Investment Strategies

Content

Abstract

Keywords

Content

List of Figures

List of Tables

Abbreviations

1 Introduction
1.1 Motivation and Aim of the Thesis
1.2 Structure of the Thesis

2 Modern Portfolio Theory
2.1 Literature Review
2.2 Optimal Portfolio
2.3 Efficient Market Hypothesis

3 Applied Methodology
3.1 Data
3.2 Risk, Return and Correlation
3.3 Time Frame
3.3.1 Ex Post
3.3.2 Ex Ante
3.4 Portfolio Strategies
3.4.1 Crossover Simple Moving Average
3.4.2 Equally Weighted Portfolio
3.4.3 Minimum Variance Portfolio
3.4.4 Certainty Equivalence Tangency Portfolio
3.4.5 James Stein Estimator
3.4.6 Black Litterman Model

4 Empirical Results
4.1 Risk, Return and Correlation
4.2 Strategies in an Ex Post Framework
4.3 Strategies in an Ex Ante Framework
4.3.1 Crossover Simple Moving Average
4.3.2 EQW, MVP and CET
4.3.3 James Stein Estimator
4.3.4 Black Litterman Model
4.4 Constraints of the observed Magnitudes
4.4.1 Transaction Costs
4.4.2 Liquidity
4.4.3 Market Situation during the Sample Period

5 Conclusion and Further Research

Appendix

References

Abstract

This thesis analyzes different investment strategies that are applied to an international equity portfolio. The evaluated strategies are: the Crossover Simple Moving Average, the Equally Weighted Portfolio, the Minimum Variance Portfolio, the Certainty Equivalent Tangency Portfolio, the James Stein Estimator and the Black Litterman Model. From the viewpoint of a European investor it will be shown whether the final performance parameters are mainly due to returns of foreign markets or through the exchange rate development. Beside the analysis in mean-variance terms, it will be evaluated how robust the results are over time. The investigation is carried out from an ex post- as well as from an ex ante perspective. In order to examine the time window of a strategy, the in- and the out of the sample periods are varied. The empirical investigation indicates that – the relative young more sophisticated approaches are superior to the traditional strategies, the impact of the exchange rate development cannot be ignored in an equity portfolio, nearly no conclusion can be drawn in the context of a superior in- and out of the sample period.

Keywords

Investment Strategies – Modern Portfolio Theory – Optimal Portfolio – International Portfolio Management – Active Asset Management – Exchange Rate Risk

List of Figures

Figure 2-1: The efficient frontier and the highest reachable utility function of an investor. 16

Figure 3-1: Ex post time frame

Figure 3-2: Ex ante time frame

Figure 3-3: Intuitive explanation of the BLM

Figure 3-4: Normal distribution

Figure 4-1: Exchange rate development of foreign currencies against the Euro

Figure 4-2: MVP and CET ex post risk- and return parameters

Figure 4-3: MVP and CET ex ante risk- and return parameters

Figure 4-4: MVP and CET ex ante risk- and return parameters

Figure 4-5: MVP and CET ex ante risk- and return parameters

Figure 4-6: Portfolio weights of the MVP and CET

Figure 4-7: Portfolio weights of the MVP and CET

Figure 4-8: Portfolio weights of the MVP and CET

Figure 4-9: JSE ex ante risk- and return parameters

Figure 4-10: JSE ex ante risk- and return parameters

Figure 4-11: JSE ex ante risk- and return parameters

Figure 4-12: Portfolio weights of the JSE

Figure 4-13: Portfolio weights of the JSE

Figure 4-14: Portfolio weights of the JSE

Figure 4-15: BLM ex ante risk- and return parameters

Figure 4-16: Portfolio weights of the BLM

Figure A-1: CSMA ex ante return parameters of individual markets

Figure A-2: CSMA ex ante return parameters of individual markets

Figure A-3: CSMA ex ante risk parameters of individual markets

Figure A-4: CSMA ex ante risk parameters of individual markets

List of Tables

Table 4-1: Average returns of different international equity markets

Table 4-2 : Average variances of different international equity markets

Table 4-3: Average correlation coefficients without the exchange rate

Table 4-4: Average correlation coefficients with the exchange rate

Table 4-5: Average EQW, MVP and CET ex post risk- and return parameters

Table 4-6: Average portfolio weights of the MVP and CET

Table 4-7: CSMA ex ante risk- and return parameters

Table 4-8: Number of involved transactions of the CSMA

Table 4-9: Average EQW, MVP and CET ex ante risk- and return parameters

Table 4-10: Average JSE ex ante risk- and return parameters

Table 4-11: Average BLM ex ante risk- and return parameters

Abbreviations

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1 Introduction

Nowadays the merits of international portfolio diversification are widely acknowledged in the academic literature. The risk reduction of an international portfolio can be achieved because the correlations between international asset markets are rather low compared to a portfolio which entirely consists of national securities. Hence, international investment strategies are superior compared to strategies which invest solely in a local market since they are able to generate a greater return for a certain risk, or less risk for a given return.

Beside the advantages of international diversification, the investment in other currencies bears an additional uncertainty that arises through foreign exchange rate fluctuations. However, the development of the exchange rate is not solely a one-sided downside risk; it is also a chance of a higher return since the movement can be in favor of a position. In other words, exchange rate changes have different effects on investors of different currencies. Even if the domestic return is much lower than in other countries, it might be the case that an investment in another state will result in a lower return because of the exchange rate development. Therefore, the residence and the therewith-associated currency of an investor is crucial for the result of an international diversified portfolio.

In order to analyze the two risk drivers of an international diversified portfolio separately, the results of the investment strategies are calculated in two ways - with and without the exchange rate development. This method allows evaluating whether exchange rate movements are dispensable or if currency fluctuations are significant for international equity portfolios and therefore the exchange rate risk should be hedged.

The choice of the investment strategy should be compatible with the needs, the expectations and the personality of an investor. In many papers utility theory is used to determine an investor’s optimal investment strategy. These approaches use utility functions to figure out which strategy fits best to an investor. The methodology of this paper is from another angle since no individual utility functions are incorporated, it is different in the way that an empirical study will be carried out to demonstrate the relevant performance parameters of different investment strategies and thereafter it can be chosen which strategy is most appropriate for an individual investor.

The investment strategies which will be applied are the Crossover Simple Moving Average, the Equally Weighted Portfolio, the Mean Variance Portfolio, the Certainty Equivalence Tangency Portfolio, as well as the James Stein Estimator and the Black Litterman Model. In order to gain from the merits of low correlated markets, an international portfolio is set up which comprises ten indices from all over the globe that range from countries with low GDP per head to those with relative high levels.

1.1 Motivation and Aim of the Thesis

There is a large amount of evidence, which indicates that international investment strategies are superior to those strategies which entirely invest in the domestic market. Most of the studies regarding this topic are carried out from the viewpoint of a US investor.^[1] The author did not find any recent evidence in the context of an international diversified portfolio which consists entirely of equities from the perspective of a European investor. For this reason, this thesis investigates from a European point of view how different investment strategies performed. In other words, this thesis aims to demonstrate, in mean-variance terms, how each considered strategy performed over the last 8 years as different variables are allowed to vary. The main goals are to answer the following questions:

- How did individual investment strategies perform over the last 8 years?
- Could this performance also have been achieved when the exchange rate development was considered? (viewpoint from a Euro-based investor)
- Which out of the sample length was optimal for each strategy?
- How robust are the results of each strategy?
- Does a superior strategy exist or is one strictly dominated?

1.2 Structure of the Thesis

The paper is divided into three major sections. Section 1 contains a literature review in the context of Modern Portfolio Theory (MPT) and the investment strategies that are considered in this thesis, describes how the optimal portfolio is constructed by using MPT and offers a discussion of the general validity of investment strategies. Section 2 provides reasons for the choice of data, explains the fundamentals of the calculation of the portfolio risk- and return parameters and shows how the resulting magnitudes for each investment strategy are calculated. Section 3 presents the empirical results of the analysis. In this section, the performance and the robustness without- and with the exchange rate development will be demonstrated. After the three major sections, the final part summarizes the main findings and provides ideas which further research seems promising and fruitful.

2 Modern Portfolio Theory

After having discussed the most relevant papers in the context of Modern Portfolio Theory (MPT) and the considered strategies of this thesis, it will be shown how MPT can be used to determine an investor’s optimal portfolio. Thereafter, questions of the general validity of the MPT approach are put forward by using the Efficient Market Hypothesis.

2.1 Literature Review

Markowitz (1952) created the fundament for MPT by explicitly stating the portfolio selection problem and by providing a solution for it. Markowitz explained the desirable effects from portfolio diversification and proposed the expected return-variance relationship. A various amount of papers confirmed empirically that with a broad portfolio diversification an investor can increase the expected return for a given level of risk, while on the other side, he can reduce the risk level for a given expected return, see Elton and Gruber (1977), Solnik (1995), Gerke, Mager and Röhrs (2005), Leon Li (2007).

Several papers build upon Markowitz ideas and explain how the MVP and CET can be used as portfolio strategies. Some important references are Sharpe (1970), Merton (1972) and Fama (1976). Recent models that extend the basic framework of Markowitz provide solutions for its shortcomings. For example, on one side, Stein estimation methods help to obtain more accurate estimates, while on the other side, models like the BLM allow the incorporation of additional information rather than just the first two moments.

The estimation literature shows that portfolio optimization procedures can be improved. Portfolio optimization based on estimates of least-squares estimators lead to the problem of estimation risk which arises from the difference between the observed estimates and the true unobservable parameters. Different studies such as Solnik (1982) or Board and Sutcliffe (1992) show that the problem of estimation risk is significant. Different articles have found that the portfolio optimization is extremely sensitive to the estimates of the mean returns. For this reason, papers such as Best and Grauer (1992) and Chopra and Ziemba (1993) suggest applying more robust estimation methods. The James Stein estimation procedure is more efficient since the method is able to provide smaller mean squared errors. This is achieved by pooling information across different series and therefore leads to the so called shrinkage estimator that shrinks the individual assets means towards a global grand mean, see for example Jorion (1986), Grauer and Hakansson (1998) or Kempf and Memmel (2002).

The motivation of Black and Litterman (1992) was to introduce a new model which can be applied in practical asset allocation. Their model allows taking additional information into account and provides a solution which does not produce an extreme portfolio weight allocation. The model is based on Markowitz’s (1952) mean-variance optimization, on Sharpe’s (1964) CAPM, and on Black’s (1989) global CAPM. The very recent literature shows several papers which deal with the Black Litterman Model in different ways. A few very important references, which seem to be the most established ones in this relative young field, are Bevan and Winkelmann (1998), He and Litterman (1999) and Satchell and Scowcroft (2000). The mentioned papers are very tough. For a more substantial discussion of the method see Jay Walters’ (2008) The Black Litterman Model: A Detailed Exploration or Thomas M. Idzorek’s (2002) A Step-By-Step Guide to the Black-Litterman Model. Their papers try to enable as many readers as possible to follow, since their derivations are explicitly explained and in small steps.

2.2 Optimal Portfolio

Modern Portfolio Theory proposes how a risk-averse investor should construct an optimal portfolio by considering the trade-off between market risk and expected return.^[2] The theory is based on the concepts of Markowitz (1952, 1959) and quantifies the merits of diversification. Out of an opportunity set of assets, an investor creates an efficient frontier of optimal portfolios. Markowitz’s portfolio selection model shows that each portfolio on the efficient frontier is optimal since it provides the maximum possible expected return for a given level of risk or with the lowest risk for a given expected return.^[3] Therefore, a rational investor will hold a portfolio on the efficient frontier which has risk-return characteristics that are consistent with an investor’s preferences. The following section concerns two parts – the efficient frontier and utility functions.

The efficient frontier can be derived by either maximizing the return for each risk level or by minimizing the risk for each return as it is done in this Thesis.^[4] Therefore, the weight vector X (x1,…,xN) has to be chosen in a way that the variance is minimized subject to:

Abbildung in dieser Leseprobe nicht enthaltenand Abbildung in dieser Leseprobe nicht enthalten

Where the first equation simply means that the expected return is held constant and the second that all endowment is invested in the considered investment opportunity set. Thus we are looking for the smallest possible variance for a given Abbildung in dieser Leseprobe nicht enthalten, so the minimization problem becomes:

illustration not visible in this excerpt

Whereas

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

In which ri denotes the vector of the expected returns, R the fixed portfolio return,^[5] C a vector of the magnitudes which are invested in xi, Abbildung in dieser Leseprobe nicht enthalten is simply a vector of ones, Abbildung in dieser Leseprobe nicht enthalten the variance-covariance matrix and therefore Abbildung in dieser Leseprobe nicht enthaltenis the portfolio variance. By using the Lagrange Multiplier method we can solve this system of equations and obtain the following first order conditions:^[6]

illustration not visible in this excerpt

Whereas the Lagrange Multiplier is Abbildung in dieser Leseprobe nicht enthalten. By defining Abbildung in dieser Leseprobe nicht enthalten , the upper equation from (1) can be written as:

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

Substituting this equation into the lower one from (1) and defining Abbildung in dieser Leseprobe nicht enthaltenyields:

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

Bringing the terms together and solving the equations for the portfolio variance leads to:

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

Assume H takes the form of:

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

If Abbildung in dieser Leseprobe nicht enthalten, a, b and c can be defined in the following way:^[7]

illustration not visible in this excerpt

Substituting the equations into the portfolio variance, equation (2) becomes:

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Multiplying the terms out and substituting for the determinant of H the term Abbildung in dieser Leseprobe nicht enthalten, yields to the following parabola:

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Equation (3) is the expression for the efficient frontier. By taking the square root of that formula the expression is obtained which is used to describe the risk-return relationship on Abbildung in dieser Leseprobe nicht enthaltendiagram. However, the taking of a square root leads to a parabola because a positive number and its negative counterpart squared precede to the same result. Therefore, to each given risk level belong two different expected return parameters.^[8] Since we assume a risk inverse investor, solely the expected returns above the MVP are considered which in turn means that the efficient frontier is only the part from the parabola that is above the MVP.^[9]

Through the minimization of the variance, we can calculate for a given return the corresponding lowest possible risk parameter but however, from equation (3) the optimal portfolio weights cannot be calculated directly. The following steps explain how the formula is derived that allows the calculation of the optimal portfolio of the efficient frontier.

Since the determinant of H is equal to Abbildung in dieser Leseprobe nicht enthalten, the term Abbildung in dieser Leseprobe nicht enthaltenfrom equation (3) can be written as:

illustration not visible in this excerpt

Substituting the last term in equation (3), and dividing the left hand side by 1/c while the right hand side by c/c2 yields:

illustration not visible in this excerpt

dividing by Abbildung in dieser Leseprobe nicht enthalten yields to the following hyperbola:

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Since Abbildung in dieser Leseprobe nicht enthaltencan be canceled with the denominator of the subtrahend Abbildung in dieser Leseprobe nicht enthalten, the asymptotes become Abbildung in dieser Leseprobe nicht enthalten and the center of the hyperbola is Abbildung in dieser Leseprobe nicht enthalten, so Abbildung in dieser Leseprobe nicht enthalten.^[10] Finally, the equation Abbildung in dieser Leseprobe nicht enthaltenwhich comes from the 2nd and 3rd equation from (2) yields to:

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Equation (4) allows calculating the optimal portfolio weights for each entered portfolio return R. For each return, the minimal possible risk level is achieved since the formula produces points on the efficient frontier.

Having determined the efficient frontier, a second part which regards to the investor’s preferences is necessary to determine the optimal portfolio. Utility theory can be used since it enables to measure and to compare the relative satisfaction of investment alternatives. An investor assigns a score, the so called utils, to an investment according to two variables – the risk and the expected return.

In the case of the expected return, the attribute of nonsatiation applies which means that X+1 units of wealth is always associated with higher utility then X units of wealth. In short, more being preferred to less. Therefore, if an investor has to choose between certain assets which have the same risk, he or she will always choose the one with the highest expected return. Since utility increases as wealth increases, the first derivative of the utility function with respect to wealth must be positive.^[11]

The second property of a utility function is about the assumption of an investor’s sensation towards risk. A rational economic agent is risk averse and only enters risky actions if a kind of compensation is received. Likewise in this paper it is assumed that all investors are risk averse. The risk aversion implies that an investor only substitutes a risk-free alternative against a speculative prospect if the involved risk premium is positive. The investor dislikes if an outcome is uncertain and therefore assigns lower welfare to a risky asset compared to a risk free one. The greater the risk, the larger the objection. Since utility decreases as the risk increases, the first derivative of the utility function with respect to wealth must be negative.

Bringing the two components together, the tendency towards the expected return and risk, the shape of the utility function must be convex in a Abbildung in dieser Leseprobe nicht enthaltendiagram. Assume that the utility function takes the following form:^[12]

illustration not visible in this excerpt

in which W denotes the wealth and Abbildung in dieser Leseprobe nicht enthalten is the absolute risk aversion parameter. Since it is a one period model it is assumed that Abbildung in dieser Leseprobe nicht enthalten is constant. The greater Abbildung in dieser Leseprobe nicht enthalten, the larger becomes the importance of the variance and therefore, the greater is the risk aversion.^[13] By rewriting equation (5) in terms of Abbildung in dieser Leseprobe nicht enthalten yields:

illustration not visible in this excerpt

Again, the same notions than before apply. To obtain the optimal portfolio weights the utility function has to be maximized subject to the budget constraint. The restriction implies that exactly Abbildung in dieser Leseprobe nicht enthalten is invested in the portfolio. Therefore, the Lagrange function takes the following form:

illustration not visible in this excerpt

Taking the first order conditions results in:

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Solving the upper equation from (6) for X yields:

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Substituting this expression into the lower equation from (6) and take the definitions for b and c of matrix H results in:

Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten Abbildung in dieser Leseprobe nicht enthalten

The combining of the last two equations yields to:

Abbildung in dieser Leseprobe nicht enthalten

Equation (7) shows how much an investor should invest in each of the assets in the investment opportunity set in order to maximize his or her utility. As the investor becomes more risk averse, Abbildung in dieser Leseprobe nicht enthalten becomes larger which gives the first term less weight relative to the second. As Abbildung in dieser Leseprobe nicht enthalten approximates infinity, the first term disappears, which shows that the expected return parameters are not anymore considered in the utility function.

By considering the efficient frontier and the utility function the system can be solved and the optimal portfolio weight vector can be obtained. It is optimal because on that portfolio the highest reachable utility function is just tangential to the efficient frontier and therefore the utility of an investor is maximized.^[14] This is shown in the figure 2-1.^[15]

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Figure 2-1: The efficient frontier and the highest reachable utility function of an investor. The optimal portfolio occurs at the point where the two functions are tangential to each other.

2.3 Efficient Market Hypothesis

The Efficient Market Hypothesis (EMH) serves the reason to question the general validity of the later presented investment strategies. If financial markets are perfectly efficient in the sense that all relevant information is immediately correctly incorporated in the market prices, then there remains no unexploited profit opportunity.^[16] Choosing specific assets or indices cannot result in continuous abnormal profits since the portfolio manager does not have more information than the market. Therefore, in the case that the market is in this manner efficient, the moving away from a passive portfolio to an active managed portfolio cannot systematically result in abnormal returns and hence the later presented investment strategies cannot be superior to a passive portfolio management.

Papers concerning market efficiency are often based on Eugene Fama’s (1970) influential article Efficient Capital Markets. According to Fama, the efficiency of a market can be characterized as weak, semi-strong or strong. The weak form efficiency implies that all past information concerning the prices and returns of assets are incorporated in the actual security price. Hence, if the market would be weakly efficient then the applied strategies in this thesis (except from the Black Litterman Model) would not be able to achieve abnormal returns since these strategies, if at all, take solely the first two moments. The semi-strong efficiency builds upon the weak form but it is an enhancement since it assumes that in the actual security prices all public information is incorporated as well. In the case that the market would be semi-strong efficient then neither of the considered investment strategies would be able to outperform the market. The last possibility, the strong form of market efficiency states that no information, whether public or private, can be exploited because security prices fully reflect all information that is in the system.^[17]

All three different forms are based on competition. In a market with low barriers and high profits, new competitors will enter until no abnormal profits can be systematically realized. Since security trading has low barriers and information is widely available to investors, the economic expression ‘perfect competition’ seem to fit well to security pricing. Therefore, the EMH rests on the idea of a ‘random walk’.^[18] If prices follow a random walk then the flow of information is instantaneously and information is immediately reflected in security prices. Hence tomorrow’s price movement will only depend on tomorrow’s news and are absolutely independent from the past. As news comprises by definition new events, news is unpredictable and thus, future price movements must be unpredictable.

However, in reality there is no such thing as perfect competition. Different financial markets have a different aggregate supply and demand. Further, information is not freely and instantaneously available to all investors. There are market participants like banks and other companies which have information advantages; and interestingly, it is not surprising, when these firms report their abnormal profits, that nobody argues that it is a violation of the Efficient Market Hypothesis.

3 Applied Methodology

The theory is explained in a short mathematical way.^[19] Since numerous papers deal with the MVP- and the CET strategy, their methodology is only covered very briefly. The thesis focuses on JSE and the BLM as these methods are becoming more and more popular nowadays and further since these methods are not as much analyzed in the context as it is done in this thesis.

3.1 Data

The main idea behind the choice of the time period and the data that was collected can be characterized with four points:

- Relative up-to-date sample with the characteristics of the 21st century to clarify conclusions for the near future.
- Internationally markets, geographically representing different locations.
- Solely equity indices. The intake of bonds or derivatives would probably have had improved the performance, but however, the results of the applied strategies would have to be divided into different factors.
- To analyze whether the portfolio weights are composed mainly of equity markets or of developed markets, the equity indices range from countries with a relative low GDP per head to those which have a relative high GDP per head.

Fulfilling these criteria, the chosen sample consists of 2261-daily observations from the 31st December 1999 until the 1st September 2008 covering a period of more than 8 years. The collected data concentrates on four different types of time series.

- Stock Indices
- Currency Markets
- Euribor
- IFO Business Climate Index

The included 10 countries are being listed in a consecutive order with the countries of the lowest GDP per head first:^[20] India (IBOMBSE 100), China (SHANGHAI SE), Brazil (BRIPX 50), South-Africa (DJSA 30), Chile (IGPAGEN), Russia (MSCI RUSSIA), New Zealand (DJNZ), Japan (NIKKEI 225), Germany (DAX 30) and the United States (DOW JONES IND). The countries are representative for each bigger region. Germany pleads the European market, the United States and Japan as classical investment aims, New Zealand for an exotic developed market. Since the potential investment possibilities of India, China and Brazil were already considered before the beginning of analyzed time frame, South Africa can be seen as an exotic emerging market. At the beginning of the year 2000 when the sample starts, each of the markets was already characterized by a stable economy with a developed financial system or with circumstances which are prosperous for an improving legal and political situation.

The discussion on the exchange rates does not include pegged exchange rate systems; it is entirely based on the assumption that exchange rates are freely floating. However, it is worth mentioning that a part of the analyzed time frame contains two pegged currencies. From 1997 until 2005 the Chinese currency was fixed to the USD at a peg of 8.27 Renmimbi per USD. At the 21st of July 2005 the fixed exchange rate system was abolished which caused an immediate one-off revaluation of the RMB. The effect of this beg was that the exchange rate fluctuations between the Euro and the RNB were caused by the Euro-USD development until mid 2005.

The MSCI World Index, which consists of 23 indices of developed countries, was used as a benchmark. Several of the considered investment strategies require some kind of risk free rate of interest. Since the analysis is from the viewpoint of a European investor, the 3-Month Euribor has been chosen as the risk free interest rate.^[21]

For the analyst forecasts of the Black Litterman Model the IFO Business Climate Index has been taken. The forecasts are the unadjusted expectations about the business expectations for the next 6 month time.^[22]

3.2 Risk, Return and Correlation

In this thesis solely the arithmetic means have been calculated to determine the return parameters. The return at the end of a holding period is the average of the daily returns which are calculated as in equation (8).^[23]

illustration not visible in this excerpt

whereas P is the price of particular index i and t a certain point in time. In order to obtain the actual return for a European investor a composition has to be done between the stock market return and the currency return. This is demonstrated in equation (9).^[24]

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On its analogy the exchange rate of an index at a point in time is St,i. Equation (9) can be shown in form of equation (10).^[25]

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Equation (10) shows that the return of a European investor can be decomposed in the local return of Rt,I and the et,I which is the exchange rate return of the local currency against the Euro that is calculated from the spot exchange rate St,i. Therefore, the expected total return is a joint result of the expected return of the local index plus the expected exchange rate return plus a cross term.^[26]

The overall risk of an investment in another currency can be calculated from the individual risk of an asset and the risk which arises through the exchange rate. The following approach is often used in empirical studies to calculate the effects of international diversification and is based on the paper from Eun and Resnick (1994).^[27]

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Equation (11) shows that variance of an asset in terms of another currency is the sum of four estimated parameters.^[28]

The so far calculations determined the risk- and return parameters for an asset in the local market and in terms of another currency. In order to evaluate the performance of an international multi-asset portfolio it is necessary calculate the combined risk- and return parameters. Equation (12) demonstrates how the portfolio return parameter is calculated. It is the summation of the return of an asset times its portfolio weight.^[29]

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where Rp is the total portfolio return, xi is the weight which is allocated to index i and N represents the total number of included assets. The calculation of the portfolio variance is more complex since beside the allocated weight, the correlation between the individual assets has to be taken into account.^[30] As shown in equation (13) the portfolio variance increases as the correlation is positive and decreases as the included assets move countercyclical.^[31] Therefore, from an investor’s point of view, it is desirable if the covariance terms are small or even negative since in this case the advantage of diversification can be used and hence the portfolio risk is reduced.

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3.3 Time Frame

A data set can be divided between the in the sample and the out of the sample period. The in the sample period is taken to estimate certain parameters and then based on these parameters, particular variables are optimized. The out of the sample data is reserved to test whether those optimized variables are a good fit in the out of the sample period as well. Therefore, the time frame of the in the sample- and out of the sample data can be arranged in two different ways – in an ex post and in an ex ante framework. The following paragraphs describe these two types.

3.3.1 Ex Post

The ex post framework implicitly assumes that there is no incomplete information when a decision is made. In this approach the decision is made with complete information before complete information becomes available. It is retrospective. The ex post models look back and analyze which decision would have been optimal based on the occurrences that have actually came to pass. Therefore, the framework takes the information of the present into account to form a decision of the past which in turn results in zero pricing errors.^[32] Therefore, an ex post analysis enables the evaluation of the potential that an investment strategy has.^[33] In other words, the actual future developments are captured to analyze the potential scope that a strategy has. Figure 3-1 demonstrates an ex post framework graphically.

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Figure 3-1: Ex post time frame. The blue lines demonstrate the measurement of the optimal portfolio weights, while the red lines stand for periods on which the performance is measured. The in the sample period is the same than the out of the sample period. The optimal portfolio weights are calculated in a period and then these weights are used to measure the performance of that period.

3.3.2 Ex Ante

In reality the optimal portfolio weights of a time window are only revealed after the time window has passed. Therefore, the performance results of an ex post model are not to be obtainable in practice because changes happen and since not all information is complete, the forecast cannot be systematically correct.^[34] The question is whether the results are also desirable when the portfolio optimization is entirely based on historical information.^[35] An ex ante approach solely uses information that is based on the time before the point in time when a decision is made. The in the sample period is used for the calculation of the optimal portfolio weights. Once the optimal portfolio weights are obtained they are applied to the out of the sample period in which the performance is measured. Afterwards the in the sample and the out of the sample period are rolled one step further. The process continues over the whole range of the data set.

This approach is closer to reality because it shows how a strategy could have performed when applied over a certain period. The part of the results which is ex post is explicitly mentioned, in all other cases the approach is ex ante. An established method to analyze the performance of investment strategies is the ex ante back testing procedure.^[36] Figure 3-2 demonstrates the method.

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Figure 3-2: Ex ante time frame. The blue lines demonstrate the measurement of the optimal portfolio weights, while the red lines stand for periods on which the performance is measured. The in the sample period is one period before the out of the sample period. The optimal portfolio weights are calculated in a period and then these weights are used in the next period to measure the performance in that period.

3.4 Portfolio Strategies

Since the literature comprises a large amount of articles which deal with the methodology of the EQW, the MVP and the CET, the explanation of their applied methodology is solely roughly explained while the emphasis lies on the, not as widely grazed, more sophisticated strategies such as the JSE and the BLM. The MVP, the CET as well the JSE and the BLM build up on the basics of Markowitz. The comparison of these strategies with a benchmark which was chosen to be the MSCI World might seem to be insufficient since in this way the strategies are solely judged against a passive portfolio. Therefore, besides comparing the results solely with the MSCI World, a moving average strategy has been applied to evaluate if the same dataset enables a strategy which is active and dynamic to outperform the more sophisticated modern portfolio theory strategies.

The optimization process of the different strategies involves the setting of constraints. Two restrictions that are applied to all considered strategies have to be mentioned.

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The first constraint implies that the exact amount of the endowment is invested in at least one of the investment alternatives. The constraint forces that the system of equations is solved in a way that all endowment is invested even if the returns are negative and that the total number of weights does not succeed the total endowment. The second restriction excludes the possibility of short selling up to a certain extend. No short or long position can be greater in its absolute magnitude than 3 times the value of total endowment. If this restriction would be violated, some mathematically optimized solution would show extreme long and short positions. Even if in theory the portfolio performance might improve when the restriction is ignored, it is not feasible in reality because of the immense transaction costs which would be involved when changing the extreme portfolio weights and further, from a legal point of view there are often restrictions as well. In the cases in which one of the optimized portfolio weights is greater than 3 in its absolute value, an algorithm adjusts the optimal weights that no absolute value is greater than 3.

[...]

^[1] See Li Leon (2007), p. 1867.

^[2] See Markowitz (1952), p. 79.

^[3] See Breuer, Gürtler and Schuhmacher (1999), pp. 40-50.

^[4] For a more detailed derivation of the efficient frontier see Feldman and Reisman (2003) or for a highly math-based derivation see Merton (1972).

^[5] Remember that in order to minimize the variance, the analyst has to fix the expected return on a certain level to calculate the smallest possible variance for that expected return.

^[6] Elton et al. (2003) use the same approach but instead of using a Lagrangian function to minimize the variance, the function is used to maximize the Sharpe ratio, pp. 100-106.

^[7] The first impression for these definitions might be confusing. The setting up of an example with a variance-covariance matrix, a return vector and a vector of ones and then solving the system of equations helps to understand the procedure.

^[8] Except for the MVP.

^[9] See Deutsch (2005), p. 106.

^[10] Beside a math-based derivation, Markowitz (1952) showed diagrammatically that the centre of the hyperbola is the portfolio with the minimum variance, pp. 85-87.

^[11] For a detailed analysis of the mathematical properties of utility functions see Chiang and Wainwright (2005), pp. 373-378.

^[12] For a more detailed discussion of the absolute risk aversion parameter see Breuer, Gürtler and Schuhmacher (1999), pp. 29-36.

^[13] If the investor is risk averse then >0, neutral then=0 and a risk seeking investor would have a which is smaller than zero. For a diagrammatical demonstration of the three types of the behavior towards risk see Gast (1998), p. 89.

^[14] This approach ignores the risk free rate of interest. By including riskless investments such as government bonds the system needs first to be solved for a straight line which is tangential to the efficient frontier starting at the risk free rate of interest and then in the second step it can be solved for the optimal portfolio by considering the utility function. Hence leveraging or de-leveraging with the risk free rate of interest takes place.

^[15] For a more detailed diagrammatical derivation of the investor’s optimal portfolio see Breuer, Gürtler and Schumacher (1999), pp. 55-61.

^[16] See Mishkin (2004), p. 152.

^[17] See Fama (1970), pp. 409-410.

^[18] See Mishkin (2004), p. 154.

^[19] Due to the limited scope of the thesis, no extensive derivation of a strategy can be covered.

^[20] The ranking of the GDP per head level was done on the 23.09.2008, source https://www.cia.gov/library/publications/the-world-factbook. The daily observations of the equity indices have been collected from Datastream of Thomson Financial Limited.

^[21] Source: Datastream of Thomson Financial Limited.

^[22] See the methodology of the Black Litterman Model for a explanation of the data usage.

^[23] See Campbell, Lo and MacKinlay (1997), p. 254.

^[24] See Bugár and Maurer (1997), p. 66.

^[25] See Bugár and Maurer (1997), p. 66.

^[26] Bugár and Maurer (1997) show the individual contribution of the three factors to the total return for an Hungarian investor.

^[27] See Eun and Resnick (1994), p. 145.

^[28] For a decomposition which shows by how much percentage the 4 components contribute to the total risk see Bugár and Maurer (1997) for a Hungarian investor or Maurer and Mertz (1999) for a German investor.

^[29] See Elton et al. (2003), p. 53.

^[30] See Markowitz (1952), p. 81.

^[31] Whereas xj is the weight which is allocated to index j. For a detailed derivation of this formula see Elton et al. (2003), pp. 56, 57.

^[32] See Cochrane (2005), p. 123.

^[33] See Bugár and Maurer (1997), p. 72.

^[34] Chocrane (2005), pp. 123-125, explains why often ex post results are presented and argues that “the only solution is to impose some kind of discipline in order to avoid dredging up spuriously good in-sample pricing”.

^[35] See Glen and Jorion (1993), p. 1882.

^[36] See for example: Eun and Resnick (1994), Glen and Jorion (1993) or Maurer and Mertz (1999).