International diversification through country index funds and...

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International diversification through country index funds and ETFs

By Yan Zhao♦

First draft Sept, 2007

This draft: Aug, 2008

Abstract

This paper studies the international asset allocation by comparing various portfolio strategies

on country index funds and ETFs. We compare the performance of simple portfolio

construction strategies, such as the equally-weighted, capitalization-weighted, GDP-weighted

and dividend-weighted portfolios, with optimal portfolio strategies based on mean-variance

analysis: the global minimum variance portfolio, the mean-variance efficient portfolio and the

Bayes-Stein shrinkage portfolio.

We find simple portfolio strategies perform similarly to optimal portfolio strategies based on

mean-variance analysis in developed countries while outperform optimal portfolio strategies

in emerging markets. Which shows the loss from simple rather than optimal

diversification is smaller than the loss from model estimation errors. Among the simple

portfolio strategies, the most commonly used capitalization-weighted portfolio suffers from

price fluctuations and systematically over-weights the over-valued countries and

under-weights the under-valued countries; to overcome this shortcoming, we propose the

portfolio based on the GDP weights since the GDP is the most straightforward fundamental at

the country level and a relative stable measure of the size of each country in the global

economy. A delta strategy based on the difference of a country’s GDP share and capitalization

share has predictive power. We further show the gains from diversification can be achieved in

portfolios consisting of as few as 3 developed countries or 4 emerging markets.

♦ International Business School, Brandeis University, [email protected]

mailto:[email protected]

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1. Introduction

Mean-variance efficient portfolios play an important role in portfolio allocation

problems. An investor who cares only about the mean and variance of portfolio

returns should hold an optimal portfolio on the mean-variance efficient frontier

(Markowitz (1952)). To implement this optimal portfolio in practice, an investor has

to estimate the mean and variance of a portfolio’s returns via their sample analogues.

However, due to estimation errors portfolios constructed using these estimators are

extremely unstable and the optimal weights on each asset fluctuate substantially over

time, and thus these unstable portfolios perform poorly out of sample (Littlemand

(2003), Brandt (2004), Roon(2004), DeMiguel, Garlappi and Uppal (2007), and

Demiguel and Nogales (2007)).

The instability of the mean-variance portfolios can be explained partly by the

well-documented difficulties associated with estimating mean asset returns. For

instance, Jagannatham and Ma (2003) state that ‘the estimation error in the sample

mean is so large that nothing much is lost in ignoring the mean altogether’. For this

reason, the global minimum variance portfolio, which relies solely on estimates of the

covariance matrix and is not as sensitive to estimation errors, also plays an important

role in asset allocation decision.

Another prominent role in the asset allocation decision is played by the shrinkage

estimation approach. The idea of shrinkage estimation is attributed to James and Stein

(1961) and is designed to handle errors in estimating expected returns by shrinking the

sample means toward a grand mean. The estimator thereby reduces the extreme

estimation errors that may occur in the cross section of individual means.

Despite considerable efforts to handle the estimation errors, the out-of-sample

performance of portfolios based on mean-variance analysis is still not consistently

better than a simple equally-weighted portfolio, for instance, DeMiguel, Garlappi and

Uppal (2007) compare fourteen models of optimal portfolio choice using seven

empirical datasets, and find none of them consistently outperform the equally-

weighted portfolio. Thus, keeping portfolio strategies simple is very important

empirically.

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The most widely used simple portfolio strategy is the capitalization-weighted

portfolios, such as the MSCI world index and MSCI EAFA index, which is well

accepted by practitioners and academics. The benefits of the traditional capitalization

weighted index are numerous (Schoenfeld (2006), Arnott, Hsu, and Moore (2005)): it

is self-rebalanced, and thus incurs low trading costs, it has high investment capacity

and by design it is a market clear investment. However, the capitalization-weighted

portfolio is flawed in a fundamental way: it systematically over-weights the

over-valued stocks and under-weights the under-valued stocks as stock markets

fluctuate;

A neutral way to overcome these shortcomings is instead to use GDP as the weighting

scheme. Not only the GDP is the most straightforward fundamental at the country

level, but also it reflects the size of each country in the global economy. A country’s

GDP is highly correlated with the market size and is a relatively more stable measure.

As documented in Hamza, Kortas, LHer and Roberge (2005, 2006), the GDP is a

better weighting scheme than capitalization when investing globally.

Other weighting schemes, such as the dividend weighting scheme is inspired by

Arnott, Hsu and Moore (2005), who construct portfolios based on metrics of firms’

book values, revenues, dividends, incomes and others. Using data for the U.S data

from 1962 to 2004, they report the portfolio weighted by dividends outperform the

S&P500 in a variety of scenarios. Hsu and Campollo (2005) show that portfolios

based on dividends also outperform the capitalization-weighted portfolio in 23

developed countries besides the U.S from 1984 to 2004. Estrada (2006) further shows

the superior performance of the dividend-weighted portfolio in 16 developed

countries over a period from 1973-2006.

The objective of this paper is to investigate the portfolio construction issue at country

level. We compare the performance of the above optimal portfolio strategies based on

mean-variance analysis: the mean-variance efficient portfolio, the global minimum

variance portfolio, and the Bayes-Stein shrinkage portfolio with the simple portfolio

strategies of investing in each of the countries equally or according to their relative

shares in the capitalization, GDP, or dividends using 43 country index funds and

ETFs.

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By comparing the above different portfolio strategies, we have the following findings.

First, we find simple portfolio strategies perform similarly to, if not better than, the

optimal portfolio strategies based on mean-variance analysis in developed countries

while outperform the optimal portfolio strategies in emerging markets. Which shows

the gain from optimal diversification is offset by estimation errors, and the loss from

simple rather than optimal diversification is smaller than the loss from model

estimation errors.

Secondly, among the simple portfolio strategies, the most commonly used

capitalization-weighted portfolio suffers from price fluctuations and systematically

over-weights the over-valued countries and under-weights the under-valued countries.

To overcome this shortcoming, we propose a portfolio weighting scheme based on the

GDP weights. Our empirical results illustrate that GDP-weighted portfolios

outperform capitalization-weighted portfolios. We claim that it is because GDP is the

most straightforward fundamental at the country level and measures the size of each

country in the global economy.

Thirdly, we further show the gains from diversification can be achieved in portfolios

consisting of as few as 3 developed countries or 4 emerging markets. In the case of

portfolios consisting of a smaller number of countries, the estimation errors are less

severe and thus, the optimal portfolios have better out of sample performances.

Finally, we propose on the one hand, GDP reflects a country’s fundamentals and is

more stable, on the other hand, a country’s market capitalization fluctuates due to

various noises and it should come back to fundamentals sooner or later. We then claim

that a delta strategy which puts delta weight (delta weight=GDP weight – CAP weight)

on each country will have some predictive power. If the delta weight is positive, we

believe the country’s ETF is undervalued and will buy delta shares of that country,

similarly, if delta weight is negative, which shows the country’s ETF is overvalued,

and we will short delta shares of that country. This long-short market neutral strategy

works well in mean-reverting markets like emerging markets, while does not work in

a trending market like developed countries in 1980.

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The remainder of the paper is organized as follows. Section 2 describes the data and

methodology. Section 3 reports the empirical performance of different portfolios.

Section 4 is the robustness check and Section 5 concludes.

2. Sample selection, portfolio strategies and evaluating methodology

2.1 Data

Quarterly nominal GDP in dollar terms and monthly dividend and market

capitalization data are taken from Datastream. Monthly return series are calculated

using Morgan Stanley Capital International (MSCI) indexes in dollar terms, and

accounting for both capital gains and dividends. Finally, the U.S. 3-month T-bill rates

(risk-free rates) are also from the Datastream. The sample period runs from Dec 1979

to Dec 2006 for 23 developed countries, and from Dec 1994 to Dec 2006 for 20

developing countries. The sample periods are selected to cover as long a history as

possible with data from Datastream database.

Table 1 and table 2 show the summary statistics of the return series for both 23

developed countries and 20 emerging markets. The distributions of returns series are

quite different in developed countries and in emerging markets. In general, emerging

markets have higher volatilities and higher excess kurtosis, suggesting that the return

distributions are more fat-tailed than a normal distribution and more outliers.

2.2 Mean-variance portfolio strategies

2.2.1 Mean-variance efficient portfolio (MVE)

In the mean-variance framework of Markowitz (1952), we consider an investor whose

preferences are fully captured by the mean and variance of a portfolio. At each time t,

the investor selects tω to maximize the following utility subject to no short-selling

constraints; the no short-selling constraints are mainly for the purpose of reducing the

extreme weights on each country:

Max ttttt ωωγ

µω Σ− ''

2 subject to 11' =tNω and 0>=itω ………………………..(*)

Where tµ denotes the N-vector of expected excess returns (over the risk free asset)

on the N risky assets available for investors at time t. tΣ is the corresponding N x N

variance matrix of returns. tω is the vector of portfolio weights invested in the N

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risky assets at time t. N1 defines a N x1 vector of ones. Finally, γ denotes an

investor’s risk aversion, in our following analysis, γ is chosen to be 2, results for

other values of γ are discussed in robustness check.

To implement this classic model, we solve the above maximization problem with the

mean and covariance matrix of asset returns replaced by their sample counterparts,

t

^

µ and t

^

Σ , respectively. The sample mean and covariance matrix is calculated using

all the observations up to time t, that is, we do not drop the earliest observation when

we add a new observation. Note that this portfolio strategy completely ignores the

estimation errors.

2.2.2 Bayes-Stein shrinkage portfolio (BS)

Shrinkage estimation is pioneered by Stein (1955) and James and Stein (1961), the

shrinkage estimator shrinks the sample means toward a grand mean. In our analysis,

we follow the estimator proposed by Jorion (1985, 1986), who takes the grand mean

to be the mean of the global minimum variance portfolio. In addition to shrinking the

estimate of the mean, Jorion also accounts for estimation error in the covariance

matrix through traditional Bayesian estimation method. More specifically, following

Jorion (1986), the estimator we use for expected returns is

MIN

tttt

BS

t

^^^^^

)1( µφµφµ +−=

Where

)()'()2(

2

^^1^^^

^

MIN

ttt

MIN

tt

t

MN

N

µµµµ

φ

−Σ−++

+=

−

For 10^

<< tφ the Jorion estimator shrinks the sample means toward the mean of the

global minimum variance portfolio MIN

t

^

µ . The estimator thereby reduces the extreme

estimation errors that may occur in the cross-section of individual means. The

implement in Jorion (1986) also accounts for estimation errors in the covariance

matrix by a traditional Bayesian estimation. The following expression for the

covariance matrix is utilized in the portfolio construction.

NtN

NN

t

t

t

t

BS

t

MMM 11

11

)1(

)1

1(1^

'

'

^

^

^

^^

−

Σ++

+

+

+Σ=Σ

τ

τ

τ

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t

tt M

^

^

^

1 φ

φτ

−

= Where M is the length of estimation window up to time t.

The Bayes-Stein portfolio is constructed by maximize the utility function (*) in

section 2.2.1 using BS

t

^

µ and

BS

t

^

Σ , which combines a shrinkage approach and a

traditional Bayesian estimation, and hence, is known as the Bayes-Stein portfolio.

2.2.3 Global minimum variance portfolio (MIN)

The global minimum variance portfolio of risky assets is constructed to minimize the

variance of the portfolio variance subject to no short-sale constraints, that is,

Min ttt ωω Σ' subject to 11' =tNω and 0>=itω

To implement this strategy we only use the estimate of the covariance matrix of asset

returns, and in this paper, we use the sample covariance matrix using the observation

up to time t and completely ignores estimates of the expected returns.

2.3 Simple portfolio strategies

The most straight forward simple strategy is to hold a portfolio that equally invests in

each of the countries. We call it equally-weighted portfolio (EQUAL). This strategy

completely ignores all estimates of the data. A second simple strategy is the

capitalization-weighted portfolio (CAP). To construct the capitalization-weighted

portfolio, the weight on each country is set according to their relative market

capitalization at time t, the relative market capitalization is calculated as the market

capitalization of each country relative to the sum of the capitalization of all countries.

In a similar fashion we also construct the GDP-weighted portfolio (GDP) and the

dividend-weighted portfolio (DIV). We finally examine a composite strategy by

equally weighting the CAP, GDP and DIV weights (COMBO). To calculate the GDP,

CAP and DIV weights, we use the trailing 5-year averages to reduce the excessive

volatility in portfolio weights would result from using current data. For the GDP data,

we use the data lagged by a year to account for all the seasonally adjustment and

make sure the data are available at the time of portfolio rebalance. The

capitalization-weighted portfolio bears close resemblance to the MSCI benchmarks,

but they are not identical, the MSCI benchmark indices are weighted by float not

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aggregate capitalization and are rebalanced annually. In addition, the

capitalization-weighted portfolios in this paper use trailing 5-year average

capitalization to reduce the excessive volatility and are rebalanced quarterly.

Finally, all the portfolios are rebalanced at the end of each quarter, and returns for

these portfolios are calculated on a monthly basis. For reference, the performance of

the MSCI world index and MSCI emerging markets index in the same sample period

are also reported.

2.4 Methodology for evaluating performance

The measurements for evaluating performances of different portfolios are the

followings:

Alpha is calculated as ))(()( fmft RREbetaRRE −−− using the CAPM model,

where MSCI index is used as a proxy of the market return, and risk-free rate is the

annualized U.S. 3-month T-bill rate.

Sharpe ratio is defined as excess returns over the risk-free rates divided by their

standard deviation. In formula: )( t

ft

RStd

RRoSharpeRati

−=

Certainty-equivalent return (CEQ) is the risk-free rate that an investor is willing to

accept rather than adopting a risky portfolio strategy, we compute the CEQ as the

following: VarMeanCEQ2

γ−= , where Mean and Var are the mean and variance of

the portfolio return over risk-free rates. γ is the risk aversion of investors, which is

set equal to 2 in our analysis.

Turnover rate is the amount of trading required to implement each portfolio strategy.

We compute the portfolio turnover, defined as the average sum of the absolute value

of the trades across all available assets, as indicated by the following formula:

|))1()((|1

1

'

1

+−= ∑∑==

twtwT

Turnover i

T

t

i

T

t

Where )1( +twi is the weight at time t+1; while )(' twi is the weight just before the

rebalance, it is different from the weight at time t, because changes in asset prices

have caused a change in the relative weights in the portfolio

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In addition to reporting the raw turnover rates for each strategy we also report the

transaction cost generated by the turnover rate. Following Balduzzi and Lynch

(1999), DeMiguel, Garlappi and Uppal (2007) , we assume a 50 basis points

transaction cost per one-way transaction, the total transaction cost is calculated as

turnover*0.5%*2.

3. Empirical Results

3.1 Performance comparison

The performance comparison is done separately in developed countries and emerging

markets. Two simple reasons that we do not construct portfolios in a combined

universe; the first one is that the sample periods for developed countries (1979-2006)

and emerging markets (1994-2006) are different, significant data would be lost if we

combine the two universes. Secondly, the weights on emerging markets will be

overwhelmed by the weights on developed countries in the capitalization-weighted

and GDP-weighted portfolios.

3.1.1 Developed countries

In this section, we compare empirically the performance of portfolios constructed

based on the mean-variance analysis with portfolios based on simple strategies listed

in the above section. Table 3 reports the performance of all portfolios for 23

developed countries, and Figure 1 shows the performance of these portfolios in a

mean-variance space.

To compare the performance of different strategies, we first look at the Sharpe ratio of

different portfolios. By design, the ex-post mean-variance efficient portfolio should

have the highest Sharpe ratio when there are no estimation errors. The magnitude of

the difference between the ex-post mean-variance efficient Sharpe ratio and the

ex-ante mean-variance efficient Sharpe ratio measures the loss from model estimation

errors when implementing the mean-variance strategy. Similarly, the difference

between the ex-post mean-variance efficient Sharpe ratio and the Sharpe ratios of

simple portfolio strategies measures the loss from simple rather than optimal

diversification when there are no estimation errors.

We first look at the loss from model estimation errors. We can see the difference

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between ex-post mean-variance efficient Sharpe ratio and ex-ante efficient Sharpe

ratio, as indicated in Table 3, is as big as 0.57 (1.07-0.50), which means an investor

cannot fully capture the theoretical gains from optimizing models due to estimation

errors. The poor ex-ante performance of mean-variance strategies is because the

ex-ante portfolio weights vary substantially from the ex-post optimal weights, as

shown in table 6. While the ex-ante global minimum variance portfolio seems to have

a better Sharpe ratio compared to the ex-ante mean-variance efficient Sharpe ratio

(0.73 vs. 0.50), which shows that by ignoring the estimates of expected returns

altogether but exploiting the information about correlations only does lead to better

performance. Since the ex-ante mean-variance efficient portfolio completely ignores

the estimation errors, the Bayes-Stein portfolio which explicitly accounts for

estimation errors might also lead to a better performance in terms of Sharpe ratio. And

indeed, the Sharpe ratio of the Bayes-Stein portfolio is 0.72, which is also higher than

the ex-ante mean-variance efficient Sharpe ratio of 0.50.

Other performance measures, like Portfolio Alpha and CEQ, tell a similar story as

above. By construction, the ex-post mean-variance efficient portfolio has the highest

Alpha and CEQ, but ex-ante, they all suffer greatly from estimation errors. For

instance, the CEQ of the mean-variance efficient portfolio drops from 21.41% ex-post

to 12.13% ex-ante, which is even lower than the CEQs of all simple strategies (except

for the capitalization-weighted portfolio).

Secondly, to look at the loss from simple rather than optimal diversification when

there are no estimation errors we compare the difference between ex-post

mean-variance efficient Sharpe ratio and the Sharpe ratios form simple portfolio

strategies. For example, the Sharpe ratio of the equally weighted portfolio is 0.75,

indicating a 0.32 loss (1.07-0.75).

And in general, simple strategy portfolios have Sharpe ratios in a range of 0.49-0.75,

while the ex-ante optimal portfolios have Sharpe ratios in a range of 0.50-0.73. This

similar range of the Shape ratio shows that simple portfolio strategies are comparable

to if not better than optimal strategies based on the mean-variance analysis because

they do not suffer from the model estimation errors.

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Finally, if we think of practical issues when implementing these different portfolios,

an importance concern is the portfolio turnover rates, which directly affect transaction

costs and net returns. From Table 3 we can see that the turnover rates of all ex-ante

optimal portfolios are much higher than the simple portfolio strategies, since those

ex-ante optimal portfolio weights are extremely sensitive to model inputs.

The various performance measures analyzed above illustrate that simple portfolios

suffer less from model estimation errors and have lower turnover rates, thus to keep

portfolio strategy simple is very important empirically. If we take a close look at the

different simple strategies, we can notice that the most commonly used capitalization-

weighted portfolio delivers the lowest Alpha, Sharpe ratio, CEQ and net returns

among all strategies. Since all five simple strategies consist of the same 23 MSCI

country ETFs, the difference in the performance stems solely from the different

weights given to these benchmarks in each ETF. Table 6 presents the average weights

on each country.

The first explanation of the better performance of other simple portfolios over the

capitalization-weighted portfolio is that the capitalization-weighted portfolio can be

highly concentrated when one major country or region outperforms others for a

sustained period of time. For instance, during the 1980s Japan market bubbles, the

capitalization-weighted portfolio put around 40% weights on Japan from 1989 to 1994,

as showed in the figure 3. And when the Japan bubbles burst in late 1990s, the

performance of the capitalization-weighted portfolios drops. To exclude the Japan

effect, we reconstruct all portfolios using 22 countries excluding Japan, the results are

show in Table 4. Table 4 shows once we exclude Japan, the performance of the

capitalization-weighted portfolio improves, for instance, alpha increases to 2.03%

excluding Japan from -0.17% including Japan; and Sharpe ratio increases to 0.62 from

0.49. A second explanation of the better performance of other simple portfolios over

the capitalization-weighted portfolio is that they are less affected by the price

fluctuations, since the capitalization-weighted portfolio systemically over-weights the

over-valued assets and under-weights the under-valued assets as prices fluctuate;

that’s why once we controlled the Japan effect, the capitalization-weighted portfolio

still underperforms other simple portfolios.

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A neutral way to overcome the shortcomings of the capitalization-weighted portfolio

is instead to use GDP as the weighting scheme. Not only the GDP is the most

straightforward fundamental at the country level, but also it reflects the size of each

country in the global economy. In addition, GDP is a relatively more stable measure.

As show in figure 3, while the capitalization-weighted portfolio highly concentrated

on Japan in 1980s, the GDP-weighted portfolio gives a very stable weight on Japan

around 15-20%. And indeed, from the different measure of performance, the

GDP-weighted portfolio delivers a higher Alpha, Sharpe ratio and CEQ while

maintaining a similar turnover rate compared to the capitalization-weighted portfolio.

Other weighting schemes also help to overcome the shortcomings of the

capitalization-weighted portfolio, such as portfolio based on dividends paid. However

this strategy must be implemented with cautions, because in some countries,

dividends may not be regularly paid, and investors may not relate dividends to stock

prices closely.

Finally, the equally-weighted portfolio seems to deliver the best alpha, Sharpe and

CEQ at the cost of slightly higher turnover rates among the simple strategies, but

again, cautions must be used when using this strategy because of the small country

effect. As show in table 6, the equally-weighted portfolio systematically puts more

weights on smaller countries where the returns are higher. This superior performance

of the equally-weighted portfolio is consistent to the findings of DeMiguel, Garlappi

and Uppal (2006) and Hamza, Kortas, LHer and Roberge (2005, 2006).

3.1.2 Emerging markets

Stories are getting more interesting when we turn to the emerging markets. Table 5

reports the performances of all portfolios for 20 emerging markets, and Figure 2

shows the performances of these portfolios in a mean-variance space.

We still first take a look at the loss from model estimation errors by comparing Sharpe

ratios of different portfolios; and we can see in general, the loss from estimation erros

is big. The ex-post mean-variance efficient Sharpe ratio is 1.20, while the ex-ante

mean-variance efficient Sharpe ratio is only 0.49, suggesting there are substantial loss

of 0.71 from estimation errors. Again, this poor ex-ante performance is because of the

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deviation from the ex-post optimal weights assigned to each county, which is shown

in table 7. The Bayes-Stein portfolio which accounts for estimation error by shrinking

the sample mean to a common grand mean does have improvement with a Sharpe of

0.72. By ignoring the estimates of expected returns altogether but exploiting the

information about correlations, the ex-ante global minimum variance portfolio also

seems to perform very well which has a Sharpe ratio of 0.76.

We now turn to the loss from simple rather than optimal diversification when there are

no estimation errors. Comparing Sharpe ratios of simple strategies to the portfolios

based on mean-variance analysis, we find the simple strategies yield higher Sharpe

ratios compared to ex-ante optimal portfolios; the simple portfolios have Sharpe ratios

in a range of 0.78-0.89, while the ex-ante mean-variance efficient portfolios have

Sharpe ratios in a range of 0.49-0.76. The higher range of Sharpe ratios of simple

strategies shows that errors in estimating means and covariance destroy all the gains

from optimal relative to simple diversification, investors are better off using the

simple strategies.

Other performance measures, like Portfolio Alpha and CEQ, tell a similar story as

above. For instance, the Alpha of the ex-ante mean-variance efficient portfolio is less

than half of the ex-post, and CEQ of the ex-ante mean-variance efficient portfolio is

around one-third of the ex-post. And simple portfolios in general have higher alpha

and CEQ than the ex-ante optimal portfolios. Therefore, the simple portfolios

generally achieve better performances than the ex-ante optimal portfolios based on

mean-variance analysis due to the much smaller loss suffered from model estimation

errors.

Finally, we consider the practical issues of the portfolio turnover rates. The range of

the turnover rates of simple portfolios (71.24% to 85.37%) is actually lower than the

range of ex-ante optimal strategies (84.44% to 95.77%), lower turnover rates thus lead

to higher net returns; which further shows the investors are better off using the simple

strategies. Therefore, to keep portfolio strategy simple is more attractive in emerging

markets.

Compared to other simple portfolios, the superior performance of the GDP-weighted

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portfolios comes from different weights put on each country. In emerging markets,

country sizes are positively related to equity market returns, GDP weights

systematically put more weights on bigger countries where returns are higher. It is

shown in table 7, the GDP does assign much greater weights on the high returns

countries, examples are Russia, Brazil and Mexico. In contrast, country sizes are

negatively related to equity market returns among developed countries, the

equally-weighted portfolio works better than other simple portfolios because equally-

weighted portfolio puts relative more weights on small countries where the returns are

high.

Finally, the Dividend-weighted portfolio also achieves good performance. However,

in the emerging markets, due to the fast growing and different culture, companies may

choose to reinvest most of the profits instead of paying dividends to fully capture the

growth opportunity; in addition, because of the culture differences, dividends may be

paid ac hoc. Thus, stock prices in the emerging markets may not closely relate to

dividends; and the fluctuation of dividends may not reflect the company’s

fundamental and hence the dividends might not serve as weighting scheme.

3.2 How many countries are needed to diversify efficiently?

The above performances are all based under assumption of investing in 23 developed

countries and 20 emerging markets. The question is what is the minimum number of

countries needed to be included in a portfolio in order to achieve similar performance?

In this section, we show 3 largest developed countries according to their market

capitalization are efficient to achieve similar performance in Table 3 as discussed

above; and for emerging markets, as few as 7 largest countries in terms of

capitalization or 4 largest countries in terms of GDP are efficient.

To determine how many countries are needed at least to diversify efficiently in

developed countries, we plot portfolio variances against number of countries included

in each portfolio in figure 4. First I rank the countries based on their capitalization

values, the x-axis can be read as: top 1 country in terms of cap (which is the US), top

2 countries in terms of cap (which are the US and Japan), top 3 countries in terms of

cap (the US, Japan and the UK). The y-axis is the equally-weighted,

capitalization-weighted and GDP-weighted portfolio risks (STD). Figure shows the

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portfolios consisting 3 countries achieve the lowest portfolio risk.

To further evaluate whether the performance of portfolios consisting of 3 developed

countries is comparable to the performance of portfolios consisting of 23 developed

countries, we report the performance measures in table 8.

The first thing to note is that because of the small number of countries (N=3), the

number of moments to be estimated is small, thus, estimation errors are less severe.

Comparing the ex-ante mean-variance efficient Sharpe ratio and ex-post

mean-variance efficient Sharpe ratio, the loss is 0.12 (0.62-0.49) which is much

smaller than in the case of 23 countries. The global minimum variance portfolio and

the Bayes-Stein shrinkage portfolio achieve Sharpe ratios of 0.52 and 0.56

respectively and the loss from estimation errors is even smaller in those two portfolios.

The CEQ measure shows the same story, the average CEQ of ex-ante optimal

portfolios are about only 1% less than those of the ex-post optimal portfolios

Secondly, in general, the performance of the mean-variance portfolios is slightly

better than those of simple portfolios. This is as expected, as the number of assets

decrease, estimation errors decrease, thus investors are more likely to capture the

gains from optimization.

Finally, although the benefit from diversification is smaller due to the small number of

countries, net returns of different portfolios are still comparable to the net returns in

the case of 23 countries. And therefore, in develop countries the risk-sharing gain can

be achieved in portfolios consisting of as few as three.

We then look at the emerging markets; stories are quite different in emerging markets.

Figure 5 ranks the emerging markets based on their capitalization values, it shows the

largest 7 countries in terms of capitalization achieves the lowest portfolio risk. These

countries are Taiwan, Korea, Brazil, South Africa, India, Mexico and Malaysia.

Similarly, Figure 6 ranks the emerging markets based on their GDP values, it shows

the largest 4 countries in terms of GDP achieves the lowest portfolio risk. These

countries are Brazil, Mexico, Korea and India. Unlike in the case of developed

countries, where the US, the UK and Japan are the largest countries in terms of

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capitalization across all time period; in the universe of emerging markets, the largest

countries in terms of capitalization and GDP slightly vary across different time period.

Thus, to make our portfolios implementable, the portfolios include the largest

countries at the time of rebalance.

Table 9 and Table 10 show the performance of the different portfolios in case of 7 and

4 countries. Because of the small number of moments to estimate, the estimation error

problem is less severe and the loss from optimization is less, which is indicated by the

difference of Sharpe ratios between ex-ante and ex-post optimal portfolios. For

instance, the difference of Sharpe ratio between the ex-ante mean-variance efficient

Sharpe ratio and the ex-post mean-variance efficient Sharpe ratio is only 0.17, which

is much smaller than in the case of 20 countries.

Secondly, consistent to the previous results, GDP-weighted portfolios achieve better

performances than the ex-ante optimal portfolios; GDP-weighted portfolios have

higher Sharpe ratios, CEQ, Alphas than the ex-ante optimal portfolios, while at the

same time, maintain lower turnover rates and thus, higher net returns.

And finally, portfolios consisting of as few as 4 or 7 countries achieve comparable

gains from risk-sharing in the case of 20 emerging markets. And actually portfolios

consisting of 4 biggest countries in terms of GDP achieve higher gross returns, alpha,

CEQ, and lower transaction costs, thus higher net returns than the portfolios of 20

emerging markets.

3.3 Delta-strategy

If we believe GDP reflects fundamentals and is stable, and the market capitalization

fluctuates due to noisy prices, but should come back to fundamentals sooner or later.

Then a delta strategy will have some predictive power. The delta strategy puts a delta

weight (=GDP weight – CAP weight) on each country. If the delta weight is positive,

we believe the country’s ETFs is undervalued and will buy delta shares of that country,

similarly, if delta weight is negative, which shows the country’s ETFs is overvalued,

and we will short delta shares of that country. Thus, the delta strategy is a long-short

market neutral strategy.

- 17 -

By construction this delta strategy should work well in emerging markets, where the

GDP weight of a country can be very different from the capitalization weight,

investors can benefit from buying undervalued ETFs at quarter t and selling them at

t+1 when prices of the ETFs grow back to fundamentals and selling overvalued ETFs

at quarter t and buying them back at t+1 when prices of the ETFs fall back to

fundamentals. And indeed, table 11 shows the delta strategy yields a net return of as

high as 29.58% with Sharpe ratio of 1.29 in emerging markets.

However, this delta strategy may not work well in a trending market like the

developed countries in the 1980s, where the prices come back to fundamentals at a

relatively slow speed. As we can see the delta strategy only yield 0.71% gross return

in developed countries, and the Sharpe ratio is negative. This poor performance of the

delta strategy is because the long-side and short-side portfolios grow almost at the

same speed due to the bubble in Japan in 1980s. As shown in figure 6, through out the

period from January 1981 to June 1995, the weights on Japan in the short-side of the

portfolio are always above 40%; and more than 80% from October 1985 to December

1990; in these periods the short-side portfolio grows even faster than the long-side

portfolio due to the Japan bubbles. As expected, in a trending market where the prices

do not convert to fundamentals in a long time, investors will keep selling Japan ETFs

at low prices and buying it back at high prices, which hurts the performance of the

delta strategy. If we exclude the Japan from our investment universe, the performance

improves significantly: the gross return increase to 6.06% and Sharpe ratio to 0.38.

4. Robustness tests

4.1 Annually rebalance and GDP(PPP)-weighted portfolios

The benefit of rebalancing is well acknowledged, while the question of how often is

the portfolio to be rebalanced is still an open question. The above performance at

based on quarterly rebalance, in this section we also report the performance of

annually rebalanced portfolios. Since we do not have enough annual data points for

the optimal portfolios based on mean-variance analysis, only the performance of

simple portfolios is presented. We also consider an additional simple portfolio which

is weighted on the purchasing power parity GDP (GDP(PPP)). The purchasing power

parity takes into account the relative cost of living and the inflation rates of the

countries, rather than using just exchange rates which may distort the real differences

- 18 -

in income. We could not include the GDP(PPP)-weighted portfolio in the quarterly

rebalanced portfolio analysis because the GDP(PPP) data are only available at annual

frequency. Table 12 and table 13 report the annual performance of simple portfolios in

23 developed countries and 20 emerging markets.

The performance of annually rebalanced portfolios are very similar to that of quarterly

rebalanced portfolios and the main results of this paper are the same. The

equally-weighted portfolio achieves the highest Sharpe ratio and net returns in 23

developed countries. The GDP-weighted and GDP(PPP)-weighted portfolios have the

best Sharpe ratios and returns in 20 emerging markets, consistent to the results

discussed in section 3.1. The GDP(PPP)-weighted portfolio has better Sharpe ratio

and returns than the GDP-weighted portfolio, indicating additional benefit by taking

into consideration of inflation rates and relative cost of living.

4.2 Difference risk aversion levels

For all the asset allocation strategies considered above, we only report results for the

case in which risk aversion is equal to 2. We also consider the following levels of risk

aversion: γ = {1, 3, 5, 10}. The risk aversion levels only affect the performance of

mean-variance portfolio and the Bayes-Stein shrinkage portfolio. We find that the

results are not very different across risk aversion levels, only the CEQ return are

affected, because there is no optimization in the simple strategies, as risk aversion

increases the CEQ of the simple strategies drops.

5. Conclusion

This paper compares different optimal portfolio strategies with simple portfolio choices using

data from country index funds and ETFs and find how many counties are needed at least to

diversify globally.

We have the following contributions. We find the gains from optimal diversification is offset

by estimation errors, thus performance of the simple portfolios are very close to the optimal

portfolios based on mean-variance analysis in developed countries while outperform the

optimal portfolios in emerging markets.

Among the simple portfolio strategies, the most commonly used capitalization-weighted

- 19 -

portfolio suffers from price fluctuations and systematically over-weights the over-valued

countries and under-weights the under-valued countries. To overcome this shortcoming, we

propose the GDP weighting scheme based on the belief that the GDP is the most

straightforward fundamental at the country level and it is a relatively stable measure of the

size of each country in the global economy.

We further show the gains from diversification can be achieved in portfolios consisting of as

few as 3 developed countries or 4 emerging markets, in the case of portfolios consisting of

smaller number of countries, the estimation errors are less severe and thus, the optimal

portfolios have better out of sample performances.

Finally, we also find that a delta strategy based on the difference of a country’s GDP share and

capitalization share has a predictive power.

- 20 -

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- 22 -

Table 1 Summary Statistics for 23 developed countries

Table 1 shows the monthly arithmetic mean return (Mean), standard deviation (STD),

risk-adjusted return (RAR=AM/STD), minimum (Min) and maximum (Max) return of each country

ETF over the Dec/1979-Dec/2006 period. It also shows the skewness and kurtosis of each country

ETF. All data are accounted for capital gains and dividends.

Country Mean STD RAR MIN MAX Skewness Kurtosis

Australia 1.22% 6.62% 18.43% -44.51% 20.79% -1.03 6.71

Austria 1.14% 6.51% 17.49% -23.29% 28.11% 0.45 2.37

Belgium 1.35% 5.70% 23.68% -18.85% 26.76% 0.23 3.11

Canada 1.02% 5.54% 18.49% -22.04% 17.98% -0.58 2.41

Denmark 1.37% 5.48% 24.99% -13.38% 21.30% 0 0.35

Finland 1.45% 9.33% 15.55% -31.76% 33.26% 0.13 1.24

France 1.21% 6.07% 19.86% -23.18% 21.03% -0.4 1.37

Germany 1.14% 6.36% 17.86% -24.35% 23.69% -0.27 1.64

Greece 1.80% 6.83% 26.28% -19.48% 21.04% -0.28 1.23

HK 1.48% 8.87% 16.74% -43.44% 33.23% -0.35 2.89

Ireland 1.11% 5.64% 19.74% -17.67% 18.42% -0.12 0.99

Italy 1.32% 7.18% 18.35% -20.39% 30.99% 0.3 0.84

Japan 0.96% 6.63% 14.53% -19.38% 24.26% 0.31 0.56

Netherlands 1.34% 5.22% 25.58% -17.80% 15.82% -0.53 1.44

New Zealand 0.82% 6.55% 12.49% -20.03% 27.65% 0.15 1.36

Norway 1.19% 7.36% 16.17% -27.85% 22.15% -0.44 1.24

Portugal 0.73% 6.38% 11.38% -19.36% 28.41% 0.31 1.6

Singapore 1.06% 7.52% 14.10% -41.34% 25.84% -0.54 4.03

Spain 1.45% 6.54% 22.16% -21.61% 26.72% 0.12 1.31

Sweden 1.72% 7.17% 24.07% -22.23% 22.91% -0.13 0.64

Switzerland 1.17% 5.16% 22.62% -17.64% 16.68% -0.15 1.1

UK 1.21% 5.26% 22.96% -21.53% 19.44% -0.1 1.57

US 1.13% 4.30% 26.41% -21.22% 13.28% -0.57 2.37

- 23 -

Table 2 Summary Statistics for 20 emerging markets

Table 2 shows the monthly arithmetic mean return (Mean), standard deviation (STD),

risk-adjusted return (RAR=AM/STD), minimum (Min) and maximum (Max) return of each country

ETF over the Dec/1994-Dec/2006 period. It also shows the skewness and kurtosis of each country

ETF All data are accounted for capital gains and dividends.

Country Mean STD RAR MIN MAX Skewness Kurtosis

Argentina 1.56% 11.59% 13.43% -31.36% 52.92% 0.33 2.64

Brazil 1.81% 11.45% 15.80% -37.63% 36.78% -0.39 1.38

Chile 0.77% 6.63% 11.59% -29.10% 20.13% -0.57 2.39

Colombia 1.74% 9.91% 17.52% -23.45% 30.61% 0.07 0.4

Czech 1.75% 8.19% 21.32% -27.57% 30.08% -0.15 1.43

Hungary 2.27% 10.28% 22.07% -38.80% 46.17% 0.1 3.36

India 1.18% 8.26% 14.28% -17.61% 22.13% -0.05 -0.54

Indonesia 1.26% 14.73% 8.55% -40.54% 55.59% 0.38 2.22

Israel 1.08% 7.36% 14.64% -18.85% 26.90% -0.12 0.97

Korea 1.32% 12.77% 10.32% -31.26% 70.60% 1.27 5.98

Malaysia 0.54% 9.84% 5.53% -30.20% 50.05% 0.83 6.02

Mexico 1.63% 8.69% 18.81% -34.25% 19.14% -0.83 1.64

Peru 1.53% 8.18% 18.68% -33.62% 36.11% -0.12 3.65

Philippines 0.00% 9.68% -0.02% -29.22% 43.39% 0.57 2.89

Poland 1.51% 10.58% 14.29% -34.82% 40.21% 0.27 1.56

Russia 3.48% 17.76% 19.57% -59.23% 61.13% 0.17 1.7

South Africa 1.08% 7.93% 13.59% -30.51% 19.45% -0.69 1.41

Taiwan 0.41% 8.79% 4.69% -21.73% 29.25% 0.4 0.66

Thailand 0.27% 12.76% 2.12% -34.01% 43.23% 0.37 1.75

Turkey 2.54% 16.89% 15.04% -41.24% 72.30% 0.57 2.1

- 24 -

Table 3. Portfolio performance in 23 developed countries quarterly rebalanced

Table 3 summarizes the quarterly performance of equally-weighted portfolio (EQUAL), the

GDP-weighted portfolio (GDP), the capitalization-weighted portfolio (CAP), the dividend-weighted

portfolio (DIV), composite portfolio (COMBO), global minimum variance portfolio(MIN),

mean-variance efficient portfolio (MVE), Bayes-Stein shrinkage portfolio (BS) and the MSCI world

index (MSCI_world) for 23 developed countries over the Dec/1979-Dec/2006 period. The

arithmetic mean of annualized gross return (Gross Return), standard deviation (STD) and beta

with respect to the MSCI world market index are reported in the table. Ex-post excess returns are

measure by Alpha, which is calculated using the CAPM model. Sharpe Ratio is the excess return

over risk-free rate divided by the standard deviation. The certainty-equivalent return (CEQ) is the

risk-free rate that an investor is willing to accept rather than adopting a risky portfolio strategy.

Turnover rate is the amount of trading required to implement each portfolio strategy. Assuming a

half percentage transaction cost per one-way transaction, the total Transaction Cost is calculated

as turnover*0.5%*2. Net return is Gross return net transaction cost.

Gross

Return STD Alpha Beta

Sharpe

Ratio CEQ Turnover

Transaction

Cost Net

Return

EQUAL 17.48% 15.40% 4.79%

0.90

0.75 15.11% 49.45% 0.49% 16.99%

GDP 14.39% 14.29% 1.15%

0.98

0.59 12.34% 41.37% 0.41% 13.97%

CAP 13.16% 14.86% -0.17%

0.99

0.49 10.95% 41.02% 0.41% 12.75%

DIV 14.58% 14.01% 1.49%

0.96

0.62 12.62% 36.71% 0.37% 14.22%

Combo 14.14% 14.17% 0.82%

0.99

0.58 12.13% 39.54% 0.40% 13.74%

MIN_exante 16.35% 13.91% 3.71%

0.90

0.73 14.41% 69.81% 0.70% 15.65%

MVE_exante 16.96% 21.97% 5.81%

0.70

0.50 12.13% 103.57% 1.04% 15.92%

BS 17.90% 16.65% 5.89%

0.81

0.72 15.13% 75.49% 0.75% 17.15%

MIN_expost 15.65% 12.62% 2.81%

0.92

0.77 14.06% 39.25% 0.39% 15.25%

MVE_expost 24.40% 17.31% 12.98%

0.73

1.07 21.41% 52.81% 0.53% 23.87%

MSCI_world 13.41% 14.34% 0.00%

1.00

0.52 11.35% 13.41%

- 25 -

Table 4. Portfolio performance in 22 developed countries excluding Japan quarterly

rebalanced





index (MSCI_world) for 22 developed countries excluding Japan over the Dec/1979-Dec/2006

period. The arithmetic mean of annualized gross return (Gross Return), standard deviation (STD)

and beta with respect to the MSCI world market index are reported in the table. Ex-post excess

returns are measure by Alpha, which is calculated using the CAPM model. Sharpe Ratio is the

excess return over risk-free rate divided by the standard deviation. The certainty-equivalent

return (CEQ) is the risk-free rate that an investor is willing to accept rather than adopting a risky

portfolio strategy. Turnover rate is the amount of trading required to implement each portfolio

strategies. Assuming a half percentage transaction cost per one-way transaction, the total

Transaction Cost is calculated as turnover*0.5%*2. Net return is Gross return net transaction

cost.

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost Net

Return

EQUAL 17.75% 15.57% 5.19%

0.89

0.76 15.33% 49.03% 0.49% 17.26%

GDP 15.58% 14.49% 2.78%

0.92

0.67 13.48% 37.13% 0.37% 15.21%

CAP 14.80% 14.24% 2.03%

0.91

0.62 12.77% 33.50% 0.34% 14.46%

DIV 15.08% 14.27% 2.34%

0.91

0.64 13.04% 34.21% 0.34% 14.74%

Combo 15.24% 14.27% 2.44%

0.92

0.65 13.20% 35.01% 0.35% 14.89%

MIN_exante 16.44% 14.07% 3.93%

0.88

0.75 14.46% 67.79% 0.68% 15.77%

MVE_exante 16.59% 22.41% 5.64%

0.67

0.48 11.57% 103.08% 1.03% 15.56%

BS 17.98% 16.84% 6.03%

0.80

0.72 15.15% 74.00% 0.74% 17.24%

MIN_expost 16.13% 12.87% 3.53%

0.89

0.79 14.47% 37.36% 0.37% 15.76%

MVE_expost 24.41% 17.31% 12.99%

0.73

1.07 21.41% 52.80% 0.53% 23.88%

MSCI_world 13.41% 14.34% 0.00%

1.00

0.52 11.35% 13.41%

- 26 -

Table 5. Portfolio performance in 20 emerging markets quarterly rebalanced




mean-variance efficient portfolio (MVE), Bayes-Stein shrinkage portfolio (BS) and the MSCI

emerging market index (MSCI_emerging) for 20 emerging markets over the Dec/1994-Dec/2006









cost.

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 25.11% 25.27% 7.78%

0.96

0.84 18.73% 85.34% 0.85% 24.26%

GDP 25.45% 24.19% 7.96%

0.97

0.89 19.60% 73.90% 0.74% 24.71%

CAP 20.99% 21.92% 3.34%

0.99

0.78 16.19% 69.25% 0.69% 20.30%

DIV 21.89% 22.46% 4.38%

0.98

0.80 16.84% 71.24% 0.71% 21.17%

Combo 23.04% 22.47% 5.45%

0.98

0.85 17.99% 71.61% 0.72% 22.33%

MIN_exante 19.48% 20.45% 3.37%

0.88

0.76 15.30% 95.17% 0.95% 18.53%

MVE_exante 19.66% 32.27% 4.55%

0.80

0.49 9.25% 95.77% 0.96% 18.70%

BS 23.80% 27.80% 6.58%

0.96

0.72 16.08% 84.44% 0.84% 22.96%

MIN_expost 22.65% 17.07% 6.29%

0.89

1.10 19.74% 66.71% 0.67% 21.99%

MVE_expost 33.87% 25.01% 18.79%

0.80

1.20 27.62% 93.05% 0.93% 32.94%

MSCI_emerging 17.84% 21.11% 0.00%

1.00

0.66 13.38% 17.84%

- 27 -

Table 6 Portfolio Weights in 23 developed countries

Table 6 shows the average weight of each country in the 10 portfolio strategies over the

Dec/1979-Dec/2006 period. The order of the countries is according to their average monthly

returns from high to low. Concentrate rate is the Herfindahl-Hirschman index.

Country

Monthly

return EQUAL GDP CAP DIV Combo MIN_exante MVE_exante BS MIN_expost MVE_expost

Greece 1.80% 4.35% 0.70% 0.33% 0.47% 0.50% 0.00% 0.00% 0.34% 0.00% 0.00%

Sweden 1.72% 4.35% 1.16% 0.74% 0.82% 0.90% 0.67% 20.91% 8.64% 0.00% 27.52%

HK 1.48% 4.35% 0.53% 1.52% 2.43% 1.49% 0.04% 21.98% 5.96% 0.00% 11.58%

Finland 1.45% 4.35% 0.53% 0.48% 0.54% 0.51% 1.33% 3.72% 1.49% 0.00% 0.00%

Spain 1.45% 4.35% 2.50% 1.31% 1.95% 1.80% 5.27% 0.00% 2.67% 0.00% 3.45%

Denmark 1.37% 4.35% 0.71% 0.30% 0.23% 0.35% 6.48% 7.31% 8.56% 6.72% 27.08%

Belgium 1.35% 4.35% 1.09% 0.56% 0.79% 0.82% 3.94% 0.84% 3.98% 3.86% 8.40%

Netherlands 1.34% 4.35% 1.66% 2.04% 3.19% 2.34% 3.43% 4.02% 4.12% 0.00% 0.00%

Italy 1.32% 4.35% 5.32% 1.74% 2.07% 3.06% 0.91% 2.84% 3.23% 0.56% 0.00%

Australia 1.22% 4.35% 1.80% 1.40% 2.46% 1.89% 4.38% 0.00% 4.28% 5.76% 0.00%

France 1.21% 4.35% 6.62% 2.98% 4.31% 4.63% 0.00% 0.00% 3.73% 0.00% 0.00%

UK 1.21% 4.35% 5.61% 8.83% 15.69% 10.01% 3.28% 0.00% 3.19% 4.68% 0.00%

Norway 1.19% 4.35% 0.69% 0.21% 0.24% 0.38% 0.00% 1.18% 5.35% 0.00% 0.00%

Switzerland 1.17% 4.35% 1.20% 1.77% 1.44% 1.47% 1.15% 3.63% 3.43% 5.17% 2.52%

Austria 1.14% 4.35% 0.85% 0.13% 0.11% 0.23% 12.79% 11.16% 9.96% 11.10% 19.44%

Germany 1.14% 4.35% 9.07% 3.76% 3.52% 5.49% 0.00% 0.00% 3.01% 0.00% 0.00%

US 1.13% 4.35% 37.42% 41.35% 46.50% 42.13% 37.47% 5.56% 5.11% 46.71% 0.00%

Ireland 1.11% 4.35% 0.43% 0.22% 0.28% 0.27% 0.04% 0.00% 1.79% 0.00% 0.00%

Singapore 1.06% 4.35% 0.27% 0.56% 0.54% 0.46% 1.51% 3.08% 6.83% 0.36% 0.00%

Canada 1.02% 4.35% 3.20% 2.46% 2.80% 2.84% 7.05% 0.00% 2.85% 0.00% 0.00%

Japan 0.96% 4.35% 17.95% 26.95% 8.99% 17.91% 5.61% 8.77% 2.81% 10.00% 0.00%

New_Zealand 0.82% 4.35% 0.24% 0.13% 0.33% 0.24% 1.93% 5.00% 7.66% 3.18% 0.00%

Portugal 0.73% 4.35% 0.47% 0.21% 0.30% 0.30% 2.73% 0.00% 1.04% 1.90% 0.00%

Concentration 4.35% 19.38% 25.61% 25.63% 22.84% 17.94% 12.94% 5.79% 25.62% 20.92%

- 28 -

Table 7 Portfolio Weights in 20 emerging markets

Table 7 shows the average weight of each country in the 10 portfolio strategies over the

Dec/1994-Dec/2006 period. The order of the countries is according to their average monthly

returns from high to low. Concentrate rate is the Herfindahl-Hirschman index.

Country

Monthly

return EQUAL GDP CAP DIV Combo MIN_exante MVE_exante BS MIN_expost MVE_expost

Argentina 1.56% 5.00% 5.95% 2.22% 3.81% 4.06% 0.41% 0.41% 4.96% 0.00% 0.00%

Brazil 1.81% 5.00% 15.98% 11.66% 20.00% 15.53% 0.15% 0.15% 4.22% 0.00% 0.00%

Chile 0.77% 5.00% 1.73% 3.60% 4.91% 3.40% 3.83% 3.83% 3.84% 9.90% 0.00%

Colombia 1.74% 5.00% 2.07% 0.70% 0.73% 1.17% 16.18% 16.18% 3.97% 8.47% 14.02%

Czech 1.75% 5.00% 1.43% 0.93% 1.54% 1.31% 16.31% 16.31% 4.43% 13.75% 23.09%

Hungary 2.27% 5.00% 1.20% 0.89% 0.86% 0.98% 0.00% 0.00% 10.47% 0.00% 28.37%

India 1.18% 5.00% 9.09% 8.18% 6.15% 7.68% 9.27% 9.27% 4.42% 9.32% 0.00%

Indonesia 1.26% 5.00% 3.90% 2.32% 2.12% 2.48% 0.00% 0.00% 3.00% 0.00% 0.00%

Israel 1.08% 5.00% 2.44% 2.53% 4.04% 3.05% 24.29% 24.29% 5.88% 27.50% 0.00%

Korea 1.32% 5.00% 11.45% 11.74% 8.52% 10.43% 1.62% 1.62% 4.16% 0.00% 0.00%

Malaysia 0.54% 5.00% 1.95% 6.94% 7.08% 6.29% 1.48% 1.48% 3.43% 4.66% 0.00%

Mexico 1.63% 5.00% 11.50% 7.97% 5.32% 8.37% 0.00% 0.00% 4.41% 0.00% 0.00%

Peru 1.53% 5.00% 1.28% 0.58% 0.87% 0.92% 5.13% 5.13% 4.72% 10.85% 5.68%

Philippines 0.00% 5.00% 1.74% 1.99% 1.11% 1.62% 2.58% 2.58% 2.95% 6.74% 0.00%

Poland 1.51% 5.00% 3.91% 1.58% 0.93% 2.14% 0.00% 0.00% 5.52% 0.00% 0.00%

Russia 3.48% 5.00% 7.06% 5.79% 3.73% 5.52% 0.00% 0.00% 10.27% 0.00% 27.70%

South Africa 1.08% 5.00% 2.94% 8.42% 10.98% 7.40% 16.74% 16.74% 4.06% 5.88% 0.00%

Taiwan 0.41% 5.00% 6.86% 15.87% 11.23% 11.18% 2.01% 2.01% 4.08% 2.94% 0.00%

Thailand 0.27% 5.00% 3.17% 2.89% 2.90% 2.96% 0.01% 0.01% 2.61% 0.00% 0.00%

Turkey 2.54% 5.00% 4.34% 3.20% 3.15% 3.52% 0.00% 0.00% 8.61% 0.00% 1.14%

Concentration � 5.00% 8.27% 8.66% 9.34% 8.05% 15.41% 15.41% 5.95% 14.30% 23.36%

- 29 -

Table 8. Portfolio performances in 3 developed countries quarterly rebalanced











Turnover rate is the amount of trading required to implement each portfolio strategies. Assuming

a half percentage transaction cost per one-way transaction, the total Transaction Cost is

calculated as turnover*0.5%*2. Net return is Gross return net transaction cost.

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 13.64% 14.70% 0.54%

0.96

0.53 11.48% 38.80% 0.39% 13.25%

GDP 12.54% 14.28% -0.56%

0.96

0.46 10.50% 36.44% 0.36% 12.18%

CAP 12.09% 15.21% -1.08%

0.97

0.41 9.77% 38.53% 0.39% 11.70%

DIV 13.55% 13.81% 0.54%

0.95

0.55 11.64% 32.24% 0.32% 13.23%

Combo 12.78% 14.15% -0.43%

0.97

0.48 10.78% 35.71% 0.36% 12.43%

MIN_exante 13.79% 13.97% 0.67%

0.96

0.56 11.84% 35.86% 0.36% 13.43%

MVE_exante 14.55% 17.69% 2.06%

0.88

0.49 11.42% 54.61% 0.55% 14.00%

BS 13.63% 14.70% 0.53%

0.96

0.52 11.47% 38.94% 0.39% 13.24%

MIN_expost 13.81% 13.80% 0.72%

0.96

0.57 11.90% 32.96% 0.33% 13.48%

MVE_expost 14.87% 14.54% 2.31%

0.89

0.62 12.75% 30.59% 0.31% 14.56%

MSCI_world 13.41% 14.34% 0.00%

1.00

0.52 11.35% 13.41%

- 30 -

Table 9 Portfolio performances in 7 developed countries quarterly rebalanced














cost.

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 20.83% 21.14% 3.54%

0.96

0.80 16.36% 71.91% 0.72% 20.11%

GDP 24.95% 25.28% 11.32%

0.70

0.83 18.56% 67.75% 0.68% 24.27%

CAP 19.68% 22.11% 5.91%

0.71

0.71 14.79% 62.93% 0.63% 19.05%

DIV 21.93% 23.29% 8.34%

0.69

0.77 16.51% 64.97% 0.65% 21.28%

Combo 21.74% 22.73% 8.00%

0.71

0.78 16.57% 63.99% 0.64% 21.10%

MIN_exante 18.36% 18.14% 2.87%

0.83

0.80 15.07% 83.75% 0.84% 17.52%

MVE_exante 21.91% 22.89% 5.48%

0.90

0.79 16.67% 91.13% 0.91% 21.00%

BS 20.82% 21.13% 3.69%

0.95

0.80 16.35% 74.55% 0.75% 20.07%

MIN_expost 17.51% 17.83% 1.96%

0.84

0.76 14.34% 64.50% 0.64% 16.87%

MVE_expost 26.93% 23.87% 10.35%

0.91

0.96 21.23% 64.22% 0.64% 26.29%

MSCI_emerging 17.84% 21.11% 0.00%

1.00

0.66 13.38% 17.84%

- 31 -

Table 10 Portfolio performances in 4 developed countries quarterly rebalanced














cost.

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 26.38% 24.86% 9.88%

0.90

0.90 20.20% 73.31% 0.73% 25.65%

GDP 28.30% 26.74% 14.90%

0.68

0.91 21.15% 66.37% 0.66% 27.64%

CAP 26.02% 25.61% 12.50%

0.69

0.86 19.46% 63.01% 0.63% 25.39%

DIV 28.13% 28.65% 15.03%

0.66

0.84 19.92% 65.24% 0.65% 27.47%

Combo 26.81% 26.45% 13.43%

0.68

0.87 19.82% 64.05% 0.64% 26.17%

MIN_exante 23.61% 23.31% 8.69%

0.79

0.84 18.18% 78.69% 0.79% 22.82%

MVE_exante 21.39% 22.98% 5.05%

0.89

0.76 16.11% 89.55% 0.90% 20.49%

BS 26.38% 24.86% 9.85%

0.91

0.90 20.20% 75.31% 0.75% 25.62%

MIN_expost 24.25% 22.68% 8.64%

0.84

0.90 19.11% 63.06% 0.63% 23.62%

MVE_expost 27.50% 24.68% 10.95%

0.91

0.96 21.41% 63.89% 0.64% 26.86%

MSCI_emerging 17.84% 21.11% 0.00%

1.00

0.66 13.38% 17.84%

- 32 -

Table 11 Portfolio performances of Delta strategies

Table 11 summarizes the quarterly performance of the Delta strategy in 23 developed countries

over the Dec/1979-Dec/2006 period and 20 emerging markets over the Dec/1994-Dec/2006




gross return divided by the standard deviation. The certainty-equivalent return (CEQ) is the





�

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

Develop 0.71% 14.66% -3.80%

(0.19)

(0.36) -1.44% 37.68% 0.38% 0.34%

Develop_exJapan 6.06% 16.05% 1.49%

(0.18)

0.38 3.48% 27.60% 0.28% 5.78%

Emerging 29.58% 22.85% 21.52%

0.18

1.12 24.36% 66.66% 0.67% 28.92%

- 33 -

Table 12. Portfolio performance in 23 developed countries annually rebalanced

Table 12 summarizes the annually performance of equally-weighted portfolio (EQUAL), the













�

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 17.32% 23.32% 4.56%

0.88

0.49 11.88% 36.03% 0.36% 16.96%

GDP 15.27% 19.77% 2.46%

0.89

0.47 11.36% 27.82% 0.28% 14.99%

PPP 16.05% 20.11% 3.28%

0.88

0.50 12.00% 27.38% 0.27% 15.78%

CAP 14.12% 18.73% 1.14%

0.91

0.44 10.61% 25.16% 0.25% 13.87%

DIV 14.85% 16.46% 2.25%

0.86

0.54 12.14% 22.60% 0.23% 14.63%

MSCI_world 13.41% 14.34% 0.00% 1.00 0.52 11.35% � � 13.41%

- 34 -

Table 13. Portfolio performance in 20 emerging markets annually rebalanced

Table 13 summarizes the annually performance of equally-weighted portfolio (EQUAL), the













cost.

�

Gross


Sharpe

Ratio CEQ Turnover

Transaction

Cost

Net

Return

EQUAL 25.27% 29.31% 3.33%

0.99

0.73 16.68% 69.09% 0.69% 24.58%

GDP 26.31% 33.96% 4.31%

0.99

0.66 14.78% 67.97% 0.68% 25.63%

PPP 28.66% 35.57% 6.71%

0.99

0.70 16.01% 69.12% 0.69% 27.97%

CAP 22.30% 30.19% 0.24%

1.00

0.61 13.18% 58.96% 0.59% 21.71%

DIV 21.86% 31.43% -0.04%

0.99

0.57 11.98% 57.22% 0.57% 21.29%

MSCI_emerging 17.84% 21.11% 0.00% 1.00 0.66 13.38% � � 17.84%

- 35 -

Figure 1 Efficient Frontiers for 23 developed countries

Figure 1 places the performance of equally-weighted portfolio (EQUAL), the GDP-weighted

portfolio (GDP), the capitalization-weighted portfolio (CAP), the dividend weighted portfolio (DIV),

composite portfolio (COMBO), global minimum variance portfolio(MIN), mean-variance efficient

portfolio (MVE), Bayes-Stein shrinkage portfolio (BS) and the MSCI world index (MSCI_world) for

23 developed countries over the Dec/1979-Dec/2006 period in the mean-variance space.

- 36 -

Figure 2 Efficient Frontiers for 20 emerging markets

Figure 2 places the performance of equally-weighted portfolio (EQUAL), the GDP-weighted

portfolio (GDP), the capitalization-weighted portfolio (CAP), the dividend weighted portfolio (DIV),

composite portfolio (COMBO), global minimum variance portfolio(MIN), mean-variance efficient

portfolio (MVE), Bayes-Stein shrinkage portfolio (BS) and the MSCI emerging market index

(MSCI_em) for 20 emerging markets over the Dec/1994-Dec/2006 period in the mean-variance

space.

- 37 -

Figure 3 Time series weights on Japan in GDP-weighted and capitalization-weighted portfolios

- 38 -

Figure 4 Portfolio variance against number of countries in 23 developed countries

Figure 4 ranks the countries based on their capitalization values, and the x-axis is read as: top 1

country in terms capitalization (which is the US), top 2 countries in terms of capitalization (which

are the US and Japan), top3 countries in terms of capitalization (the US, Japan and the UK). The

y-axis is the equally-weighted, capitalization-weighted and GDP-weighted portfolio risk (STD).

- 39 -

Figure 5 Portfolio variance against number of countries in 23 developed countries

Figure 5.1 ranks the countries based on their capitalization values, and the x-axis is read as: top 1

country in terms capitalization, top 2 countries in terms of capitalization etc. The y-axis is the

equally-weighted, capitalization-weighted and GDP-weighted portfolio risk (STD).

Figure 5.2 ranks the countries based on their GDP values, and the x-axis is read as: top 1 country

in terms GDP, top 2 countries in terms of GDP etc. The y-axis is the e equally-weighted,

capitalization-weighted and GDP-weighted portfolio risk (STD).

- 40 -

Figure 6 Delta strategy short sell side weight on Japan

Figure 6 shows the short sell side weight on Japan of the delta strategy over time.

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