UNIVERSITY OF AMSTERDAM
Do long/short equity hedge funds show
persistent performance in times of crises
?
Faculty Economics & Business
Bachelor’s Thesis Finance and Organization
June 2018
Student name: Elizaveta Mikhaleva Student number: 11087293
Study specialization: Finance and Organization Supervisor name: Philippe Versijp
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Statement of originality
This document is written by Student Elizaveta Mikhaleva who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
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Abstract
In this research, the performance of long/short equity hedge funds is examined during the period of 1997-2012. This time period contains two major crises: the Dot com bubble (2000-2002) and the subprime crisis (2007-2009). The main focus of this research is to find whether the long/short hedge funds outperform the market during crises and whether this performance is persistent. The Fama-French five-factor model (1993) with an addition factor, the market volatility, is applied to test the performance. In this research, it is found that long/short equity hedge funds do not outperform the market during the overall time period examined (1997-2012), however a significant market outperformance is found during 2003-2006 and 2010-2012.
Although, the long/short hedge funds do not outperform the market, they do show persistence in their performance.
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Table of contents
1. Introduction ... 5
2.Literature Review ... 6
2.1. Hedge fund history ... 6
2.2. Hedge fund characteristics ... 6
2.3. Hedge fund performance in existing literature ... 7
2.4. Hedge fund strategies ... 8
2.5. Long/short equity performance in existing literature ... 9
2.6. Market volatility ... 11
3.Methodology ... 11
3.1. Data ... 11
3.2. Biases and solutions ... 12
3.2.1. Survivorship bias ... 13
3.2.2.Self-selection bias ... 13
3.2.3.Backfill bias ... 13
3.3. Time period ... 13
3.4. The model... 14
3.5. Hypotheses and expectations ... 15
3.6. Descriptive Statistics ... 16
4.Results and Analysis ... 17
4.1. Correlations ... 17
4.2. Testing hypotheses ... 19
4.2.1.Regression output 1997-2012 ... 19
4.2.2.Hedge fund beta ... 20
4.2.3.Testing hypothesis 1 ... 20
4.2.4.Testing hypothesis 2 ... 23
5.Conclusion ... 24
Appendix ... 27
Appendix 1 – Correlation tables ... 27
Appendix 2 – Regression outputs ... 28
References ... 33
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1. Introduction
The hedge fund industry has grown significantly over the past decades (Agarwal & Naik, 2005). According to the Hedge Fund Research (HFR)1, the total assets under management of hedge funds have grown by more than $930 million during the period from 1990 to 2004. Also, the number of operating hedge funds has grown tremendously during this time (Agarwal & Naik, 2005). Nowadays, the hedge fund industry keeps growing and gains its reputation among more and more investors (Agarwal & Naik, 2005).
Despite such development and popularity of this industry, there is limited information on this sector that is publicly available (Agarwal & Naik, 2005). Lack of transparency and difficult access to the information on the hedge funds make it harder to examine the industry (Agarwal & Naik, 2005). Nevertheless, there are a number of studies made which examine performance, influencing factors and risks associated with hedge fund investments. Studies by Ackermann et.al (1999), Agarwal and Naik (2000, 2005), Liang (1999), Stulz (2007) and Fung and Hsieh (2004, 2011) mainly focus on whether hedge funds bring diversification benefits as they claim to, whether they outperform the equity market and whether this
performance persists. However, studies made so far yield contradicting results due to the use of different methodologies, databases and performance measures. Therefore, a question arises: if hedge funds have such a contradicting nature, why does this industry keeps on growing? Perhaps, hedge funds do outperform the market significantly, even in times of the financial distress and this effect persists throughout the years. Subsequently, this leads to the research question of this thesis: do long/short equity hedge funds show persistent
performance in times of crises?
This thesis differs from the existing literature in various aspects. Firstly, this thesis focuses on one particular strategy, long/short equity and not on several common strategies. This hedge fund strategy has the largest share on the hedge fund market (Agarwal & Naik, 2005). Fung and Hsieh (2011) examine long/short equity hedge funds. However, the time period they use is different from one in this thesis. The time period examined in this thesis is from January 1997 to December 2012, which contains two crises: the Dot com bubble (2000-2002) and the subprime crisis (2007-2009). Moreover, not only the whole period is examined, but also five sub-periods. To test hypotheses and subsequently to answer the
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Hedge Fund Research (HFR) - one of the main advisory firms for hedge funds – www.hedgefundresearch.com. Source: Agarwal, V., Naik, N.Y. (2005). Hedge funds. Foundations and Trends in Finance, 1(2), 103-169.
6 research question, the Fama-French five-factor model (1993) is applied, but a new additional factor, the market volatility, is added.
This thesis is structured in the following way. Section 2 presents the brief history on hedge funds, their characteristics and performance. This section also gives an overview on of previous literature on the performance of long/short equity hedge funds. Section 2 is
concluded with an explanation of the market volatility and its importance as an independent variable in this thesis. Section 3 presents the methodology and the data. Section 4 shows the results. Finally, in section 5 conclusions are made.
2. Literature Review
2.1.Hedge fund history
Hedge funds industry has grown significantly over the past years and gained its reputation not only among investors, but also among regulators (Agarwal & Naik, 2005). Although, the number of hedge funds has been growing by around 25% annually since the late 1980s, they have existed for almost 60 years now (Ackermann et al., 1999). It is believed that the first hedge fund was found by A.W. Jones around 1959 (Ackermann et al. 1999; Agarwal & Naik, 2005; Fung & Hsieh, 2011). To “hedge out the market risk” (Agarwal & Naik, 2005, p.106), Jones took both long and short positions in equities (Fung & Hsieh, 2011). Taking long positions in the
undervalued securities and then shorting the overvalued ones brought Jones a return of 670% during the ten-year-period (Agarwal & Naik, 2005). The strategy Jones used to get such high returns is also known as long/short equity strategy (Agarwal & Naik, 2005), which is the focus of this thesis.
2.2.Hedge fund characteristics
There exists an outstanding amount of literature discussing the distinctive features of hedge funds. Hedge funds are usually compared to the mutual funds. The basic idea of a hedge fund is similar to one of mutual fund – investment pooling (Bodie, Kane & Marcus, 2011). However, hedge funds can not only take long positions as mutual funds do, but also have the right to take short positions in both liquid and illiquid assets, to use derivatives and to borrow (Stulz, 2007). Typically, hedge funds are limited partnerships consisting of no more than 100 advanced investors, thus this exempts them from the Investment Company Act2 and explains why they disclose minimum information in contrast to mutual funds (Ackermann et al., 1999). Another crucial difference between hedge funds and mutual funds is the incentive structure (Stulz, 2007).
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Investment Company Act (1940) requires transparency and predictability of the strategy to protect unsophisticated investors (Bodie et. al, 2011).
7 With minimum investment requirement ranging from $250,000 to $1 million (Bodie et al., 2011), hedge fund managers usually get a management fee of around 1% and around 14% of profits annually (Ackermann et al., 1999). In contrast to hedge fund managers, managers of mutual funds are rarely entitled to the incentive compensation (Stulz, 2007).
Although hedge funds have a lot of beneficial characteristics, such as flexibility, diversification and incentive fees (Ackermann et al, 1999), they still lack transparency. According to Fung and Hsieh (2004), lack of transparency of hedge funds makes it difficult for analysts to use the information on the performance of hedge funds and to determine its effect on the overall portfolio.
2.3.Hedge fund performance in existing literature
Performance of hedge funds is highly researched. As researchers use different methods, databases and performance measures, the results from their studies can contradict each other. Nevertheless, there are a number of researches which conform to each other. These are some of the conclusions on performance of hedge funds from the studies used in this thesis.
Ackermann et al. (1999) examine performance of hedge funds during the period of 1988 to 1995. They find that hedge funds continuously “outperform mutual funds, but not the standard market indices”, such as S&P500 market index (Ackermann et al. 1999, p. 833). In their opinion, such outperformance is explained by the incentive fees and investment flexibility (Ackermann et al., 1999). Findings by Ackermann et al. (1999) also conclude that, despite the fact that hedge funds are unable to outperform market indices, they are still considered as profitable investments.
Liang (1999) uses a model consisting of eight factors for equity, currency and commodities, debt and cash to find the differences in betas of mutual and hedge funds. Liang (1999) makes a conclusion that hedge funds are less correlated with the equity market than mutual funds are, thus they have lower betas than mutual funds do.
For their research, Agarwal and Naik (2000) use a sample from January 1994 to September 1998. The aim of their research is to compare the performance of directional and non-directional hedge fund strategies during the market downturns and upturns (Agarwal & Naik, 2000). They found that non-directional strategies underperform the S&P500 market index during the good times, but they also lose significantly less during the bad times (Agarwal & Naik, 2000). In comparison to non-directional strategies, directional perform better during market up-moves and worse during the down-moves, but also move with the market (Agarwal & Naik, 2000).
Fung and Hsieh (2004) focus on explaining hedge fund portfolios’ returns. To do so, Fung and Hsieh (2004), combine seven asset-backed style factors to create the benchmarks for hedge funds. Common risks in hedge funds are captured by these benchmarks (Fung & Hsieh, 2004).
8 The risk factors in the model by Fung and Hsieh (2004) are: equity market, equity size, bond market, credit spread and trend-following factors for bonds, currency and commodities. They find that all these risk factors are able to explain most of the return movements in the portfolios (Fung & Hsieh, 2004).
Bollen (2011) investigates whether the data available on hedge funds provides all the essential information for both investors and regulators. The investigation period is from January, 1994 to September, 2008 (Bollen, 2011). Bollen (2011) finds that risk exposure of hedge funds
significantly changes by the end of the investigation period. He concludes that higher
transparency of hedge funds leads to easier risk access for both investors and regulators (Bollen, 2011).
In their research, Cai and Liang (2012) use linear dynamic model to investigate performance of hedge funds and their “risk-shifting behavior” during the period of 1994-2008.They conclude that risk factors introduced by Fung and Hsieh (2004) are “appropriate to capture hedge fund returns with the dynamic model” (Cai & Liang, 2012, p. 66). They also look at the changes of alpha3 for each hedge fund style (Cai & Liang, 2012). Their findings conform to the previous studies (Fung & Hsieh, 2011) which show that alpha tends to decrease over time (Cai & Liang, 2012).
In this thesis, it is expected that non-directional strategy, such as long/short equity will outperform the market during the crisis periods as in the research by Agarwal and Naik (2000). Morevover, it is expected that long/short equity hedge funds will continue performing better than the market, but the outperformance may decrease by the end of the time period examined (Cai & Liang, 2012). It may also appear that hedge funds will not outperform the market as in
Ackermann et al.(1999). However, as another time period is examined in this thesis, this is not an expectation.
2.4.Hedge fund strategies
There exist a number of strategies used in the hedge fund industry. There are two main categories of hedge funds strategies: directional, which are the bets on one sector delivering higher returns than the others and non-directional, which are simply exploiting errors in security valuations (Bodie et al., 2011). There are ten strategies that are commonly used in the hedge fund industry. According to one of the most used hedge fund indexes (Stulz, 2007), Credit Suisse Hedge Fund Index (2016), also used in this thesis, these strategies are the following:
Convertible Arbitrage: main idea of this strategy is to profit from first buying the convertible securities and then selling the underlying stock when a pricing error occurs.
9 Emerging Markets: this strategy involves investing in currencies, debt, equity and other
instruments of developing countries, such as countries of Latin America and Southeast Asia.
Dedicated short bias: to profit from this strategy, an investor should take more short than long positions in equities.
Market Neutral: funds following this strategy usually take both long and short positions in the stock while trying to minimize their systematic risk exposure.
Event Driven: this strategy enables hedge funds to deliver high returns when market or corporate events, such as mergers, bankruptcies, spin-offs, etc. occur. Hedge funds would profit by
exploiting security pricing errors caused by such events.
Fixed Income Arbitrage: by leveraging both long and short positions in “related fixed income securities” (Credit Suisse, 2016), this type hedge funds are able to profit.
Global Macro: to profit from this strategy, one has to be able to anticipate price movements in international markets for currency, commodities and equities and to be flexible in holding positions “in any market and with any instrument” (Credit Suisse,2016).
Managed Futures: this strategy usually involves global investments in “listed equity, bonds, commodity futures and currency markets” (Credit Suisse, 2016). Managers of funds exploiting this strategy usually use “systematic trading programs which rely on historical prices and market trends” (Credit Suisse, 2016).
Multi-strategy: managers exploiting this strategy try to deliver positive returns continuously by diversifying the capital. Diversification decreases risks and volatility in returns.
Long/short Equity: this strategy usually involves taking both long and short positions in equities and hedging the risk across industries and regions. This thesis investigates the following strategy and its performance throughout the years.
2.5.Long/short equity performance in existing literature
Long/short equity strategy was previously studied by a number of researchers. According to Stulz (2007), long/short equity is one of the most popular strategies used. This finding coincides with the one by Agarwal and Naik (2005). Equity hedge, which is included in long/short equity style, had the largest market share in 2004 (Agarwal & Naik, 2005). Nowadays, long/short equity continues to be one of the leading strategies in the hedge fund industry (Credit Suisse, 2016). Long/short equity dominating the hedge fund market throughout the years is represented in graph 1. Therefore, this particular strategy is studied in this thesis. Specifically, it studies
10 whether long/short equity hedge funds outperform the market (represented by S&P500) and whether this performance persists throughout the years, including years of crises.
There exist studies which examine performance of hedge funds during the Dot com bubble (Capocci et al., 2005) and the subprime crisis (Bollen, 2011; Schaub & Schmid, 2013). These studies examine the performance of the common strategies4. In contrast, this thesis focuses on investigating only one strategy.
In previous studies, including those carried out by Capocci et al. (2005), Schaub and Schmid (2013) and Bollen (2011) the time period examined is usually from 1994 to 2008. This time period includes the two crises – the Dot com bubble (2000-2002) and the subprime crisis (2007-2009). In this thesis, these two crises are also included. However, as the main goal of this thesis is to find persistence in performance, it focuses on examining sub-periods before, during and after the crises. Thus, there is a reason to add the time period of 2010-2012, as post-the-subprime-crisis period.
In their research, Fung and Hsieh (2011), examine long/short equity hedge funds, their risks and performance during 1994-2008. They use the Fama-French three factor model (1992) as the basis for their model and add extra risk factors. One of their findings is that long/short hedge funds tend to have lower exposure to the systematic risk when compared to mutual funds (Fung & Hsieh, 2011). Fung and Hsieh (2011) also find that only 15% of long/short equity hedge funds have significant alphas at the significance level of 5%. They also make a conclusion about persistency of alpha over time. Fung and Hsieh (2011) find that only less than 20% of the
long/short equity hedge funds they examine have “significant, persistent, positive alpha” and that alpha tends to decay with time (Fung & Hsieh, 2011, p. 568).
In their studies, Capocci et al. (2005) and Cai and Liang (2012) conclude that long/short equity hedge funds outperform the market before, during and after both the crises discussed in this thesis.
Subsequently, a question arises, whether hedge funds are able to maintain their performance consistently, whether their performance is persistent (Agarwal & Naik, 2005). In their research, Brown, Goetzmann and Ibbotson (1999) conclude that there is no evidence of persistence in the performance of hedge funds. Agarwal and Naik (2000) find that outperformance persists, but only for the short term. This coincides with the findings in the research by Fung and Hsieh (2004, 2011), who conclude that outperformance persists, but decreases with time. A conclusion can be made that the outcomes of the previous studies are contradicting, thus there is one more reason to examine persistence in performance of long/short equity hedge funds.
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Graph 1: Historical sector weights of hedge fund strategies (source: https://lab.credit-suisse.com/#/en/index/SECT/SECT/constituents )
2.6.Market volatility
Market volatility uncertainty can strongly influence the investment decisions (Agarwal, Arisoy & Naik, 2017). This uncertainty can also be risky for hedge funds and affect their returns (Agarwal et al., 2017). This may occur due to a number of reasons. Firstly, market volatility can have a strong effect on hedge funds as they usually take speculative and leveraged positions (Cao & Jayasuriya, 2011). Secondly, when a shock hits the economy, the market becomes more
uncertain and volatile (Agarwal et al., 2017). Subsequently, this leads to “difficult-to-access situations” (Agarwal et al., 2017, p. 492) and to a significant investment withdrawal (Agarwal et al., 2017; Cao & Jayasuriya, 2011). Lastly, use of the market timing strategy by the managers of hedge funds can be also influenced by market volatility (Cao & Jayasuriya, 2011; Karmel, 2005). Therefore, including the market volatility in the model adds value to the research on the
performance of hedge funds.
3. Methodology 3.1.Data
The quantitative data used in this thesis is obtained from the different sources. Most
importantly, the data for the monthly long/short equity hedge fund returns is collected from the Credit Suisse Hedge Fund Index (2016). Credit Suisse Hedge Fund Index is one of the leading asset-weighted hedge fund indices, which contains more than 9,000 funds in its database (Credit Suisse, 2016). According to Credit Suisse (2016), the value-weighted method they use provides a
12 more accurate description of investments than equal-weighted index does. The reason for that is the value-weighted index adjusts automatically in case of changes in share prices and/or financial market events (Credit Suisse, 2016). Thus, economic changes are represented more accurately (Fung & Hsieh, 2004). For this reason, this index will be used to study the performance of hedge funds in the times of both equity and debt crises and in the periods of recovery. Credit Suisse Hedge Fund Index is a reliable data source to obtain the returns of hedge funds as it has certain requirements for hedge funds entering the database (Credit Suisse, 2016). The requirements are minimum $50 million assets under management (AUM), a track record of a minimum of 12 months and current audited financial statements (Credit Suisse, 2016).
Before looking for persistence in the performance of hedge funds, it is important to study how they perform in comparison to the market. In this thesis the equity market for this comparison is S&P500 index. This index is largely used as a benchmark of hedge fund performance in the studies by Ackermann et.al (1999), Fung and Hsieh (2004) and Fama and French (1993). The market monthly returns are obtained from Kenneth R. French Data Library (2018)5. The risk-free rate, which is the 1 month U.S. Treasury Bill is also obtained from that source.
Data for the other factors of the model used in this thesis is obtained from Kenneth R. French Data Library (2018), FRED St. Louis and Cboe.
3.2.Biases and solutions
There are a number of biases that occur when working with the data from all databases for hedge funds (Agarwal & Naik, 2005). It is important to take these biases into account and possibly find solutions to them to get more accurate research results. There could occur measurement bias and multi-period sampling bias (Agarwal & Naik, 2005). The possible
solution for the measurement bias, according to Fung and Hsieh (2002), is to use a fund of funds. However, the hedge fund database used for this thesis does not include fund of funds into its index. Therefore, another solution is needed. Taking into account the fact that total period examined in this thesis is 16 years (192 months) and 4 sub-periods are 36 months each and 1 sub-period is 48 months long, it will not lead to such biases. Fung and Hsieh (2000) find that multi-period bias is very small when a minimum 36-month period return history is used.
Therefore, this thesis identifies three, most important, biases, such as survivorship bias, self-selection bias and backfill bias (Agarwal & Naik, 2005).
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3.2.1. Survivorship bias
Survivorship bias occurs when unsuccessful funds stop reporting the information to the
database on their performance and then leave it (Bodie et al., 2011). However, the results of the studies in this thesis should not be significantly affected by the survivorship bias. According to the Credit Suisse Hedge Fund Index (2016), it minimizes the effect of the survivorship bias by not removing funds in the process of liquidation. In this way it captures “all of the potential negative performance before a fund ceases to operate” (Credit Suisse, 2016).
3.2.2. Self-selection bias
Hedge funds disclose the information on their performance willingly (Ackermann et al., 1999). Therefore, they choose to report this information only in case of a good performance, leading to self-selection bias (Agarwal & Naik, 2005). However, self-selection bias is limited (Agarwal & Naik, 2005). As soon as hedge funds have reached their own AUM goal, they can choose to no longer report on their performance to the database (Agarwal & Naik, 2005). In their research, Fung & Hsieh (2000) come to the conclusion that this bias is not significant accounting for these reasons.
3.2.3. Backfill bias
Backfill bias, also known as instant history bias, occurs when hedge funds choose to report to the database only after their successful performance (Bodie et al., 2011). This can cause an upward bias and then the historic performance of included hedge fund is not represented accurately (Bodie et al., 2011).
3.3.Time period
This thesis investigates the performance of long/short equity hedge funds during the period from January 1997 to December 2012. This period is then sub-divided into five sub-periods: 1997-1999, 2000-2002, 2003-2006, 2007-2009 and 2010-2012. Two of these sub-periods are periods of the economic downturns. These sub-periods are the Dot com bubble (2000-2002) and the subprime crisis (2007-2009). The total period studied is subdivided into five sub-periods to examine the performance of long/short equity hedge funds before, during and after crises and to find whether performance persists throughout the periods. During the time period examined there occurred a lot of events which could have affected the financial market. The examples of such events are the burst of the Dot com bubble, the terrorist attack on the 9/11, wars in Western and Central Asia and the subprime crisis. These events could potentially have an effect on the performance of hedge funds. On the other hand, hedge funds could exploit these market
inefficiencies and still deliver higher returns than the market (Capocci et al., 2005; Cai & Liang, 2012).
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3.4.The model
To evaluate the performance of hedge funds, researches use different models and different performance measures. Most of the researches base their studies on common models, for example, the Fama-French model (1992) or the Carhart model (1997).
Fama and French (1992) extend the Capital Asset Pricing Model (CAPM) by Sharpe (1964) and Lintner (1965). They add their own variables, such as SMB (small minus big), which is the difference between the returns of the small and large stocks and HML (high minus low) which is the difference between the returns on high and low-book-to-market stocks.
In 1997 Carhart introduces his model as an extension to a three factor model by Fama and French (1992) with an additional factor, momentum effect. According to Bodie et al. (2011, p. G-8) momentum effect exists when “poorly performing stocks and well-performing stocks in one period continue that abnormal performance in following periods”.
In their research Fung and Hsieh (2004) use a seven factor model to study the performance of hedge funds. This model is also an extension to one by Sharpe (1964) and includes equity oriented, bond oriented and trend following risk factors. Fung and Hsieh (2004) conclude that 57% of all hedge funds examined in their studies, contain all the seven risk factors.
Despite the fact that these models are widely used in the research on the performance of hedge funds, another model is used in this thesis. In 1993 Fama and French introduce a five factor model, which is an extension to their model in 1992. They add two additional factors. The first factor is “TERM’ and the second is ‘DEF’ which are able to capture changes in returns on bonds and stock. ‘TERM’ is the difference between monthly returns on long-term government bond and one month U.S. Treasury bill rate measured at the end of previous month (Fama & French, 1993). ‘DEF’, the default factor, is the difference between the monthly returns on a market portfolio of long-term government bonds and long-term government bond monthly returns (Fama & French, 1993). This factor is similar to one in the model by Fung and Hsieh (2004), the credit spread. Shifts in the financial market positively affect the probability of default and thus increase risk in returns (Fama & French, 1993). As this thesis focuses on studying the
performance of hedge funds during the period with a number of such economic shifts, this factor is a valuable addition to the model.
Another factor will also be added to this model – the market volatility factor. This factor was not used in any of the models described in this thesis. The reason for an additional factor is that hedge funds usually “take speculative positions that can be heavily influenced by financial
15 market volatility” (Cao & Jayasuriya, 2011, p. 1692). Times of crises can be considered as times of high volatility in the market, thus this additional factor is worth including in the model.
Overall, the model used in this thesis to answer the research question is the following:
RL/S – Rf = α + β1(Rmkt – Rf) + β2(SMB) + β3(HML) + β4(TERM) + β5(DEF) + β6(VOL) + ei
The dependent variable, RL/S – Rf , is the monthly return on long/short equity hedge funds minus
monthly risk-free rate. Alpha (α) is the Jensen’s alpha (Jensen, 1968) and represents the ability of a fund to deliver abnormal returns (Fung & Hsieh, 2011). Rmkt – Rf is the monthly equity
market return minus monthly risk-free rate. SMB and HML were earlier described in this section and data for these variables was obtained through Kenneth R. French Data Library (2018). Variables TERM and DEF were also discussed earlier in section 3.4. The data for these two factors is retrieved from FRED St. Louis. VOL is the market volatility factor. The data for this factor is obtained from VIX Index (Cboe)6. The error term in this model is ei.
3.5.Hypotheses and expectations
To answer the research question, first, a question whether long/short equity hedge funds outperform the market should be addressed. Therefore, the first hypothesis is: long/short equity hedge funds outperform the market:
H0: α = 0 (no outperformance)
H1: α ≠ 0 (outperformance)
Expectations for this hypothesis are that long/short equity hedge funds will outperform the market during 1997-2002 and also during 2003-2012. These expectations are based on the findings by Cappocci et al. (2005) and Cai and Liang (2012).
In order to find persistence in the found performance of long/short equity hedge funds, alphas from each year from 1997 to 2012 are compared. Thus, the second hypothesis addressed in this thesis is: alpha changes significantly during the time periods examined:
H0: αt = αt+1 (alpha does not change - persistence in alpha)
H1: αt ≠ αt+1 (alpha changes– no persistence in alpha)
16 In order to do so, firstly, alphas from each year are gathered. The values for alphas are found under ‘_cons’ (intercept) in the regression output for each year. Secondly, the differences between each year’s alphas are calculated. Thirdly, the differences between alphas are squared and summed up. Finally, taking into account that both alphas and differences between them are normally distributed (Keller, 2009) the sum of the squared alphas is compared to a value in the table of the chi-squared distribution. It is also assumed that alphas are independent (Keller, 2009). The value in the table for the chi-squared distribution can be found in the intersection cell of the degrees of freedom and level of significance. If the squared value of differences in alphas is less than the value in the table, then the null hypothesis is not rejected (Keller, 2009) and persistence in alphas is proved.
It is expected that the performance of long/short equity hedge funds persists during 1997-2012 time period. This expectation is consistent with the findings by Fung and Hsieh (2004, 2011), who conclude that hedge fund outperformance (alpha) tends to persist. However, alpha also decreases with time (Fung & Hsieh, 2011). This means that as time passes, hedge funds are no longer able to add value as they used to before (Agarwal & Naik, 2005).
3.6.Descriptive Statistics
In table 1, the descriptive statistics for all the variables in the model are presented for the total time period examined of 1997-2012. Table 1 shows the number of observations, means, standard deviations and minimum and maximum values of the variables. Looking at the minimum and maximum values in table 1, one can observe that the excess monthly returns of the long/short equity hedge funds are higher than those of the equity market. The minimum and maximum values for the hedge funds variable are -0, 1186 (-11,86%) and 0, 1257 (12, 57%) respectively. For the market, the minimum value is -0, 1723 (-17,23%) and the maximum is 0, 1135 (11,35%). This is consistent with the observation that hedge funds outperform the market. The difference between the minimum and maximum value of the volatility variable is relatively large. This can be explained by the fact that the total time period examined in this thesis includes such events as the burst of the Dot com bubble, wars, the subprime crisis and others. All these events could have affected the financial market and thus increase the volatility.
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Variable Observations Mean Std. Dev. Min Max
R(L/S)- Rf 192 0,0054641 0,0293612 -0,1186 0,1257 Rmkt - Rf 192 0,0041589 0,0484485 -0,1723 0,1135 SMB 192 0,0026203 0,0370567 -0,1728 0,2214 HML 192 0,0026036 0,0343868 -0,111 0,129 TERM 192 0,041317 0,0109673 0,015267 0,064555 DEF 192 0,0256265 0,0085867 0,01451 0,06014 VOL 192 0,012594 0,1802378 -0,2836847 1,023157
Table 1: Descriptive statistics for the period 1997-2012.
4. Results and Analysis 4.1.Correlations
In table 2, the correlation matrix is presented between the variables examined in this thesis. The correlation matrix helps to examine how strong is the relationship between two variables (Bewick, Cheek & Ball, 2003). The correlation values lie in the range of [-1;1], where a strong negative relation between two variables is denoted by -1 and a strong positive relation is denoted by 1(Bewick et al., 2003). It is useful for the research to examine the correlation matrix first, as it can assist not only in identifying relationships between variables, but also in explaining the results of the regression (Bewick et al., 2003). However, it is important to note that correlation does not suggest causality (Bewick et al., 2003).
For the period 1997-2012, the highest correlation is observed between the monthly excess returns on long/short equity hedge funds and the monthly excess returns on the market. The correlation between two variables is 0.7401. This means that hedge funds’ performance moves together with the market during the period of 1997-2012 (Bewick et al., 2003).
Another high correlation is seen between the volatility and the returns on hedge funds and also the volatility and the market returns. However, these correlations are negative: -0.5262 for hedge funds and -0.5798 for the market. This means that volatility has an inverse effect on these two variables (Keller, 2009). Thus, when the market volatility increases, the market returns will be lower. Usually investors would expect higher returns to compensate for higher risk, meaning that in times of higher risk come higher returns (Bodie et al., 2011). However, the opposite is found in this thesis. Nevertheless, these finding are supported in a number of studies, for example, by Fama and Schwert (1977) and Nelson (1991).
18 The third highest correlation in Table 2 is the correlation between variables DEF and TERM. The correlation is negative and has a value of -0.5722. This result makes sense, as in times of financial distress, thus in time of higher volatility, the credit spread gets wider, which leads to a higher probability of default (Tang & Yan, 2010). Thus, from the inverse relationship between DEF (the credit spread) and TERM it can be concluded that when the credit spread gets wider, the long-term government bonds returns decrease (assuming that the short term rate remains at its level). Subsequently, this leads to a change in the interest rate, what TERM variable explains in the model. That is the reason why the correlation between DEF and TERM is negative.
R(L/S)-Rf Rmkt-Rf SMB HML DEF TERM VOL
R(L/S)-Rf 1.00 Rmkt-Rf 0.7401 1.00 SMB 0.4840 0.2323 1.00 HML -0.3572 -0.1584 -0.3318 1.00 DEF 0.0880 -0.0210 0.0083 0.0786 1.00 TERM -0.1403 -0.1144 0.0407 -0.1044 -0.5722 1.00 VOL -0.5262 -0.5798 -0.2269 0.0425 0.0373 -0.0023 1.00
Table 2: Correlation matrix of the variables R(L/S) - Rf, Rmkt-Rf, SMB, HML, DEF, TERM and VOL for the
period 1997-2012.
The strong relations discussed above, between: (1) the hedge fund return and the market return, (2) the volatility and these two types of returns and (3) the credit spread and change in interest rates, persist when examining the sub-periods7:
1. Excess long/short equity hedge fund return and the excess market return: in the sub-1997-1999, the correlation is relatively stronger and is equal to 0.8447. Considering the sub-periods of 2003-2006 and 2007-2009, the correlations do not differ significantly from the correlation of the overall time period. However, the lowest correlation (0.4744) is found during the period of the equity crisis (2000-2002) and the highest correlation (0.9437) is found during the period of post debt crisis (2010-2012). This means that long/short equity hedge funds move together with the market and their strong relation is mostly noticeable in the times just before the equity crisis and after the debt crisis.
According to Agarwal and Naik (2005), if the fund’s returns are uncorrelated with market index returns, then this fund is considered to be market neutral. Based on the correlation
19 coefficients in table 2 and on the findings by Agarwal and Naik (2005), it can be
concluded that long/short equity hedge funds are not market neutral.
2. Volatility vs. performance of the long/short equity hedge fund vs. performance of the market: the correlation between these variables does not vary greatly during the five sub-periods. Nevertheless, during 2000-2002 a significantly lower correlation (-0.3751) is found between the volatility and the hedge funds returns. Also, during 2007-2009, the correlation between these two variables is -0.7254, which is relatively larger than the one from the overall period. Therefore, it means that there was a stronger effect (Bewick et al., 2003) of the volatility on the returns of the long/short equity hedge funds during the debt crisis than during the equity crisis.
3. DEF vs. TERM: during the five sub-periods, the correlation between the DEF and TERM keeps getting stronger and reaches its maximum (negative) value (-0.9519) during the period of the Dot com bubble in 2000-2002 and its second highest value (-0.8767) during the subprime crisis in 2007-2009. As the correlation value for the period 2000-2002 is almost (minus) 1, there exists problem of the multicollinearity. Multicollinearity occurs when two variables are highly correlated with each other (Keller, 2009). Multicollinearity affects “the interpretation of the coefficients” (Keller, 2009, p. 714).
4.2.Testing hypotheses
4.2.1. Regression output 1997-2012
In order to identify the relationship between the dependent and independent variables (Bewick et al., 2003) and most importantly, to estimate alphas, regression analyses are used. The
regression outputs assist in testing the hypotheses and in answering the research question
(Bewick et al., 2003). The results of the regression for total period of 1997-2012 are presented in table 3. There are 192 observations in the regression for 1997-2012, which equal to the total number of months examined. The regression has R-squared equal to 0.6992 which means that 69, 92% of the variance in the dependent variable (R(L/S)-Rf) can be explained by the independent
variables, which are represented by the risk factors (Stock & Watson, 2015).
The independent variables Rmkt - Rf, SMB and HML are statistically significant at the
significance level of 0,1%. They are also significant at significance levels of 1% and 5%. This means that these risk factors have an influence on the excess returns of long/short equity hedge funds (the dependent variable). This outcome is in line with the findings by Fung & Hsieh (2004, 2011).
20 The independent variable VOL is statistically significant at the 1% significance level. This leads to the conclusion that the market volatility factor has an influence on the long/short hedge funds excess returns. This conforms to the reason why this variable was added to the model (Agarwal et al., 2017; Cao & Jayasuriya, 2011; Karmel, 2005).
4.2.2. Hedge fund beta
Beta of the long/short equity hedge funds is equal to 0.3441. This value is considerably above zero, but also is considerably lower than 1. According to Agarwal and Naik (2005) if the hedge fund beta is close to zero, this means this fund is market neutral. If beta is not close or does not equal to zero, then this hedge fund is “different from the original hedge fund definition” (Agarwal & Naik, 2005, p. 110). This is, however, not a surprising result. According to Patton (2009), there are a lot of hedge funds which claim to be market neutral, however in practice they are not. The lowest betas are observed during 2000-2002 (0.2647) and during 2003-2006
(0.2897). This conforms to the prior study by Liang (1999), where he finds low hedge fund betas with the U.S. equity market. During the sub-periods 1997-1999, 2007-2009 and 2010-2012 the hedge fund betas equal to 0.4895, 0.4010 and 0.4772 respectively.8 This shows that long/short equity hedge funds are not market neutral (Agarwal & Naik, 2005). However as the values are still considerably lower than 1, it still means that long/short equity hedge funds are less sensitive to the changes in the market than the market itself (Bodie et al., 2011).
4.2.3. Testing hypothesis 1
In Table 3, a coefficient for alpha can be found in the first column under ‘Intercept’. The value of alpha during 1997-2012 is equal to -0.0001586. The p-value is 0.985. This means that it is insignificant at the significance level of 5%, thus the null hypothesis is not rejected. This implies that long/short equity hedge funds do not outperform the market during the total time period of 1997-2012. The outcome of this regression for alpha does not conform to the expectations previously made and does not support hypothesis 1.
21 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.3441485 (0.0346614)*** 0.000 SMB 0.2070218 (0.0440431)*** 0.000 HML -0.1601646 (0.0463621)*** 0.001 TERM 0.2268512 (0.1372535) 0.100 DEF -0.1964431 (0.1706088) 0.251 VOL -0.0216525 (0.0078095)** 0.006 Intercept -0.0001586 (0.0085331) 0.985 R-squared 0.6992 Prob>F 0.0000 Number of observations 192
Table 3: Robust regression output for the period 1997-2012. The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
The regression analysis for the time period of 1997-2012 shows that long/short equity hedge funds do not outperform the market during this time period. To get a better understanding why this is the case, the performance estimators (alphas) for the sub-periods of 1997-1999, 200-2002, 2003-2006, 2007-2009 and 2010-2012 are analyzed. Alphas are found from the regression outputs for each sub-period under ‘_cons’ (the intercept)9. The minimum alpha is observed during 2007-2009 and has a value of -0.009422. The maximum alpha is observed during 2000-2002 and has a value of 0.1208732. The summary on the alpha coefficients and p-values for five sub-periods is presented in Table 4.
22
Table 4: Coefficients and p-values from the regressions on the five sub-periods. Significance levels: *=0.05. 1. 1997-1999
The coefficient of the intercept is 0.024072. The p-value is 0.679. Alpha is statistically insignificant at any significance level. Therefore, the null hypothesis is not rejected. The conclusion is that long/short equity hedge funds do not outperform the market during the period of 1997-1999. This is inconsistent with the expectations imposed.
2. 2000-2002 (the Dot com bubble)
The same conclusion can be made for the sub-period of 2000-2002, as the p-value is 0.450 and thus null hypothesis is not rejected, as alpha is not statistically significant. These findings do not support hypothesis 1.
3. 2003-2006
For the period of 2003-2006 coefficient for alpha is equal to 0.0607757. The p-value is 0.019 and thus it is statistically significant at the significance level of 5%. Therefore, the null hypothesis is rejected, which leads to the conclusion that long/short equity hedge funds outperform the market during 2003-2006. This is consistent with the expectations and with the findings by Cai and Liang (2012).
4. 2007-2009 (the subprime crisis)
Long/short equity hedge funds do not outperform the market during the subprime crisis: p-value is 0.836 (very close to 1) and thus it is statistically insignificant.
Time Period Coefficient p-value
1997-1999 0.024072 0.679
2000-2002 0.120873 0.450
2003-2006 0.060776* 0.019
2007-2009 -0.009422 0.836
23
5. 2010-2012
The p-value for the alpha coefficient is equal to 0.043, which means it is statistically significant at the significance level of 5%. Therefore, the null hypothesis is rejected. This leads to the conclusion, supported by Cai and Liang (2012) that long/short equity hedge funds outperform the market during 2010-2012.
Overall, there is no outperformance of the market by long/short equity hedge funds. This is not in line with the prior literature reviewed in this thesis. The reasons for this can be:
1. The use of the different database – most of the prior studies use other databases than Credit Suisse Hedge Fund Index. Most of the times, a combination of various databases is used.
2. The use of the different methodology – the model used in this thesis is the five-factor model (Fama & French, 1993) and an additional factor, the market volatility (VOL). This type of model is not observed to be used in the prior studies on the performance of hedge funds.
Although outperformance is not found during the total period examined and the findings do not support hypothesis 1, there is prior literature supporting these findings. According to Ackermann et al. (1999) hedge funds do continuously outperform the mutual funds, however not the market. In their work, Fung and Hsieh (2011) find that about 15% of long/ short equity hedge funds outperform the market and that less than 20% of those funds show persistent performance.
4.2.4. Testing hypothesis 2
In this thesis, the following is found on the persistence of alpha. As discussed in the
methodology, alphas are gathered from each year regression outputs. Thus, there are 16 values of alpha, which corresponds to the total number of years examined in this thesis. Then, the
difference between alphas is calculated. Finally, each alpha difference is squared and the sum of all squared alpha differences is calculated. The sum of the squared alpha differences is equal to 3,267. Next, this value is compared to one found in the Chi-squared distribution table. Degrees of freedom (df) equal to 1 and the significance level is 5%. Thus, looking at the intersection in the table of 1 df and χ20.05, the value is found to be 3,841 which is larger than 3,267. This means,
that the null hypothesis is not rejected. Therefore, alpha does persist during 1997-2012. This finding is in line with the expectations imposed in this thesis and supports hypothesis 2. Alpha values for each year, differences between alphas and the squared values of the differences are presented in the table 5.
24 Year Alpha value Difference Squared difference
1997 -0,1805 1998 -0,9608 -0,7803 0,6089 1999 0,2792 1,2401 1,5377 2000 0,0155 -0,2637 0,0695 2001 -0,3666 -0,3821 0,1461 2002 -0,7299 -0,3634 0,1321 2003 0,0672 0,7972 0,6355 2004 0,1035 0,0363 0,0013 2005 -0,0137 -0,1172 0,0137 2006 0,0849 0,0987 0,0097 2007 0,2154 0,1304 0,0169 2008 0,1197 -0,0957 0,0092 2009 0,0119 -0,1078 0,0116 2010 0,0898 0,0779 0,0061 2011 -0,0891 -0,1788 0,0319 2012 0,1033 0,1924 0,0370 SUM 3,2673
Table 5: Testing hypothesis 2. The difference between values of alphas are calculated with the formula: difference = αt+1 - αt. The values in the table are rounded off to 4 decimal places.
The final results of this research are that long/short equity hedge funds do not outperform the market during 1997-2012, however the performance does persist during this time period.
5. Conclusion
The main goal of this thesis was to examine whether the performance of long/short equity
hedge funds persists during the period of 1997-2012, which includes two major crises– the Dot com bubble (2000-2002) and the subprime crisis (2007-2009). In order to answer the research question, another question was addressed first. This question is whether long/short equity hedge funds are able to exploit the market inefficiencies and to outperform the market. To evaluate the performance a six factor model was used. The basis for this model was a five factor model by Fama and French (1993). An extra factor, the market volatility, was added to this model. To examine whether the performance of long/short equity hedge funds persists, the total period was
25 divided into five sub-periods. To find persistency, alphas were gathered from 16 yearly
regressions, and their differences were squared and summed up in order to compare the resulted value to the value in the Chi-squared distribution table.
The high correlation values between the excess returns of the long/short equity hedge funds and the market excess returns indicate that long/short hedge funds are not market neutral (Agarwal & Naik, 2005; Patton, 2009). This is also confirmed by the values of hedge fund betas which are considerably above zero, meaning that long/short equity hedge funds are not market neutral (Agarwal & Naik, 2005; Patton, 2009). Despite the fact that these hedge funds are not market neutral, they are still less sensitive to the market changes than the market itself (Bodie et al., 2011) during the overall period examined. Looking at the relation between the hedge funds returns and the volatility, it can be concluded that volatility has a stronger effect on the hedge funds than on the market. Specifically, this occurs during 1997-1999, 2007-2009 and 2010-2012.
The results of this thesis show that long/short equity hedge funds do not outperform the equity market over the total time period examined (1997-2012). Looking at the performance of hedge funds in each sub-period, it can be concluded that the only times long/short equity hedge funds outperform the market are during 2003-2006 and 2010-2012.
The findings on performance of long/short equity hedge funds during the overall period in this thesis are not in line with the initial expectations based on the findings by Capocci et al. (2005) and Cai and Liang (2012), who state that hedge funds outperform the market during 1997-2002 and during 2003-2012, respectively. However, the findings in this thesis conform to the findings by Ackermann et al. (1999), who find that hedge funds do not tend to outperform the market. Also, these findings can be supported by Fung and Hsieh (2011) who find that only 15% of the long/short equity hedge funds they examine tend to have a significant alpha at the 5%
significance level.
The findings on the persistence of long/short equity hedge funds during the overall period, are in line with the expectations imposed in this thesis and can be supported by Fung and Hsieh (2011) who conclude that alphas tend to persist, however decrease with time.
One of the limitations of this thesis could be the database used. This thesis might not represent the full picture on the performance of hedge funds, as only one database was used due to the hard access to other databases. The model used to answer questions in this study does not include all the possible risk factors which could influence the performance of long/short equity
26 hedge funds. Despite the fact that the solutions for the possible biases are included in the study, there are still other biases that could occur and affect the results.
Therefore, the further research should take into account these limitations and try to find solutions to them, for example, by combining several accessible databases on hedge funds and including more possible risk factors. Then the performance of the hedge funds can be described more in detail and contribute to the research in this industry.
Testing the persistence of long/short equity hedge funds’ performance, it was assumed that alpha values are drawn independently (Keller, 2009) for the simplicity of the test. As this is not particularly the case, it is advised for further research to use other tests on persistence to obtain more accurate results.
27
Appendix
Appendix 1 – Correlation tables
1997-1999 2000-2002 2003-2006 2007-2009 2010-2012 VOL -0.5011 -0.4739 -0.1712 0.2106 0.0085 -0.1317 1.0000 DEF 0.1673 0.1449 0.0718 -0.4688 -0.7272 1.0000 TERM 0.0561 -0.0251 0.1808 0.1490 1.0000 HML -0.6949 -0.5374 -0.3688 1.0000 SMB 0.4820 0.1653 1.0000 RmRf 0.8447 1.0000 RLSRf 1.0000 RLSRf RmRf SMB HML TERM DEF VOL
VOL -0.3751 -0.6332 -0.3111 0.1643 -0.0637 0.0931 1.0000 DEF -0.1423 0.0066 -0.1450 -0.1191 -0.9519 1.0000 TERM 0.0840 0.0256 0.0784 0.0699 1.0000 HML -0.6057 -0.5021 -0.6332 1.0000 SMB 0.7506 0.1770 1.0000 RmRf 0.4744 1.0000 RLSRf 1.0000 RLSRf RmRf SMB HML TERM DEF VOL
VOL -0.5981 -0.6333 -0.3694 0.1138 0.1824 -0.1306 1.0000 DEF 0.0967 0.1688 0.2982 -0.2773 -0.5432 1.0000 TERM -0.3312 -0.2456 -0.3881 0.2960 1.0000 HML 0.0412 -0.0779 -0.2341 1.0000 SMB 0.5941 0.5779 1.0000 RmRf 0.7606 1.0000 RLSRf 1.0000 RLSRf RmRf SMB HML TERM DEF VOL
VOL -0.7254 -0.6413 -0.3064 0.0089 0.2489 -0.0618 1.0000 DEF -0.1074 -0.1333 0.1249 -0.1175 -0.8767 1.0000 TERM -0.0248 -0.0249 -0.1753 0.0404 1.0000 HML -0.0145 0.4610 0.2866 1.0000 SMB 0.1758 0.3246 1.0000 RmRf 0.7716 1.0000 RLSRf 1.0000 RLSRf RmRf SMB HML TERM DEF VOL
VOL -0.6223 -0.5912 -0.1826 -0.2385 0.0805 -0.1587 1.0000 DEF -0.0106 0.0485 -0.2113 -0.1678 -0.8351 1.0000 TERM 0.0310 0.0172 0.2317 -0.0473 1.0000 HML 0.2861 0.3217 0.2121 1.0000 SMB 0.5599 0.5777 1.0000 RmRf 0.9437 1.0000 RLSRf 1.0000 RLSRf RmRf SMB HML TERM DEF VOL
28
Appendix 2 – Regression outputs
1997-1999 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.4895015 (0.0732423)*** 0.000 SMB 0.2777523 (0.0675717)*** 0.000 HML - 0.3863595 (0.12855)** 0.005 TERM 0.0318703 (0.7305053) 0.966 DEF -0.9297883 (1.189) 0.441 VOL -0.0335962 (0.019689) 0.099 Intercept 0.024072 (0.0575264) 0.679 R-squared 0.8818 Prob>F 0.0000 Number of observations 36
The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
29 2000-2002 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.2647355 (0.0874197)** 0.005 SMB 0.3448262 (0.0687846)*** 0.000 HML 0.0273387 (0.1036378) 0.794 TERM -1.343645 (1.771841) 0.454 DEF -2.060299 (2.48018) 0.413 VOL 0.029188 (0.0285139) 0.314 Intercept 0.1208732 (0.1578621) 0.450 R-squared 0.7021 Prob>F 0.0000 Number of observations 36
The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
30 2003-2006 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.2896929 (0.0768268)*** 0.001 SMB 0.1679317 (0.0854659) 0.056 HML 0.1993254 (0.0765378)* 0.013 TERM -1.132033 (0.4965022)* 0.028 DEF -0.5317708 (0.3483125) 0.135 VOL -0.0316556 (0.0170691) 0.071 Intercept 0.0607757 (0.0248451)* 0.019 R-squared 0.6957 Prob>F 0.0000 Number of observations 48
The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
31 2007-2009 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.4010446 (0.0774415)** 0.000 SMB -0.0469418 (0.1193434) 0.697 HML -0.2877413 (0.098996)** 0.007 TERM 0.3279561 (0.914182) 0.722 DEF 0.005083 (0.376014) 0.989 VOL -0.0265092 (0.0189664) 0.173 Intercept -0.0094221 (0.0450046) 0.836 R-squared 0.7905 Prob>F 0.0000 Number of observations 36
The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
32 2010-2012 R(L/S)-Rf Coefficient P>|t| Rmkt - Rf 0.4772402 (0.0466298)*** 0.000 SMB 0.0415594 (0.0857827) 0.632 HML -0.1034266 (0.0697196) 0.149 TERM -0.5811131 (0.2973799) 0.060 DEF -2.038451 (0.9525628)* 0.041 VOL -0.0147362 (0.0063265)* 0.027 Intercept 0.0732361 (0.0345753)* 0.043 R-squared 0.9117 Prob>F 0.0000 Number of observations 36
The standard errors of the coefficients are given in the parentheses. Significance levels: *=0.05, **=0.01, ***=0.001.
33
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