The idea of comparing the performance of different risky investments, for example investment funds, on a quantitative basis dates back to the beginnings of the asset management industry and has been an important field of research in finance since then. Performance measures serve as valuable quantitative evidence for the portfolio manager's performance as well as for the evaluation of investment decisions ex post. Based on the idea of the capital asset pricing model proposed by Treynor (1961), Sharpe (1964), and Lintner (1965), Treynor (1965) developed the first quantitative performance measure intended to rate mutual funds, the Treynor Ratio. Since then, a large number of performance measures with very different characteristics have been developed, for example by Sharpe (1966), Jensen (1968), Treynor & Black (1973), Sortino & Price (1994), and Israelsen (2005). In addition to their power of rating investments ex post, their ability to predict future performance has been thoroughly analyzed by Grinblatt & Titman (1992), Brown & Goetzmann (1995), Carhart (1997), and others. Besides academia, the driving force behind the development of more sophisticated performance measures has always been the investors. This is understandable, as "the truly poor managers are afraid, the unlucky managers will be unjustly condemned, and the new managers have no track record. Only the skilled (or lucky) managers are enthusiastic" (Grinold & Kahn, 2000, p. 478).
By combining and applying the results of previous research to a new sample of nearly 10,000 mutual funds that invest in different countries and asset classes, this thesis clarifies its central research question: Is the Information Ratio a useful and reliable performance measure? In order to answer this central question, it has been split up into the following subparts: What are the characteristics of a useful and reliable performance measure? What actually is "good" performance? Is the "good" performance a result of luck or of skilled decisions and does it persist over time? How does the Information Ratio compare to other performance measures, and what are its strengths and weaknesses? This empirical study aims at answering all of these questions and provides a framework for performance evaluation by use of the Information Ratio.
Table of Contents
List of Figures
List of Tables
List of Abbreviations
1 Introduction
1.1 Motivation and Objective
1.2 Course of the Investigation
2 Theoretical Overview
2.1 Methods of Fund Performance Measurement
2.1.1 Characteristics of a Reliable Performance Measure
2.1.2 The Treynor Ratio
2.1.3 The Sharpe Ratio
2.1.4 Jensen’s Alpha
2.1.5 The Sortino Ratio
2.1.6 The M² Measure
2.1.7 The Omega Measure
2.2 The Information Ratio
2.3 Sources of Active Returns: How to Beat the Benchmark
2.4 Agency Problems Related to Performance Measures
3 Data Description and Sources
3.1 Mutual Fund Selection
3.2 Benchmark Selection
3.3 Descriptive Statistics
4 Empirical Study on Selected Performance Measures
4.1 Is the Information Ratio a Reliable Measure of Performance?
4.2 The Information Ratio Versus Other Measures
4.3 The Art of Selecting the Benchmark
4.4 Does Data Frequency Matter?
4.5 Other Influences on Performance Measures
4.6 Performance Persistence: Outperformance by Luck or Skill?
4.7 Summary of Empirical Results
5 A Practical View on Performance Measurement
6 Conclusion
List of References
Appendix A
Appendix B
List of Figures
Figure 1: Investment Opportunity Set Based on Information Ratios
Figure 2: Index Development of Several Security Types
Figure 3: Box Plots of Equity US and Equity Germany Fund Information Ratios
Figure 4: Selected Performance Measures for Equity US Funds
Figure 5: Fund Rankings of Selected Performance Measures
Figure 6: Fund Rankings Using Modified Information Ratios
Figure 7: Factor Decomposition of Rankings Based on Information Ratios
Figure 8: Development of Major Large Cap US Equity Indices
Figure 9: The Effect of Benchmark Selection on the Information Ratio
Figure 10: Ranking Differences Caused by Different Benchmarks
Figure 11: Comparison of Rankings Based on Different Data Frequencies
Figure 12: Performance Persistence of Equity US Funds
Figure 13: Framework for Performance Evaluation – Year 2008
Figure 14: Active Share Versus Tracking Error
Figure 15: Distribution of Information Ratios Based on Different Data Frequencies
List of Tables
Table 1: Sample Size of the Fund Dataset Grouped by Fund Classification
Table 2: Overview of Benchmark Indices
Table 3: Descriptive Statistics of Fund Returns
Table 4: Information Ratios of Different Fund Categories
Table 5: Test Statistics for the Difference of Threshold Values of Equity US Funds
Table 6: Test Statistics for the Difference of Threshold Values of US Funds
Table 7: Inconsistency of the Information Ratio for Negative Alphas
Table 8: zStatistics for Significant Difference of the Information Ratios
Table 9: Information Ratios in Relation to Fund Launch Years
Table 10: Number of Top 25% Ranks over Lifetime
Table 11: Performance Persistence of Equity US Funds Over Time
Table 12: Information Ratio – Threshold Values for 1st Quartile Funds (very good)
Table 13: Information Ratio – Threshold Values for 2nd Quartile Funds (good)
Table 14: Test Statistics for Information Ratios of Equity US Funds
Table 15: Test Statistics for Information Ratios of Selected US Funds
Table 16: Distribution Properties of Performance Measures for Equity US Funds
Table 17: Sharpe Ratio – Threshold Values for 1st Quartile Funds (very good)
Table 18: Sharpe Ratio – Threshold Values for 2nd Quartile Funds (good)
Table 19: Sortino Ratio – Threshold Values for 1st Quartile Funds (very good)
Table 20: Sortino Ratio – Threshold Values for 2nd Quartile Funds (good)
Table 21: Correlation of the Information Ratio With Other Performance Measures
Table 22: Return Distribution of the S&P 500 Index (Timeframe: 1998 until 2008)
Table 23: Performance Persistence of Equity and Equity Small Cap Funds
Table 24: Performance Persistence of Corporate Bond Funds
List of Abbreviations
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1 Introduction
1.1 Motivation and Objective
“I do not want a good General,
I want a lucky one.”
(Napoleon Bonaparte)
In contrast to Napoleon, investors typically do not want to pick a lucky person to administer their funds, but both Napoleon and the investor face a similar problem: how to separate the lucky from the skilled. Historic data shows that five out of one hundred portfolio managers achieve an outstanding performance just by luck, and statistics also reveal that luck – in most cases – does not persist over time. The lucky managers will, however, always cite their superior skills as a reason for their success, while the unsuccessful ones will place the blame on bad luck. By assessing all active managers on the two dimensions luck and skill, four groups are created. The separation of the skilled and lucky from the unskilled but lucky managers and the separation of the skilled but unlucky from the unskilled and unlucky managers is of special interest to all stakeholders in the investment industry. It is, therefore, the investor’s task to apply understandable guidelines, preferably on a quantitative basis, when it comes to evaluating a portfolio manager. On the other hand, it is the fund administration’s task to judge the performance of its managers objectively and to transfer the results into a variable remuneration scheme or to decide about the replacement of a certain manager. (Grinold & Kahn, 2000, pp. 478480)
The idea of comparing the performance of different risky investments, for example investment funds, on a quantitative basis dates back to the beginnings of the asset management industry and has been an important field of research in finance since then (Jensen, 1968, p. 389). Performance measures serve as valuable quantitative evidence for the portfolio manager’s performance as well as for the evaluation of investment decisions ex post (Treynor, 1965, p. 63). Based on the idea of the capital asset pricing model (CAPM) proposed by Treynor (1961), Sharpe (1964), and Lintner (1965), Treynor (1965) developed the first quantitative performance measure intended to rate mutual funds, the Treynor Ratio. Since then, a large number of performance measures with very different characteristics have been developed, for example by Sharpe (1966), Jensen (1968), Treynor & Black (1973), Sortino & Price (1994), and Israelsen (2005). In addition to their power of rating investments ex post, their ability to predict future performance has been thoroughly analyzed by Grinblatt & Titman (1992), Brown & Goetzmann (1995), Carhart (1997), and others. Besides academia, the driving force behind the development of more sophisticated performance measures has always been the investors. This is understandable, as “the truly poor managers are afraid, the unlucky managers will be unjustly condemned, and the new managers have no track record. Only the skilled (or lucky) managers are enthusiastic” (Grinold & Kahn, 2000, p. 478).
By combining and applying the results of previous research to a new sample of nearly 10,000 mutual funds that invest in different countries and asset classes, this thesis clarifies its central research question: Is the Information Ratio a useful and reliable performance measure? In order to answer this central question, it has been split up into the following subparts: What are the characteristics of a useful and reliable performance measure? What actually is “good” performance? Is the “good” performance a result of luck or of skilled decisions and does it persist over time? How does the Information Ratio compare to other performance measures, and what are its strengths and weaknesses? This empirical study aims at answering all of these questions and provides a framework for performance evaluation by use of the Information Ratio.
The Information Ratio, developed in 1973 by Treynor & Black, is one of the most important performance measures in the investment management industry (Grinold, 1989, p. 31). It is a ratio for the excess return of a portfolio relative to a specified benchmark divided by the volatility of the excess returns. The measure, therefore, is able to show how much additional return has been generated per unit of additional risk, which is important information in the field of active management (Treynor & Black, 1973). Besides the interesting characteristics of the Information Ratio, it is of special interest because it is founded on two different theoretical frameworks. While the first framework goes back to the founders of the Information Ratio, the second framework closely connects it to the fundamental law of active management, which was developed by Grinold (1989). The fundamental law of active management is a central framework for active managers and provides insight on how to use the rationale behind the Information Ratio to construct active portfolios for predefined risk budgets. Additionally, the Information Ratio has not yet been analyzed in an extensive empirical study across different asset classes and countries, which is therefore a supplementary motivation for this paper.
The empirical study is based on return data of nearly 10,000 funds in the timeframe from January 1, 1998 until December 31, 2008 and yields some important results, which are summarized very briefly in this paragraph. Generally, funds have been categorized according to their investment universe in 13 distinct classes, for example “Equity US” or “Money Market EUR”. In order to judge the value of a certain performance measure, a quartilebased grading system with the four categories “very good”, “good”, “below average”, and “poor” has been developed. Threshold values have been calculated that separate the “very good” quartile from the “good” quartile, and so on. Using this method, the threshold values of the Information Ratio are found to vary over time and also across different asset classes, so that it becomes necessary to recalibrate the framework annually. The quality and reliability of the Information Ratio is dependent on certain factors of the data selection process. Firstly, only one benchmark should be used for all funds in a fund category in order to allow for better comparability and the selection of this benchmark can heavily influence the threshold values. The benchmark should optimally cover a large proportion of the market that is within the investment universe of the respective fund. Secondly, data frequency should be as high as possible, for example, daily or weekly. Monthly data does not accurately represent the true volatility of returns within a calendar year. Thirdly, nonnormally distributed fund returns can affect the usability of the Information Ratio. For example, money market funds show strong nonnormal returns, and, therefore, cannot be reliably evaluated with the Information Ratio. There are, however, other measures available that take higher moments of return distributions into account. In order to separate lucky managers from skilled ones, the track record plays an important role, as luck generally is not persistent over time. The final framework evaluates the performance of the active manager based on the quartilebased grade of the Information Ratio, penalizes low active weights using an additional measure and incentivizes persistent (skilled) performance by looking at the manager’s track record.
1.2 Course of the Investigation
Following the introduction and the motivation for the topic, Section 2 lays out the theoretical foundations of this paper. Firstly, Subsection 2.1 explains different methods of fund performance management by describing the characteristics of reliable performance measures in part 2.1.1, and continues by presenting six widelyused ratios to evaluate fund performance in the mutual fund industry in parts 2.1.2 to 2.1.7. Each performance measure is explained briefly and its advantages and disadvantages are outlined in order to get a good overview of the rationale behind these measures. As the Information Ratio is at the center of interest of this study, it is explained in detail in Subsection 2.2. In order to better understand the motivation behind active management, Subsection 2.3 describes the fundamental law of active management. This leads to a better understanding of the relevant parameters that influence the level of excess returns and clarifies the theoretical framework of the Information Ratio from a different perspective. Subsection 2.4 presents agency problems in the fund management industry in general and special issues that are related to the Information Ratio.
Section 3 elaborates on the composition and characteristics of the dataset that is used in the empirical study by explaining the selection of mutual funds (3.1) and benchmark indices (3.2), as well as by showing descriptive statistics of the different fund categories (3.3).
The empirical study, which is the central part of this thesis, is presented in Section 4. It starts in Subsection 4.1 by testing the Information Ratio for stability over time and across different fund categories and continues in Subsection 4.2 by comparing the ranking order of the Information Ratio against several other performance measures. Subsections 4.3 and 4.4 provide information about the robustness of the Information Ratio against the selection of different benchmarks and data frequencies. Other influences that could possibly affect the quality of the Information Ratio, such as nonnormality of returns or survivorship bias inherent in the dataset, are described and analyzed in Subsection 4.5. In order to separate lucky from skilled managers, the persistency of good Information Ratios over time has been researched in Subsection 4.6. The empirical part concludes with a summary and the development of a specific performance evaluation framework detailed in Subsection 4.7.
Section 5 sheds light on the experiences and opinions of several practitioners with respect to performance measurement in general and the use of the Information Ratio in particular. This view will complement the results of the empirical analysis.
The thesis concludes with Section 6, where all findings are summarized and starting points for future research are presented.
2 Theoretical Overview
2.1 Methods of Fund Performance Measurement
2.1.1 Characteristics of a Reliable Performance Measure
Before providing an overview of some of the most important performance measures, it is necessary to characterize the properties of a reliable and “good” performance measure. Treynor (1965, p. 64) had two requirements in mind when developing the first performance measure. Firstly, the ratio should provide the same value as long as the performance of the manager does not change, even in unfavorable market conditions. Secondly, the ratio has to incorporate the specific preferences of investors’ risk aversion. According to Hübner (2007, p. 65), there are two factors that determine the quality of a performance measure: stability and precision. A stable measure is robust with respect to the selection of asset pricing models and should not vary strongly in terms of its classification over time, that is if a “very good” Information Ratio is above 0.5, optimally, this should also be true in all subsequent years. Additionally, the performance measure should be precise, which means that it should be able to provide the “true” ranking of funds based on the investor’s preferences. The Information Ratio will be tested on both factors in the empirical part of this paper. Chen & Knez (1996, pp. 511513) explain that a performance measure has to evaluate the service that is provided to the investor. Does the active manager really “enlarge the investment opportunity set faced by the investing public and, if so, to what extent” (Chen & Knez, 1996, p. 512)? They introduce the idea of an “admissible performance measure”, which is characterized by four criteria. Firstly, the ratio has to assign a performance of zero to the passive benchmark portfolio. Secondly, the ratio has to be a linear function in order to allow for good comparability and to ensure that outperformance can be attributed to superior information. Next, the performance measure has to be continuous so that funds with an equal performance receive the same performance value. Lastly, the function of the ratio has to be nontrivial. As an addition, Chen & Knez (1996, p. 514) prefer measures that assign higher performance values to superior funds and lower performance values to inferior funds.
As outlined in detail within the introduction, the development of the first performance measures dates back to the proposal of the CAPM by Treynor (1961), Sharpe (1964) and Lintner (1965). While Treynor (1965) was the first to introduce a meaningful performance measure, immediately after, this field of research was extended by Sharpe (1966) and Jensen (1968). The Information Ratio was introduced by Treynor & Black (1973). Other important and widely used measures were developed by Sortino & Price (1994), Modigliani & Modigliani (1997), and Keating & Shadwick (2002). All of these measures are still widely used in the fund management industry, although some of them have been developed more than 40 years ago. Therefore, the measures will be presented and characterized briefly in the following subsections (Hübner, 2005, p. 415). As the Information Ratio is in the center of interest of this thesis, it will be explained and analyzed in detail in a separate section (cf. Chapter 2.2).
2.1.2 The Treynor Ratio
Treynor (1965) developed the Treynor Ratio (TR) based on the idea of the CAPM that had been proposed just shortly before. Treynor (1965, pp. 6465) introduces the so called “characteristic line” for each investment fund, which basically is a regression line showing the relationship between the fund’s returns and the benchmark’s returns. The slope of the line is called β and characterizes the fund’s volatility in relation to the volatility of the benchmark. A β = 2.0 for example means that the respective fund will change its rate of return by 2% if the benchmark rate of return changes by 1%. The intercept of the characteristic line is called α and expresses the average outperformance or underperformance of the fund in relation to the benchmark. The Treynor Ratio is defined according to Equation 1:
illustration not visible in this excerpt
where rP is the return of the fund or portfolio, rM is the return of the benchmark for the respective market and β is the regression beta as explained above. The Treynor Ratio therefore measures relative return in relation to relative risk or, to put it in other words “portfolio performance per unit of systematic risk” (Hübner, 2005, p. 416). While Treynor (1965, p. 69) initially defined rM as the rate of return of the market benchmark, that is a stock index, the ratio has later been calculated with rM equal to zero or the riskfree rate (Hübner, 2005, p. 418).
In practical applications the Treynor Ratio is beneficial in cases where investors have to select one out of many actively managed investment funds (Hübner, 2007, p. 65). However, it has certain drawbacks which limit its practical use. Firstly, if the β is close to zero, which can be the case for selected funds, the ratio will go to infinity. Secondly, it is unstable and imprecise for marketneutral funds, such as certain hedge fund strategies. Thirdly, if the β is negative, the ratio even provides positive values for funds with a negative alpha (Hübner, 2005, p. 416, 2007, p. 65).
2.1.3 The Sharpe Ratio
Introduced by Sharpe (1966), the Sharpe Ratio (SR), which was initially called rewardtovariability ratio, is meant as an extension of the Treynor Ratio. While Treynor (1965) strived to only evaluate fund performance ex post, Sharpe (1966) explicitly aimed at predicting future performance with his measure and also by using the Treynor Ratio (p. 119). The Sharpe Ratio has been discussed heavily in literature, and its theoretical foundation was also extended twice by the founder in Sharpe (1975) and Sharpe (1994). Using Equation 2, the Sharpe Ratio can be easily calculated:
illustration not visible in this excerpt
where rP is the return of the fund or portfolio, rf is the return of the riskfree rate and σP is the standard deviation of the fund or portfolio. The formula clearly highlights the differences between the Sharpe Ratio and the Treynor Ratio. The Treynor Ratio only considers the systematic part of the risk of a mutual fund but does not take into account the diversifiable risks. In contrast to this, the Sharpe Ratio uses the total risk in its denominator. Therefore, the Sharpe Ratio is also able to highlight the risks inherent in an inappropriately diversified fund (Sharpe, 1966, p. 128). These characteristics advise the use of the Sharpe Ratio if one investment portfolio is to be chosen as the single investment of a particular investor. In this case, only total risk counts. Therefore, style portfolios will generally not be evaluated by the use of Sharpe Ratios. A style portfolio consists of a defined group of asset that shares similar characteristics, such as value stocks or small cap stocks. These types of funds should not be the core asset within an overall asset allocation strategy and, therefore, not be evaluated with a performance measure that looks at total risk (Hübner, 2007, p. 65).
In terms of practical applications, the Sharpe Ratio has several drawbacks. Horowitz (1966) had already discovered that the performance predictability capabilities of this ratio were rather limited when correcting for the fund’s objectives. Another problem arises if returns that are used to calculate the Sharpe Ratio are not normally distributed. In this case, it is not possible to easily compare Sharpe Ratios that are based on returns with different distribution characteristics without further adjustments (Mahdavi, 2004, p. 47). A different, yet important, problem can arise from the estimation of returns and volatilities, which are the two input factors of the Sharpe Ratio. Lo (2002) found that the Sharpe Ratio for a hedge fund could be overstated by as much as 65% and provide inaccurate rankings. Additionally, the Sharpe Ratio can provide false rankings if the numerator becomes negative, that is the fund performance is below the riskfree rate (Scholz, 2006, p. 347). Several improvements have been proposed to compensate for this issue by Israelsen (2003; 2005), Scholz & Wilkens (2006), and others. Despite these drawbacks, the Sharpe Ratio is as widely used as it is easy to calculate and provides useful ranking information based on Markowitz’ (1952) meanvariance framework.
2.1.4 Jensen’s Alpha
Jensen (1968) introduced the Jensen’s Alpha measure, which is based on the CAPM and closely related to the Treynor Ratio. The Jensen’s Alpha uses the CAPM in order to find the rate of return a portfolio or investment fund should yield based on its beta. The alpha itself is then a measure of outperformance or underperformance relative to the return that would be expected based on the CAPM. Therefore, Jensen’s Alpha only accounts for systematic risk and ignores diversifiable risk. Graphically, the alpha measure is the vertical distance between the fund’s return and the security market line in an expected return/systematic risk diagram (Moy, 2002, p. 227). Jensen’s Alpha can be derived according to Equation (3):
illustration not visible in this excerpt
where rP is the rate of return of the portfolio or investment fund, βP is the beta of the portfolio, rf is the riskfree rate and rM is the market or benchmark rate of return. In contrast to the Sharpe Ratio, Jensen’s Alpha measures the return above or below the benchmark at the fund’s risk level. The rankings of both measures will therefore differ depending on the level of unsystematic risk inherent in the respective funds. When combining both measures, it is actually possible to find funds with a high level of unsystematic risk as these funds will show low Sharpe Ratios but high alpha measures (Moy, 2002, pp. 227228). These funds lack diversification.
Although the alpha measure is quite popular in practice, it has some major weaknesses (Grinblatt & Titman, 1993, p. 47). According to Kothari & Warner (2001), the power of this measure is quite low as the CAPM has been proven not to be true in reality and because it is extremely difficult to detect excess returns when the fund’s style differs from the valueweighted benchmarks. Additionally, the alpha measure is very sensitive to the β and can be misleading if there is little correlation between the fund and the benchmark (Moy, 2002, p. 229). It can also yield biased results when evaluating funds based on market timing strategies (Grinblatt & Titman, 1993, p. 47). Extensions of Jensen’s Alpha use the three factor model proposed by Fama & French (1993) or other assetpricing models, but do also have drawbacks in practical applications (Kothari & Warner, 2001, p. 2009).
2.1.5 The Sortino Ratio
In contrast to the assumptions of standard portfolio theory, investors do not judge upsiderisk and downsiderisk equally. Logically, all investors prefer positive over negative returns but the widely used standard deviation measure does not differentiate between positive volatility (leading to additional positive returns) and negative volatility (leading to additional negative returns). Both movements are weighted equally, although this is not consistent with the investor’s preferences (Estrada, 2006, p. 117). While the idea of adverse volatility had already been proposed by Levy (1968, p. 45), it took decades until Sortino & Price (1994) operationalized this idea and developed the Sortino Ratio (SoR), an extension of the Information Ratio, which will be presented in Section 2.2. Together with the Sortino Ratio, the idea of downside deviation is introduced and explained. Downside deviation is a measure for negative volatility, and it is calculated based only on returns that are below the minimum acceptable return (MAR). All returns above the MAR do not increase the volatility; they actually decrease it (Sortino & Price, 1994, p. 61). The Sortino Ratio can be calculated based on Equation 4:
illustration not visible in this excerpt
and where rP is the return of the fund or portfolio, rM is the return of the benchmark for the respective market and MAR is the minimum acceptable return. The MAR is often equal to the benchmark’s return (rM) of the same period or the riskfree rate. The σdown is simply the downside deviation that was mentioned previously. Compared to the Information Ratio, the Sortino Ratio is the same except that the tracking error volatility is calculated based only on returns that are below the minimum acceptable return.
In terms of practical applications, the Sortino Ratio can be unreliable or impossible to calculate if there are no or only very few returns below the MAP within the observation period. This could lead to the denominator being equal to zero in an extreme case or to a false estimate of the downside risk if there are only few returns below the MAP present. On the positive side, Chaudhry & Johnson (2008) found that the Sortino Ratio actually is a measure with high predictive power when it comes to returns that are positively skewed. Pedersen & Satchell (2002) came to a similar conclusion for asymmetric returns in general.
2.1.6 The M² Measure
Based on the idea of the Sharpe Ratio, Modigliani & Modigliani (1997) developed the M² performance measure or the measure of riskadjusted performance. While the interpretation of the value of most other performance measure, including the Sharpe Ratio, on a standalone basis is difficult and lacks intuition, Modigliani & Modigliani (1997) overcome this difficulty. Their measure adjusts the risk (standard deviation) of an investment portfolio by mixing it with the riskfree rate so that it exactly matches the standard deviation of the respective benchmark, which in most cases is a market index. This adjustment allows the investor to actually compare the outperformance or underperformance of a particular fund versus its benchmark on a riskadjusted basis. This means, the M² measure can be directly interpreted as riskadjusted outperformance (for positive M²) or underperformance (for negative M²) in terms of percentage points. It, therefore, supports the economic intuition of investors and is easily understood (Modigliani & Modigliani, 1997, pp. 4547). The M² measure is calculated according to Equation 5:
illustration not visible in this excerpt
where rf is the rate of return of the riskfree asset, rP is the rate of return of the portfolio or fund, rM is the return of the benchmark, σM is the standard deviation of the benchmark, and σP is the standard deviation of the portfolio or fund. According to Muralidhar (2000, p. 64) there are four different categories of funds to be distinguished: funds showing outperformance on an absolute and riskadjusted basis, funds showing outperformance only on an absolute basis, funds showing underperformance on an absolute and riskadjusted basis, and funds showing underperformance only on an absolute basis. Only funds that can be classified into the first category are really superior to an investment into the benchmark.
2.1.7 The Omega Measure
The Omega Measure introduced by Keating & Shadwick (2002) is a relatively new measure within the field of performance measurement. It is the only measure being presented within this thesis that is able to correctly deal with nonnormally distributed returns and does not require any kind of utility function in order to correctly rank different mutual funds by investors’ preferences. Omega is directly based on the cumulative distribution function of the returns and is therefore able to incorporate all higher moments of this distribution. The ratio introduces a loss threshold and weights possible gains or losses relative to this threshold by their probability. Equation 6 formalizes this relationship:
illustration not visible in this excerpt
where F(x) is the antiderivative of the cumulative distribution function of the returns that is defined in the interval between [ a, b ] and l is the loss threshold that has to be specified exogenously (Keating & Shadwick, 2002, p. 71). In order to rank different funds, their Omega Measure has to be calculated for a given loss threshold. The fund with the highest ratio is considered the best fund – it has the highest probability for returns above the threshold. However, the ranking order can change for different loss thresholds (Keating & Shadwick, 2002, p. 78). Additional research has yet to be performed in order to judge the advantages and disadvantages of this measure in the empirical environment.
2.2 The Information Ratio
“The information ratio is an important – perhaps the single most important – measure of investment performance. Investment managers will desire to have an investment strategy with the highest possible information ratio.” (Grinold, 1989, p. 31)
As indicated by Grinold’s (1989) quote, the Information Ratio (IR) is a popular and widely used performance measure. It was developed by Treynor & Black (1973) and initially called “appraisal ratio”. The Information Ratio indicates how much additional excess return over the benchmark can be obtained per additional unit of residual risk. Therefore, it is able to quantify how much value is added or destroyed by the active manager. As an example, the active manager usually has some expectations about future stock price developments. He will use this information to overweight or underweight certain stocks relative to the market portfolio, and thereby incur additional risks (relative to the market). Through the use of the Information Ratio, the investor is able to see how much additional return has been generated by the active manager in relation to the additional risks he had to incur in order to implement his superior information by overweighting or underweighting certain stocks. This is also the reason why this ratio is called Information Ratio – it measures the quality of the manager’s superior information (Goodwin, 1998, pp. 3435).
According to Treynor & Black (1973), the Information Ratio can be calculated as shown in Equation 7:
illustration not visible in this excerpt
where rP is the rate of return of the portfolio or fund, rM is the return of the benchmark and σER is the volatility of the excess return, that is the standard deviation of the α. The rationale behind the Information Ratio is very closely related to the investor’s utility function as shown by Jacobs & Levy (1996, p. 12). They explain that investors of active funds are not riskaverse in the common sense but rather regretaverse. Regretaversion means that they generally accept the risk of a passive investment in this asset class but – depending on the excess returns of the active fund – regret their decision to invest in an active fund and not in the passive alternative. According to the derivation of Jacobs & Levy (1996, p. 12), the utility of the investor rises with increasing excess returns and decreases with increasing residual risk. The relationship between residual risk and utility depends on to the investor’s individual regret aversion. Equation 8 mathematically illustrates the utility function:
illustration not visible in this excerpt
where U is the investor’s utility, α is the excess return, λ is the regretaversion coefficient and ω is the residual risk. Based on this idea, investors are able to use the Information Ratio or, to be precise, the residual risk measure in order to limit the fund universe based on their personal risk preferences. Alternatively, the investor can restrict the active manager by setting a maximum residual risk limit that he is willing to bear. In practice, the residual risk is sometimes called tracking error volatility or simply tracking error. In this study, the term tracking error always refers to tracking error volatility, that is the residual risk or the standard deviation of the excess returns. Investors should, however, be aware that restrictions in terms of tracking error can lead to losses in overall utility (Israelsen & Cogswell, 2007, pp. 419420; Jacobs & Levy, 1996, p. 13). The term tracking error can be misleading and would be better changed into “differential from benchmark” according to Israelsen & Cogswell (2007), as a high tracking error is not negative per se. They found within their study that funds with a low tracking error show a higher beta, similar standard deviation and lower alpha when compared to funds with high tracking errors (Israelsen & Cogswell, 2007, p. 424). Grinold & Kahn (2000) had previously proposed the exante use of the Information Ratio for risk budgeting purposes. Figure 1 shows the portfolio possibility lines for different target Information Ratios in an excess return/residual risk framework. It can be stated that different Information Ratios allow for different opportunities. While Portfolio 1 is only achievable for a fund with a target ratio of 1.0, Portfolio 2 can be reached with a lower target ratio of 0.5. Based on this framework, the portfolio manager can only increase his alpha by increasing the tracking error for a given target Information Ratio (Grinold & Kahn, 2000, pp. 118119).
Figure 1: Investment Opportunity Set Based on Information Ratios
illustration not visible in this excerpt
Source: Adapted from Grinold & Kahn (2000, p. 118)
In terms of applications, the Information Ratio is used when it comes to selecting an actively managed portfolio for an investor who currently holds passive portfolios, such as index funds (Hübner, 2007, p. 65). It is also important to note that the Information Ratio does not provide any guidance with respect to asset allocation decisions. An actively managed bond fund with an Information Ratio of 0.5 is not automatically inferior to an actively managed equity fund with an Information Ratio of 1.0, as the measure does not incorporate correlations and levels of risk tolerance of the investor. Therefore, it should only be used to compare investment portfolios within the same style and asset universe (Goodwin, 1998, p. 41). However, Grinold & Kahn (2000, p. 114) mention that according to their research a top quartile manager has an Information Ratio of 0.5 and an exceptional manager should achieve a value of 1.0 or above. They believe this classification is true for all asset classes and time horizons with only slight deviations. Jacobs & Levy (1996, p. 12) also found an Information Ratio of 0.5 or above to be “very good” without restrictions to asset classes. Goodwin (1998, p. 41) analyzed the distribution of Information Ratios for samples of funds with different investment universes and found significantly different results across fund categories. This seems to be more plausible than the findings of both other studies and therefore different ranges of Information Ratios are expected in the empirical analysis when evaluating funds that invest in different asset classes and countries.
2.3 Sources of Active Returns: How to Beat the Benchmark
A general issue in the field of active portfolio management is the identification of sources for excess returns above the benchmark, the socalled alpha. While this paper cannot elaborate on this topic in detail, it was found to be essential to present the key concepts. As will be explained later in this section, the Information Ratio is very closely related to two determinants of active returns. This section will improve the theoretical understanding of the Information Ratio as well as explain the rationale behind actively managed funds.
Grinold (1989) developed the fundamental law of active management, a framework for active managers that is well known in the fund management industry (Staub, 2007, p. 358). Grinold & Kahn (2000) explain that investors select among different opportunities based on their personal preferences, which for actively managed funds “point toward high residual return and low residual risk” (p. 5). This concept, in fact, is very closely related to the theory of the Information Ratio (cf. Chapter 2.2) and also an important part of the theoretical fundament of this paper. Achieving high Information Ratios is not only favorable in terms of performance measurement, but also closely related to successful active management. To put it in other words, successful active managers will automatically achieve superior Information Ratios and the input factors of the Information Ratio can provide valuable guidance when taking investment decisions. Grinold (1989) identified two factors that lead to high Information Ratios. The first factor is the skill of the manager to correctly predict the residual return of each security in his investment universe. This factor is called the Information Coefficient (IC) and measures the correlation between the actual alpha and the forecasted alpha. If a manager is always right in his forecasts, he will achieve an IC of 1, while a manager without any skill will get an IC of 0 (Wander, 2003, p. 37). The second factor describes the number of independent investment decisions that are taken per year and is called breadth. Clearly, if a highly skilled manager takes 100 (in contrast to 10) investment decisions per year, his Information Ratio should and actually will be higher (Grinold & Kahn, 2000, pp. 147150). The fundamental law of active management illustrates the relationship between Information Ratio, Information Coefficient, and breadth to be as follows:
illustration not visible in this excerpt
where IR is equal to the Information Ratio and IC symbolizes the Information Coefficient. It should be noted that this relationship is only an approximation. As illustrated by Equation 9, the Information Ratio can be doubled by doubling the IC, that is the skill of the manager; by quadrupling the breadth, that is the number of independent investment decisions; or by using a combination of both actions (Grinold & Kahn, 2000, p. 148). However, the crucial point is the correct forecasting of residual returns, which should be a key skill of all active managers. Quantitative or qualitative models may be used in order to predict future returns. While some managers specialize in security selection of US equities, for example, other managers are good at forecasting returns of certain asset classes. Depending on the number of independent bets a manager takes, different skill levels are required in order to achieve a “good” or “very good” Information Ratio (Wander, 2003).
Irrespective of Grinold’s (1989) framework, believers of the efficient markets hypothesis would always question the value added of actively managed funds (Bodie, Kane, & Marcus, 2005, p. 378). Based on the strong form of the efficient markets hypothesis, all information, public and private, is reflected in the current stock prices. Therefore, it is impossible for active managers to add any value. In this case, active returns are generated simply by chance. The semistrong form of the efficient markets hypothesis suggests that all publicly available information is already incorporated in the current stock prices. In order to generate positive excess returns, active managers would have to have insider information, which would be illegal. Only the weak form of the efficient markets hypothesis, which states that past price information is contained in the current stock prices, would allow active managers to add value by performing economic or fundamental analysis (Grinold & Kahn, 2000, p. 481). While Jensen (1968) and additional researchers in the 1970s suggested that active management was inferior to passive investments, this view changed within the 1980s. It became generally accepted that there were inefficiencies in the market that could be successfully exploited. This is particularly true for markets with lower liquidity like small cap stocks, real estate, alternative investments, and emerging markets. However, when taking fees for active management and tax effects into account, it is today from an academic point of view still unclear whether actively managed funds are able to consistently outperform the market. At least on average they are not able to do so (Baks, Metrick, & Wachter, 2001, pp. 4546; Malkiel, 1995). This is also confirmed by the empirical results of this paper as presented in Section 4. Based on detailed analyses, Wermers (2000) found that active funds were able to outperform the market during 1975 through 1994 by 1.3% per year. On average, 0.6% can be attributed to higher average returns in relation to the characteristics of securities held by a fund, and the remaining 0.7% is due to stock picking abilities. However, when taking fees into account, the active funds underperform the market by 1.0%. Almost 1.6% of the 2.3% return differences is caused by fund expenses and transaction costs. Still, certain drawbacks of passive investments should be kept in mind before taking an investment decision: passive funds will never be able to outperform the market; they are always fully invested (even in severe market conditions), and they modify their holdings to match the benchmark without taking into account any opportunities that might be available in the market (Kjetsaa, 2004, p. 103).
2.4 Agency Problems Related to Performance Measures
The investment management business is a very typical environment in which agency problems in the form of moral hazard can arise. This is due to the fact that the manager is able to easily hide certain information, and monitoring activities are very costly, if not impossible at all for the investor. Therefore, it is crucial to closely align the interest of the fund manager, the agent, with the interest of the principal, the investor, by the use of performance measures and investment guidelines (Golec, 1992, pp. 8283; Stoughton, 1993, pp. 20092010). Initially developed by Ross (1973) and Holmstrom (1979), principalagent problems are in many cases resolved by introducing variable payment schemes that are designed to increase the agent’s wealth only if the agent acts in the principal’s interest. The simplest form of a performancebased contract utilizes a fixed part, which is paid in all cases, and a variable part, called bonus, that has call optionlike characteristics and depends on the level of success of the manager (Grinblatt & Titman, 1989a, pp. 808809). More complex forms of performancebased compensations include caps on the maximum level of the variable payment, certain trigger levels that have to be reached before a bonus will be paid, and penalties for inferior performance (Grinblatt & Titman, 1989a, pp. 810811). The implementation of a performancebased payment scheme is the responsibility of the fund administration, which also has to select appropriate performance measures. These actions will help to limit information asymmetries and align the interest of the investors with that of the manager.
While these general problems are mostly clear at first sight, additional issues can arise because of the use of a particular performance measure. A problem, mainly related to the Information Ratio, will be illustrated based on the fundamental law of active management. In this example, which has been adapted from Grinold & Kahn (2000, pp. 149150), a target Information Ratio of 0.5 is assumed. This target can be achieved in many different ways; three of them will be illustrated in the following. Firstly, a manager who has market timing skills that result in quarterly information (four per year) about returns of different securities markets will need an IC of 0.25 in order to reach his goal: [illustration not visible in this excerpt]. Another manager is good at securities selection. He tracks 100 companies and adjusts his investment fund on a quarterly basis. This manager has to achieve an IC of just 0.025 for an Information Ratio of 0.5: [illustration not visible in this excerpt]. A third manager is specialized on two companies and revises his view on each company 200 times a year. The skill level required for this manager will therefore also be 0.025, as he is taking 400 bets every year: [illustration not visible in this excerpt]. As outlined above, the Information Ratio assigns all three managers the same Information Ratio and, therefore, equal performance. While this paper does not intend to discuss the efforts and achievements of the three managers in the example, an extreme case scenario in the form of a thought experiment will demonstrate the agency problem. If one imagines a portfolio manager who takes just one active investment decision per year and whose correlation between forecasted and actual returns is 0.1, this manager will achieve an Information Ratio of 0.1: [illustration not visible in this excerpt]. As will be shown in the empirical part of this paper, an Information Ratio of 0.1 for an Equity US fund would in most years be assigned a “good”, and in some years even a “very good” rating, although the manager did practically nothing (cf. Appendix A, Table 12 and Table 13). The Information Ratio can therefore incentivize strategies that are unfavorable to investors. This is also confirmed by Cremers & Petajisto (2007), who found that closet indexers, which are actively managed funds that in fact are taking few active bets or very closely track the benchmark’s weighting of stocks, managed about 30% of all assets in 2003 (p. 3). This is very annoying for investors, as they could buy index funds with lower management fees instead. It seems that performance measures, which use the tracking error as risk measure would need a second dimension that observes the active weights of the fund (Cremers & Petajisto, 2007, p. 8). In order to tackle this issue, Cremers & Petajisto (2007, pp. 67) proposed the Active Share measure that is easy to calculate and able to quantify the active holdings of a mutual fund in relation to the corresponding benchmark. It is calculated according to Equation (10):
illustration not visible in this excerpt
[...]

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