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International diversification in the modernized economy
Bachelor Thesis Finance Faculty of Economics and Business
University of Amsterdam
Abstract
In this thesis, the effectiveness of international diversification will be tested for current and past market conditions in order to evaluate the changes of its effectiveness over time. The results of this research will be compared with the results from different studies, in order to get a better understanding of these changes over time. Globalization and economic freedom have rapidly changed among developed countries over the years and financial markets have become more integrated with one another than before. This suggest that returns on investments of different countries have become more dependent on each other. This raises the question whether or not the advantage of diversifying into foreign and domestic securities rather than exclusively diversifying into domestic securities has changed with respect to time. Fifteen developed countries will be used and compared in multiple time-series ranging from 1985 till 2015. The potential reduction of the minimum risk by adding international securities to the portfolio in the period 1985-1990 was 29,14% and for the period 2010-2015 this was 14,38%. The results are consistent with the expectations as argued in this thesis.
Name: Job Stoffelen
Student Number: 10645802
Program: Economics and Business (BSc ECB) Track: Economics and Finance
Supervisor: Philippe Versijp Date: 2017
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Statement of originality
This document is written by Job Stoffelen who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Table of Contents
1. Introduction ………3 2. Literature Review ………...6 2.1 Portfolio Selection ..………...6 2.2 Diversification ..………....9 2.3 Correlation .….………....13 2.4 Conclusion……….……14 3. Methodology ... 16 3.1 Data ... 16 3.2 Time period ... 17 3.3 Hypotheses ... 18 3.4 Research method ... 18 4. Descriptive Statistics ... 20 5. Results ... 23 5.1 Minimum-variance portfolio ... 235.2 Sharpe ratio portfolio ... 24
5.5 Discussion: Comparison between models ... 25
6. Conclusion ... 26
References ... 28
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1. Introduction
In the current economic environment, investors have experienced how unpredicted political changes can affect their portfolio returns. The news article ‘What has gone wrong in Brazil?’ evaluates the scandals surrounding the presidency of Brazil and the conviction of a major government-run oil company for bribery have led to a recession of the Brazilian economy in 2015 (BBC, 2016). Together with the strong depreciation of the Brazilian real against the US dollar many companies were in trouble. As the US is the biggest trading partner of Brazil, the depreciation of the Brazilian real (Bloomberg, 2016) led to an increase of short-term import cost. An unexpected shock like this is likely to have a negative effect on portfolio’s that only hold domestic securities, as uncertainty over the future of Brazil’s economy among investors grow. However, even when unexpected shocks like the scandals in Brazil affected the Brazilian market, other countries were not affected as much, even the closely related countries. The depreciation actually softens the effect as export goods from Brazil became cheaper. So, the scandals were only very hurtful to those who were exclusively invested in Brazilian securities. Some domestic invested investors have possibly reduced their loss by holding different securities, i.e. obligations and stocks from various industries. This method of reducing risk is known as diversification. But those invested in foreign securities on top of that are likely to avoid even more of the negative effect. This is known as international diversification and it was proven by Levy and Sarnat (1970) that this method significantly diversifies risk of a portfolio.
The world around us has changed dramatically since the findings of Levy and Sarnat in 1970. Globalization and economic freedom have developed over the years and technological
advancements have changed the fundamentals of the world economy. Data from the World Bank shows the rapid development of world trade, which is substantiated by the results from Hummels (2007) on the developments of transportation costs. That also reflect the impact of the technological advancement that took place over the years. These developments on
international trade are an indicator of the increasing globalization among developed countries. This increasing international trade due to lower trading cost has made companies from
different countries work more tidily together. These lower trading cost also resulted from more openness on regulation. The economic freedom is high among developed countries as shown by the index of economic freedom from the website of The Heritage Foundation (2015). These changes partly explain the increasing integration of markets among developed countries. Another indicator of these developments is the collaboration of the European
5 countries, which started to work together more closely after the second world war. Ultimately, this resulted in the creation of the Monetary Union and the implementation of a widely used common currency, the Euro. This lead to a highly significant increase of European market integration which was proved by Hardouvelis, Malliaropulos and Priestley (2006).
Given these changes on market integration and knowing that market integration has a positive effect on the correlation between markets the following can be explained. The increasing market integration among developed countries has increased the correlation between their market. This correlation between developed countries is known as the inter-country correlation, used by Levy & Sarnay (1970). When this inter-country correlation increases between two countries, returns on the securities from these countries will be more correlated. This makes it harder for investors to diversify their portfolio by the method of international diversification, as prices of different countries are more likely to move in the same direction. In summary, this means that the increased market integration among developed countries leads to more inter-country correlation over time which results in less gain from international diversification. Therefore, it is arguable whether this method is still as effective under current market conditions as it was under past market conditions for developed countries.
This thesis will add to the body of scientific knowledge regarding international diversification by evaluating and comparing the current against past advantages of this method. In order to evaluate whether this method is still as advantageous given the dramatic economic
environment changes. The research will be similar to that of Levy and Sarnat (1970), but reviewed in a very different economic environment. Levy and Sarnat conducted their research from 1951 till 1967, while this thesis conducts its research from 1985 till 2015. For this last timeframe the effectiveness of international diversification will be tested under current and past market conditions in order to see if the effect changed over time.
Given the findings of Levy and Sarnat (1970) and the different global changes as argued above we suspect that the mentioned changes will result in a more integrated economy among developed countries. This means that the inter-country correlations among developed
countries is higher than before, which implies that international diversification will be less effective under current market conditions.
6 At first a literature review will be presented about portfolio selection, diversification and correlation. For portfolio selection three different methods will be evaluated. For
diversification, a distinction will be made between domestic diversification and international diversification. For correlation, the relation between inter-country correlation and the
volatility of a portfolio will be explained. Secondly, the methodology of this thesis will be explained by evaluating the specifics about the data used, time-period chosen, hypotheses formulated and research method applied, in order to enable the reader to replicate the results of this research. Thirdly, a set of descriptive statistics will be provided concerning the
variables used for all the calculations that were made. Fourthly, the results of the calculations regarding the optimal portfolio model will be evaluated and used in order to test the
hypothesis stated in the methodology. At last, a conclusion will be provided on the research of this thesis regarding the application of the optimal portfolio model that was used in order to find the optimal portfolio in order to show the differences in the potential gains of
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2. Literature review
This section entails the literature review on international diversification and the development of inter-country correlations over the years. Firstly, the theoretical framework will be
presented in order to explain the theory regarding international diversification, which assumes that a significant gain can be achieved by adding foreign securities to a portfolio which only consist of domestic securities. Secondly, an explanation of the theory regarding portfolio selection. Then both diversification for a domestic portfolio and diversification for adding foreign securities to this portfolio will be explained. Finally, the literature on inter-country correlation will be presented and an explanation of this term will be provided.
2.1 Portfolio selection
“There are two stages of selecting a portfolio, the first stage starts with observation and experience and ends with beliefs about future performance of available securities. The second stage starts with these relevant beliefs about future performances and ends with the choice of the portfolio” (p.77). This idea was written by Markowitz in his paper on portfolio selection (1952), in this paper he explains a method for creating the optimal portfolio. One of the first methods ever for portfolio selection was created by Williams (1938) who based the selection of securities based on an anticipated return which was based on future expectations.
Discounting the future dividends of a company back to the present and comparing the
findings with current market prices. Although different methods exist, the method applied by Markowitz became the foundation for modern portfolio selection as this method was found to be the most precise.
The modern portfolio selection progress is still build upon the work of Markowitz (1952) and extended by Litner (1965). Litner explains further explains the optimization process for
constructing the optimal portfolio. Showing mathematically how an investor should constructs its portfolio. But also given an graphical model which explains the market equilibrium of an investor.
8 Figure 1: Market equilibrium model by Litner and Sharpe
Reprinted from: ‘International Diversification of Investment Portfolios’ Levy H. and Sarnat M., 1970, pp. 671
This graph shows the efficient frontier of portfolio A, if the investor would prefer to minimize risk, the investor will choose the point on line A that aligns with the lowest standard
deviation. However, it is important to understand that every investor is unique and will have its own preferences. A portfolio can be optimized by using the unique utility function of an investor, but this method is not required for the research of this thesis. As this thesis is not interested in the preference of investors but only in the effectiveness of international
diversification and this is better tested using two different methods of optimizing a portfolio. First, the three portfolios are evaluated below.
1. Maximum utility portfolio 2. Minimum-variance portfolio 3. Maximizing the Sharpe ratio
Starting with the ‘maximum utility portfolio’ seems logical, as an investor will most likely want to maximize its own utility. Determining the utility function can be hard or even
impossible and is different for any individual, making the results incomparable. It is necessary to use methods of portfolio selection that give more comparable results.
The second method is the ‘minimum-variance method’, which minimizes the variance of a portfolio. The variance is used to determine the risk of a portfolio, therefore minimizing the variance will minimize the risk. This method states that the investor is not interested in the returns of its investments, which is a rather extreme condition. Only institutional investors and banks might prefer lower risk over higher returns. But these extreme conditions are usable when conducting research on diversification as only this factor will be of interest to the
9 investor, making the results more conclusive on the effect of diversification.
As stated by Litner (1965, p589) the objective of diversification is to produce the best portfolio, the one with the most favorable combination of risk and return. Therefore, looking at the third method: ‘maximizing the Sharpe ratio’. The Sharpe ratio was first introduced by Sharpe (1966) and is formulated as follows.
(1) Sharpe Ratio =
𝑟𝑝−𝑟𝑓
𝜎𝑝
Which will give the optimal return per weighted unit of risk. This does not always mean that the investors utility function is maximized as investor are likely to be risk averse. But it can be seen as the mathematically optimal way of selecting the securities of a portfolio as it will give best combination of risk and return.
2.2 Diversification
Diversification is a very old word and its meaning is still similar to that written in the bible. It states: ‘But divide your investments among many places, for you do not know what risks might lie ahead’ (Ecclesiasticus 11.2). It was not until the work of Markowitz (1952) that the modern application of the term became important in the financial world. Investors were already applying this method but due to Markowitz there was also a mathematical framework for calculating the volatility of a portfolio from historical data. This framework showed that holding a portfolio consisting of multiple securities can reduce risk without or with little loss in return on a portfolio. Different empirical studies prove this idea and Statman (1987) in his paper questioned to which extend the effect of diversification on the risk of a portfolio was significant. In the graph below the answer to this question is represented.
Figure 2: Diversification of risk for a domestic portfolio
10 As shown in the graph a great deal of reduction in risk is created when adding the first extra securities to the portfolio. When the portfolio consists of 100 stocks it is clear that adding more securities to the portfolio doesn’t affect the risk of the portfolio as much, as
diversification is no longer advantageous. Statman (1987) used the S&P 500 to conduct his research which means he only used American securities. His portfolio is therefore exclusively diversified into domestic securities.
Many empirical studies examined the advantages of international diversification. The first to extend this literature to an international environment was Grubel (1968). Grubel concluded that growing international relations created by international diversification would lead to complete new kinds of world welfare gains. It was not until Levy and Sarnat (1970) first tested the potential gain from international diversification. Levy and Sarnat showed that countries that are not highly correlated with one another hold a high potential gain when adding securities from these countries to the portfolio. From their studies Solnik (1974) showed the significant advantage for investing internationally rather than domestically. Again, proving the findings of Levy and Sarnat (1970) that this was due to low correlations between stock prices of different countries. The extent of the effect of adding foreign securities to a portfolio is presented by the following graph.
Figure 3: The potential of International Diversification
Reprinted from: B. Solnik, “Why not diversify internationally rather than domestically”, Financial Analyst Journal, 1974, pp. 48-54.
11 From the graph, it can be seen that the potential gain from adding international securities to a portfolio. The black line represents the domestic portfolio consisting of U.S. stocks is more or less the same as the one from Statman (1987), both showing the potential of holding multiple stocks domestically. And the blue line represents an international portfolio with even less percent risk. Still this does not fully conclude that international diversification is an effective method as additional risk is taken due to regulation and extra cost associated with foreign investments. These barriers withhold investors from investing in foreign securities and tend to prefer holding domestic securities. Even though the potential gain from diversifying into foreign securities as well is significant. It seems as if investors have a preference for domestic securities and this phenomenon known as home-bias was proven by Coval and Moskowitz (1999). From there research it was shown that investor invest a majority of their investment in their home-country. This is idea was again tested by Cooper, Sercu and Vanpée (2013), trying to explain this behavior of investors and showing that this is due to different factors of which the most important seem to be extra cost, regulation and hedging demand.
Investing in foreign securities creates the following two anomalies: ‘exchange rate risk’ and ‘country risk’. ‘Exchange rate risk’ is the risk that is created as foreign securities have to be bought with a foreign currency, an investor will have to exchange his money first and when the security is sold the investor will have to convert the money back into the originally used currency. The exchange rate is in most cases not fixed and can be very volatile leading to either a gain from investing in foreign securities or a loss, on top of the return from the investment itself. The chance of losing value due to exchange rate fluctuations is known as the exchange rate risk. However, exchange rate risk can be hedged by using future contracts. For example, an investor can buy a forward contract, which means the investor signs a contract now so that the investor can later exchange the foreign currency of his investment against an agreed upon exchange rate. This means that the investor has two options, hedging or no-hedging for currency risk. However, which of the two is better. The following graph from Solnik (1995) provides an overview of hedging against no-hedging among developed countries.
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Reprinted from: B. Solnik, “Why not diversify internationally rather than domestically”, Financial Analyst Journal, 1974, pp. 48-54.
From this graph, it is clear that the difference between both methods is not of great impact on the risk of the portfolio. Even though international hedge is better in this example there are many different perspectives from which the effectiveness of hedging can be evaluated. Eun and Resnick (1988) found significant evidence for hedged investment strategies to outperform unhedged investment strategies due to highly correlated exchange rate changes against the US dollar. This is supported by the findings of Campbell, Serfaty De Medeiros and Viceira (2010), whom tested seven major developed-currencies such as the dollar, euro, Japanese Yen, Swiss Franc, Pound Sterling, Canadian Dollar and Australian Dollar and found a
substantial reduction in risk for optimal currency hedging. But in conflict De Roon, Eiling and Gerard (2010) showed that the cost of hedging can be significant as well and therefore lead to a reduction of the Sharpe ratio. The study showed improved performance of average return and the Sharpe ratio for being exposed to currency risk. These studies show the uncertainty of currency fluctuations for developed countries and that there is no conclusive solution for it. The preference of the investor will be the decisive factor when it comes to currency hedging. ‘Country risk’ is risk specific to a country and can be split into two kinds: ‘political risk’ and ‘economic risk’. Political risk resolves from the political changes in a country, like changes of the regulations on trade, taxation and/or business activity. But political risk can also resolve from the political instability of a country, which creates uncertainty among investors, like unexpected regulations changes or corruption inside the political system. ‘Economics risk’ is risk associated with a countries financial conditions, the ability to repay its debt, the ability to steer inflation and the ability to reduce unemployment. These two kinds of risk can be
13 dependent on each other, making it difficult to allocate a proportion of the country risk to each kind. However, expectation about country risk can be made and it is likely that securities reflect this information. Even though country risk is specific for a country it is necessary to understand that the financial activities in one country influence the financial activities of another country. Meaning that this risk is not only subject to the country itself as it can
indirectly influence the returns on foreign securities as well. The extent to which this spillover effect is effective is given by the correlation between these countries.
2.3 Correlation
Correlation is a statistical term and describes to what extend two variables have a relationship with one another. If we assume the mathematical solution of Markowitz (1952) on portfolio selection, correlation is vital for determining the volatility of a security. In order to formulate the hypothesis of this thesis it is necessary to make assumption on how this correlation is expected to develop. As both papers from Levy and Sarnat (1970) and Solnik (1974) argued that the potential gain of international diversification is due to the correlation between countries.
In the formula for volatility of a portfolio from the paper of Markowitz (1952) there is one term which represents the correlation between the securities of the portfolio. Knowing that the securities that will be used in this thesis are indexes reflecting a domestically diversified portfolio, representing the performance of a countries economy. It can be concluded that the correlation between these index prices represents the relationship between the economies of these countries, which is also known as the inter-country correlation.
The inter-country correlation for example will show the relation between the price of a portfolio consisting of securities from country A and the price of a portfolio consisting of securities from country B. If the correlation is higher or lower than zero there is correlation between these countries, if it is equal to zero there is no correlation between these countries and would therefore be independent. When these countries are dependent there has to be correlation, for instance in the form of trade. The correlation will then be between zero and one, resulting in potential gain from international diversification. When the correlation is one both countries will be fully dependent on one another, resulting in zero potential for
diversifying internationally. To summarize, negative correlation leads to very high potential gains, a correlation between zero and one will lead to high potential gains and a correlation of
14 one will mean there is no potential gain from international diversification. From this it can be concluded that as the inter-country correlation of country A and B gets closer to one the potential gain from international diversification becomes smaller and smaller, which was first discussed by Grubel (1968). Another example would be the study of Stulz (1999) on the relationship between globalization and cost of capital. Here the correlation between a small country and the world is evaluated as shown in the following graph.
Figure 4. Impact on the volatility of the world market portfolio, 𝜎𝑊𝑜𝑟𝑙𝑑 after, of adding a country to the world market portfolio.
Reprinted from Stulz, R. M. (1999). ‘Golbalization, corporate finance, and the cost of capital. Journal of applied corporate finance’, 12(3), pp. 65.
We assume that the volatility of the world market portfolio before adding a country is 15%. The old market portfolio represents 90% of the new one, so that the new country has a weight of 10% in the new market portfolio. We express, 𝜎𝑊𝑜𝑟𝑙𝑑 𝑎𝑓𝑡𝑒𝑟, after as a function of the volatility of the small country market portfolio, 𝜎𝑆𝑚𝑎𝑙𝑙 𝑐𝑜𝑢𝑛𝑡𝑟𝑦, and the correlation between the small Small country country market portfolio and the world market portfolio before the addition, 𝜌𝑆𝑚𝑎𝑙𝑙 𝑐𝑜𝑢𝑛𝑡𝑟𝑦,𝑊𝑜𝑟𝑙𝑑 𝑏𝑒𝑓𝑜𝑟𝑒.
Figure 4 from Stulz (1999) gives a clear example of the effect of inter-country correlation. It examines the effect on the volatility of the returns, standard deviation, due to adding a country to the world portfolio. The correlation between the small country and the world before the country ranges from minus one to plus one. From the graph, it is clear that this correlation has a positive effect on the volatility of the world portfolio after the country is added. Meaning that the risk of the portfolio worsens when this correlation increases. This is logical because a
15 higher correlation would mean that the returns of the world portfolio before and the added country are more dependent on each other, which means that the prices move in the same direction. This results in less potential gain from diversification.
2.4 Conclusion
In summary, this literature review explained the ideas of Markowitz (1952) on how a portfolio selection is made. Further examined are the three different methods: ‘maximizing utility portfolio’, ‘minimum-variance portfolio’ and ‘maximizing the Sharpe ratio’, each of this method has its unique conditions. After which the concept of diversification and the potential gain from international diversification were evaluated. Explaining that the potential gain from international diversification is likely to have changed over time. As the developments of globalization and economic freedom have changed the inter-country correlation among developed countries as market became more integrated than before.
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3. Methodology
In this section the methodology used in this thesis will be presented and explained by dividing the methodology part into labeled subsections that are titled: data, time period, hypotheses and research method. The purpose of this section is to enable the reader to replicate the results of this thesis. On top of the explanation provided in this section, formulas and tables will be used with descriptions in order to give the reader a better understanding of the research methods used.
3.1 Data
In order to conduct the research of this thesis data has been gathered with regard to the MSCI index. Data regarding the MSCI index has been converted to US Dollars by using the
Datastream option in the search engine. Thomson Reuters Datastream and Wharton Research Database Service System (WRDS) have therefore been used in order to gather all the data. No missing data gaps had to be filled, however there was data missing for Korea before 1985, this data has however not been used in the research of this thesis. The following selection of fifteen developed countries has been made: Japan (JP), United Kingdom (UK), United States (USA), Australia (AUS), Germany (DE), Korea (KOR), Canada (CAN), France (FR),
Switzerland (CH), Sweden (SE), Belgium (BE), The Netherlands (NL), New Zealand (NZ), Spain (ES) and Norway (NO). Of which all representing higher developed countries that run stabilized economies and can therefore be defined modernized. This statement is supported by the HMI index of 2014 from the website of the United Nations Development Programme. And the index on economic freedom as well as the GDP per capita of each country retrieved from the website of the Foundation of Heritage and the website from the World Bank (2015). A table of all the data for each corresponding country is given below.
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Country Name World Rank
HMI Index 2016 Score Economic Freedom GDP per Capita (PPP) Belgium 44 68,4 $42.973 Canada 6 78,0 $44.843 France 75 62,3 $40.375 Germany 17 74,4 $45.888 Japan 22 73,1 $37.390 Korea, South 27 71,7 $35.277 Netherlands 16 74,6 $47.355 New Zealand 3 81,6 $35.152 Norway 32 70,8 $66.937 Poland 39 69,3 $25.105 Spain 43 68,5 $33.711 Sweden 26 72,0 $45.986 Switzerland 4 81,0 $58.087 United Kingdom 10 76,4 $39.511 United States 11 75,4 $54.597 World 175** 60,7* $10.058 High Income - - $39.717 Middle Income - - $4.776 Low Income - - $618
*Indicates the total countries in the index of the economic freedom, **Indicates the average of all the scores.
3.2 Time period
In this thesis, data for the MSCI index for all countries has been gathered starting in 1-1-1982 until 1-12-2016. Periods of 5 years have been used for constructing the portfolios, ranging from 1985 to 1990, 1990 to 1995, 1995 to 2000, 2000 to 2005, 2005 to 2010 and 2010 to 2015. The portfolios of each period represent the portfolio for the years of that time-period, so the optimal portfolio computed by the minimum-variance method of 1985 to 1990 represent the same portfolio for the years 1985, 1986, 1987, 1988, 1989 and 1990.
18 3.3 Hypothesis
On the basis of foregoing research and the literature review, it is expected that the gain from international diversification among developed countries under current market conditions is significantly less than the gain from this method when applied under market conditions in the past. The hypothesis is described as:
(1) H0: The gain from diversifying a portfolio internationally rather than domestically is lower in current market conditions than market conditions in the past.
(2) H1: The gain from diversifying a portfolio internationally rather than domestically is unchanged by current market and past market conditions.
The hypothesis will be tested for the two different methods, the minimizing variance method and the maximizing Sharpe ratio method, which will be used in order to compute the optimal portfolio.
3.4 Research method
In order to examine the potential gain from international diversification among developed countries. The method as applied by Sarnat and Levy (1970) will be used as guideline in order to make the results comparable.
First, For the whole period of 1982 till 2015 the monthly rate of return for each country was defined as the percentage change in the dollar value of its MSCI index.
(2) 𝑟𝑖 =𝑃𝑖(𝑡)−𝑃𝑖(𝑡−1)
𝑃𝑖(𝑡−1)
From this the monthly rate of return the average rate of return for a given security was computed.
(3) 𝑟̅ = 1
𝑁∑ 𝑟𝑖(𝑡) 𝑁
𝑡=1 .
The variance of the security will then be determined by a summation of the excess return that can easily be conducted from 1 and 2:
19 (4) 𝜎𝑖2 = 1 𝑁∑ (𝑟𝑖− 𝑟̅𝑖) 2 𝑁 𝑡=1
For calculating the variance of a portfolio that consist of three assets the following formula can be used.
(5) 𝜎𝑝,𝑥2 = 𝑉𝑎𝑟(𝑅𝑝,𝑥) = 𝑥𝐴2𝜎𝐴2+ 𝑥𝐵2𝜎𝐵2+ 𝑥𝐶2𝜎𝐶2+ 2𝑥𝐴𝑥𝐵𝜎𝐴𝐵+ 𝑥𝐴𝑥𝐶𝜎𝐴𝐶 + 𝑥𝐵𝑥𝐶𝜎𝐵𝐶
Here, the problem arises that adding one security to the portfolio there are twice as many covariance terms when compared to a portfolio consisting of two securities. This can easily be resolved by using matrix notation.
(6) 𝜎𝑝,𝑥2 = 𝑉𝑎𝑟(𝑥′𝑅) = 𝑥′ ∑ 𝑥 = (𝑥 𝐴, 𝑥𝐵, 𝑥𝐶) ∙ ( 𝜎𝐴2 𝜎𝐴𝐵 𝜎𝐴𝐶 𝜎𝐴𝐵 𝜎𝐵2 𝜎𝐵𝐶 𝜎𝐴𝐶 𝜎𝐵𝐶 𝜎𝐶2 ) ( 𝑥𝐴 𝑥𝐵 𝑥𝐶 ) = 𝑥𝐴2𝜎 𝐴2 + 𝑥𝐵2𝜎 𝐵2+ 𝑥𝐶2𝜎𝐶2+ 2𝑥𝐴𝑥𝐵𝜎𝐴𝐵+ 𝑥𝐴𝑥𝐶𝜎𝐴𝐶 + 𝑥𝐵𝑥𝐶𝜎𝐵𝐶
Applying this to a fifteen by fifteen matrix will give the volatility of the portfolio. This matrix is computed by using the ‘MMULT’ command in excel. Now we can use the solver add-on in excel which simply changes the weight that is invested in each index with the restriction that all weights add up to 100%. The formula as provided in the literature review, formula 1, will be maximized using the solver tool in excel. And for the formula of the volatility of the portfolio as given by formula 6 will be minimized using the solver tool in excel. This will give us the optimal portfolio, as described by Markowitz (1959), for each method for each time-period. A table with the weights of the optimal portfolio for each time-period and for each method is provided in the appendix, where the MSCI index for each country is denoted by the corresponding country’s code.
For each method for each time-period the optimal portfolio will be compared to the domestic diversified portfolio, which only invest in the home-country, by using the same formula that was used in order to compute the monthly return. Dependent on which method is applied, the following formula is used.
(7) 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 = −𝜎𝑜−𝜎𝑑
20 (8) 𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒 =𝑆ℎ𝑜−𝑆ℎ𝑑
𝑆ℎ𝑑 (Sharpe ratio method)
The small letter ‘o’ stands for the optimal portfolio while the small letter ‘d’ represents the exclusively domestic diversified portfolio. The exclusively domestic diversified portfolio is the same for each period but different for each country. The so-called home-country
represents the country which is tested, if this would be for instance Japan, this would mean that the domestic portfolio consists for 100% of the MSCI Japan index. The same goes for all the other countries. The minus sign in front of the fraction in formula 7 is used in order to show the following: the proportion of the volatility of the domestic portfolio that can be diversified by changing to the optimal portfolio. This can be interpreted as the potential gain from international diversification given the minimum-variance method. For all the countries, a comparison will be made and the average of all these results will be computed on basis of the following formula. (9) 𝐴𝑣𝑒𝑟𝑎𝑔𝑒 = 1 𝑛∑ 𝑎𝑖 𝑛 𝑖=1 = 1 𝑛(𝑎1+ 𝑎2+ ⋯ + 𝑎𝑛)
The average of the results as computed in the way stated above will be determined using the ‘AVERAGE’ command in excel. These averages have been constructed into a figure that can be found in the section ‘results’.
For calculating the inter-country correlation from 2000 till 2015 I used the same matrix algebra to compute the covariance-variance matrix but then applied to the corresponding timeframe and with the information provided by this matrix the correlations between the countries was computed by the following formula.
(10) 𝜌𝐴𝐵 = 𝐶𝑜𝑟𝑟(𝐴, 𝐵) = 𝐶𝑜𝑣(𝐴,𝐵) (𝜎𝐴∙𝜎𝐵)
For testing the hypothesis two separate tests for each method are used. For the minimum-variance method a linear regression was conducted in STATA on two variables. The first variable represents the proportion of the domestic portfolio that can be diversified by
changing to the optimal portfolio for a specific country denoted by the country code provided in section 3.1. The second variable is time, representing the years ranging from 1985 to 2015. In order to see if this proportion as described above has changed over the years.
21 For testing the significant difference between the Sharpe ratio of the domestic portfolio
compared to the optimal portfolio the corrected version of the Jobson-Korkie test (1981) that was described by Memmel (2003) is used. The hypothesis of the test is as followed.
𝐻0: 𝑆ℎ𝑖 − 𝑆ℎ𝑛 = 0 𝐻1: 𝑆ℎ𝑖 − 𝑆ℎ𝑛 < 0
The ‘i’ represents the domestic diversified portfolio and the ‘n’ represents the optimal portfolio given the Sharpe ratio method. In order to test this hypothesis, the following steps were taken for computing the Z-value, as described by Memmel (2003).
(11) 𝑍 = 𝑆ℎ̂𝑖−𝑆ℎ̂𝑛
√𝑉̂ The asymptotic variance of the Z-value above is as follows.
(12) 𝑉 =1 𝑇[2 − 2𝜌𝑖𝑛+ 1 2(𝑆ℎ𝑖 2+ 𝑆ℎ 𝑛 2 − 2𝑆ℎ 𝑖𝑆ℎ𝑛𝜌𝑖𝑛2)]
In order to compute the expectation of the asymptotic variance the individual terms have been calculated by using the following formulas.
(13) 𝜌𝑖𝑛= 𝜎𝑖𝑛 𝜎𝑖𝜎𝑛 = 𝐶𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒 (𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐,𝑜𝑝𝑡𝑖𝑚𝑎𝑙) 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐)∗𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒(𝑜𝑝𝑡𝑖𝑚𝑎𝑙) (14) 𝑆ℎ𝑖 = 𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 (𝑑𝑜𝑚𝑒𝑠𝑡𝑖𝑐) (15) 𝑆ℎ𝑛 = 𝑆ℎ𝑎𝑟𝑝𝑒 𝑟𝑎𝑡𝑖𝑜 (𝑜𝑝𝑡𝑖𝑚𝑎𝑙) (16) 𝑇 = 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠
For each portfolio sixty observations on prices for each period of 5 years have been used. With these observations, the covariance of each period was computed by using the COVARIANCE.S command, on the returns of the domestic portfolio and the summation of the multiplication of the optimal portfolio weights with the returns, for each period and each country. The Sharpe ratio of the was domestic portfolio for each period and country was calculated by using formula 1 and for the optimal portfolio the results from the excel solver tool were used. With all this information, the asymptotic variance was computed for each period for each country and from this the Z-value for each period for each
22 country was computed as well as an average of the volatility and return of the domestic portfolio, as well as on the correlation between the optimal and domestic portfolio. For simplicity, the risk-free rate was not applied in the Sharpe ratio formula. The results can are presented in section 4.
23
4. Descriptive Statistics
This section will present descriptive statistics concerning the data used in order to construct the optimal portfolio for both methods.
Table 1: Descriptive statistics
The table below shows the proportion of the volatility of the exclusively domestic diversified portfolio that can be diversified by changing to the optimal portfolio based on the minimum-variance method as explained in section 3.3. 1985-1990 1990-1995 1995-2000 2000-2005 2005-2010 2010-2015 Japan 0,481 0,599 0,401 0,258 0,047 0,109 UK 0,451 0,402 0,038 0,075 0,188 0,225 USA 0,263 0,080 0,198 0,085 0,106 0,144 Australia 0,546 0,423 0,205 0,244 0,364 0,386 Germany 0,498 0,429 0,225 0,442 0,368 0,395 Korea 0,255 0,602 0,735 0,588 0,500 0,378 Canada 0,244 0,164 0,300 0,320 0,390 0,171 France 0,501 0,401 0,281 0,322 0,283 0,386 Switzerland 0,371 0,349 0,312 0,168 0,163 0,170 Sweden 0,441 0,550 0,392 0,543 0,393 0,369 Belgium 0,494 0,272 0,172 0,356 0,400 0,289 Netherlands 0,269 0,249 0,241 0,330 0,327 0,320 New Zealand 0,601 0,515 0,339 0,390 0,329 0,245 Spain 0,520 0,505 0,454 0,360 0,347 0,504 Norway 0,509 0,544 0,424 0,396 0,511 0,457
24 Table 2: Descriptive statistics
The table below shows the amount with which the inter-country correlation from the research of Levy and Sarnat from 1951 till 1967 increased when compared to the research from this thesis from 2000 till 2015.
Japan New Zealand United Kingdom United States
Japan 1 +0,75 +0,59 +0,88
New Zealand - 1 +0,14* +0,62
United Kingdom - - 1 +0,79
United States - - - 1
Even though the research was not conducted in the exact same way it can still be concluded that inter-country correlations have indeed become greater in the time-period 2000-2015 when compared to 1951-1967. Despite the extra-large error in this case, due to the application of two different methods, inter-country correlation show to be significantly higher in the time period 2000-2015 when compared to 1951-1967 but still only the inter-country correlation between New Zealand and the United Kingdom didn’t increase significantly. However, its inter-country correlation from 1951 till 1967 was relatively high, 0,59, when compared to the average of 0,06.
Table 3: Descriptive statistics
The table below shows the amount with which the inter-country correlation from the research of Levy and Sarnat from 1951 till 1967 increased when compared to the research from this thesis from 2000 till 2015.
Belgium France Germany Netherlands United States
Belgium 1 +0,19 +0,20 +0,22 -0,08*
France - 1 +0,48 +0,45 +0,50
Germany - - 1 +0,15 +0,41
Netherlands - - - 1 +0,31
United States - - - - 1
Again, an extra-large error was used to test whether the inter-country correlations increased significantly. Only one value did not significantly increase, however the inter-country correlation of Belgium and the United States from 1951-1967 was 0,83 on an average of 0,58.
25 Table 4: Descriptive statistics
The table below gives an overview of the regression results that were computed in the way that is explained in section 3.4. For each country, a separate regression was conducted in order to evaluate
the development of the effect of international diversification for the minimum-variance method with respect to time.
Intercept Coefficient T-score
Japan 39,17129 -0,019425*** -10,33 United Kingdom 20,42712 -0,010095*** -3,82 United States 7,909075 -0,0038797*** -3,19 Australia 12,09221 -0,0058624*** -2,73 Germany 6,527328 -0,0030655* -1,82 South Korea -4,233791 0,003089 0,72 Canada -3,36957 0,0018167 1,12 France 11,0098 -0,0053215*** -3,99 Switzerland 18,85301 -0,009297*** -12,16 Sweden 7,230268 -0,0033913*** -2,52 Belgium 6,953246 -0,0033087 -1,59 Netherlands -5,74825 0,0030185 5,79 New Zealand 25,87487 -0,0127326*** -13,02 Spain 7,846826 -0,003698*** -2,88 Norway Average 4,826566 10,35801 -0,0021761** -0,005003*** -2,18 -9,98
*Indicates significance at a 5% level, ** Indicates significance at a 2% level, *** Indicates significance at a 1% level. Average represents the average of the potential gain given by the fraction of the volatility that is diversified after changing to the optimal portfolio and these averages were regressed against the years ranging from 1985 till 2015.
26 Table 5: Descriptive statistics
The table below shows the Z-values computed by using the corrected Jobson and Korkie (1981) test of Memmel (2003) as described in section 3.4. For each country, a separate regression was conducted in
order to evaluate the development of the effect of international diversification for the Sharpe ratio method over time.
Z-value 1985-1990 1990-1995 1995-2000 2000-2005 2005-2010 2010-2015 Japan -0,832 -4,446*** -1,207 -0,426 0,000 -0,133 United Kingdom -0,767 -3,700*** -2,390*** -0,312 -4,652*** -4,753*** United states -0,496 -3,052*** -0,765 -0,375 -2,590*** -0,883 Australia -1,402* -5,042*** -1,044 -0,314 -16,837*** -0,068 Germany 0,000 -0,022 -0,632 -0,089 -8,700*** -3,014*** South Korea -14,407*** -4,144*** -1,370* -2,355*** -7,097*** -0,593 Canada -3,627*** -3,130*** -0,925 -0,137 -4,739*** -1,926** France -0,608 -0,435 -8,159*** -0,190 -3,114*** -1,571* Switzerland -0,394 -7,170*** -2,014** -0,588 -4,721*** -0,930 Sweden -1,476* -0,451 -0,034 -0,004 -27,389*** -0,439 Belgium -0,564 -4,781*** -3,571*** -0,343 -3,646*** -0,591 Netherlands -0,351 -3,149*** -2,048** -0,146 -4,196*** -0,935 New Zealand -2,784*** -2,528*** -4,166*** -0,139 -21,053*** -2,124*** Spain -2,712*** -32,437*** -2,148** -0,402 -2,651*** -64,384*** Norway -1,538* -0,297 -0,109 -0,046 -4,500*** -0,166 Average -1,641* -4,211*** -1,259 -0,209 -24,596*** -0,782
27
5. Results
In this section, the hypotheses stated in section 3.3 will be tested and evaluated based on the research methodology consisting of two methods. First the optimal portfolio will be created for each method for each time-period. These portfolio’s will be compared to the domestic portfolio, which consist of one of the country’s indexes, for each time-period. The
significance of the differences for both methods was tested, after which an evaluation of the results is given.
5.1 Minimum-variance method
The results in in table 1 show the significant potential of international diversification when an investor seeks to minimize risk. In order to show the changes of the proportion that can be diversified by international diversification over time: the average of all the countries for each time-period was computed as described in section 3.4. The graph below represents a line that connect these averages together in order to show to changes over the years.
In the graph, the trendline was created to show that the effect of international diversification is indeed weakening with respect to time. However, this does not show whether this change is statistically significant.
Therefore, the regressions on each country and the average of all countries with respect to time was conducted, as described in section 3.4, in order to evaluate the significance of the change of the fraction that of the domestic diversified portfolio that can be further diversified
0.25 0.27 0.29 0.31 0.33 0.35 0.37 0.39 0.41 0.43 0.45 1985-1990 1990-1995 1995-2000 2000-2005 2005-2010 2010-2015
Average gain of International Diversification
(Minimum-variance method)
28 by changing to the optimal portfolio based on the minimum-variance method. The regression outputs from these regressions can be found in the appendix and an overview of the results is represented in table 4 in section 4. From table 4 it is shown that 12 of the 15 developed countries that were used in this research showed a significant negative relation with respect to time for an alpha of 5%. This means that the effect from international diversification for the minimum-variance method has weakened over time, which would mean that the hypothesis stated in section 3.3 would hold. The average of all the 15 countries also showed a highly significant negative relation for an alpha of 1. Belgium shows a negative relation but the evidence is not significant enough for the hypothesis stated in section 3.3 to hold, given the one-sided p-value of 0,061 and an alpha of 5%. However, it would hold for an alpha of 10%. This slight difference could be explained by the relative higher inter-country correlation for Belgium when compared to other countries from the beginning in 1985. South Korea, Canada and the Netherlands show a positive effect and are therefore inconsistent with the hypothesis stated in section 3.3. The same reasoning as for Belgium would apply for each of these countries. For Canada, the effect is different as the optimal portfolio for the time-period 2010-2015 consists for 27,46% of the Canada index which means that there is little advantage from changing to the optimal portfolio when compared to other time-periods for this country. As Canada has a relative low volatility in this time-period when compared to the other countries, this means that the other periods are likely to be more advantageous as the last time-period which resolves in a positive relation. For South Korea, the same problem as for Canada arises as the optimal portfolio consists for 46,86% of South Korea but then for the time-period 1985-1990. In order to show that the hypothesis stated in section 3.3 holds for South Korea a second regression on the period ranging from 1990 till 2015 was conducted, the output of this regression is shown in the appendix. The result shows that there is a highly significant
negative relation with an alpha of 1%. For the Netherlands, the volatility is low for the whole time period ranging from 1985 till 2015. This could again possible be due to the high inter-country correlation that the Netherlands has from the beginning in 1985 when compared to the other countries. Summarizing the results shows significant prove for the reduction of the effect from international diversification over time. Only Canada and the Netherlands show insignificant prove for the hypothesis in section 3.3, which is could be explained by the high inter-country correlation of these countries and/or the relative constantly low risk that these countries bare.
29 5.2 Maximizing Sharpe ratio method:
The potential gain from international diversification by maximization the Sharpe ratio was computed in the same way as the minimum-variance, however this showed significant errors as returns greatly forced the Sharpe ratio to differ for different time-periods. The results showed highly significant advantages ranging from 0% to 12406% with an average of 792%. 0% meaning that the domestic portfolio is better than the optimal portfolio for this method, which means the investor will not change to the optimal portfolio as the investor will be worse off in that case. The average shows highly significant improvement of the portfolio by adding foreign securities to it, however no conclusive arguments can be made about the development of market integration among developed countries. For different time-period there is no logical path as there was for the minimum-variance method.
The vertical axis represents the multiplication of the Sharpe ratio from the domestic portfolio which will give the Sharpe ratio of the optimal portfolio. In example, if the multiplication is 2,87 as it is for the time-period 2000-2005 it means that the Sharpe ratio of the domestic portfolio can be increased by this factor when the investor changes to the optimal portfolio, meaning that the investor gets a 187% higher Sharpe ratio. The horizontal axis represents the different time-periods. The graph shows the random path that the difference between the Sharpe ratio of the optimal portfolio and the Sharpe ratio of the exclusively domestic invested portfolio has. The trend shows an increasing path but there is no literature explanation or explanation from the results of this thesis. Logically, it’s the effect of the returns of a the different indexes on the Sharpe ratio and is due to the economic environment that created
2.00 4.00 6.00 8.00 10.00 12.00 14.00 16.00 1985-1990 1990-1995 1995-2000 2000-2005 2005-2010 2010-2015
Average gain from International Diversification
(Maximum Sharpe ratio method)
30 these returns. For instance, we see a high Sharpe ratio potential gain for 2005-2010, which was the period of the internet bubble which resulted in extra high returns. In the period after we see a reduction of the Sharpe ratio potential gain which was due to the economic crisis starting in 2008. In order to see if there is a significant difference between the Sharpe ratios of the domestic portfolio and the optimal portfolio the Jobson and Korkie (1981) test, which was corrected by Memmel (2003), was conducted as described in section 3.4.
The results from the Jobson and Korkie test provide a statistical prove, which can be used to evaluate whether the Sharpe ratio of the optimal portfolio is significantly greater than the domestic portfolio for each country for each time-period. It’s important to understand that part of the findings result from the fact that the countries index is part of the optimal portfolio, which means that the Sharpe ratios are likely to be indifferent in that case. Therefore, a test on the average of all the domestic portfolios for each time-period was computed as described in section 3.4. The result as shown in table 5 show that the Z-value is further from zero at the first three time-periods when compared to the fourth and sixth time-periods which would mean that the Sharpe ratio of the optimal portfolio became closer to the Sharpe ratio of the domestic portfolio over the years. The fifth time-period shows an error, which could be explained by the economic bubble that created abnormal returns. This would mean that the Sharpe decreased over time which would be consistent with the hypothesis stated in section 3.3. The same holds when the amount of countries that show a significant difference is divided by the amount of countries that is not part of the optimal portfolio for each given time-period. This gives us the following factors: 8/13, 12/13, 8/13, 1/12, 13/14, 6/13. From these factors and knowing that fifth time-period is an error, it is shown that the difference between the Sharpe ratios of the domestic and the optimal portfolio is significant for less and less countries over time.
31 5.3 Discussion
The results from the minimum-variance method the hypothesis is accepted and it is clear that there is less potential gain from international diversification under current market conditions than market conditions in the past. The same hypothesis is however rejected for the maximum Sharpe ratio method as there is no conclusive result that showed a reduction of the Sharpe ratio over time. This can be clarified by the fact that return of a portfolio is not affect in the same way as the variance of a portfolio is by the increasing inter-country correlation as argued in this thesis. In conflict the return on a portfolio is subject to many other factors as well. These factors are likely to have a much greater effect on the portfolio return than the effect of higher inter-country correlations.
32
6. Conclusion
The effectiveness of diversifying risk through international diversification is proven to still be an effective method among developed countries under current market conditions. A possible conclusion could be that country risk, which is specific for each country, can be diversified. In this research the effectiveness of minimum-variance and the Sharpe ratio method is shown to diminish over time. It is argued that this resolves from the increasing globalization and economic freedom developments that led to more market integration among developed countries. This means that the inter-country correlation has increased with respect to time and this is proven for some developed countries by comparing data from the study of Levy & Sarnat (1970) with the data provided by this thesis. Whether this increasing inter-country correlation is the reason for the diminishing effectiveness of international diversification is only evaluated in the literature review of this thesis, in order to find evidence for the
correlation between both effects further research would be required. That the effectiveness of international of international diversification diminishes over time is proven to hold given the results from this thesis but it would be interesting to conduct this research again in the future to see if the hypothesis will hold again.
The potential of adding foreign securities to an exclusively domestic invested portfolio is proven to be very effective for the return on the portfolio. The Sharpe ratio highly increases in for most countries when the optimal international portfolio is created. Some of the MSCI indexes had really high returns for certain time-periods and the optimal portfolio takes advantage of this by investing fully in the corresponding index. This however is not
conclusive for diversifying risk as the risk of these optimal portfolios is relatively high when compared to most of the exclusively domestic invested portfolios.
In this thesis, there is no research conducted with respect to these developing countries. The reason for this is that developing countries are generally more regulated, hold higher risk and experience periods with extreme volatility. Making the results inconsistent with the results from the developed countries. Further research on the potential gain from adding foreign securities of developing countries would be interesting for extending this thesis.
33 Last, this thesis did not provide any research on the exchange rate risk that is taken when foreign securities that are traded in a different currency then that of the home-country. In the literature review an argumentation on exchange rate risk has been provided. However, there is no conclusion on whether an investor should hedge this risk or not. It would be interested to extend this research by making a correction for exchange rate risk and to see whether hedging or not-hedging would be better.
34
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36
Appendix:
_cons 39.17129 3.761182 10.41 0.000 31.47881 46.86377 Time -.019425 .0018806 -10.33 0.000 -.0232712 -.0155788 JP Coef. Std. Err. t P>|t| [95% Conf. Interval] Total 1.19013024 30 .039671008 Root MSE = .09365 Adj R-squared = 0.7789 Residual .254348789 29 .008770648 R-squared = 0.7863 Model .935781454 1 .935781454 Prob > F = 0.0000 F( 1, 29) = 106.69 Source SS df MS Number of obs = 31_cons 20.42712 5.280639 3.87 0.001 9.627 31.22724 Time -.010095 .0026403 -3.82 0.001 -.015495 -.004695 UK Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .754101135 30 .025136705 Root MSE = .13149 Adj R-squared = 0.3122 Residual .501364846 29 .017288443 R-squared = 0.3351 Model .252736289 1 .252736289 Prob > F = 0.0006 F( 1, 29) = 14.62 Source SS df MS Number of obs = 31
_cons 7.909075 2.430589 3.25 0.003 2.937962 12.88019 Time -.0038797 .0012153 -3.19 0.003 -.0063653 -.0013942 USA Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .143549471 30 .004784982 Root MSE = .06052 Adj R-squared = 0.2345 Residual .106219435 29 .003662739 R-squared = 0.2600 Model .037330035 1 .037330035 Prob > F = 0.0034 F( 1, 29) = 10.19 Source SS df MS Number of obs = 31
_cons 12.09221 4.288566 2.82 0.009 3.321105 20.86331 Time -.0058624 .0021443 -2.73 0.011 -.0102479 -.0014769 AUS Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .415909803 30 .01386366 Root MSE = .10678 Adj R-squared = 0.1775 Residual .330677863 29 .011402685 R-squared = 0.2049 Model .08523194 1 .08523194 Prob > F = 0.0106 F( 1, 29) = 7.47 Source SS df MS Number of obs = 31
37
_cons 6.527328 3.372304 1.94 0.063 -.3698079 13.42446 Time -.0030655 .0016861 -1.82 0.079 -.0065141 .000383 DE Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .227778314 30 .00759261 Root MSE = .08397 Adj R-squared = 0.0714 Residual .204472292 29 .007050769 R-squared = 0.1023 Model .023306022 1 .023306022 Prob > F = 0.0794 F( 1, 29) = 3.31 Source SS df MS Number of obs = 31
_cons -4.233791 6.6178 -0.64 0.527 -17.76871 9.30113 Time .0023674 .0033089 0.72 0.480 -.0044 .0091348 KOR Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .801323233 30 .026710774 Root MSE = .16478 Adj R-squared = -0.0165 Residual .787423378 29 .02715253 R-squared = 0.0173 Model .013899855 1 .013899855 Prob > F = 0.4800 F( 1, 29) = 0.51 Source SS df MS Number of obs = 31
_cons -3.369457 3.247334 -1.04 0.308 -10.011 3.272086 Time .0018167 .0016237 1.12 0.272 -.001504 .0051375 CAN Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .197783797 30 .006592793 Root MSE = .08086 Adj R-squared = 0.0083 Residual .189598479 29 .006537879 R-squared = 0.0414 Model .008185318 1 .008185318 Prob > F = 0.2724 F( 1, 29) = 1.25 Source SS df MS Number of obs = 31
_cons 11.0098 2.665256 4.13 0.000 5.558744 16.46087 Time -.0053215 .0013326 -3.99 0.000 -.008047 -.002596 FR Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .197948389 30 .00659828 Root MSE = .06636 Adj R-squared = 0.3325 Residual .127719992 29 .004404138 R-squared = 0.3548 Model .070228397 1 .070228397 Prob > F = 0.0004 F( 1, 29) = 15.95 Source SS df MS Number of obs = 31
38
_cons 18.85301 1.529143 12.33 0.000 15.72556 21.98046 Time -.009297 .0007646 -12.16 0.000 -.0108607 -.0077333 CH Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .256396473 30 .008546549 Root MSE = .03807 Adj R-squared = 0.8304 Residual .042041397 29 .001449703 R-squared = 0.8360 Model .214355076 1 .214355076 Prob > F = 0.0000 F( 1, 29) = 147.86 Source SS df MS Number of obs = 31
_cons 7.230268 2.696066 2.68 0.012 1.716193 12.74434 Time -.0033913 .001348 -2.52 0.018 -.0061483 -.0006343 SE Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .159211707 30 .005307057 Root MSE = .06713 Adj R-squared = 0.1508 Residual .130689946 29 .00450655 R-squared = 0.1791 Model .028521761 1 .028521761 Prob > F = 0.0177 F( 1, 29) = 6.33 Source SS df MS Number of obs = 31
_cons 6.953246 4.157665 1.67 0.105 -1.550133 15.45663 Time -.0033087 .0020788 -1.59 0.122 -.0075603 .000943 BE Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .337948539 30 .011264951 Root MSE = .10352 Adj R-squared = 0.0486 Residual .310799259 29 .010717216 R-squared = 0.0803 Model .02714928 1 .02714928 Prob > F = 0.1223 F( 1, 29) = 2.53 Source SS df MS Number of obs = 31
_cons -5.74825 1.043456 -5.51 0.000 -7.882357 -3.614143 Time .0030185 .0005217 5.79 0.000 .0019515 .0040856 NL Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .042173103 30 .00140577 Root MSE = .02598 Adj R-squared = 0.5198 Residual .019576236 29 .000675043 R-squared = 0.5358 Model .022596867 1 .022596867 Prob > F = 0.0000 F( 1, 29) = 33.47 Source SS df MS Number of obs = 31
39
_cons 25.87487 1.955643 13.23 0.000 21.87514 29.87461 Time -.0127326 .0009778 -13.02 0.000 -.0147325 -.0107328 NZ Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .470819502 30 .015693983 Root MSE = .04869 Adj R-squared = 0.8489 Residual .068763852 29 .002371167 R-squared = 0.8539 Model .402055649 1 .402055649 Prob > F = 0.0000 F( 1, 29) = 169.56 Source SS df MS Number of obs = 31
_cons 7.846826 2.568315 3.06 0.005 2.594033 13.09962 Time -.003698 .0012841 -2.88 0.007 -.0063244 -.0010716 ES Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .152512527 30 .005083751 Root MSE = .06395 Adj R-squared = 0.1956 Residual .118598017 29 .004089587 R-squared = 0.2224 Model .03391451 1 .03391451 Prob > F = 0.0074 F( 1, 29) = 8.29 Source SS df MS Number of obs = 31
_cons 4.826566 1.993184 2.42 0.022 .7500463 8.903085 Time -.0021761 .0009966 -2.18 0.037 -.0042143 -.0001378 NO Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .08317263 30 .002772421 Root MSE = .04963 Adj R-squared = 0.1116 Residual .071429214 29 .002463076 R-squared = 0.1412 Model .011743415 1 .011743415 Prob > F = 0.0372 F( 1, 29) = 4.77 Source SS df MS Number of obs = 31
_cons 10.35801 1.002638 10.33 0.000 8.307383 12.40863 Time -.0050033 .0005013 -9.98 0.000 -.0060286 -.003978 AVERAGE Coef. Std. Err. t P>|t| [95% Conf. Interval] Total .080157397 30 .002671913 Root MSE = .02497 Adj R-squared = 0.7667 Residual .018074629 29 .000623263 R-squared = 0.7745 Model .062082768 1 .062082768 Prob > F = 0.0000 F( 1, 29) = 99.61 Source SS df MS Number of obs = 31
