Institutional investors face serious challenges due to the current low interest rate environment and their payment obligations. Life insurance companies in Germany have invested the majority of their assets in fixedincome securities. Thus, asset managers struggle to earn adequate returns as a consequence of low yields of highgrade sovereign bonds. As a consequence, institutional investors are forced to rethink their asset allocation, which is vital for the investment success.
This thesis aims to evaluate asset allocation strategies in the light of the low interest rate environment that enable investors to generate adequate riskadjusted returns. A sample portfolio is developed that is broadly diversified, has exposure to alternative investments and applies the Risk Parity approach. The performance is evaluated over different evaluation periods on a riskadjusted basis and in comparison to other asset allocation strategies. As a result, the sample portfolio outperforms the current asset allocation of German life insurers and naïve diversification. However, portfolios with significant exposure to private equity or stocks outperform the sample portfolio in terms of Sharpe ratio but require a higher risk tolerance. Nevertheless, the sample portfolio achieves a satisfactory risk and return profile and is well balanced in terms of risk contribution.
Table of contents
ACKNOWLEDGEMENTS
LIST OF FIGURES
LIST OF ABBREVIATIONS
ZUSAMMENFASSUNG
ABSTRACT
1 INTRODUCTION
1.1 BACKGROUND AND PROBLEM STATEMENT
1.2 PURPOSE OF THIS MASTER THESIS
2 THEORETICAL BACKGROUND OF MODERN PORTFOLIO THEORY
2.1 RELATION BETWEEN RISK AND RETURN
2.2 DIVERSIFICATION & EFFICIENT FRONTIER
2.3 THE MEANVARIANCE PORTFOLIO AS AN OPTIMUM?
2.4 EFFICIENT MARKET HYPOTHESIS
2.5 EXTENSIONS OF MODERN PORTFOLIO THEORY
2.6 CRITICAL EVALUATION OF MODERN PORTFOLIO THEORY & CAPITAL MARKET THEORY
3 LOW INTEREST RATE ENVIRONMENT AND THE IMPLICATIONS ON INSTITUTIONAL ASSET MANAGEMENT
3.1 OVERVIEW OF ASSET MANAGEMENT
3.2 CAUSES AND IMPLICATIONS OF THE LOW INTEREST RATE ENVIRONMENT
3.3 STATUS QUO OF INSTITUTIONAL ASSET MANAGEMENT
3.3.1 German insurance companies
3.3.2 U.S. Endowment funds
3.4 SELECTED INVESTMENT APPROACHES
3.4.1 Active vs. Passive
3.4.2 CoreSatelliteinvesting
3.4.3 Risk Parity approach
3.5 DO ALTERNATIVE ASSETS IMPROVE ASSET ALLOCATION OF INSTITUTIONAL INVESTORS?
4 DEVELOPMENT OF A SAMPLE PORTFOLIO FOR AN INSTITUTIONAL INVESTOR
4.1 PRACTICAL IMPLEMENTATION OF THE SAMPLE PORTFOLIO
4.1.1 Investment Summary
4.1.2 Objectives and Limitations
4.1.3 Investment Selection and Benchmarks
4.1.4 Asset Allocation & Portfolio Construction with SmartFolio
4.1.5 Portfolio Management with Netfolio
4.2 PERFORMANCE EVALUATION
4.2.1 Overview of Performance Evaluation
4.2.2 Key Figurebased Analysis
4.2.2.1 RiskReturn Profile
4.2.2.2 Risk and Return Ratios
4.2.3 Performance Analysis in the light of the Development on Capital Markets
4.3 BACKTESTING THE PERFORMANCE
4.4 OVERALL RANKING OF THE PERFORMANCE
4.5 CRITICAL EVALUATION OF THE SAMPLE PORTFOLIO IMPLEMENTATION AND THE METHODOLOGY
5 CONCLUSION AND OUTLOOK
REFERENCES
Acknowledgements
I would like to thank all the people and companies who contributed in some way to the work in this thesis. First, I thank my academic advisor, Professor Lepelmeier for the motivation, the immense knowledge in the field of asset management and the sophisticated discussions about my investment approach and the methodology.
My sincere thanks also goes to the companies that supported me with their asset allocation and portfolio management software. In particular, Boris Gnedenko from SmartFolio, Daniela Bachmann, Roland Strasser and the team of Alphasys IT Services AG that provided me with their Netfolio tool and supported me in setting up the portfolios. Without their precious software and the support it would not have been possible for me to develop a sample portfolio and to evaluate that many asset allocation strategies.
Moreover, I thank my fellow students and friends for proofreading my thesis, especially Sassy, Si mon, Korni, Lisa, Felix, Stegi, Luise, Jan, Mr. Öhm, Linda and Julia. Your help was very useful to me.
Last but not least, I would like to thank my family and my girlfriend for supporting me throughout writing this thesis and in my life in general. Your love and support means a lot to me.
List of figures
FIGURE 1: TRADEOFF BETWEEN RISK, RETURN AND LIQUIDITY
FIGURE 2: RELATIONSHIP BETWEEN EXPECTED RETURN AND STANDARD DEVIATION OF RETURN
FIGURE 3: RISK REDUCTION THROUGH PORTFOLIO DIVERSIFICATION
FIGURE 4: THE ATTAINABLE SET AND THE EFFICIENT SET OF PORTFOLIOS
FIGURE 5: THREE LEVELS OF EFFICIENT MARKETS
FIGURE 6: KEY INTEREST RATES BY FED AND ECB BETWEEN 2001 AND MID2015
FIGURE 7: DEVELOPMENT OF MAJOR 10YEAR SOVEREIGN BONDS
FIGURE 8: ASSET ALLOCATION OF LIFE INSURANCE COMPANIES IN GERMANY AT YEAREND 2013
FIGURE 9: ASSET ALLOCATION OF HARVARD AND YALE ENDOWMENT FUNDS 2014/ 2015
FIGURE 10: RISKRETURN CHART OF DIFFERENT ASSET CLASSES BETWEEN THE YEARS 19912008
FIGURE 11: RISKRETURN ESTIMATES OF DIFFERENT ASSET CLASSES BY YALE ENDOWMENT FUND
FIGURE 12: CORRELATION OF RETURNS OF DIFFERENT ASSET CLASSES ON A THREEYEAR AND ON A TENYEAR BASIS
FIGURE 13: EFFICIENT FRONTIERS WITH AND WITHOUT ALTERNATIVE INVESTMENTS
FIGURE 14: ASSET ALLOCATION OF THE SAMPLE PORTFOLIO
FIGURE 15: OVERVIEW OF ASSET CLASSES AND FINANCIAL INSTRUMENTS USED FOR THE SAMPLE PORTFOLIO
FIGURE 16: OVERVIEW OF THE SAMPLE PORTFOLIO IN SMARTFOLIO
FIGURE 17: PARAMETER ESTIMATION SETTINGS IN SMARTFOLIO
FIGURE 18: PORTFOLIO OPTIMIZATION SETTINGS IN SMARTFOLIO
FIGURE 19: SAMPLE PORTFOLIO IN SMARTFOLIO AFTER OPTIMIZATION
FIGURE 20: PORTFOLIO OVERVIEW IN NETFOLIO
FIGURE 21: RISKRETURN CHART FOR THE SHORT TERM INTERVAL
FIGURE 22: RISK AND RETURN RESULTS FOR THE EVALUATION PERIOD 2ND MARCH TO 17TH JULY 2015
FIGURE 23: ASSET CLASS PERFORMANCE ATTRIBUTION
FIGURE 24: PERFORMANCE ANALYSIS OF THE SAMPLE PORTFOLIO FOR THE SHORT TERM PERIOD
FIGURE 25: RISK AND RETURN PROFILE FOR THE INDIVIDUAL YEARS 20122014
FIGURE 26: RISK AND RETURN PROFILE FOR THE FULL SAMPLE PERIOD
FIGURE 27: ASSET CLASS PERFORMANCE ATTRIBUTION FOR THE FULL SAMPLE PERIOD
FIGURE 28: PERFORMANCE RANKING ACCORDING TO THE SHARPE RATIO
List of abbreviations
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Zusammenfassung
Institutionelle Investoren stehen aufgrund des aktuellen Niedrigzinsumfelds und ihrer Zahlungs verpflichtungen vor großen Herausforderungen. Deutsche Lebensversicherer sind größtenteils in festverzinslichen Renten investiert. Da die Renditen sicherer Staatsanleihen auf historische Nied rigstände gesunken sind, fällt es ihnen schwer, eine ausreichende Verzinsung zu erwirtschaften. Folglich sind institutionelle Investoren gezwungen ihre Asset Allocation zu überdenken, da diese fundamental für den Anlageerfolg ist. Das Ziel dieser Arbeit ist, Anlagestrategien vor dem Hinter grund des Niedrigzinsumfelds zu bewerten, die es Investoren ermöglichen, ausreichende risikoad justierte Renditen zu erzielen. Dafür wird ein Musterportfolio erstellt, das breit diversifiziert ist, auch in Alternative Investments investiert und den Risk Parity Ansatz nutzt. Die Performance wird über verschiedene Zeiträume sowie im Vergleich zu anderen Anlagestrategien betrachtet. Es wird festgestellt, dass das Musterportfolio besser abschneidet als die aktuelle Vermögensallokation der Deutschen Lebensversicherer oder Naive Diversifikation. Portfolios mit einem hohen Anteil an pri vate equity oder Aktien erreichen eine höhere Sharpe ratio, bedürfen jedoch einer höheren Risi kotoleranz. Nichtsdestotrotz erzielt das Musterportfolio ein zufriedenstellendes Rendite /Risikoprofil und ist sehr ausgeglichen, was den Risikobeitrag der einzelnen Investments betrifft.
Abstract
Institutional investors face serious challenges due to the current low interest rate environment and their payment obligations. Life insurance companies in Germany have invested the majority of their assets in fixedincome securities. Thus, asset managers struggle to earn adequate returns as a consequence of low yields of highgrade sovereign bonds. As a consequence, institutional in vestors are forced to rethink their asset allocation, which is vital for the investment success. This thesis aims to evaluate asset allocation strategies in the light of the low interest rate environment that enable investors to generate adequate riskadjusted returns. A sample portfolio is developed that is broadly diversified, has exposure to alternative investments and applies the Risk Parity ap proach. The performance is evaluated over different evaluation periods on a riskadjusted basis and in comparison to other asset allocation strategies. As a result, the sample portfolio outper forms the current asset allocation of German life insurers and naïve diversification. However, portfolios with significant exposure to private equity or stocks outperform the sample portfolio in terms of Sharpe ratio but require a higher risk tolerance. Nevertheless, the sample portfolio achieves a satisfactory risk and return profile and is well balanced in terms of risk contribution.
1 Introduction
1.1 Background and Problem Statement
Institutional investors face tremendous challenges with regard to the significantly low interest en vironment of the recent years and their longterm promise of guaranteed returns. Insurance companies and pension funds in Germany have invested the majority of their assets in fixed income securities and heavily suffer from low interest rates. Even though the interest rates on sovereign and corporate bonds declined for over 20 years as a secular trend, the movement ac celerated when the U.S. real estate bubble burst in 2008 and caused a Global Financial Crisis (GFC). All major central banks worldwide, in particular the Federal Reserve System (Fed), Bank of England, Bank of Japan as well as the European Central Bank (ECB) cut their base interest rate and provided injections of liquidity into the financial system.^{1} The aim was to avoid a meltdown of the financial sector and to limit the consequences for the real economy. As a result of the unconven tional ultraeasy monetary policy that refers to direct purchases of government debt and covered bonds, the bond yields dropped to historic lows: e.g. 10Y German government bonds yielded at its lowest point in April 2015 at 0.049 % p.a. and around 30% of all government debt in the Eurozone trade on a negative interest rate since investors are hunting for “safe assets”.^{2}
Despite lower guaranteed rates on new contracts, the margins of life insurance companies were squeezed. Considering an average technical interest rate^{3} of around 3.2% p.a., institutional investors struggle to earn adequate returns for meeting their payment guarantees by only investing in a bondheavy portfolio.^{4} Many institutional investors still hold substantial amounts of high coupon bonds and benefit from valuation reserves, however, the realization of writeups on account of increasing bond prices is a temporary phenomenon. As Deutsche Bundesbank reminds, the business model of life insurers or pension funds reaches a critical level when the returns on investments fall below the promised returns.^{5} Consequently, the longer the low interest rate environment persists, the higher is the pressure on institutional investors as new investments or reinvestments on the bond markets hardly achieve acceptable returns.
Moreover, individual board members of Deutsche Bundesbank are already warning of the risks posed by maintaining low interest rates as it paves “the way for new asset bubbles, thereby sow ing the seeds of the next crisis”.^{6} In light of the sovereign debt crisis in Europe, the low interest rates, high liquidity and potential price exaggerations on certain markets are a future threat to fi nancial stability.
With regard to the challenging environment of low interest rates and increased uncertainty about future development of asset classes, institutional investors are forced to rethink their asset alloca tion. Even though, German insurance companies and pension funds consider increasing their allo cation to equities, real estate and alternative investments, they lag behind U.S. institutional inves tors, which have always had higher exposures to equities and nontraditional investments. On av erage, life insurance companies in Germany hold almost 89% fixedincome securities in their port folios and lack of both adequate returns and diversification.^{7} In order to generate better returns, investors face two broad choices:
1. Increase the allocation to equities
2. Increase the allocation to alternative investments
However, many investors recently became cautious towards equities due to the severe draw downs of stock indices during the GFC. The Standard & Poor’s 500 index (S&P 500) showed a max imum loss of over 55%^{8} and the Deutsche(r) Aktien Index (DAX) lost a staggering 40% solely in 2008.^{9} Moreover, investors discovered that their portfolios’ risk levels were far higher than as sumed, demonstrating remarkable deficiencies in their return/risk forecasting and pricing models. It has been shown that meanvariance optimization leads to portfolios concentrated in terms of weights. Hence, it seems necessary to implement proper risk management techniques that take parameter estimation errors into account when constructing a portfolio in order to achieve down side protection.
The second approach to generate higher returns is to increase the allocation to alternative in vestments such as hedge funds, private equity, commodities or infrastructure investments, which promise yield enhancement and greater diversification. Minor steps already have been taken to allocate more outside of traditional fixed income; however, greater effort seems required in order to be successful on a longterm basis. Other institutional investors, in particular the Harvard and Yale endowment funds, allocate the majority of their assets to real and alternative investments and thereby achieved superior riskadjusted returns over the last decades. According to latest re search, asset allocation explains approximately 90% of the investment returns and similar returns to the endowments funds can be achieved by constructing a diversified index portfolio.^{10}
In summary, institutional investors in Germany face serious challenges on account of the low interest rate environment with regard to their heavy fixedincome asset allocation. Given a sustainably low interest rate, the question raises which strategies:
1. enable institutional investors to generate adequate returns to fulfill the promised returns for policyholders
2. improve the riskreturn profile
3. improve diversification, even in volatile markets.
1.2 Purpose of this Master Thesis
The aim of this thesis is to analyze and evaluate asset allocation strategies in light of the current low interest rate environment that enable institutional investors to generate adequate returns, to achieve downside protection and to fulfill their promise of guaranteed returns. As indicated in the problem statement, institutional investors struggle to earn sufficient returns for their policyhold ers and are forced to rethink their asset allocation. Therefore, this thesis evaluates in particular, whether a broadly diversified portfolio including higher exposure to alternative investments and a riskbased investment approach are beneficial from a risk and return perspective and ought to be applied by institutional investors.
This paper is built upon the hypothesis that asset allocation is vital for long term investment suc cess and holding a diversified portfolio including alternative assets as well as using a Risk Parity approach is superior for institutional investors. The proposal is based on the idea that estimating expected returns and covariances of asset classes for Markowitz’ meanvariance optimization is associated with high levels of uncertainty. Moreover, it is assumed that holding a broadly diversi fied mix of assets ensures that a portfolio is not overexposed to single asset classes and the inves tor benefits from increased diversification. The Risk Parity strategy incorporates that risk contribu tion of asset classes differs among each other and Markowitz’ optimization is very sensitive to es timation errors. Thus, constructing the sample portfolio with equalweighted risk (volatility) con tribution is supposed to provide robustness to the portfolio and superior returns for the institu tional investor.
Consequently, the motivation for this thesis is to examine whether institutional investors should consider higher allocations to alternative assets and if a Risk Parity strategy is beneficial from a risk and return point of view. In particular, the following three research questions shall be an swered:
1. Are the riskadjusted returns of a broadly diversified portfolio including alternative assets higher than traditional asset allocation strategies, in particular a balanced stock and bond portfolio or the current asset allocation of German life insurers?
2. Are the riskadjusted returns of a Risk Parity strategy superior to naïve diversification?
3. Are the riskadjusted returns of the Yale Model higher than traditional asset allocation approaches?
In order to answer these questions, a €10 billion sample portfolio is developed, which will be managed and monitored over a timeframe of five months during preparation of this thesis. Addi tionally, backtesting of the last three years will be performed in order to generate increased evi dence of the results. A final performance evaluation of the portfolio will take place in considera tion of risk and return aspects, with regard to the major developments on capital markets and in comparison to selected benchmarks. The portfolio will be broadly and globally diversified includ ing alternative investments as they promise an attractive riskreturn profile and greater diversifi cation due to the low correlation to traditional investments. The portfolio will be constructed us ing passive investments, like Exchange Traded Funds (ETFs), which attempt to track specific mar ket indices as closely as possible. The reason for passive investments is that most active fund managers are not able to outperform the market (at least not after management fees and trans action costs) and ETFs offer very low management fees in contrast to common mutual funds.
At first, this thesis provides the theoretical background of Modern Portfolio and Capital Market Theory (CMT). After a critical evaluation of the concepts, the current low interest rate environ ment and the implications on institutional asset management are analyzed. In particular, how se rious German insurance companies suffer from the historical low interest rates, while endowment funds such as Yale benefit from their large allocation to alternative investments. As adequate re turns are required for the payment obligations of insurance companies, selected investment ap proaches that promise yield enhancement and an improved risk and return profile are provided. Furthermore, a sample portfolio based on these promising investment approaches is developed: broad diversification, passive investing, Risk Parity and exposure to alternative investments. After explaining the construction and the final asset allocation, the performance of the sample portfolio is evaluated on a riskadjusted basis and compared to different benchmarks. Finally, in the conclu sion the advantages for institutional investors will be evaluated.
2 Theoretical Background of Modern Port folio Theory
2.1 Relation between Risk and Return
Portfolio theory, which is also known as Modern Portfolio Theory (MPT), was introduced in 1952 by Harry Markowitz in a Journal of Finance article. It has dominated the discussion of asset alloca tion models.^{11} MPT explores the optimal allocation of wealth between different assets in an in vestment portfolio and provides the basis for understanding the relationship between risk and expected return on securities.^{12} By following his methodology, the investor aims to either maxim ize expected return for a given level of risk or to minimize risk for a given level of expected re turn.^{13}
Return and risk of investments are closely linked; any action an investor performs, or is concerned with, is tied directly or indirectly to return and risk.^{14} Thus, for investing decisions and perfor mance evaluations, risk and expected return should always be considered jointly. As can be seen in figure 1, the investor, in fact, faces a tradeoff between return, risk and liquidity. However, li quidity is not considered in MPT since it assumes a perfectly competitive capital market as a sim plification without taxes, liquidity issues and transaction costs.^{15} In general, assets with higher ex pected returns are riskier.
illustration not visible in this excerpt
Figure 1: Tradeoff between risk, return and liquidity Source: SK Finance (2015).
Theoretical Background of Modern Portfolio Theory 7
For portfolio optimization, the expected returns from financial assets must be estimated. Howev er, since this is done under conditions of uncertainty, probability functions are used. Furthermore, investors need to be aware of the risk they are assuming, gain understanding how their invest ments can be influenced by different market outcomes and be prepared for the consequences of their decisions.^{16} Risk can be defined as the possibility that the actual result differs from the ex pected outcome of the investment and is often measured using the dispersion or variability of re turns, called the variance:^{17}
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By taking the square root of the variance, the standard deviation can be calculated: [illustration not visible in this excerpt]
Standard deviation measures the dispersion around the arithmetic mean in a normally distributed function, which is used more often in practice. This could be explained with the fact that it is, un like the variance, expressed in the same unit as the underlying data. As a rule of thumb, the larger the dispersion, the larger is the variance or standard deviation.^{18} Generally speaking, these risk measurements imply both positive and negative differences, but since most investors are con cerned that the actual outcome is less than expected (the socalled downside risk), semivariance that only takes unfavorable outcomes into account, might be an appropriate alternative.^{19}
More importantly, MPT requires investors to estimate lots of unknown input data, in particular future returns and the correlation structure of those as well as the standard deviation of returns on a portfolio. Considering that 11.175 correlation coefficients for a portfolio of 150 financial assets are required, estimation is extremely complex and leads to major breakthroughs in simplifying inputs to the portfolio model, which will be covered in chapter 2.5.^{20}
What factors influence the risk level of a financial instrument? Assuming that risk means variabil ity of returns, there are several risk factors that determine the rates of return. In particular, inter est rate risk, market risk, inflation risk, business risk, financial risk, liquidity risk as well as ex change rate and country risk influence the returns of financial assets. Changes in the interest rate as an example affect securities inversely.^{21} Consequently, security prices increase when interest rates decreases and bonds react more directly to changes than common stocks, especially bonds with long durations. All relevant risk factors need to be considered in portfolio optimization; how ever, this master thesis will especially focus on interest rate risk and to smaller amounts on mar ket and country risk as the author intends to develop a global diversified portfolio in the challeng ing low interest rate environment.
The reason for investors to hold more risky assets is the additional return expected. This risk premium can be considered as the compensation for taking the mentioned risk factors and can be calculated between any two classes of financial assets, such as longterm and short term bonds (in particular liquidity, interest and inflation risk) or risky corporate bonds versus AAAsovereign bonds with little risk of default.^{22} A major key point in MPT is that portfolio risk is a specific characteristic and unlike the return, not the weighted average mean of the individual security risks.^{23} Under certain circumstances such as a low or negative correlation, a security might have a much smaller risk when held in a portfolio than held by itself.
The author believes that asset allocation is most important for portfolio returns and that a global diversification across different markets and asset classes as well as exposure to alternative investments is best to ensure superior longterm returns.
2.2 Diversification & Efficient Frontier
The common phrase “Don’t put all your eggs in one basket” captures the benefits of asset allocation.^{24} As investors add financial assets to their portfolios, the exposure to particular risks becomes smaller and the portfolio risk can be substantially reduced. The Law of Large Numbers states that, the larger the sample size, the higher is the probability that the sample mean is close to the expected value of the population.^{25} However, when diversification is performed properly, it is free of costs and does not discourage from opportunities; e.g. the market will not pay a premium for taking an unnecessary burden by failing to diversify.^{26}
What we learn from MPT is that the correlation is the key element of portfolio diversification, stating that the lower the correlation coefficient between the assets, the greater the diversifica tion. According to Jones, the correlation coefficient can be defined as “a statistical measure of the extent to which two variables are associated” and the returns on any two securities move togeth er.^{27} It is bound by the values of +1.0 (perfect positive correlation) and 1.0 (perfect negative or inverse correlation); a value of 0.0 refers to no linear relationship. When stock returns are not re lated to each other and the sources of risk are independent, the correlation will be low or even negative leading to the effect of diversification: the portfolio variance becomes smaller than arithmetic mean of the individual risks and the investor benefits from reduced risk.^{28} Put differ ently, investors can reduce their exposure to individual asset risk by investing into a diversified portfolio of assets. The maximum diversification can be achieved when the correlation is perfectly negative (case C) as shown in figure 2.
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Figure 2: Relationship between expected return and standard deviation of return for various correlation coefficients
Source: Elton, E. J. (2011), p.77.
Another important measure in this risk context is the covariance as it measures the absolute (unlike the correlation coefficient that measures the relative) amount two variables tend to covary or move together. The variance of a simple portfolio consisting of two securities A and B can be calculated as follows using the covariance p:^{29}
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The actual comovement is highly important because it impacts the portfolio’s standard devia tion.^{30} When transferring this finding into the general nsecurity case, it becomes clear that port folio risk is a function of each individual risk and the covariance between the individual assets’ re turns. When adding another financial asset to a large portfolio, not only the asset’s individual risk but also a covariance between the new financial asset and every other asset in the portfolio is added.^{31} The risk of a large and welldiversified portfolio is mainly attributable to the influence of the covariances, which increases disproportionately while the importance of each individual fi nancial asset decreases.^{32} As a consequence, the firm’s specific or unsystematic risk can be diver sified in a large portfolio whereas the covariance terms which can be seen as systematic or mar ket risk is nondiversifiable (see figure 3).^{33}
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Figure 3: Risk reduction through portfolio diversification Source: Knecht, M. (2013), p. 63.
Taking a multisecurity case with different return and risk expectations as an example, when an investor changes the portfolio weights of the securities, the results are many different portfolio returns and variances, which can be aggregated to an attainable set of portfolios.^{34} The attainable set (or opportunity set) is therefore the entire set of portfolios that could be formed from a group of securities. Moreover, the attainable set contains some efficient portfolios, which are defined as ones that have the largest expected return for a given level of risk or the smallest portfolio risk for a given level of expected return.^{35} Given the meanvariance portfolios, it is possible to plot the ef ficient frontier as illustrated in figure 4, which demonstrates the tradeoff between expected port folio return and portfolio risk. The global minimum variance portfolio is the one that implies the smallest risk. The bottom segment to portfolio B is dominated by the upper segment as these portfolios provide a higher return at the same risk level.
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Figure 4: The attainable set and the efficient set of Portfolios Source: Jones, C. P. (2014), p. 206f.
Basically, according to Markowitz’s approach investors must determine the riskreturn opportuni ties available before selecting their optimal portfolio lying on the efficient frontier. Rational inves tors will prefer efficient portfolios since these portfolios are optimized and promise the best tradeoff between risk and return. However, it is important to note that Markowitz’ model does not provide one optimal portfolio, but investors are required to select the most appropriate based on their preferences. The concept of indifference curves explains how the selection takes place. The curves for riskaverse investors are upwardsloping as they expect a higher return when ac cepting a higher risk.^{36} Thus, the shapes of the curves depend on the investors’ risk preference. Assuming that investors are riskaverse and rational, each indifference curve provides the same level of utility to a certain investor as it represents the combinations of expected return and risk that are equally desirable. A higher indifference curve is superior to the investor since it offers a higher expected return for the same risk. Thus, the optimal portfolio provides the investor the greatest possible utility and is lying on the investor’s highest indifference curve that is tangent to the efficient frontier.^{37}
2.3 The MeanVariance Portfolio as an Optimum?
As mentioned previously, Markowitz’s meanvariance approach does not provide the investor with one optimum portfolio, but portfolio selection depends on the investor and his or her indi vidual return and risk preference. With regard to the difficulties in estimation of input parameters as well as the definition of the investor’s preference function, the question arises whether the meanvariance analysis as part of the Markowitz model is best for an investor or if there are more suitable alternatives.
According to Frahm and Wiechers, the benefits of quantitative asset allocation strategies in gen eral are questionable and part of continued discussions in literature. Considering the parameter estimation complexity, applying quantitative methods in estimating the input parameters suffers from estimation errors as a main problem.^{38} Additionally, the meanvariance analysis is highly sensitive to input parameters.^{39} Hence, there is a strong momentum away from meanvariance optimization and towards more robust alternatives such as the minimumvariance (see figure 4) or heuristic approaches, in particular risk budgeting strategies. The minimumvariance strategy for example targets on minimizing the overall portfolio return variance and similarly to a risk budget ing portfolio does not depend (explicitly) on estimations of the expected asset returns. Contrary to the Markowitz model, only the risk dimension is considered and the expected returns are as sumed to be identical for all assets since the performance dimension is usually too complicated to forecast.^{40}
What is meant by a heuristic approach and what are examples of risk budgeting strategies? First of all, heuristic refers to experiencebased and trialanderror techniques that are applied in find ing reasonable solutions for the portfolio problem without claiming to find optimal ones.^{41} A sim ple approach and alternative to the Markowitz model is the equally weighted portfolio (some times referred to as naïve diversification), which allocates the same weight to all securities that the investor seeks to hold. This 1/N rule of thumb remarkably reduces the sensitivity to input pa rameters.^{42} Beyond, DeMiguel et al. even argue that the input parameter uncertainty is so severe that under real conditions not even sophisticated extensions to the Markowitz approach are able to consistently outperform the naïve diversification strategy. According to their findings on the U.S. equity market, it takes minimum 3000 months (equals 250 years) for portfolios based on the meanvariance strategy to outperform naïve diversification, depending on the size of the portfo lio.^{43}
Risk Parity is another example of heuristic approaches. The basic idea is that the portfolio consists of different assets providing equal risk to the portfolio. Initially, some professionals used the term Risk Parity when asset weights were inversely proportional to asset class volatility.^{44} However, this early definition overlooks the effect of correlation between assets in a portfolio, but multiasset cases show that correlation assumptions play a critical role in portfolio construction. In 2006, Qian established a more sophisticated definition, which considers the asset weights to be adjusted so that each asset has the same contribution to the portfolio risk.^{45} Thus, it is often referred as an “equal risk contribution” portfolio (ERC portfolio) as it is equally weighted in terms of risk, not in terms of asset weights.^{46} Although this approach was already applied before 2008, it significantly gained popularity due to the GFC in 2008 as well as an increased risk aversion by investor’s who used to underestimate the enormous drawdown risk of equities.^{47}
Chaves et al. found out that Risk Parity strategies are superior on a consistent basis to wellestablished approaches such as mean or minimumvariance portfolios, both in terms of Sharpe ratio and also with regard to diversification.^{48} However, it must be pointed out that according to Maillard et al. the already mentioned naïve portfolio nonetheless outperforms the Risk Parity approach as he derives that the ERC portfolio is located between meanvariance and the naive diversification in terms of volatility.^{49} In summary, Risk Parity portfolios appear to be robust and attractive alternatives to the Markowitz model.
2.4 Efficient Market Hypothesis
With regard to investors’ asset allocation and the question whether an active or passive investment approach is superior, it does matter how fast capital markets react on news and how well security prices reflect information.^{50} Because of its major impact and implications, it seems reasonable to take a closer look on the concept of efficient markets.
An Efficient Market (EM) can be defined as one which “fully and correctly reflects all relevant in formation in determining security prices”^{51} and on which it is impossible to generate economic profits by arbitrage. As information is the key to valuate stock prices, the concept assumes that investors incorporate all relevant information into prices for their investment decision:^{52}
1. All known information such as past (income statement from previous annual reports) and current information (stock splits, adhoc announcement of an acquisition, dividend an nouncement).
2. Information that can reasonably be inferred (expected economic growth for next year or if investors believe the central banks to decrease interest rate next week).
Thus, the Efficient Market Hypothesis (EMH) states that investors are not able to earn abnormal profits as competition and arbitrage drives stock prices to the equilibrium and therefore ensures fair prices. If major information about a security would not be included in its price, arbitrageurs could exploit mispricing and earn excess returns without bearing a risk.
According to Fama, there are three different levels of the EMH:^{53}
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Figure 5: Three levels of Efficient Markets
Source: Fama, E. F. (1970), p. 383417.
It is important to note that if an investor believes in the strong form of EMH, the weak and the semistrong form are also encompassed. The strong form as the extreme means that no one  not even a company insider  is able to make profit from using insider information, which can be considered as unrealistic and is not held by many people.^{54} Other reasons are that the strong form assumes a high amount of rational investors, no transaction costs and information is costless. In 1991, Fama published a subsequent article in the Journal of Finance which deals with a weaker and more economic version of the efficiency hypothesis. But the bottom line stays the same: it is financially not worth to act on any information.^{55}
According to Jones, there is much evidence that supports the market efficiency argument, despite the counterarguments and the increasing prominence of behavioral finance.^{56} The weakform re fers to the random walk theory and means that stock prices follow random trends and price changes are independent. Although statistical tests demonstrated a small dependence in price changes, when considering transaction costs (inter alia costs for analysis as well as brokerage costs), excess returns disappear. Therefore, most market observers accept weak efficiency level while rejecting the strong form and feel that the capital market is, to a high degree, semistrong efficient. An important paradox in this context is that investors that do not believe the EMH and attempt to find and exploit market inefficiencies help to make the market more efficient.^{57}
Moreover, latest studies have shown that only around 1.5% of equity funds were able to outper form the S&P 500 Index on a fiveyear basis^{58} and even with regard to emerging markets, where more inefficiencies could be assumed, 90% of all fund managers underperformed their bench marks.^{59} Additional studies confirm that most fund managers are not able to beat the market as S&P Capital IQ Fund Research reports that in 2014, 80% of largecap mutual funds underper formed the S&P 500 and index investing is better for emerging markets, too.^{60} Another author compares Templeton’s Emerging Market fund with different index funds from Vanguard and Di mensional Funds. The underperformance becomes even clearer the longer the evaluation periods are (one, three and five years).^{61} Even though the EMH does not preclude investors from outper forming the market, it points out that it is hard to achieve and ex ante the few outperformers cannot be identified.^{62} It remains to be said that institutional investors and in particular insurances or pension funds who hold portfolios with a longterm investment horizon, should consider taking index funds into account.
2.5 Extensions of Modern Portfolio Theory
Even though the key ideas of the Markowitz model such as the riskreturn tradeoff or diversification still remain valid, the limitations of portfolio theory and its restrictive assumptions led to several extensions.^{63} These extensions reduce the input needed in solving for the optimum portfolio, provide frameworks that better recognize investor’s investment preferences and improve the pricing of capital assets. This chapter shall give a brief overview of major extensions.
The capital asset pricing model (CAPM) as part of capital market theory (CMT) was developed in the mid1960s and allows investors to determine the relevant risk of an individual financial asset. Furthermore, it provides the relationship between risk and the expected returns from the invest ment and is the simplest form of an equilibrium model.^{64} Note that CMT makes several assump tions, like homogeneous expectations of numerous investors, no transaction costs or personal in come tax, no inflation and the same riskfree rate for borrowing or lending money. According to CAPM, all investors will select the same risky portfolio of assets, called the market portfolio, and mix it with riskfree assets. Furthermore, investors should earn a premium when buying a risky as set such as a stock and the higher the risk associated with this stock, the higher the risk premium should be.^{65}
The risk premium is determined by a ß measure of risk that is related to the covariance of the return i from any security with the market portfolio M.^{66} Since it is more convenient to use a standardized and relative measure of risk, the following equation for Beta  which takes the variance of the market portfolio into account  has pushed through:
illustration not visible in this excerpt
COV = Covariance between the asset’s return and the market portfolio’s return
illustration not visible in this excerpt
A Beta of 1.0 refers to an alignment with the market portfolio and provides the same return as the overall market. A portfolio with a Beta lower than 1.0 would be expected to be less volatile than the market (resulting in a lower return) and vice versa. The conclusions of the CAPM are therefore that return and risk are positively related and the risk of a security comes from its effect on portfolio risk.^{67} The expected risk of return on an asset is therefore the function of the two components riskfree rate (RF) and risk premium:
illustration not visible in this excerpt
On account of the restrictive assumptions of the CAPM, the Arbitrage Pricing Theory (APT) was developed as an alternative and a more general appeal by Ross in 1976.^{68} Unlike the CAPM, the APT does not assume a singleperiod investment horizon, the absence of taxes, as well as a risk free rate for borrowing and lending.^{69} It is a multifactor model and posits a positive relationship between expected return and risk, similar to the CAPM. However, the expected return is affected by a variety of risk factors and is therefore not dependent on an underlying market portfolio as the CAPM is.^{70} Based on the law of one price, the model assumes that no arbitrage is possible and that two identical assets cannot sell at different prices. As a multifactor model, APT incorporates broad economic forces that influence security returns, for example deviations of the factors’ ex pected values or “surprises” in inflation, growth rates of the economy, the term structure of in terest rates, defaultrisk premium and general market risk.^{71} Most researchers argue that using three to five of the mentioned factors is appropriate to approximate asset prices.^{72}
Beyond asset pricings, other major extensions to MPT aim at simplifying the inputs for portfolio optimization and to reduce the calculations to determine optimal portfolios. The oldest and prob ably most widespread simplifications are single and multiindex models (SIM and MIM). The Mar kowitz model requires investors to determine expected returns and variances for each security as well as correlation between each possible pair of securities.^{73} As an example, when a life insur ance company holds 150 financial securities, it is required to estimate (N(N1)) / 2 = (150(1501)) / 2 = 11.175 correlation coefficients without any simplification. Moreover, the calculations increase rapidly with an increasing number of assets. In contrast, a SIM simplifies matters by comparing all securities to a single benchmark and consequently, the effort can be considerably reduced to 3N + 2 = 3*150 + 2 = 452 estimates.
Observations of security markets reveal that security returns are influenced by common underlying factors. Thus, a SIM assumes that securities tend to move together and are influenced by one macroeconomic force. To keep things simple, this factor can be represented by the return on a market index or a broad market portfolio such as the S&P 500 or the Morgan Stanley Capital International (MSCI) World. The expected return on an individual security can be described as linearly related to a stock market index:^{74}
illustration not visible in this excerpt
The return of the security is a combination of a company specific return component represented by the alpha and a market related component. The beta in the calculation takes into account how sensitive a security’s return is to the return on the market. Subsequently, the risk of a security becomes the sum of the market related (systematic risk) and the specific risk (unsystematic risk).^{75} The estimations of alpha and beta are regularly received from regression analysis of historical data and subject to adjustments in order to further improve the predictive ability. As betas are not stable over time and in order to avoid sampling errors based on historical biases, Blume suggests to address this matter by adjusting beta of individual securities towards their mean value.^{76} But since portfolio betas are measured with less error and change less than individual betas, historical betas on portfolios are superior prognosticators for the future.^{77}
Lastly, it must be noted that numerous research projects attempt to further improve the models and to extend the SIM to a more accurate multiindex model (MIM).^{78} MIM take several inde pendent variables into account such as inflation, economic or industry growth, interest rates, ex change rates etc. that have significant influence on security returns as the SIM can be seen as an oversimplification. However, the more factors are considered, the more complex the model be comes as it requires additional parameter estimations for its application. However, this contra dicts investors’ need to simplify and reduce the inputs. Nevertheless, all models provide frame works that assist in making portfolio analysis more efficient and pragmatic than the traditional Markowitz approach.^{79}
2.6 Critical Evaluation of Modern Portfolio Theory & Capital Market Theory
Despite the rapid success and their status as one of the most important and influential economic theories about finance and investment, both MPT and CMT face criticism and suffer from several drawbacks and limitations.^{80} They are considered not to mirror the real world in many ways, which is mainly based on their restrictive assumptions. This chapter provides a critical appraisal of both frameworks after listing the major issues with MPT:^{81}
1. Asset returns are regularly assumed to be normally distributed (“Gaussian bell curve”)
2. Determining efficient portfolios heavily relies on estimating risk, return and correlation inputs
3. The correlation between assets are presumed to be fixed and constant forever; securities are truly independent so that an investor can diversify completely
4. Certain inputs of risks or future distributions are assumed to be known
5. Investors are assumed as rational “homo economicus”
6. The risk is represented by volatility only
7. MPT does not take taxes or transaction costs into account
8. All investors are considered to be price takers that cannot influence prices
9. All investors have access to the same information at the same time
Empirical observations of the capital market indicate that security returns are prone to extreme events and the normal distribution is not an appropriate description of reality.^{82} According to Mandelbrot, the tails of the Gaussian bell curve are too thin as there are large swings of three to six standard deviations from the mean that occur considerably more often than predicted by the normal distribution.^{83} Thus, the Markowitz model underestimates risk. Instead, both skewness and kurtosis as the fat tails should be considered in order to make the portfolio more robust for extreme scenarios. Moreover, when determining the efficient frontier, there are numerous esti mates required, but as the expected return and variance of the securities cannot be known with certainty, estimation errors may influence adversely the quality of the resulting portfolio, espe cially when the market is crashing.^{84} More robust estimation techniques may help, but it is not possible to completely avoid estimation errors.
Beyond, the Markowitz model assumes that by considering several asset classes and different re gions with low correlation, the effect of diversification can be achieved. But in times of enormous market stress such as the GFC, both the correlation between different asset classes remarkably increased and the promised diversification did not occur. This resulted in substantial losses as al most any security dropped in price.^{85} According to behavioral finance, this phenomena leading to more severe crises is even reinforced by herding behavior of investors that do not act rational an ymore.^{86} MPT’s assumption that investors are rational must be questioned when considering some examples of irrationalities such as the tulips bubble in seventeenthcentury in the Nether lands, the dotcom bubble in the late 1990s or the housing bubble until 2007 in the U.S.^{87}
The assumption concerning investors having equal information at the same time refers to the EMH that prices fully reflect all available information. In reality, this assumption does not hold true as the financial market contains insider trading, information asymmetries and some investors evaluating information differently since their investment time horizon differs.^{88} At least, it takes some time for new information to spread and to be considered for investment decisions. Another point stressed by Grossman and Stiglitz is that the market cannot be perfectly efficient as there would be no incentive anymore for investors to unearth the information that get rapidly reflected in security prices.^{89} However, a clear answer concerning validity of EMH cannot be given. On the one hand, there are many different studies that provide evidence of inefficiency and anomalies in various security markets and irrational investors will always be present.^{90} On the other hand, there is some degree of efficiency that abnormal returns after transaction costs disappear and it is financially not worth to act on information.
Both simple factor models CAPM and SIM remain popular and are widely used in practice due to simplicity and utility in a variety of situations. Nonetheless, a large number of input estimates are still required and the necessity to rely on historical data to predict the future is another draw back.^{91} The core criticism to CAPM relates to the fact that the assumption of a true market portfo lio is unobservable as it must contain all risky assets, including those that are not traded.^{92}
In practice, a stock index is often chosen as an approximation of the market portfolio. However, it must be noted that this substitution is not innocuous as the results of empirical tests are dependent on the index chosen.^{93} Another shortcoming is that CAPM claims a linear positive relation between beta and expected return. Nevertheless, empirical studies show that low beta stocks may offer a higher return than the CAPM would predict. The evidence of this socalled low volatility anomaly is based on several studies by both academic and practitioner authors, who contradict the key statement of the CAPM that investors need to bear higher risks in terms of beta when seeking higher returns.^{94} Moreover, further studies demonstrate additional market anomalies such as the size and value effect that cannot be explained by CAPM.^{95}
The SIM substantially reduces inputs required for portfolio analysis as it assumes that the correla tion structure between different securities relies only on one macroeconomic factor.^{96} Conse quently, the existence and selection of a suitable index factor is crucial. At the same time, the simplicity of the single index approach causes a number of weaknesses as more factors may be needed in practice. Many researchers came to the conclusion that there are forces beyond the market that cause financial assets to move together. Over 40 years ago, King already provided ev idence on the existence of industry influences on stock movements.^{97} Thus, a more realistic model should allow the systematic risk to be described by several factors, leading to multiindex mod els.^{98} At least a second factor should be considered, for example industry date as mentioned above or the interest rate, which was substantially decreased since beginning of the GFC in 2008 and has a major impact on asset prices and asset allocation.^{99} This will be further analyzed in chapter 3.2.
[...]
^{1} Cf. Holzhausen, A. (2013), p. 11.
^{2} Cf. Warner, J. (2014), n. pag.
^{3} Life insurance companies used to promise guaranteed returns for their new contracts. The average technical interest rate can be seen as the industry’s average minimum return in order to meet obligations and to service outstanding policies.
^{4} Cf. Kablau, A. / Weiß, M. (2014), p. 4.
^{5} Cf. Kablau, A. / Weiß, M. (2014), p. 2.
^{6} Cf. Bloomberg (2012), n. pag.
^{7} Cf. GDV (2014), n. pag.
^{8} Cf. GVC Management (2013), n. pag.
^{9} Cf. Manager Magazin (2008), n. pag.
^{10} Cf. Kaplan, P. D. / Ibbotson, R. G. (2000), p. 32.
^{11} Cf. Francis, J. C. / Kim, D. (2013), p. 2.
^{12} Cf. Rabin, J. (2013), p. 947.
^{13} Cf. Rabin, J. (2003), p. 947.
^{14} Cf. Jones, C. P. (2014), p. 165.
^{15} Cf. Garz, H. / Günther, S. / Moriabadi, C. (1997), p. 19.
^{16} Cf. Jones, C. P. (2014), p. 157.
^{17} Cf. Jones, C. P. (2014), p. 155.
^{18} Cf. Jones, C. P. (2014), p. 179
^{19} Cf. Capinski, M. J. / Kopp, E. (2014), p. 9.
^{20} Cf. Elton, E. (2011), p. 130ff.
^{21} Cf. Jones, C. P. (2014), p. 156.
^{22} Cf. Jones, C. P. (2014), p. 159.
^{23} Cf. Jones, C. P. (2014), p. 181.
^{24} Cf. Baker, H. Kent / Filbeck, G. (2013), p. 208.
^{25} Cf. Jones, C. P. (2014), p. 184
^{26} Cf. Houthhakker, H. S. (1996), p. 153.
^{27} Jones, C. P. (2014), p. 186.
^{28} Cf. Garz, H. / Günther, S. / Moriabadi, C. (1997), p. 37.
^{29} Cf. Garz, H. / Günther, S. / Moriabadi, C. (1997), p. 41.
^{30} Cf. Jones, C. P. (2014), p. 189.
^{31} Cf. Jones, C. P. (2014), p. 194f.
^{32} Cf. Garz, H. / Günther, S. / Moriabadi, C. (1997), p. 46.
^{33} Cf. Knecht, M. (2013), p. 63.
^{34} Cf. Jones, C. P. (2014), p. 191ff.
^{35} Cf. Jones, C. P. (2014), p. 205.
^{36} Cf. Jones, C. P. (2014), p. 207.
^{37} Cf. Jones, C. P. (2014), p. 207f.
^{38} Cf. Frahm, G. / Wiechers, C. (2011), p. 3.
^{39} Cf. Merton, R. C. (1980), p. 323361.
^{40} Cf. Roncalli, T. (2013), p. 98f.
^{41} Cf. Roncalli, T. (2013), p. iii.
^{42} Cf. Roncalli, T. (2013), p. iii.
^{43} Cf. DeMiguel, V. / Garlappi, L. / Uppal, R. (2007), p. 1915ff.
^{44} Cf. Ang, A. (2014), p. 101.
^{45} Cf. Qian, E. (2006), p. 41ff.
^{46} Cf. Roncalli, T. (2013), p. iii.
^{47} Cf. Grimm, R. (2014), p. 287.
^{48} Cf. Chaves, D. B. / Hsu, J. C. / Li, F. / Shakernia, O. (2011), p. 108ff.
^{49} Cf. Maillard, S. / Roncalli, T. / Teiletche, J. (2010), p. 1ff.
^{50} Cf. Jones, C. P. (2014), p. 319.
^{51} Dinçer, H. (2013), p. 177.
^{52} Cf. Jones, C. P. (2014), p. 319.
^{53} Fama, E. F. (1970), p. 383  417.
^{54} Cf. Jones, C. P. (2014), p. 325.
^{55} Cf. Fama, E. F. (1991), p. 1575  1617.
^{56} Cf. Jones, C. P. (2014), p. 341f.
^{57} Cf. Jones, C. P. (2014), p. 342.
^{58} For the fiveyear period ending June 30, 2014
^{59} Cf. S&P (2014), p. 3.
^{60} Cf. CNBC (2014), n. pag.
^{61} Cf. CBS News (2014), n. pag.
^{62} Cf. Jones, C. P. (2014), p. 345.
^{63} Cf. Bellalah, M. (2009), p. 5.
^{64} Cf. Kevin, S. (2008), p. 194.
^{65} Cf. Jones, C. P. (2014), p. 239.
^{66} Cf. Booth et alt., P. (2004), p. 155.
^{67} Cf. Jones, C. P. (2014), p. 247f.
^{68} Cf. Sharifzadeh, M. (2010), p. 85.
^{69} Cf. Jones, C. P. (2014), p. 250.
^{70} Cf. Focardi, S. M. / Fabozzi, F. J (2004), p. 88.
^{71} Cf. Jones, C. P. (2014), p. 252f.
^{72} Cf. Bühler, W. / Hax, H. / Schmidt, R. (1999), p. 167.
^{73} Cf. Elton, E. J. (2011), p. 130f.
^{74} Cf. Kevin, S. (2008), p. 183.
^{75} Cf. Kevin, S. (2008), p. 183.
^{76} Cf. Blume, M. E. (1975), p. 785795.
^{77} Cf. Elton, E. J. (2011), p. 142f.
^{78} Cf. Strong, R. (2008), p. 143.
^{79} Cf. Kevin, S. (2008), p. 187.
^{80} Cf. Bezjak, F. (2010), p. 37.
^{81} Cf. Chatterjee, R. (2014), p. 118; Mindel, N. M. / Sleight, S. E. (2009), p. 117f.; Crundwell, F. (2008), p. 360.
^{82} Cf. Rice, M. / DiMeo, R. A. / Porter, M. (2012), p. 25.
^{83} Cf. Mandelbrot, B. / Hudson, R. L. (2010), p. 12f.
^{84} Cf. Fabozzi, F. J / Focardi, S. M. / Kolm, P. N. (2006), p. 304.
^{85} Cf. Mindel, N. M. / Sleight, S. E. (2009), p. 117f.
^{86} Cf. Bruce, B. R. (2010), p. 148; Cf. Lehner, O. M. / Harrison R. (2014), p. 118.
^{87} Cf. Mindel, N. M. / Sleight, S. E. (2009), p. 117f.
^{88} Cf. Chourmouziadis, K. (2010), p. 40.
^{89} Cf. Grossman, S. J. / Stieglitz, J. E. (1980), p. 405.
^{90} Cf. Chourmouziadis, K. (2010), p. 41.
^{91} Cf. Baker, H. K. / Powell, G. (2009), p. 351.
^{92} Cf. Amenc, N. / Le Sourd, V. (2005), p. 129.
^{93} Cf. Dash, A. P. (2009), p. 96.
^{94} Cf. Blitz, D. C. / van Vliet, P. (2007), p. 102113; Baker, N. L. / Haugen, R. A. (2012), p. 2f.
^{95} Cf. Fama, E. F. / French, K. R. (1993), p. 3f.
^{96} Cf. Strong, R. (2008), p. 140f.
^{97} Cf. King, B. (1966), p. 139f.
^{98} Cf. Glantz, M. / Kissell, R. (2013), p. 176.
^{99} Cf. DeFusco, R. A. / McLeavey, D. W. / Pinto, J. E. / Runkle, D. E. (2011), p. 474f.
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 Benjamin Güttler (Author), 2015, Asset allocation strategies in the current low interest rate environment, Munich, GRIN Verlag, https://www.grin.com/document/335117

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