CA’ FOSCARI UNIVERSITY OF VENICE
Master’s Degree Programme
Amministrazione Finanza e Controllo
(Business Administration – LM 77)
Final Thesis
Manipulation and manipulation-free Performance
Measures for Hedge funds
Supervisor
Ch. Prof. Barro Diana
Academic Year 2017/2018 Graduand Lincetti Leonardo Matriculation Number 839222
Manipulation and manipulation-free Performance
Measures for Hedge funds
ABSTRACT
Performance Measures manipulation assumes a key role in the alternative investment context. Performance Measurement Manipulation consists in the managers ability to alter the performance measurement by using several techniques do not add value to the fund, but only enhance the final bogus results.
The objective of this research is to understand, through an ex-post analysis, if the performance measures analysed, divided in manipulation and manipulation-free ones, may have been manipulated.
For answering to this question, we consider hedge fund monthly returns for 11 strategies from EDHEC-risk Institute database. From these data, performance measures have been calculated and we speculate, by using a rank correlation of the different measurement rankings, if these ones may have been manipulated or artificially modified. In other words, if fund managers have been encouraged to take advantages by employing measures, which are manipulable, and altering the fund performance.
INDEX
Introduction
1Chapter 1
Performance measures in Hedge Funds
1.1 Performance measures overview 3
1.2 Sharpe Ratio 5
1.3 Treynor Ratio 6
1.4 Information Ratio 7
1.5 Modigliani Index 8
1.6 Jensen Alpha Index and its extensions 8
1.6.1 Extension to Jensen’s Alpha: Brennan model 9 1.6.2 Extension to Jensen’s Alpha: Black’s zero-beta model 10 1.7 Hedge funds distributional returns and alternative performance
measures
10
1.8 Sortino Ratio 12
1.9 Omega Ratio 13
1.10 The relevance of performance measures to hedge funds context 13
Chapter 2
Performance measures Manipulation
2.1 Introduction to Performance Measurement Manipulation 17
2.2 Different methods of manipulation 17
2.3 Static manipulation: conditioning information 19
2.4 Static manipulation: the use of derivatives 20
2.5 Dynamic manipulation 21
2.6 MPPM: manipulation Proof Performance Measures and Morningstar Rating System
context
3.1 Hedge funds strategies in a real context 29
3.1.1 Convertible arbitrage 30
3.1.2 CTA Global 32
3.1.3 Distressed securities 34
3.1.4 Emerging markets 35
3.1.5 Event driven 37
3.1.6 Fixed income arbitrage 38
3.1.7 Global Macro 40
3.1.8 Long-Short Equity 41
3.1.9 Relative value 43
3.1.10 Shortselling 44
3.1.11 Funds of funds 46
3.2 Comparison between performance measures for 1997-2018 and 2013-2018 returns
48
Chapter 4
Ex post analysis of a possible performance measurements
manipulation
4.1 Introduction to the ex-post analysis 51
4.2 Performance measurements using rank correlation 52
4.3 Data and empirical results 55
4.3.1 Investor point of view 55
4.3.2 Manager fund point of view 67
4.4 Concluding remarks 70
Conclusions
73Introduction
The dicothomy between manipulable and manipulation-free performance measures has always belonged to the alternative investment world, since the theme of performance measures manipulation holds a significant role in the fund managers’ activities. The evaluation of hedge funds is difficult for the non-normal distribution of their returns, which usually are asymmetric and leptokurtic.
Performance Measurement Manipulation concerns about the managers ability to modify the performance measurements, which are manipulable, by using several techniques do not add value to the fund, but only enhance the final bogus results.
The composition tries to understand if calculated performance measures have been manipulated. Initially, we downloaded an aggregate sample of hedge fund monthly returns, divided in 11 fund strategies, from EDHEC-Risk Institute database and from these data we obtained performance measures. Thus, since we do not know the type of hedge funds involved, an ex-post analysis has been created through a correlation of the different performance measures rankings, following the models of Eling and Schuhmacher (2007) and Nguyen (2009) papers, in order to understand if the choice of performance measures is relevant and if there is persistence in performance in different time sections. Later, the objective was to identify possible anomalies due to manipulation attempts by fund managers.
The essay is organized as follows. The first chapter starts with a briefly description about the literature that is behind this topic: Chen and Knez (1996) describe the conditions about the admissibility of the performance measures for a correct evaluation about fund managers. The chapter continues with an overview about some of the main traditional performance measures, like Sharpe Ratio, Treynor Ratio, Sortino Ratio, Modigliani Index, which are usually employed for the evaluation of mutual funds and they mainly consider the first two moments of the return distribution; therefore, this typology of performance measurements may be manipulated by fund manager for attracting investors. After a briefly explanation about the higher moments of the return distribution, the Omega Ratio is pointed out, since it includes the first four moments of the distribution and it represents the manipulation-free performance measure in our analysis.
The second chapter introduces the theme of the performance measures manipulation. The topic is explained through a description of the different approaches, in detail two static methods and a dynamic one. Successively, the manipulation proof performance measures (MPPM) are showed, elaborated by Goetzmann et al. (2007), and the representation of the Morningstar rating system, which is used for evaluating funds.
The third chapter describes fund returns, derived from the most significant strategies for hedge funds, such as Convertible Arbitrage, CTA Global, Distressed Securities, Emerging Markets, Event Driven, Fixed Income Arbitrage, Global Macro, Long Short Equity, Relative Value, Shortselling, Funds of Funds. For each strategy the main statistical parameters and the Jarque-Bera normality test are presented. Later, a comparison between calculated performance measures is made.
The fourth chapter starts with an introduction to the ex-post analysis and it continues with a briefly description of literature about rank correlation among performance measures. Successively, the two rank correlation methods chosen are showed, respectively Spearman Rho and Kendall Tau. Finally, data and empirical results are pointed out, divided respect to a manager fund or an investor view point.
Chapter 1
Performance measures in Hedge Funds
1.1 Performance measures overview
Performance measures are fundamental and essential to give a clear overview about hedge funds, and about the whole investment instruments world too. According to Chen and Knez (1996), performance measures are based on the conception of admissibility, which represents the necessary condition so that the service provided by fund managers could be correctly evaluated.
A general performance function is considered admissible if and only if:
1) Passive investors portfolios1 have value equal to 0; it is consistent with the principle that the performance is positive when referred to a portfolio which contains superior information2.
2) It is linear, in other words a linear combination of a passive portfolios will never have a performance value different from 0.
3) It is a continuous function in the way that 2 funds whose returns are not separable from one another, will have similar performance values.
4) It gives a fair and non-trivial result, that is the alfa (extra-performance) with respect to the naïve3 alternative will have to been proportional to the performance of the bargaining activity.
1 A performance measure is considered non-admissible when it does not respect all the conditions above mentioned. Furthermore, it is also non-admissible when it does not distinguish among normal and superior information.
2 Normal information are those that are publicly available, while superior information are those that permit to obtain an extra-performance than the naïve alternative.
3 Naive is considered the benchmark. Benchmark, or market index, is an artificial basket of securities which represents a specific market pace. It is useful for measuring the market pace.
5) It is positive, in a way that it assumes a positive value if the extra-performance stands out greater than 0 with odd equal to 1.
Following Basile and Savona (2007), since hedge fund returns usually lack the normality requirement, it is very arduous to estimate the probability associated with an event, using the first two moment of the distribution4. Most of the performance measures assume that funds performances have a normal distribution, which permits to evaluate a fair and correct estimate of the expected loss. However, in the hedge funds case there are several abnormalities linked to the performance distribution.
First, the fund systemic risk is not recognizable with only one indicator, for instance the standard deviation, but it is necessary to examine every single investment with the respective associated risk; in the peculiar case of hedge funds, due to their nature and their non-normal returns distribution, other risk measures are often used. Second, if a static index is used to evaluate the performance, all the dynamic variables, in the market changing context, are ignored. Third, since static index is used to evaluate the fund performance, all the long-term risks, such the liquidity risk, the credit risk and the event risk are not measurable in an accurate way and they may generate possible errors of performances evaluation. Fourth, the risk measurement is incomplete if the preferences of investors are not considered, in other words if an economic big picture is not contemplated.
Nevertheless, classical measures are usually used to evaluate hedge funds performances as general convention, although these ones belong to alternative investments. The main performance measures will be presented in the next section, following Astolfi et al. (2006) description, and pointing out their features and their shortcomings referred to the hedge funds use.
Performance Measures 5
1.2 The Sharpe Ratio
The Sharpe Ratio (SR) is a risk-adjusted performance measure and it is composed by the difference between the average return of a portfolio and the risk-free rate per unit of volatility or total risk.
Figure 1.1: The CML and the frontier of Risky Assets.5
The SR is described in equation:
𝑆𝑅 =(𝑅𝑝̅̅̅̅ − 𝑅𝑓) σp
(1.1)
Where: 𝑅𝑝
̅̅̅̅= Expected portfolio return 𝑅𝑓= Risk Free rate
σp= Portfolio standard deviation
The Sharpe ratio represents the benefit that an investor achieves in terms of the expected excess return for unit of risk exposure. According to the CAPM6, this indicator stands for the slope of the Capital Market Line7 (CML) and it reveals the trade-off between return and risk. The greater the Sharpe Ratio, the greater is the slope of the CML and in this way the portfolio combination will be better.
According to Basile and Savona (2007), Chen and Knez (1996), the Sharpe Ratio belongs to the non-admissible indices. The SR is a risk-adjusted index and the utility, connected to the usage of these measures, is limited to portfolios which do not have superior information. It focuses only on the managers’ capacity of diversification and on the ability of positioning on the efficient frontier. In this way, fund managers simply position their-selves along the CML and they are not able to calculate the excess return compared to the benchmark. Furthermore, since hedge funds returns do not have a normal distribution, contrary to mutual funds, the application of the Sharpe Ratio does not permit to evaluate any distortions caused by kurtosis and skewness into the calculation of the volatility.
1.3 Treynor Ratio
The Traynor Ratio (TR) is the ratio among the difference between the excess portfolio return with respect to the risk-free rate divided by the beta of the portfolio.
This ratio is defined as:
𝑇𝑝 =(𝑅𝑝̅̅̅̅ − 𝑅𝑓)
𝛽p (1.2)
Where:
Rf = risk-free rate
βp = Beta of the portfolio 𝑅𝑝
̅̅̅̅= Expected portfolio return
6 Capital Asset Pricing Model, Sharpe and Lintner 1964.
7 The capital market line (CML), in the CAPM, represents the line that connects the risk-free rate of return with the tangency point on the efficient frontier of optimal portfolios that offer highest expected return for a defined level of risk or the lowest risk for a given level of expected return.
Performance Measures 7
It differs from the Sharpe Ratio for the denominator: in the Treynor Ratio it is the beta while in the SR it is the standard deviation. The beta represents the portfolio systematic risk or market risk, whereas the standard deviation considers the total volatility of the portfolio. Since this ratio speculates on the perfect portfolio diversification, the systematic risk is the only one considered because the specific risk will be removed by the diversification effect. Since beta represents the price sensibility of a share to the market changes, Traynor Ratio is used to evaluate performance of an equity portfolio.
1.4 Information Ratio
The Information Ratio (IR) is defined as the ratio between the difference of the excess portfolio return with respect to the benchmark divided by the Tracking Error Volatility, the standard deviation obtained from the difference between the portfolio return and the benchmark return. The ratio is illustrated in the equation:
𝐼𝑅 =(𝑅𝑝̅̅̅̅ − 𝑅̅̅̅̅)𝑏
TEV (1.3)
Where: Rp
̅̅̅̅= Expected portfolio return Rb
̅̅̅̅ = Expected benchmark return
TEV = Tracking error volatility between the portfolio and the related benchmark
The Information Ratio provides the total amount of the excess portfolio return compared to the benchmark for each related unit of risk expressed by the TEV, and it permits to evaluate the fund manager ability of overperforming the benchmark respect to the risk undertaken. The Information Ratio measures the asset allocation activity respect to the benchmark chosen, whereas the Sharpe Ratio calculates the fund performance in absolute terms.
1.5 The Modigliani Index
The Modigliani Index, also known as M2 or RAP8, is a risk-adjusted performance measure and it is represented by:
𝑀2 = 𝑅𝑓+ (𝑅̅̅̅̅ − 𝑅𝑝 𝑓) ∗ 𝜎𝑏 𝜎𝑝 (1.4) Where: 𝑅𝑝
̅̅̅̅= Expected portfolio return 𝑅𝑓= Risk Free rate
σp= Portfolio standard deviation
𝜎𝑏= Benchmark (market) standard deviation
Modigliani Index permits to identify the excess return of the financial activity with respect to the risk-free one if the portfolio has the same standard deviation of the market.
Modigliani index is like the Sharpe Ratio but unlike this one, it multiplies the excess return by the standard deviation of the market: the purpose is to underline which fund has better results with respect to the level of risk.
1.6 Jensen Alpha Index and its extensions
Jensen Alpha is based on the CAPM model and it is defined as follow:
𝛼 = 𝑅̅̅̅̅ − ( 𝑅𝑝 𝑓+ 𝛽𝑝∙ (𝑅𝑚− 𝑅𝑓)) (1.5)
Where:
𝑅̅̅̅̅=Expected portfolio return 𝑝 𝑅𝑓= Risk Free rate
𝛽𝑝= Beta of the portfolio 𝑅𝑚=Market return
8 Risk Adjusted Performance
Performance Measures 9
The Jensen Alpha represents the excess return of a security or a portfolio of securities respect the expected return that this portfolio would have produced, based on its beta or systematic risk. 𝑅𝑚 stands for the market return and it, obtained from a benchmark, is composed by all available securities, based on the mean-variance parameter. Jensen Alpha uses the benchmark like a landmark and it considers only the systematic risk. The Beta, as a measure of the risk taken, is proportional to the value of alpha, since last one takes into consideration only the systematic risk. The choice of the benchmark determines the result; for this reason, one of the shortcomings of this index is that it usually does not show the fair manager performance, for instance when managers apply a market timing strategy.9
1.6.1 Extension to Jensen’s Alpha: Brennan model
One of the main assumptions of the CAPM model is that taxation is absent, investors hold the same risky assets portfolio; in this way the investors are indifferent to receiving income as capital gain or dividends. Nevertheless, in the real context taxes exist and they influence the combination of risky assets portfolios. Brennan (1970) proposed a modified CAPM, considering these fundamentals. The Alpha is defined:
𝑅𝑚− 𝑅𝑓 = α + βp∗ ((Rm− 𝑅𝑓)g − T ∗ (𝐷𝑚− 𝑅𝑓)) + T ∗ (𝐷𝑝− 𝑅𝑓) (1.6)
And T = Td − Tg
1−Tg
Where:
Td= The average taxation rate for dividends Tg= The average taxation rate for capital gains Dm= The dividend yield of the market portfolio
Dp= The weighted sum of the dividend yields of the assets in the portfolio Rp
̅̅̅̅ = Expected portfolio return
Rf = Risk Free rate βp= Beta of the portfolio
Rm= Market return
1.6.2 Extension to Jensen’s Alpha: Black’s zero-beta model
Black’s zero-beta model is characterised by:
𝛼 = 𝑅̅̅̅̅ − (𝑅𝑝 𝑧+ 𝛽𝑝∗ (𝑅𝑚− 𝑅𝑧)) (1.7)
Where: 𝑅𝑝
̅̅̅̅=Expected portfolio return 𝑅𝑧= Return with zero beta 𝛽𝑝= Beta of the portfolio 𝑅𝑚=Market return
Black implements a model replacing the risk-free asset with a portfolio with a zero-beta. He claims that it is possible to take short positions on the risky assets, rather than lending or borrowing money at risk-free rate.
1.7 Hedge funds distributional returns and alternative performance
measures
According to Lhabitant (2006), it is necessary to analyze also other two moments of the distribution: skewness and kurtosis, in addition to mean and variance, in order to have a right perception of performance measures, associated to hedge funds.
The skewness represents the third moment of the distribution and it specifies that the probability of loss is different from the probability of gain as compared to variance, which
simply adds the two probabilities. The skewness is illustrated by the Fisher index:
𝑀3 = 1 𝑛∑ ( 𝑅𝑡− 𝐸(𝑅𝑡) 𝜎 ) 3 𝑛 𝑡=1 (1.8)
Performance Measures 11
Where:
n= number of observations σ= portfolio standard deviation 𝑅𝑡= portfolio return
𝐸(𝑅𝑡)= expected value of portfolio return
If M3 > 0 it means that the skewness is positive, and the distribution is asymmetrical to the right.
If M3 < 0 it means that the skewness is negative, and the distribution is asymmetrical to the left.
The kurtosis represents the fourth moment of the distribution and it points out the deviation from the normal distribution, in other words it measures extreme values in either tails. A large kurtosis in a distribution shows tail data exceeding the tails of the normal distribution, whereas a low kurtosis in a distribution shows tail data which is less utmost than those of normal one. A high kurtosis of the distribution returns means that investor will obtain more extreme returns, both positive or negative, than those achieved with the normal distribution.
The kurtosis is expressed by the Pearson index :
𝑀4 = 1 𝑛∑ ( 𝑅𝑡− 𝐸(𝑅𝑡) 𝜎 ) 4 𝑛 𝑡=1 (1.9)
The Kurtosis for a normal distribution is M4= 3.
If M4> 3 the distribution has heavier tails than normal distribution. If M4< 3 the distribution has lighter tails than normal distribution.
In the hedge funds context, the kurtosis, expressed by the Pearson index, is usually higher than 3, so it means that returns obtained tend to be different from their expected value, caused by the probability of occurrence of extraordinary events. In proof of this, Kat (2003) analyzes hedge funds returns, which are taken in account from a bundle using different strategies, and he shows that these returns usually have a remarkable degree of kurtosis, proving that hedge funds possess by their nature a large probability of big losses or gains. For this reason, the
performance measures mentioned above are not suitable for measuring the hedge funds performance. Moreover, as reported by Keating and Shadwick (2002), the two first moments, i.e. mean and variance, are inappropriate for describing the hedge funds returns and it was necessary to develop other peculiar and characteristic performance measures which requires superior moments, such as Omega ratio and Sortino ratio.
1.8 Sortino Ratio
The Sortino Ratio is like the Sharpe Ratio and the Traynor Ratio, but at denominator it uses the downside deviation, measured by downside volatility.
It has defined as follow:
𝑆𝑅 =𝑅̅̅̅̅ − R𝑝 f
𝜎𝐷 (1.10)
Where 𝜎𝐷=√
∑ni=1min〔0,(Rp − Rf)〕^2
n
σD = Downside deviation computed using only negative deviations from the reference 𝑅𝑓 Rp
̅̅̅̅ = Expected portfolio return Rf= Risk Free rate
n = number of observations
The downside deviation only considers downside deviation from a reference value, in other words it is below a certain threshold which can be equal to zero or lower than the investors’ expectations of desirable return. The downside deviation10 is more feasible to evaluate the hedge funds level of risk or all those instruments which do not have a normal returns distribution. In fact, hedge funds often must face with returns which deviate from the average and therefore it is more interesting to understand which part of the total volatility comes from negative returns. The Sortino Ratio stands for the investor’s premium reward in exchange for bearing additional units of risk.
10 Downside deviation is a variation of standard deviation, which measures the deviation of only bad volatility. It measures how large the deviation in losses is and focusing only in negative returns.
Performance Measures 13
1.9 Omega Ratio
According to Keating and Shadwick (2002), describing a distribution of returns by using only the first two moments produces a level of inaccuracy in the performance measurement. Omega Ratio embraces all the superior moments of a return’s distribution. It is a ratio between the probability to obtain a gain higher that a determined threshold and the probability to obtain a loss. It is defined as : 𝛺𝑟 =∫ (1 − 𝐹(𝑥))𝑑𝑥 𝑏 𝑟 ∫ 𝐹(𝑥)𝑑𝑥𝑎𝑟 (1.11) Where:
F(x) is the cumulative probability distribution for the returns r = the threshold value selected
a and b = intervals of possible returns
As described by Le Sourd (2007), the measure is composed by dividing returns into gains and losses with respect to a return threshold, which represents the minimum acceptable return for an investor, and so taking in account the probability weighted ratio of returns below or above the division. Omega Ratio is used to classify managers performance and the ranking is based on the interval of returns and it embraces all higher moment effects. Depending on the additional information used, Omega will generate significant different classification of portfolios if compared with other measures like Sharpe Ratio or Jensen Alpha.
In brief, it may be particularly suitable for all those instruments which do not have a normal return distribution, as in the case of hedge funds.
1.10 The relevance of performance measures to hedge funds
context
As reported in the first paragraph of this chapter, most of the performance measures utilized
for funds are risk-adjusted and they are non-admissible11. The utility of these measures is limited to the evaluation of portfolios, which are not endowed with superior information and
11 See section 1.
that are only based on the fund manager ability of portfolio securities diversification. For this reason, it is impossible to assess the fund extra return with respect to the naïve alternative. Following Kat (2003), since risk-adjusted measures, used for mutual funds, present in the denominator the risk endured, it is important to understand if the same scheme is applicable to the case of hedge funds context. Although in risk-adjusted measures the denominator can be composed with different risk evaluator, the measure more used for evaluating funds is the Sharpe Ratio and other measures, like Treynor Ratio and Sortino Ratio, represent a sort of modification of the index created by Sharpe.
Moreover, the Sharpe Ratio can be manipulated using different techniques, through which fund managers improve the fund performances: this theme will be analysed and examined in depth in the next chapter.
According to Basile and Savona (2007), kurtosis, skewness and autocorrelation create an underestimation of the risk of the portfolio and at the same time an overestimation of the hedge funds performances. As Chen and Knez (1996) reported, Sharpe Ratio does not permit to distinguish the superior information from normal ones, which are publicly available and they produce similar returns to the benchmark.
By using derivatives in the hedge funds context, fund managers can alter the risk-return ratio: they increase their positions in derivatives and this behaviour provokes a growth about the risk exposure. There are three main reasons which constitute drawbacks of the use of Sharpe Ratio as performance measures for hedge funds. First, the impossibility to evaluate the service provided by a fund manager in a fair way. Second, the underestimation of the funds risk, due to derivatives practices. Third, the impossibility to distinguish the performance obtained from the choices of investment implemented. For these causes, the usage of Sharpe Ratio as performance measure may be not enoughhy in the case of hedge funds.
Chapter
2
Performance Measurement
Manipulation
2.1 Introduction to Performance Measurement Manipulation
In this chapter there will be a deep focus on the Performance Measurement Manipulation and the analysis of the related techniques used by fund managers in order to distort the reality; in other words, since investors are mainly interested in maximizing their profits, they choose and select those managers which guarantee the best level of outcome earned. In practice, as the appraisal on managers is basically grounded in the consideration of inputs provided in exchange for the outputs received, managers are incentivized to take actions that improve these measures.
Performance Measurement Manipulation consists in the managers ability to modify and manipulate the several funds’ performance measures by using several techniques, which do not effectively add value to the fund, but they only enhance the final bogus results.
2.2 Different methods of manipulation
Since Markovitz12 has elaborated the modern portfolio theory, based on the efficient portfolio composition13, the mean variance analysis appeared, becoming a landmark for the whole financial world: asset pricing, portfolio analysis, corporate finance and investment performance measurement.
Moreover, there is another basic method for evaluating the portfolio selection that includes the employment of a single scalar measure which involves both risk and return. As it was mentioned in the previous chapter, there are a lots of scalar measures examples which can be
12 Markovitz, Harry M., 1959, Portfolio Selection: Efficient Diversification of Investments, John Wiley, New York, 1959.
13 The portfolio composition concerns the optimization of expected return and expected risk, which is expressed in term of volatility.
used for the portfolio evaluation, such as: Sharpe Ratio, Treynor ratio, Information Ratio, Alpha Jensen and many others. The usage of single scalar measures permits fund managers to manipulate results. According to Goetzmann et al. (2007), performance measures undergo several shortcomings. First, performance measures are founded in the belief that the fund returns have a normal or a lognormal distribution, but in the reality fund managers can modify the relative returns distribution by using derivatives, dynamic trading strategies or options. Second, in practice performance measures can only be estimated and so they are subject to estimation errors and assumptions.
As Goetzmann et al. (2007) claim, there are three main methodologies to manipulate performance measures. In what follows there will be a basic description that will be developed further later. The first consists in the alteration of the return distribution in order to impact on measures, without introducing estimation errors. The second encompasses dynamic manipulation and it includes the introduction of the time variation into the return distribution, affecting those measures which include stationarity.14 The third consists on a dynamic manipulation approach, through which provoking estimating errors.
The example reported in Goetzmann et al. (2007) allows to easily understand the idea of manipulation. The case considered regards the estimation of a fund’s Sharpe Ratio over a 36-month timeline with monthly data. Hence, the manager wants to maximize only the excess value of the fund’s Sharpe ratio and the methodology consists in selling an out of money option in the first month and investing the total amount available in a risk-free asset. In the case of the option expires worthless, the total amount of the fund will be deposited into the risk-free asset for the residual 35 months. In this case, the portfolio would have a standard deviation equal to zero, a positive and greater excess return, given by the premium earned for the selling option, and the computed value of the Sharpe Ratio would be infinity.
Although both Sharpe ratio and Jensen Alpha are subjected to manipulation, both are still widely used as performance measures for evaluating funds. Goetzmann et al. (2007) focus
14 Stationarity means a stationary time series is one whose statistical properties, such as mean, variance, autocorrelation, are all constant over time.
Performance Measurement Manipulation 19
their analysis on the Sharpe ratio and they identify different typologies of manipulating performance measure (Sharpe ratio): two static method and a dynamic one.
2.3 Static manipulation: conditioning information
One of the most common approaches used for the efficient portfolio composition is based on the mean variance theory proposed by Markovitz and Sharpe. It consists in researching all the possible risk-return combinations and choosing the most suitable, dependent from the investor’s utility function and from the risk-aversion. Ferson and Siegel (2001) argue that it is possible to supplement this process by using exogenous information, called conditioning information15, which are useful for the allocation of the available capital in the different financial activities. In the paper, it is considered conditioning information also information derived from all the signals provided by market operators, which permit fund managers to enhance portfolio performance through their predictive skills, based on intuition or reserved information. In this way, it is probable to create a more efficient portfolio by using this kind of conditioning information rather than investing portfolio without the same level of information. They claim that this result is true for the no extreme values about the conditioning information, in fact as you move away from the centre of the returns distribution, the contribution provided by the signal16 will be lower. This is probably due to the reason that abnormal levels of conditioning information are interpreted with a conservative response; in other words, Ferson and Siegel claim that a portfolio performance is not a monotone weighted function of the degree of information.
In a context where it is possible to influence the investors’ choices by using signals, through the extension of the mean variance portfolio analysis, the authors create a bundle of efficient portfolio with minimum volatility.
Dybvig and Ross (1985) consider a hypothetic fund manager, who starts optimizing his portfolio without the usage of the conditioning information and after, he will insert the
15 Conditioning information is described as the situation “when the optimal solution may be a function of information to be received about the probability distribution of future outcomes”. From Ferson, Siegel, The Efficient Use of Conditioning Information in Portfolios. The Journal of Finance. Vol. LVI, No. 3 (June,2001), pp. 967.
above-mentioned information about future returns. In this way, they illustrate that a conditional efficient portfolio will not appear efficient for the uninformed investors.
Finally, Ferson and Siegel (2001) claim that an elevated level of conditioning information, referred to a specific security, will provoke both a bigger conditional expected return and a reduction of the levels of risk along the efficient frontier; however, the moving on the level of risk will always be lower, because managers will maintain conservative position for extreme signals. The conservative response to extreme signals is due to the managers’ desire to maximize performance reducing risk.
2.4 Static manipulation: the use of derivatives
Another kind of static manipulation is implemented through the usage of derivatives, in particular options. In recent years there has been a broad proliferation in use of derivatives and options for all the activities related to the portfolio management, such as leverage, hedging and asset allocation. The triggering cause is that these kinds of financial tools permit investors to achieve non-linear payoffs which reach a value in terms of returns not replicable through the usage of bonds or stocks.
Lhabitant (2000) in his article proposes an interesting opportunity for enhancing the return of stock ownership, through the usage of a covered call writing strategy, which allows to obtain an option premium and to receive an instant cash-in to offset the potential risk of future variabilities. At the same time, the use of a protective put strategy permits to protect the value of the portfolio, restricting on one side the downside risks and shielding on the other side the upside potential. Nevertheless, these typologies of strategies affect the efficient market suppositions, which establishes that the usage of options is not allowed in a risk-adjusted portfolio returns. Literature about a covered call writing is debated: Yates and Kopprasch (1980) and Grube and Panton (1978) assert that a covered call writing strategy, within a optioned portfolio, permits to achieve larger returns, although the level of risk is lower, expressed in terms of standard deviation, than a buy and hold17 one; conversely,
17 “A buy and hold is a long-term investment strategy, which entails a low turnover of the securities in the portfolio.” Definition provided from: http://www.morningstar.it.
Performance Measurement Manipulation 21
Merton et al. (1982) affirm that an option buyers have a more advantageous situation than a option sellers.
Lhabitant asserts that the investor must pay attention to two conditions. First, the optimal strategy for a traditional investor involves selling an out of the money covered calls, when this last one leads the pure stock strategy. This implies that, when a stock is a market index, some covered call strategies represent in a mean-variance space a ruling efficient frontier. Second, when the protective put position is dominated, the investor will not buy puts without knowing the movements of the market.
Finally, as Lhabitant observes, his strategy of beating the market is based on the inadequacy of the performance measures and not based on skills or abilities of the fund manager.
2.5 Dynamic manipulation
As reported by Goetzmann et al. (2007), in addition to the static manipulation of the performance measures, it is possible to implement manipulation strategies to performance measures, based on a dynamic analysis of the portfolio hold. Goetzmann considers a portfolio composed by an excess return 𝑋𝑖 with a probability 𝑝𝑖.
This portfolio is composed by a Sharpe Ratio equal to:
S = ∑ 𝑝𝑖𝑥𝑖
√∑ 𝑝𝑖𝑥𝑖2− ( ∑ 𝑝𝑖𝑥𝑖 )2
(2.1)
Thus, by maximixing the equation 2.1 and setting the constraint ∑𝑝𝑖𝑥𝑖= 𝑥̅ >0, he asserts that it is analogous to minimize the mean squared return subject to an expected return of 𝑥̅ with a zero cost as constraints. Using Lagrangian method and setting 𝑝𝑖 and 𝑝̂ , respectively 𝑖 the true and the risk neutral probabilities of state i, he obtains the maximal Sharpe Ratio portfolio as follows:
or
𝑥𝑖∗ = 𝑥̅𝑀𝑆𝑅[1 +1 − 𝑝̂ / 𝑝𝑖 𝑖 𝑆𝑀𝑆𝑅2 ] Where the 𝜎𝑥218 and the 𝑆
𝑀𝑆𝑅19 are equal to:
𝜎𝑥2= 𝑥̅2[∑ 𝑝̂𝑖2 𝑝𝑖 − 1] −1 (2.3) 𝑆𝑀𝑆𝑅 = [∑𝑝̂𝑖 2 𝑝𝑖 − 1] 1 2 (2.4)
The second version of the equation 2.2 permits to select the level of portfolio’s mean excess return because, in the static approach, the leverage does not influence the Sharpe Ratio. After that, the focus moves to the analysis about the dynamic manipulation. In efficient markets Sharpe Ratio is evaluated by using statistics, based on i.i.d.20 returns, but the situation changes when the composition of the portfolio is dynamically modified.
Goetzmann (2007) imagines a circumstance in which a fund manager obtained a high or a low average return respect to the realized variance of the portfolio. Starting from this consideration, he tries to adjust his portfolio, maximizing the Sharpe Ratio, and considering the difference between past and future expected returns.
Starting from equation 2.1 and setting ∑𝑝𝑖𝑥𝑖= 𝑥̅, Goetzmann obtains the Sharpe Ratio over the entire period, putting x̅h, the historical average excess return, with a standard deviation 𝜎ℎ and x̅f, the future average excess return, with a standard deviation of 𝜎𝑓; in addition, he sets γ equal to the fraction of the total time period passed.
18 Variance of the maximal-Sharpe Ratio portfolio. 19 Maximal Sharpe Ratio.
Performance Measurement Manipulation 23 It is defined as follow: S = γx̅h+ (1 − γ)x̅f √γ(x̅h2+ 𝜎 ℎ2) + (1 − γ)(x̅f2+ 𝜎𝑓2) − (γx̅h+ (1 − γ)x̅f)2 (2.5) or = γx̅h+ (1 − γ)x̅f √γx̅h2(1 + 1 Sh2) + (1 − γ)x̅f2(1 + 1 Sf2) − (γx̅h+ (1 − γ)x̅f)2
In the second version of the equation 2.5 he shows the variance of the Sharpe Ratio over the entire period gathering the variables x̅h2 and x̅f2 , expressing the ratio 𝜎ℎ2/𝑥̅ℎ2 equal to the past Sharpe Ratio 𝑆ℎ−2 and the ratio 𝜎𝑓2/𝑥̅𝑓2 equal to the future Sharpe Ratio 𝑆𝑓−2.
Moreover, Goetzmann (2007) asserts that, for maximizing the overall Sharpe Ratio, the fund manager has to maximize the future Sharpe Ratio, 𝑆𝑓 = 𝑆𝑀𝑆𝑅 , within his future portfolio. Thus, he obtains a target mean excess return 𝑥̅𝑓∗, by maximizing the Sharpe Ratio in the 2.5, equal to: 𝑥̅𝑓∗ = { 𝑥̅ℎ(1+𝑆ℎ−2) 1+𝑆𝑓−2 𝑓𝑜𝑟 𝑥̅ℎ > 0 ∞ 𝑓𝑜𝑟 𝑥̅ℎ ≤ 0 (2.6)
The meaning of this formula can be interpreted as follow: if the fund manager has obtained a higher historical Sharpe Ratio than the future one, 𝑆ℎ > 𝑆𝑓, he should set his portfolio at an inferior level both for the mean excess return and relative variance in the future, in order to give more importance to the mean excess return and variance already realized. Vice versa, if the manager has achieved a lower historical Sharpe Ratio, 𝑆ℎ < 𝑆𝑓, than the future one, he should set his portfolio in the future by increasing the mean excess return and the relative variance than those already realized. However, if he achieved in the past negative average
excess return, he should set the future target mean excess return, 𝑥̅𝑓∗, increasing it as much as possible the level of leverage.
The hypothesis in the paper consists in altering the portfolio in a single period of measurement and therefore influencing the distribution of returns, premising that the measurement period is shorter than the rebalancing one. The example reported by the authors involves the possibility that the value of the portfolio quickly rises during the first part of the measurement period and the strategy of converting in a less aggressive position should permit to obtain a smaller final period return and keep a high degree of Sharpe Ratio. In addition, there is the possibility of smoothing returns over time in order to maintain stable the level of mean return of the portfolio but decrease its variance, in such a way that incrementing the value of Sharpe Ratio.
2.6 MPPM: Manipulation Proof Performance Measures and
Morningstar Rating System
As mentioned at the beginning of the chapter, performance measures are subjected to be manipulated through the usage of different techniques, both static and dynamic. Goetzmann et al. (2007) derive a manipulation proof performance measure (MPPM), that satisfies the following properties. First, the result’s value should be independent and not influenced by the portfolio’s dollar value. Second, the measure should exhibit a single valued result to classify each portfolio, a sort of universal score. Third, the measure should be compatible and respectful of the standard conditions of the financial market equilibrium. Fourth, an uninformed investor cannot improve his estimated result by deviating from the benchmark portfolio, while an informed investor, by exploiting the arbitrage opportunities, improves his evaluation.
Going in depth into the description of the conditions, the first one permits to remove measures that generate single fractional ranking and moreover it removes all the useless measures. The second one claims that returns are enough statistics instead of currency (dollar) losses or gains. The third and the fourth ones endow the directive. Moreover, in order that the manipulation does not be executed, scores must be time divisible for
Performance Measurement Manipulation 25
precluding the dynamic manipulation and must be coherent with an economic equilibrium; they must be concave, for eluding the scores’ growth by the usage of leverage. Finally, they must be growing in returns, to identify possible arbitrage opportunities. There is not any MPPM if all the four conditions are respected.
According to Goetzmann et al. (2007), a performance measure’s expected value, in an efficient market and without private information, can be maximized by holding some levered benchmark portfolio. He distinguishes static manipulation as the ability to modifying levered benchmark portfolio without having informational reason, whereas the dynamic manipulation consists in composing the portfolio over time, focusing on past performances achieved. Since a performance measure represents a function which combines the possible outcomes related their probability distribution, it is possible to express the estimated performance measure equal to 𝛩̂ = ((𝑟𝑡, 𝑠𝑡)𝑡=1𝑇 ), in which 𝑟𝑡 represents the all the possible returns at time t, while 𝑠𝑡 represents each time period at time t, which covers periods from 1 to T. Goetzmann asserts that a dynamic free-manipulation performance measure has to be endowed of a independence property, in sense that the modification of some of its components does not influence its ranking composed by other elements.
Thus, the unique possible MPPM obtained is equal to:
𝛩̂ ((𝑟𝑡, 𝑠𝑡)𝑡=1𝑇 ) = ϒ (1
𝑇∑𝛳𝑡(𝑟𝑡, 𝑠𝑡)
𝑇
𝑡=1
) (2.7)
The function ϒ(.) represents any increasing and concave function. Since the majority of performance measures consider each time period identically, the same principle is considered also for 𝛳𝑡 functions and for this reason, it is expressed as an average of the function of returns. The equation 2.7 represents a time-series average of assessment of the rise in utility. So, even if the manipulation does not influence the measure, the decision of the kinds of performance measures 𝛩̂ may alter the ranking of the portfolios.
Therefore, since any portfolio, which is not first order stochastically dominated, must maximize the expectation of 𝑢 (∏(1 + 𝑟𝑡)) = 𝑢 [𝑒𝑥𝑝(∑ ln(1 + 𝑟𝑡)], considering a portfolio
in a specific T period with i.i.d. returns, Goetzmann calculates the MPPM, putting the maximization as follow: 𝑢 [𝑒𝑥𝑝 (∑ln(1 + 𝑟𝑡)] = ϒ ( 1 𝑇∑ 𝛳𝑡 (𝑟𝑡, 𝑠𝑡) 𝑇 𝑡=1 ) (2.8) ∑ ln(1 + 𝑟𝑡) = ϒ𝐼(1 𝑇∑ 𝛳𝑡 𝐼 (ln( 1 + 𝑟 𝑡) 𝑇 𝑡=1 ) Where 𝛳𝑡𝐼 (x) = 𝛳𝑡 (𝑒𝑥− 1) and ϒ𝐼(𝑥) = ln(𝑢−1 [ϒ (𝑥)]).
Specifically, the second equality is verified if and only if 𝛳𝑡𝐼 and ϒ𝐼 represent linear functions or only if ϴ (x) = (1 + 𝑟)𝜕.
Finally, Goetzmann shows a specific MPPM equal to:
𝛩̂ = 1 (1 − 𝜌)𝛥𝑡𝑙𝑛( 1 𝑇∑[(1 + 𝑟𝑡) (⁄ 1 + 𝑟𝑓𝑡)] 1−𝜌 𝑇 𝑡=1 ) (2.9)
The Ф statistic represents an approximation of the portfolios’ premium return adjusted for risk, 𝑟𝑡 is the rate of return at time t and 𝑟𝑓𝑡 is the risk-free rate. T stands for the number of observations and ∆T represents the span of time between observations. The coefficient ρ represents the parameter of the risk aversion and the exponential 1-ρ over the ratio is set for considering this risk. The equation 2.9 permits, setting the logarithm divided by one minus the parameter of risk aversion ρ and by the length of the period, to guarantee that 𝛩̂ is comparable to a continuous rate of returns.
According to Maillard (2017), the MPPM is very comparable to the Morningstar Risk-Adjusted Return (MRAR), which firms usually use to evaluate the performance of different funds.
Performance Measurement Manipulation 27 𝑀𝑅𝐴𝑅 (𝛾)=(1 𝑇∑[(1 + 𝑟𝑡) (⁄ 1 + 𝑟𝑓𝑡)] −𝛾 𝑇 𝑡=1 ) −𝛾∆T1 − 1 (2.10)
The evidence shows that the parameter of risk aversion is expressed by 𝛾 and not byρ. That way, it is possible to insert the equation 2.10 for rewriting the MPPM equal to:
𝛩̂ = 𝑙𝑛 [1 + 𝑀𝑅𝐴𝑅 (ρ − 1)] (2.11)
Going in depth, the Morningstar rating system is based on a simple idea: the fund returns are evaluated in terms of costs and risk accepted by managers. It is a single scalar method, introduced in 1985, for classifying mutual funds and it assigns to them from one to five stars. The Morningstar rating, expressed with stars, permits a funds returns classification derived from elements like costs, fund’s past performance and level of risk.
Morningstar categories ensemble funds with uniform and consistent investment policies and they are based on a specific composition analysis of securities; parameters are constituted by the features of securities which compose the portfolio: capitalization and growth perspective are taken in account for mutual funds, meanwhile duration and reliability for bonds.
The commissions are deducted from funds performances, moreover the Morningstar rating considers the subscription fees: an entrance fee of 5% for mutual funds and an entrance fee of 3% for bonds. The Morningstar rating grants last 36 months performances (three years) of funds considered and it is calculated starting from the investor utility in function of fund performances; basically, this ranking is based on the risk aversion principle.21 It is possible that some funds do not have a rating and it is due to data of fund performances have less than three years or few information about fund are available.
As Winston (2005) claims, Morningstar scalar measure has several shortcomings or weaknesses and it is very simple for fund managers to implement a manipulation strategy to modify star rating. First, as mutual funds usually are not acquired in isolation but as a
21 Risk aversion consists in the choice of an investor who, between investments with a similar expected return, selects the investment with the lower risk.
portion of the portfolio, the stand-alone essence of the measure does not represent an accurate criterion to evaluate funds, since it does not consider all the interactions with the residual investor’s portfolio. Second, the wrong belief that past performances are good valuer to forecast future funds’ results. Third, leverage, like borrowing or lending at risk-free rate, is used to been rewarded by the scalar measure. Winston shows that gaming the Morningstar rating system is quite easy by using several informationless techniques and not using superior information or management skills. The several multidimensional fund performances are decreased to a single scalar number and this last one does not represent all the aspects and elements which are necessary for a good portfolio composition: a single scalar, or every scalar which is considered in separation, is subjected to errors.
Winston concludes that when someone decides to evaluate a fund performance by using a risk-adjusted scalar he must pay attention for specific elements: first, the usage of leverage, such as lending, borrowing, holding cash, deleverage through derivatives; second, the use of optionality, which consists in the employment of options.
Chapter
3
Hedge funds strategies and
performance measures in a real context
3.1 Hedge funds strategies in a real context
Fung and Hsich (1997) assert that hedge funds strategies are composed by both location and style. Location involves the asset class, such as equity, currencies and fixed income where the fund invests in, while style is related to the positions which an investor takes: for instance, taking a short or long position or keeping a market-neutral one.
Hedge funds include different strategies, linked each one to a particular approach of investment. According to Basile and Savona (2007) hedge funds strategies are divided in two macro categories: directional and non-directional (market neutral). Directional strategies are based on the provision of the market pace, realizing long positions when there is a strong possibility of a bullish market or short ones in the opposite case. Non-directional strategies are based on arbitrage opportunities, using price discrepancies of the financial activities. Following the distinction proposed by Amenc et al. (2002), strategies can be also divided in return enhancers and risk reducers. The former is related to search for high returns which provoke an increasing portfolio volatility and it includes strategies such as distressed securities, macro funds or event-driven; the latter is related to search for stable returns with a lower portfolio volatility and it involves strategies such as fixed income arbitrage and long/short funds.
Hedge fund data are downloaded from the EDHEC-Risk Institute,22 which provides hedge funds information and other services to researchers and institutional investors. Hence, there will be, for each hedge funds strategy, a brief description and a small analysis of their
22 The EDHEC-Risk Institute is an academic centre for industry-relevant financial research. In partnership with large financial institutions, it is composed by permanent professors, engineers, and support staff, and research associates and affiliate professors.
statistical parameters and their main relevant performance measures related. Data consist in aggregate monthly hedge funds returns, divided in eleven hedge fund strategies. Here, will follow the representation of each strategy with the relative bar chart of the returns distribution, the analysis of the main statistical features with a comparison between results obtained with a restricted data timeline (from July 2013 to July 2018) and a wider dataset, which includes more than 20 years (from January 1997 to July 2018).
In the following tables all the main statistical indices are showed (mean, median, std deviation, minimum, maximum, skewness, kurtosis) for each strategy and they give information about the return’s distribution; moreover, in the same tables are exhibited the value of the Jarque-Bera test and the related probability. This last one is a normality test and it allows to understand if the distribution, in this case about the strategy monthly returns, are normally distributed (Gaussian distribution). With a confidence level α of 5%, it is established a hypothesis testing, composed by two different alternatives, respectively H0 and H1.
𝐻0 stands for the null hypothesis, that is the distribution of the residuals follows a normal distribution, whereas H1 stands for the alternative hypothesis, that is the residuals do not follow a normal distribution. The Jarque-Bera test shows that if the p-value, the probability associated to the value, is bigger than the level of confidence (5%), we accept 𝐻0, vice versa we accept H1.
3.1.1 Convertible Arbitrage
Convertible Arbitrage, as mentioned in the previous section, belongs to the risk reducers strategies and it consists in the ability to take advantage of anomalies in prices of corporate segmentation, investors buy or sell these securities but they finally short the stock in a way to protect them-selves from part or all the related risk.
Making a comparison about Figure 3.1A-Figure 3.2B, the mean and the respective std deviation of the returns from 1997 to 2018 are bigger than those of the timeline from 2013 to 2018.Both the Skewness values are negative (more pronounced in the 2013-2018 returns), with asymmetrical distribution on the left. Furthermore, both the Kurtosis values are bigger than three.
Hedge funds strategies and performance measures in a real context 31 0 10 20 30 40 50 -.150 -.125 -.100 -.075 -.050 -.025 .000 .025 .050 .075 .100 .125 .150 Histogram Normal D e n s it y Convertible Arbitrage
Figure 3.1A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Convertible
Arbitrage strategy with main statistic variables table.
0 20 40 60 80 100 -.03 -.02 -.01 .00 .01 .02 .03 .04 Histogram Normal D e n s it y Covertible Arbitrage
Figure 3.1B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using Convertible
The results underline that both the returns distributions are leptokurtic and they are more concentrated toward the mean respect a normal distribution, the horizontal axis is stretched by the outliers and as well as the size of the data stands out a narrower vertical range.
Analyzing the Jarque-Bera values, both the distributions show a p-value smaller than the confidence level, that is the hypothesis of the normal distribution of the residuals must be refused.
3.1.2 CTA Global
The CTA Global23 is a fund strategy based on the capability to invest in commodity, listed financial and currency markets. The usage of this typology of strategy permits to protect the funds performances against the market downturns as well as leverage is used to enhance the influence of the market changes on the portfolios. CTA Global belongs, as the previous mentioned Convertible Arbitrage, to the risk reducers strategies.
Making the comparison between the different return distributions (Figure 3.2A and Figure 3.2B) below, we can see that mean and std deviation of the returns from 1997 to 2018 are bigger than same ones from 2013 to 2018. The skewness value for the 1997-2018 returns is positive and near to zero, whereas it is always near zero but negative for the 2013-2018 returns. The kurtosis value for both the returns distributions is near the threshold of three, but for 1997-2018 returns is below the three, that is the distribution is platykurtic; meanwhile for the 2013-2018 returns the value is a little above the three, so it is a leptokurtic distribution.
Moreover, about the Jarque-Bera normality test, both the returns distributions have a P-value bigger than the confidence level (5%), it means that the residuals of both the distributions follow a normal distribution.
Hedge funds strategies and performance measures in a real context 33 0 4 8 12 16 20 24 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 Histogram Normal D e n s it y CTA Global
Figure 3.2A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using CTA Global
strategy with main statistic variables table.
0 4 8 12 16 20 24 28 -.07 -.06 -.05 -.04 -.03 -.02 -.01 .00 .01 .02 .03 .04 .05 .06 .07 Histogram Normal D e n s it y CTA Global
Figure 3.2B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using CTA Global
3.1.3 Distressed securities
Distressed securities are based on exploiting the securities of those companies which are distressed in the financial situations, such as liquidation, bankruptcy or restructuring. These funds belong to the Return Enhancers strategies, in fact they are usually composed by high returns and an important correlation with stock and bond indices. Fund managers, who use this strategy, usually buy securities of these distressed companies at the discount price and then sell them when the same securities will be appreciated.
Observing the two datasets (Figure 3.3A/Figure 3.3B), the 1997-2018 returns present a bigger mean and a bigger standard deviation. The Skewness value is negative for both the returns, but it is more pronounced for the 1997-2018 ones (-1.34 vs -0.42). The Kurtosis value is a quite different for the two different set of returns: for 2013-2018 returns it is near three, meanwhile for the 2018 returns it is equal to eight, it means that for the 1997-2018 returns the distribution is more leptokurtic and values are more concentrated towards the mean. Finally, analyzing the Jarque-Bera values, the p-value of the 1997-2018 returns is equal to 0, that is the null hypothesis must be refused, meanwhile for the 2013-2018 returns the p-value is bigger the level of confidence, thus the null hypothesis must be accepted and the residuals follow a normal distribution.
0 5 10 15 20 25 30 -.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 Histogram Normal D e n s it y Distressed Securities
Figure 3.3A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Distressed
Hedge funds strategies and performance measures in a real context 35 0 10 20 30 40 -.03 -.02 -.01 .00 .01 .02 .03 .04 Histogram Normal D e n s it y Distressed securities
Figure 3.3B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using Distressed
Securities strategy with main statistic variables table.
3.1.4 Emerging markets
Emerging markets are strategies that permit to invest in equity or debt of emerging markets, which are characterized by a higher inflation and volatile growth. They allow to invest all the capital in individual regions or diversify it in several contexts, changing the weighting capital. Emerging markets are considered as returns enhancers strategies and are delineated by high return and a strong correlation with bond or stock indexes.
As the datasets show below (Figure 3.4A/Figure 3.4B), the 1997-2018 returns present a bigger mean and a bigger standard deviation than 2013-2018 ones. The value of the skewness is negative for both the returns, even if more marked in 1997-2018 returns (-1.19 vs -0.28). The value of the kurtosis is quite different between the two datasets: in 1997-2018 returns it is equal to 9.34 and the distribution is strongly leptokurtic, whereas in 2013-2018 returns it is equal to 3.11, that is the distribution is very similar to a normal one. Finally, about the Jarque-Bera test, the p-value of the 1997-2018 returns is equal to 0, thus the null hypothesis must be refused; on the contrary the p-value of the 2013-2018 returns is equal to 0.66, it is bigger than the level of confidence, so the residuals follow a normal distribution.
0 4 8 12 16 20 -.25 -.20 -.15 -.10 -.05 .00 .05 .10 .15 .20 .25 Histogram Normal D e n s it y Emerging Markets
Figure 3.4A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Emerging
Market strategy with main statistic variables table.
0 5 10 15 20 25 30 -.05 -.04 -.03 -.02 -.01 .00 .01 .02 .03 .04 .05 .06 Histogram Normal D e n s it y Emerging market
Figure 3.4B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using Emerging
Hedge funds strategies and performance measures in a real context 37
3.1.5 Event driven
Event driven is a strategy based on the capability to exploit pricing anomalies derived by business corporate transactions and situations, such as bankruptcy. It consists in the ability of the fund manager to evaluate the right probability of failure/success. Event driven strategy is characterized by high return and a strongly correlation with stock or bond indexes.
Making a comparison between the 2 datasets (Figure 3.5A/ Figure 3.5B), we can see that, as in the previous strategies mentioned, the mean and the standard deviation are bigger in 1997-2018 returns than 2013-2018 ones. Moreover, the value of the skewness is negative for both the datasets meanwhile the value of the kurtosis is equal to 8.18 in 1997-2018 returns, thus the distribution is leptokurtic; it is equal to 3.25 for the 2013-2018 returns and the returns distribution is like a normal one. This result is confirmed by the value of Jarque-Bera test: in 1997-2018 returns the p-value is no significant, whereas it is bigger than the α in 2013-2018 returns, so the residuals follow a normal distribution.
0 5 10 15 20 25 30 35 -.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 Histogram Normal D e n s it y Event Driven
Figure 3.5A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Event Driven
0 10 20 30 40 50 -.04 -.03 -.02 -.01 .00 .01 .02 .03 .04 Histogram Normal D e n s it y Event driven
Figure 3.5B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using Event Driven
strategy with main statistic variables table.
3.1.6 Fixed Income Arbitrage
Fixed Income Arbitrage, a Risk Reducers strategy, is based on the ability to take advantage of pricing anomalies in fixed income markets. This strategy tries to balance the interest risk by taking positions in similar securities, such as corporate or government bonds. Fixed income arbitrage is composed mainly by two classes: relative value and market neutral. The first one will be articulated in depth next sections, meanwhile the second one (market neutral) consists in a strategy based on the ability to take advantage of pricing anomalies in different sectors of the fixed income market or in the same sector but between different securities.
Making the comparison between two datasets (Figure 3.6A/ Figure 3.6B), we can see that the mean and the standard deviation are bigger in the 1997-2018 returns.
The value of the skewness is negative for both the datasets but is more pronounced in 1997-2018 returns (-3.96 vs -0.17). The value of the kurtosis is different for the two datasets: in 1997-2018 returns it is very high (29.79) and this circumstance implies that the returns do
Hedge funds strategies and performance measures in a real context 39 0 10 20 30 40 50 60 70 80 -.10 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 .12 Histogram Normal D e n s it y
Fixed Income Arbitrage
Figure 3.6A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Fixed Income
Arbitrage strategy with main statistic variables table.
0 20 40 60 80 100 120 140 160 -.012 -.008 -.004 .000 .004 .008 .012 .016 .020 Histogram Normal D e n s it y
Fix income arbitrage
Figure 3.6B: Histogram and Normal Curve of hedge funds returns from 2013 to 2018, by using Fixed Income
not follow a normal distribution (p-value is not significant). Whereas, in 2013-2018 returns the value of kurtosis is just below 3 (2.93), so since the p-value is higher than the level of confidence, the residuals follow a normal distribution.
3.1.7 Global Macro
Global Macro consists in exploiting the changes in global economy, which impact the interest rate and affect stocks, bonds and currency markets. This strategy takes advantage using leverage and derivatives for enhancing the influence of the market movements. Making the traditional comparison between 1997-2018 and 2013-2018 returns, you can see that the mean and the standard deviation are bigger in 1997-2018 returns. The value of the skewness is opposite for the two datasets: in 1997-2018 returns it is positive and equal to 0.96 while in 2013-2018 it is negative and it is equal to -0.03.
The value of the kurtosis is higher in 1997-2018 returns than 2013-2018 one (5.70 vs 3.71); this means that the 1997-2018 returns distribution is more leptokurtic and values are more concentrated towards the mean.
0 4 8 12 16 20 24 28 32 -.08 -.06 -.04 -.02 .00 .02 .04 .06 .08 .10 Histogram Normal D e n s it y Global Macro
Figure 3.7A: Histogram and Normal Curve of hedge funds returns from 1997 to 2018, by using Global Macro
