USA 0.59 6588 5.23 1.66 1.36
Venezuela 3.34 13771 7.35 1.87 3.85
Canada 1.93 5933 4.92 1.63 2.64
Austria 0.71 3248 3.24 2.22 0.36
UK 0.41 4096 3.88 1.28 1.64
Switzerland 0.03 3302 3.28 1.29 1.04
Average 0.74 4904 4.19 1.32 1.24

Outcomes for initial amount of 1000 of each investor's currency. Weights are adjusted continuously. B1-is monthly rate of return. B2-excess rate of return of the Model over 'Hold' strategy. B3-excess rate of return (if positive) of Model over Model with fixed weights. The right four columns represents the outcomes and the corresponding rates in terms of a single currency $.