Closed-end funds versus open-end funds: a comparison of performance
prior and during the financial crisis over the period 2005-2009
Floris Mathlener 10251650 Bachelors Thesis Economics and Business
Specialization: Finance and Organization Supervisor: Veliyana Malinova
Abstract
This paper implements the Fama and French Three factor model (1993) to find differences in performance between closed-end and open-end funds. This is done for the periods 2005-2008, and 2009. These are periods that represent the time before and during the financial crisis of 2008-2009. I find that open-end funds outperformed closed-end funds in the period between 2005 and 2008. Next to that I find no sign of outperformance for both kinds of funds during the financial crisis. I conclude that open-end funds hold less risk, and reach higher returns in periods of economic welfare.
Table of contents Page Abstract 1 1. Introduction 3 2. Literature review 4 2.1 Analyzing performance 4
2.2 Research on closed-end fund and open-end fund performance 7 3. Hypotheses 9 4. Data 10 5. Research Method 12 5.1 Model 12 5.2 Analyzing Performance 12 6. Empirical Results 13 6.1 Group Performance 14 6.2 Individual Performance 15 6.3 Robustness Discussion 22 7. Conclusion 23 References 25 Appendices 28
1. Introduction
Mutual fund managers continuously claim they have investment skills that beat the market returns. Mutual fund management does this by actively managing their holdings. The problem is however, that more and more studies show that mutual funds do not outperform the market. The claim of having superior investment skills by mutual fund management would then be unjustified. Mutual fund enthusiasts contradict this by saying that passive investors simply go down with the market in a period of economic depression. Passive investors would not be protected for economic shocks.
Mutual fund performance is a popular research subject. Examples are Banegas et al. (2013) who discuss mutual fund performance in the European stock markets, and Jones & Shanken (2005) who research mutual fund performance by adding a factor that they call ‘learning across funds’. These papers study the mutual fund market by focusing on open-end funds. Surprisingly little attention is given to researching how the structure of the fund affects returns.
Roughly, three types of mutual funds exist: Closed-end funds, Open-end funds, and Exchange traded funds. The question that will be answered in this thesis is whether or not
differences in returns of closed-end funds and open-end funds before and during the financial crisis in the United States of America exist.
Closed-end funds raise an amount of capital that is prescribed through an initial public offer. This happens only once, by issuing shares. The investors buy these shares as stock. Open-end fund shares mostly trade at the net asset value, which is the value of the underlying assets minus the liabilities. The shares of open-ended funds are more liquid, therefore the investment horizon of management will be short-term oriented. This can affect returns for both kinds of mutual funds.
In this thesis alphas of both closed-end and open-end funds are compared to find significant differences in returns of the two types of mutual funds. Alpha measures performance of a fund on a risk-adjusted basis compared with a benchmark index, in this case the Nasdaq Composite. For this I use the framework of the Fama & French Three Factor Model (1993), which is explained in detail in the model section.
The remainder of this thesis is organized as follows. In section 2, a literature review is provided on mutual fund performance, and differences between open-end and closed-end funds. Section 3 presents the hypotheses, with literature based backgrounds. Section 4 describes the data that is used for this research. Section 5 gives an extensive description of the used models, and how performance is analyzed. Section 6 contains the empirical findings. Last, section 7 contains
2. Literature review
In this section, I discuss literature about the subject of mutual funds. This includes mutual fund performance measures, how the types of mutual funds differ and how structural differences could affect returns.
Mutual fund performance has been measured in numerous papers. Jensen (1968) gave new insight on mutual fund trading by stating that mutual fund managers cannot outperform the market on a long term basis. That is, if the “Efficient Market Hypothesis” holds. The “Efficient Market Hypothesis” states that financial markets provide information efficiency (Fama, 1965). Following this, one cannot outperform the market on a risk-adjusted basis. This would also work on the bottom end of mutual fund performance. Funds that continuously underperform the market should vanish, but this is not the case either.
2.1 Performance measures
Mutual fund performance can be measured in numerous ways. One of the most basic ones is the “Sharpe Ratio”, developed by William Sharpe (1966). It builds on the work of James Tobin in his “Liquidity Preference as Behaviour towards Risk” (1958). There are several underlying
assumptions. First, all investors can lend and borrow at a universal risk-free rate. Next to that all investors should share the same predictions of future performance in funds or stocks. When all the preceding assumptions are met, all efficient portfolios will form a graph line in the form of:
(1) Ei = rf + rpσi
Where Ei is the expected return on a certain portfolio i, rf is the risk-free rate, rp is the risk premium on portfolio, and σi is the risk or variance of the portfolio. Investors demand a premium for the risk they take. Assumed is that investors are risk-averse, therefore rp is always positive. This means that when risk increases investors also demand higher returns. When an investor allocates his funds between a portfolio, borrowing and lending at the risk free rate he can attain a certain point on the following line:
(2) E =
r
f + ((Ei –r
f)/σi)σWhere he replaces rp from the preceding formula with ‘((Ei – rf)/σi)’, and E is the expected return of an investor, and σ his risk. ‘(Ei – rf)’ is then the excess return of a portfolio, and σi is the standard deviation, or risk. Dividing the expected excess return with the risk gives some sort of risk premium. The portfolio that maximizes this premium is optimal. This is also called the “Sharpe Ratio” (1966), which is widely known. In his research, Sharpe (1966) finds that predicted differences in
performance can be measured to some extent. This prediction is far from perfect, and can often be explained by differences in expense ratios.
The Sharpe Ratio provides good benchmarking and simple implementation. Therefore, it is widely used. The problem however, with this “Sharpe Ratio” (1966) is that not all investors are able to borrow and lend at the risk-free rate. If this is the case, the formula does not hold
Jensen (1968) provides us with one of the most common ways to measure mutual fund performance. He finds that mutual funds unperformed with respect to the S&P500 returns. Even when adding back fees. He does this by introducing Jensen’s Alpha (1968), which builds on the basic Capital Asset Pricing Model. Sharpe (1964) publishes The Capital Asset Pricing model first, but parallel work is also performed by Treynor (1961) and Lintner (1965). The model reads:
(3) Rjt – rft = βj[rmt – rft] + ejt
where ‘Rjt – rft’ is the excess return over the risk-free rate of a fund, βj is the fund-specific Beta which represents the sensitivity to market risk for a fund, ‘[rmt – rft]’ is the market excess return, and ejt is the error term. This model estimates fund required rate of returns by adding the demanded extra returns for risk. Jensen (1968) finds, that positive error terms are a sign of high excess returns for a fund. He adds a term that accounts for this positive or negative excess returns to the CAPM-model. The model then reads:
(4) Rjt – rft = αj + βj[rmt – rft] + ujt
where αj is the alpha of a fund, which represents the out- or underperformance of a fund with respect to a certain benchmark performance. This alpha term is often thought of as the ability of mutual fund managers to outperform market returns. The Alpha-term is a risk-adjusted performance measure. Jensen’s (1968) research focuses on the period between 1945 and 1964, in which he also finds underperformance with respect to the S&P500.
More recent papers find a similar mutual underperformance like Jensen. An example is Malkiel (1995) who draws the same conclusions for the period from 1971 to 1991. He states that investors are better off purchasing an index fund. Trying to select an active fund manager who provides excess returns can be costly, and increases the chance of negative average returns.
Fama and French (1993) analyze five different factors which might have had an effect on stock returns: size, leverage, earnings/price ratio, book-to-market ratio and the market premium. Their conclusion is that average returns are affected significantly by two of the five factors; size and the book-to-market ratio. The Fama and French Three Factor Model then reads:
where SMB (small minus big) represents the difference in returns between a fund with large
capitalization, and a fund with small capitalization. The capitalization of a fund is the market value of all its shares, or equity. Two portfolios are constructed; one with small stocks and one with large stocks. The difference is then the factor for a given period. HML (high minus low) is the difference in return between a fund that has of low market ratio and a fund that has a high book-to-market ratio. This factor is constructed in the same way; a portfolio with high book-to-book-to-market ratio stocks is compared to a portfolio with low book-to-market ratio stocks. The difference is then again the factor for a given period. This model is significantly better than the one factor Capital Asset Pricing model as described before (Otten & Bams, 2004).
Fama and French (2010) again show that these mutual funds underperform with respect to what you call passive investment strategies. In their latter research Fama & French (2010) use the Carhart Four Factor model. This model introduces a momentum factor to their own Fama & French Three Factor model (1992). This contributes to the model as funds that outperformed yesterday are expected to outperform today too. This factor was introduced by Carhart (1997), who analyses mutual fund performance persistence. He argues that funds which outperformed the market yesterday are more likely to do so today too. The model then reads:
(6) Rp-Rf = α + β * (Rm – Rf) + β2*SMB + β3*HML + β4 * MOMt + ɛ
Where ‘MOMt’ is the momentum factor. It represents the difference in returns at time t between winners and losers at time t-1. Carhart finds that the four-factor model that he constructed gives slightly better results than the three factor model. Still, no evidence is found that some mutual fund managers have some superior stock picking skills (Carhart, 1997).
Barras et al. provides interesting insights in his research by finding that the proportion of skilled managers of these mutual funds has diminished rapidly over the last 20 years (2010). In his research, he only includes open-end funds. Therefore, he does not draw conclusions on closed-end fund management. This could mean that closed-end funds have different proportions of skilled managers.
Cohen, Polk and Silli (2009) find that even if these skilled managers exist, we are not always able to observe them. This is mostly because of over-diversification, and pressure on management. This means that returns of a fund turn out lower because the fund invests too much. In their
research, they find four reasons for this over diversification. First, managers must diversify
according to modern portfolio theory. This is because they have rules and regulations to follow. The modern portfolio theory states that a fund should try to maximize its expected portfolio returns given amount of risk, or the other way around. Second, Cohen, Polk and Silli (2009) state that managers cannot invest all their fund capital in a security which they think is underpriced. This is because of the price effect on these securities. Third, skilled managers and unskilled managers are both risk-averse.
This is because manager skill is determined through fund performance. If skill was observable, skilled managers would be less risk-averse. Investors could see the skill of the manager and trust him or her more. Following this skilled managers will diversify more. Last, managers are often punished for high idiosyncratic risk. Investors are likely to invest in funds with a high Sharpe ratio. To attract capital fund managers should keep risk low, and this is not always optimal. This last factor only applies to open-end funds.
2.2 Research on closed-end funds and open-end funds
A wide variety of closed-end funds and open-end funds exist. Closed-end funds are all actively managed, as opposed to open-ended funds. As stated before, closed-end usually issue shares on a single occasion, whereas open-end fund shares are traded and issued continuously. This means that closed-end fund share are less liquid, and traded on exchange.
The expense ratio may be lower closed-end funds than for comparable open-ended funds due to this single issuing of shares (Greene & Hodges, 2002). The open-end mutual fund investor imposes costs on the fund by actively trading. Open-ended fund must either engage in a costly trade, or maintain a high cash position to account for mutual fund traders’ exchanges. Mutual fund management holds less equity to invest and gain returns with because of this. Greene & Hodges (2002) andNanda et al. (2000) confirm by stating that the advantage of closed-end funds is that they are not subject to these investor liquidity shocks.
As mentioned before, closed-end funds raise capital by a single initial public offer. Following this, the actual size of the fund after the IPO is very often not optimal. Nanda et al. (2000) argue that negative consequences of the inability of investors to sell their shares back to the fund arise. First, a large amount of capital may be stuck in a fund with a low quality manager. Second, a high quality manager may receive far less funding than is optimal to invest. This could make open-end funds a more profitable investment. The problem is that these agency costs often cannot be effectively measured.
Edelen (1999) states that differences in closed-end fund and open-end fund performance can be explained by differences in restrictions on the liquidity of investors. Closed-end funds do have these restrictions, whereas open-end funds do not. This should have a positive effect on returns for closed-end funds. He contradicts this by stating that closed-end funds tend to perform poorly in practice. The reasoning he gives is that there is little opportunity to monitor closed-end fund management. This is confirmed by Barclay et al. (1993), and Nanda et al. (2000) who state that a lack of monitoring opportunity for closed-end funds can lead to high agency costs, as well as underperformance. Investors in open-end funds can exit whenever they want once they detect misbehavior by management.
Mutual fund management has less opportunity to misbehave because of the structure. This contributes to the popularity of open-end funds.
Closed-end fund shares are traded on an exchange. In this way, investors can agree to buy shares of a closed-end fund at a different price than the market value of its assets (Pontiff, 1995). This could imply that two funds that are equal in every way, but one is open-ended and the other is closed-ended, have different prices.
The discount on closed-end fund share prices might be attractive for arbitrageurs. An arbitrageur is an investor who exploits price inefficiencies of stocks. They would buy shares, and then wait for the share price to converge to its net asset value (Bradley et al., 2010). The reason why this discount emerges remains a puzzle for academia and industry practitioners. Day et al. (2011) research this subject. In their paper they argue that the discount is due to an expected tax liability. Investors know that they have to pay taxes on future returns. Because of that, their future returns are lower and they will demand lower share prices. This effect is called the tax capitalization effect.
Day et al. (2011) find new evidence that expected future tax liabilities are important when explaining discounts on closed-end funds, and their implications on returns. Their methods controls for level effects and other fund-specific factors.
Berk & Stanton (2007) find that the relation between discounts and returns is highly nonlinear. They find it is unclear whether a discount has a positive or a negative effect on returns. This statement contradicts research by Chay & Trzcinka (1999), who find that closed-end funds with a higher premium have higher returns. This also means that the funds that trade at a discount underperform. Chay & Trzcinka (1999) find that, for closed-end funds, a 10% increase in the amount of premium anticipates higher excess returns ranging from 1.9% to 3% per year. The issue with this however, is causality. The premium may not cause the higher returns, but high returns may cause the premium. Stocks that receive high returns are valued higher than stocks that receive low returns, and are therefore traded at a higher price.
Brickley & Schallheim (1985) give interesting insights to this discount puzzle by finding that discounted closed-end funds which undergo a reorganization achieve significantly positive stock market returns. This means the closed-end fund turns into an open-end fund. The share price then converges to its net asset value.
Edelen (1999) argues that trading with liquidity as a motivation most likely has an adverse effect on returns of the open-ended fund. This means that underperformance that is found by other research can be due to the fact that open-end funds always need to attain a certain level of liquidity. Closed-end fund face this issue too, but in a different way. Closed-end funds cannot attract new equity due to their structure. Closed-end funds therefore often use debt to attract new funding. This leverage increases risk, but can also increase returns per share.
Mutual fund managers often have restrictions on their investment policy. These restrictions can include prohibition against derivatives, borrowing, leverage, and short sales. The holding of illiquid assets is often restricted too. Open-end funds are more strict in this (Almazan et al., 2004).
As stated before, open-end funds do not rely on leverage. Management keeps volatility, due to leverage, low. Managers does this for two reasons. First, leverage is regulated for mutual funds by the government (Almazan et al., 2004). Second, with higher debt comes higher risk. Open-end funds tend to attain less leverage than closed-end funds. This is mainly because of the demanded liquidity for open-end funds. This higher liquidity causes higher average return, but also higher volatility.
Pontiff (1997) finds that closed-end funds are more volatile than would be implied by efficient financial markets. He finds that the average closed-end fund is 64 percent more volatile than its assets. This is due to the fact that closed-end fund shares are traded on exchange.
Kosowski (2006) states that mutual fund performance is understated by most research when in recession. When looking at alpha, the risk adjusted performance measure, traditional
unconditional performance measures understate added value by active mutual fund management. This is due to the fact that risk is much higher in periods of recession. Kosowski’s research focuses on open-end funds only.
3. Hypotheses
This thesis investigates the performance of closed-end funds and open-end funds in the United States for the period from 2005-2008, and 2008-2009. The objective of this thesis is to find whether or not differences in performance exist using the Fama and French Three Factor model. No prior research on differences in return between open-end funds and closed-end funds has been done. Therefore, I base my hypotheses on separate literature about either closed-end funds or open-end funds. The first research hypothesis is expressed as follows:
H1: Closed-end funds outperformed open-end funds in the period between 2005 and 2008.
The organizational structure of closed-end funds give these funds the opportunity to take more risk than open-end funds. This follows from several features that open-end funds and closed-end funds have. This is due to the liquidity restrictions that open-end funds need to follow (Edelen, 1999; Almazan et al., 2004). Investing in more liquid assets gives lower risk, but should also lead to lower average returns. Closed-end funds’ possibility to hold leverage should give them the opportunity to take more risk, and therefore earn higher returns. Next to that, expense ratios are expected to be lower for closed-end funds. Greene & Hodges (2002) state that this is due to the cost that investors impose on the fund by actively trading.
The implication for this on returns is that less capital is available for investment, and will therefore lower returns.
The risk factor that affects returns becomes more apparent in the second hypothesis, where performance is compared in times of recession. The second research hypothesis therefore is: H2: Open-end funds outperformed closed-end funds during the financial crisis of 2008-2009.
The rationale behind this is that open-end funds invest in less risky assets. They do this because of the liquidity restrictions, as mentioned before (Edelen, 1999). This means that when the economy is thriving, open-end funds are expected to perform worse than closed-end funds. On the other hand, these open-end funds will suffer less from a recession. Second, agency costs are less of an issue for open-end funds. Barclay et al. (1993), and Nanda et al. (2000) state that a lack of monitoring opportunity for closed-end funds can lead to high agency costs, as well as underperformance. This monitoring ability is especially important in times of economic insecurity. Risk on all invested assets by the mutual fund is higher, and mutual fund managers will be tempted take more risk due to pressure to reach the goals that are set regarding returns.
Kosowski’s (2006) findings that open-end fund performance is understated in times of financial crisis supports the hypothesis as well. Although he does not include closed-end funds in is research, he proves that the effect of a financial crisis on open-end funds is not as severe as some research states.
4. Data
In this research, the sample consists of 32 open-ended funds and 32 closed-end funds. These funds are compared with respect to returns. This is done before the financial crisis (2005 -2008) and during the crisis (2008-2009). The 64 funds are selected based on a few criteria. First, I only include funds that are actively managed. Passively managed funds are difficult to compare, because these funds often just invest in one type of asset. Also, conclusions I would draw from comparing passively managed funds would be less interesting because of this narrow focus. Second, investing in equity should be the main activity. In this way I can measure active managing skills best. Also, including equity funds only justifies using a stock based benchmark like the Nasdaq Composite. Third, the funds should have been active from at least 2008. This is because the effects of the recession can only be measured on funds that were active in it. Fourth, focus of the fund should be broad market. This means that funds that only invest in one type of market or one type of equity, like small capitalization, are omitted as well. Last, all funds are U.S. based and oriented. This is because I research the U.S. mutual fund market.
Taking all the selecting criteria in mind, the sample is constructed by randomly picking U.S. based equity funds from the site of Bloomberg. Bloomberg gives a list of all the listed mutual funds in the United states. All funds in the sample are included in appendix one. The returns data on all fund returns in the sample selected from Bloomberg is then retrieved from the ‘Thomson One’ database.
Thomson One is a part of the Thomson Reuters Database. As a market proxy the Nasdaq Composite is used, as it contains stocks of small, medium, and large capitalization levels. The ‘Thomson One’ data provides weekly closing data on share prices. I use weekly data because weekly data gives a good impression of how the share price evolved. When using monthly data, one would have to go back a long time to get some significant results. If daily data would be used, the noise term would very likely be higher.
Thomson one provides me with closing share prices for each week. From this share price data I calculate weekly returns following the formula below:
(7) Rti = (Pti – Pt-1,i) / Pt-1,i
where Rti is the return of fund i at time t, Pti is the share price of fund i at time t, and Pt-1,I is the share price of fund i at time t-1, which in this case is the week before week t.
The online data library of Kenneth French provides weekly data on the values for the control variables for size and book-to-market ratio. As the risk free rate, I use one year maturity treasury bond rates for the United States. The risk-free rate is taken from the webpage of Kenneth French. I use weekly returns of mutual funds, therefore I also need weekly risk free rates. I calculate weekly risk free returns from yearly returns myself by performing the following calculation:
(8) (1 + Rw) = (1 + Ry) ^(1/52)
Where: ‘Rw’ is the weekly risk-free return, and ‘Ry’ is the yearly returns on one year treasury bonds. In this way, returns are compounded to account for ‘returns on returns’.
The data is divided into two sections for each fund. The first section consists of weekly returns of all funds for the period between 2005 and 2008. This is, until the second half of the year 2008. Literature stated that was when the financial crisis really started. This gives me 177 data points per fund. The second group consists of weekly returns per fund between 2008 and 2009. That is, from the second half of the year 2008 until the second half of 2009. This is approximately one year. After that, the official recession was over in the United States. This gives me 56 weekly data points.
One problem that could arise in the data is the survivorship bias. The ‘Thomson one’ database only provides data on funds that that are currently in the market as a mutual fund. The survivorship bias is a bias that can affect results of a regression.
It arises during the sampling of a portfolio to regress. In this way, one would omit funds that went bankrupt or stopped business. These funds mostly do not perform as good as the funds that do stay in the market. In this way you could overestimate results. In this research survivorship bias will not be a significant problem. That is, because I don’t find any evidence in existing literature that the bias is bigger for one type of mutual fund. Next to that the sample period is not very long.
5. Research method
This section describes the models and formulas that are used to measure performance of the American mutual funds, and why they are used. After that, the different types of sample categories will be explained. Last, I explain the ways to measure relative performance for closed-end funds and open-closed-end funds.
5.1 Model
To measure performance of the mutual funds I use Jensen’s alpha. All previously cited studies use some variation of a model that incorporates this alpha. Jensen (1968) introduced an active investment performance measure. This model is derived from the CAPM model. Fama and French add two control variables to the formula that increase predictability. This model gives a better representation of reality than the simple CAPM-model, which omits control variables for size and book-to-market ratio. This model is one of the most widely used models for measuring mutual fund performance. Examples are Malkiel (1995), and Fama and French (2010). The Fama and French Model as stated in the literature section is formulated as follows:
(9) Rp-Rf = α + β * (Rm – Rf) + β2*SMB + β3*HML + ɛ
To find the alpha term I need the excess fund return, the excess market return, the value of the size factor in that period, and the value for the book-to-market control variable. I then regress the excess fund return on the excess market return, the size factor and the book-to-market factor. The constant I find in regression is the alpha factor, which is the out- or underperformance of the group or fund with respect to the benchmark.
5.2 Analyzing performance
This thesis uses this Fama & French Three Factor model to analyze performance in two ways. First, I calculate grouped alphas for all 64 mutual funds. I group the returns into closed-end fund returns and open-end fund returns. Then I divide the grouped returns into two time zones; before the financial crisis and during the financial crisis. I take the capital weighted average of each category, in each time zone, per weekly data point. This means I multiply specific fund return for the given period with its capital weight.
By doing this for each fund, and then adding everything up I find the capital weighted average return for closed-end funds of open-end funds for a given weekly data point. This gives me four lists of excess returns over the market; closed-end fund returns before and during the crisis, and open-end fund returns before and during the crisis.. After that, I regress this by implementing the Fama and French Three Factor model.
Second, I calculate individual alphas for each fund. In this way I can see for each fund separately how well it performed with respect to the benchmark. I do this by regressing individual fund excess returns on the market excess return, the size factor, and the book-to-market factor. After performing regressions according to the preceding model, the individual fund alphas I find can be divided in four categories:
• Alphas of closed-end funds before the recession. • Alphas of closed-end funds during the recession. • Alphas of open-end funds before the recession. • Alphas of open-end funds during the recession
To reach the final goal of this research, which is finding differences in closed-end funds and open-end funds before and during the recession, I perform an indepopen-endent two sample T-test with unequal variances on the individual fund alphas. I do this for each comparison I make. This is also known as Welch’s T-test. In this way I can find any significant differences in mean alphas. I use the t-test with unequal variances because a variance test in Stata concludes that all variances that are researched are significantly different. The results of the variance test are discussed in the results section. The formula for the t-test reads:
(10)
where 1
is the parameter for alpha of the first compared group, and s
1² its variance, N1 is thesample size of the first group
,
2 is the alpha parameter for the second compared group,s
2² is itsvariance, and
N
2 is the sample size of the second group. 6. Empirical ResultsIn this section I describe the regression analyses and their implications. As stated before, group performance and individual performance are analyzed. This is done for the two types of funds, and for the two time periods. An overview of all the funds, and their alphas before and during the recession can be found in appendix 1. The table below gives an overview of the grouped
6.1 Group performance
To form categories in alpha, I group the returns data on a capital weighted basis. This means I calculate the weighted average for each weekly data point per type of fund and time zone. After that I regress the combined returns on the excess market returns, and the control variables for size and book-to-market ratio. By performing this I obtain one combined alpha per type of fund and time zone. See appendix 2 for regression output. The table below gives an overview of group alpha, its standard error, and the R-squared of all categories.
Table 1: Regression outcome when accounting for weights in grouped mutual fund data.
This table reports outcome of the regressions that I perform on grouped alpha data. I regress excess fund type excess returns on the market excess return, and the two control variables. The dependent variable in model (1) is the weighted excess returns of open-end funds between 2005 and 2008. The dependent variable in model (2) is the weighted excess returns of closed-end funds between 2005 and 2008. The dependent variable in model (3) is the weighted excess returns of open-end funds between 2008 and 2009. The dependent variable in model (4) is the weighted excess returns of closed-end funds between 2008 and 2009. Standard errors are reported in parentheses and are robust.
(1) open 0508 (2) closed 0508 (3) open0809 (4) closed0809
Excess Market return (Rm-Rf) 0.842*** (0.034) 0.687*** (0.050) 0.994*** (0.035) 1.352*** (0.128) Size Factor (SML) -0.020 (0.063) -0.160* (0.091) 0.074 (0.069) 0.342 (0.344) Book-to-market ratio factor (HML) 0.017 (0.013) 0.038 (0.023) -0.158*** (0.039) -0.283* (0.159) Constant 0.0006 (0.0005) -0.001 (0.0007) -0.002 (0.002) -0.002 (0.005) Obs. 177 177 56 56 R-squared 0.8977 0.7035 0.9583 0.8051
From the table I conclude the following. First, capital weighted alpha is negative in both periods for end funds. This would confirm findings by Edelen (1999), who states that closed-end funds tclosed-end to underperform due to agency costs. This result however, is not significantly different from zero when performing a simple t-test. When performing the t-test I incorporate the standard errors, which are fairly high. This means I can’t draw conclusions whether or not the alpha is significantly different from zero.
Second, open-end fund alpha is positive before the recession, and negative during the recession. This means open-end funds outperformed the market before the recession, and underperformed during the recession. Again, the problem is significance. Both alpha’s are not significantly different from zero. This again means that no real conclusions can be drawn about significance, but that is not the goal of this thesis. The goal of this thesis is to look for significant differences between the two kinds of mutual funds.
Third, I see that standard errors are fairly high. This means that the spread in performance is large. Last, I conclude that the R-squared is fairly low for closed-end funds, and high for open-end funds. For both types, R-squared increased during the recession. In some way, the variation in returns of both types of mutual funds was better explained by the market returns during the recession than before it.
6.2 Individual performance
For each fund, I regress the excess returns on the market returns, and on the two control variables according to the Fama & French Three Factor model. The output of this gives me the Jensen’s Alpha term, the Beta coefficient, the coefficient for size, and the coefficient for the book-to-market ratio. The table on the next page provides information on characteristics of the grouped individual alpha data.
Table 2: Regression output individual alpha data
This table reports results from the regression estimations that calculate individual fund alpha. For each fund, I calculate an alpha value. In this table I group the individual alpha data and provide some characteristics of the group output.
Amount of funds in group
Mean Alpha Standard Deviation of alphas in the
group
Minimum alpha Maximum alpha
Open-end funds between 2005-2008 32 0.0589% 0.09682% -0.12926% 0.24353% Open-end funds between 2008-2009 32 -0.172% 0.20529% -0.73301% 0.20274% Closed-end funds between 2005-2008 32 -0.129% 0.14562% -0.48795% 0.35164% Closed-end funds between 2008-2009 32 -0.228% 0.94878% -2.101%% 3.944%
All mean alpha’s are below zero, except for the open-end funds before the crisis. For this part, I calculate all alphas separately. Fund returns are regressed with respect to the benchmark, and the two control variables. By doing this I find fund specific alphas for the two periods. The mean alpha and standard deviation in the table above are therefore not capital weighted.
Appendix one includes a list of individual fund alphas. First, I find that with a 95% confidence level, four out of 32 closed-end funds significantly underperformed with respect to the market in the period between 2005 and 2008. When testing with a 90% confidence level, nine closed-end funds underperformed. During the recession, this number decreased to one fund at the 95% confidence level, and three funds at the 90% confidence level. This seems contra-intuitive, but can be explained by the fact that standard errors increased significantly during the crisis. In times of recession,
insecurity is high. Following this, share prices rise and drop often. Following that standard errors rise and the coefficient becomes insignificant.
Second, I find that with a 95% confidence level that five open-end funds significantly
outperform the market in the period between 2005 and 2008. One significantly underperforms. With a 90% confidence level this number increases to ten outperformers, and two underperformers. For the period between 2008 and 2009, nine out of 32 funds underperformed with a 95% confidence level. One fund outperformed significantly. With a 90% confidence level, this number of
underperformers increases to thirteen.
To prove significant differences in alpha between closed-end funds and open-end funds I perform a t-test on the mean average. This t-test exists in two forms; the test with equal variances assumed, and the test with unequal variances assumed. In the next section, I will perform a test to find whether or not I can assume equal variances.
6.2.1 Variance ratio test
To decide which t-test to use, I perform a variance ratio test to find whether or not the variances of the groups may be considered as equal or not. Next to that, I can see if there are significant differences in variance between the categories of alpha.
First, I check whether or not a significant difference in variance exists between open-end funds and closed-end funds before the recession. I perform a variance ratio test. See the table below for the outcome.
Table 3: Outcome variance test between open-end funds and closed-end funds before the crisis
This table reports the outcome of a variance test. This variance test is performed between both alpha of open-end funds and alpha of closed-end funds between 2005 and 2008. This variance test checks whether or not equal variances can be assumed.
ratio = sd(open0508) / sd(closed0508) f = 0.4421 Ho: ratio = 1 degrees of freedom = 31, 31
Ha: ratio < 1 Pr (F < f) = 0.0131 Ha: ratio ! = 1 2 * Pr (F < f ) = 0.0261 Ha: ratio > 1 Pr ( F > f ) = 0.9869
The table shows that the two-sided test outcome is significant at the 5% margin of error. This means that I can assume that variances are not equal. When looking at the one sided test outcome, I conclude that the variance in alpha of open-end funds is lower than the variance of closed-end funds. This means that the spread of alpha is bigger for closed-end funds than for open-end funds. In the figure below you can see that the box in the boxplot is wider for closed-end funds than for open-end funds. This result confirms research by Edelen (1999) and Almazan et al. (2004), who state that open-end funds take less risks when investing because of liquidity restrictions. Therefore, their variance should be lower as well. The higher risk will cause some closed-end funds to achieve significantly high returns, and other funds significantly low returns. The variance in alpha should be higher, and I prove this in my results. This results also confirm research by Pontiff (1997) who finds that closed-end fund shares are 64% more volatile than its underlying assets.
Figure 1: Boxplot individual alpha data closed-end funds and open-end funds before the crisis.
This figure show the image of a boxplot on the spread of alpha for closed-end funds and open-end funds between 2005 and 2008. The vertical line inside the two boxes represents the median, and the dots represent outliers.
Second, I check for significant differences in variance between open-end funds and closed-end funds during the recession. See the table on the next page for the outcome of the Variance Ratio test.
-.006 -.004 -.002 0 .002 .004
Table 4: Outcome variance test between open-end funds and closed-end funds before the crisis
This table reports the outcome of a variance test. This variance test is performed between both alpha of open-end funds and alpha of closed-end funds between 2008 and 2009. This variance checks whether or not equal variances can be assumed.
ratio = sd(open0809) / sd(closed0809) f = 0.0468 Ho: ratio = 1 degrees of freedom = 31, 31
Ha: ratio < 1 Pr (F < f) = 0.0000 Ha: ratio ! = 1 2 * Pr (F < f ) = 0.0000 Ha: ratio > 1 Pr ( F > f ) = 1.0000
The table shows that the P-value of the one sided test is 0.0000. The P-value is significant at all levels. This means that I can conclude that the standard deviation of open-end funds was
significantly smaller than the standard deviation of closed-end funds. Again, I find that the boxplot, which presented in figure two, is wider for closed-end funds. This again confirms results by Edelen (1999), Almazan et al. (2004) and Pontiff (1997). The figure below provides a boxplot graph. Figure 2: Boxplot individual alpha data closed-end funds and open-end funds during the crisis.
This figure show the image of a boxplot on the spread of alpha for closed-end funds and open-end funds between 2008 and 2009. The vertical line inside the two boxes represents the median. The dots represent outliers. Clearly, the spread is larger for closed-end funds.
-.02 0 .02 .04
6.2.2 Mean difference test
First, I perform the intuitive part of the empirical research. I test to find significant differences in performance before and during the financial crisis for both types of funds. I perform an independent two sample mean comparison test with unequal variances.
Table 5: Outcome mean difference test for open-end funds before and during the crisis.
This table reports the outcome of a mean difference test. This mean difference test is performed between alpha of open-end funds both before and during the crisis. This mean difference test checks whether or not significant differences in mean alpha of open-end funds before and during the financial crisis exist.
For the open-end funds, I find a t-value of 5.7486, and a p-value of 0.0000. The value is significant at the 1% level This means that the funds performed significantly better before than during the crisis. When performing the same test for closed-end funds I find something interesting. I find a t-value of 0.5886, and a p-value of 0.5602.
Table 6: Outcome mean difference test for closed-end funds before and during the crisis.
This table reports the outcome of a mean difference test. This mean difference test is performed between alpha of closed-end funds both before and during the crisis. This mean difference test checks whether or not significant differences in mean alpha of closed-end funds before and during the financial crisis exist.
diff = mean(closed0508) - mean(closed0809) t = 0.5886 Ho: diff = 0 Satterthwaite's degrees of freedom = 32.4597
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Pr(T < t) = 0.7199 Pr(|T| > |t|) = 0.5602 Pr(T > t) = 0.2801 diff = mean(open0508) - mean(open0809) t = 5.7486 Ho: diff = 0 Satterthwaite's degrees of freedom = 44.1408
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
The table shows that the difference between closed-end fund performance is not significantly lower during the recession than before it. Following this, one can state that the closed-end funds were not significantly affected by the crisis. This may confirm the theory that closed-end funds have low liquidity risk because they are not vulnerable to liquidity runs of investors (Nanda et al., 2000). The closed-end alpha means of both time zones were below zero. That means they underperformed with respect to the market.
To check if the first hypothesis holds, I perform a two sample T-test with unequal variances on closed-end and open-end alphas. This is to check for significant differences between the two types of funds in alpha before the recession.
Table 7: Outcome mean difference test for open-end funds and closed-end funds before the crisis.
This table reports the outcome of a mean difference test. This mean difference test is performed between alpha of both closed-end funds and open-end funds before the crisis. This mean difference test checks whether or not significant differences in mean alpha of open-end funds and closed-end funds before the financial crisis exist.
This time, I find a t value of 6.0542, and a p-value of 0.0000. The value is significant at the 1% level. This means that that open-end funds significantly outperform closed-end funds in times of economic growth. Following this, hypothesis one does not hold. Apparently, the increased risk-taking of closed-end funds by attaining leverage does not give higher returns. This conclusion however, is in line with research by Barclay et al. (1993), and Nanda et al. (2000) who state that closed-end funds might underperform because of high agency costs. How much of an influence these agency costs have on closed-end performance is debatable. Utility and effort are difficult to measure in a world where management cannot supervise workers at all times.
The second hypothesis is tested in the same way as the first; by performing a two sample T-test with unequal variances. Hypothesis 2 states: ‘Open-end funds outperform closed-end funds in
diff = mean(open0508) - mean(closed0508) t = 6.0542 Ho: diff = 0 Satterthwaite's degrees of freedom = 53.9271
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
Table 8: Outcome mean difference test for open-end funds and closed-end funds during the crisis.
This table reports the outcome of a mean difference test. This mean difference test is performed between alpha of both closed-end funds and open-end funds during the crisis. This mean difference test checks whether or not significant differences in mean alpha of open-end funds and closed-end funds during the financial crisis exist.
This time, the outcome is not significant. The T-value is 0.3286, and the p-value for a two sided test is 0.7445. This means that the value is not significant at any level. From this test, I
conclude there is no significant difference in performance between open-end and closed-end equity funds during a financial crisis. H2 can therefore not be adopted.
This would mean that the effects of investing in riskier assets, and attaining more risk due to leverage, had no clear benefit for closed-end funds. This contradicts research by Edelen (1999) who states that open-end funds invest in less risky assets, and hold no leverage. This reduces overall risk for the fund. Therefore, these open-end funds should be expected to outperform closed-end funds in times of recession. Also the effect of agency costs on returns is lower than expected, following findings by Barclay et al. (1993) and Nanda et al. (2000). Their findings predict that closed-end funds tclosed-end to underperform because of high agency costs.
The disadvantage of this T-test is clearly that it doesn’t take into account capital weights. All alpha’s are treated equally. Intuitively, a big fund has a bigger impact on the population than a small fund. On the other hand one can argue, that because the market for mutual funds is so big each capital weighted influence on the market can be assumed to be zero.
6.3 Robustness discussion
The results could suffer some robustness issues. However, these are not significantly large. To find the best benchmark I perform all alpha calculations with respect to four benchmarks, to try and find the best one.
diff = mean(open0809) - mean(closed0809) t = 0.3286 Ho: diff = 0 Satterthwaite's degrees of freedom = 33.8962
Ha: diff < 0 Ha: diff != 0 Ha: diff > 0
My results show that the squared’ does not change significantly when changing benchmark. ‘R-squared’ is the indicator of how well the model fits the data points.
Robustness is especially an issue for closed-end funds, who tend to have a very low value of ‘R-squared’ in the Fama & French Three Factor Model. See appendix 3 for an example of a fund that has been regressed with respect to the four benchmarks. Clearly, the ‘R-squared’ only changes about one or two percent when changing the benchmark. This means the model remains effective to some extent under different markets and market conditions. Next to that, my results are either very significant, or not significant at all. This would mean my conclusions still hold under small
differences in market situation.
For almost all closed-end funds, I find a fairly low ‘R-squared’ when regressing to find alpha. When testing a model, a high R-squared is not always an indicator of model performance. On the other hand, a very low R-squared almost definitely means you will find results that are not
trustworthy. In this case, a low R-squared could mean that the funds’ performance is not
comparable to the market return. The fund is thus greatly independent from any market proxy when considering variation in returns.
7. Conclusions
This research analyses the performance of 64 mutual funds. These funds are based in the United states, and are all actively managed. This is done over the period of 2005-2009. The time period is divided into two sections; before and during the financial crisis of 2008-2009. I do this to find significant differences in returns for both kinds of mutual funds in the given time period. By doing this I also investigate whether or not a financial crisis affects a closed-end funds and open-end funds differently.
I measure performance by calculating Jensen’s Alpha according to the Fama and French Three Factor model. The first part of the findings in this study relate to some prior mutual fund studies. Open-end funds outperform closed-end funds significantly in the period between 2005 and 2008. In this period of time, the United States’ economy was thriving. These results are in line with findings of Barclay et al. (1993), and Nanda et al. (2000), who state that closed-end funds tend to underperform due to agency costs. On the other hand, the results contradict statements made by Edelen (1999), and Almazan et al. (2004), who state that open-end funds impose liquidity
restrictions on management. This restrictions could affect investment freedom negatively and therefore affect returns negatively as well. It seems investors are better off investing in an open-end fund in times of economic welfare.
In the second part of the empirical research, I find no significant differences in returns between open-end funds and closed-end funds. These results oppose research by Edelen (1999), who states that open-end funds have higher liquidity restrictions in investments. Therefore, their investments should be less risky, and would therefore be a better choice in a financial crisis like the one in 2008-2009 than closed-end funds. The results do indicate that open-end funds suffer more from the financial crisis than closed-end funds, as open-end funds used to outperform closed-end funds in the period before.
This research does have some limitations. First, fees are not accounted for in this thesis. Properly accounting for the amount and kind of fees would make the research significantly more complicated. One type of fund might have excessive fees compared to the other, and would therefore be less attractive to invest in. Further research on this subject might consider adding the amount and type of fee to the equation, to check if the conclusions in this thesis still hold. This could generate new complications, as the effects of the fee on returns are not easily measured.
Second, this research might suffer from survivorship bias. The Thomson one database does not provide data on funds that went bankrupt. This means that a particular type of fund might have suffered more from this bias. Results would then be biased. This would especially be the case for the time period between 2008 and 2009, during the financial crisis. Literature does not provide evidence that this survivorship bias is greater for one type of fund. Therefore, I do not expect it to affect the trustworthiness of my conclusions.
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Appendices
Appendix 1: List of funds accompanied with alpha for the two time periods
This appendix reports a list of all funds in the sample, accompanied by their individual alpha value for the two time periods.
Open-‐ended Funds Alpha0508 Alpha0809 Closed-‐ended Funds closed0508 closed0809
AllianzGI Wellness Fund -‐0.0010749 -‐0.0003392 Adams Express Company -‐0.001022 -‐0.002733 Ave Maria Growth Fund 0.0004297 0.000204
Advent/Claymore Enhanced Growth &
Income Fund -‐0.0024815 -‐0.0042759 Berwyn Fund Inc. 0.0009306 0.0012949 BlackRockEco 0.0035164 -‐0.0104723 Buffalo Growth Fund 0.000138 0.0004285
Boulder Growth & Income Fund
Incorporated 0.0005775 -‐0.0059682
Bullfinch Unrestricted Series -‐0.0012926 -‐0.001983 Boulder Total Return Fund Incorporated -‐0.0004641 -‐0.0059215 Bullfinch Western New York Series -‐0.0005123 -‐0.0013108 Central Securities Corporation 0.0000319 -‐0.004858 Century Shares Trust Institutional -‐0.0001415 -‐0.0022616
Cohen & Steers Closed End Opportunity
Fund -‐0.0026326 -‐0.002573
Chesapeake Growth Fund 0.0010304 -‐0.0013398
Cohen and Steers Dividend Majors Fund
Incorporated -‐0.0023898 -‐0.0020522 Dreyfus Strategic Value Fund 0.000689 -‐0.0016667 Cornerstone Progressive Return Fund 0.0006116 -‐0.0005508
Fidelity Independence Fund 0.002101 -‐0.0054124
Cornerstone Strategic Value Fund
Incorporated NEW -‐0.001808 0.0394439 First Eagle Fund of America Class C 0.001677 -‐0.0027384
Cornerstone Total Return Fund
Incorporated NEW -‐0.0018035 0.0141818 Geneva Advisors All Cap Growth Fund -‐0.0010429 -‐0.0035152 Denali Fund Inc -‐0.0016308 -‐0.0022513 Hartford Growth Opportunities Fund -‐
Class A 0.002173 -‐0.0035557 Eagle Capital Growth Fund, Inc -‐0.0015604 -‐0.0210116 Legg Mason ClearBridge Aggressive
Growth Fund 0.0001066 -‐0.0024182
Eaton Vance Tax Advantaged Dividend
Income Fund 0.0000935 -‐0.0047469 LKCM Aquinas Growth Fund 0.000942 -‐0.0025261 Equus Total Return Inc. -‐0.0006847 -‐0.0082834 Marsico 21st Century Fd 0.0011362 -‐0.0026117 First Trust Dividend & Income Fund -‐0.0048795 0.0017538 MassMutual Select Focused Value
Fund -‐0.0001132 0.0014711 Gabelli Dividend & Income Trust -‐0.0004538 -‐0.0008045 Nicholas Fd -‐0.0010211 0.0008193 Gabelli Equity Trust Incorporated -‐0.0009836 0.0013552 Putnam Multi-‐Cap Growth Fund Class
A 0.0003486 -‐0.0018932
General American Investments
Company -‐0.000273 -‐0.0031542
Roosevelt Multi-‐Cap Fund Inv 0.0003051 -‐0.0045626
Guggenheim Enhanced Equity Income
Fund -‐0.003259 -‐0.0042369
T. Rowe Price New America Growth Fd 0.0005172 0.0000278
Guggenheim Enhanced Equity Strategy
Fund -‐0.0019507 -‐0.0150757
Telecommunications UltraSector
ProFund Service Class 0.0022864 -‐0.0038707
John Hancock Tax Advantage Dividend
Income Fund -‐0.0015562 0.00128
Thornburg Core Growth Fund 0.0007772 -‐0.0013819 Liberty All Star Equity Fund -‐0.0030146 -‐0.0041274 Touchstone Growth Opportunities
Class A 0.0006172 -‐0.0021824 Libert All Star Growth Fund -‐0.0025886 -‐0.0048819 Transamerica Growth Opportunities
Class A 0.0011696 -‐0.0004835
Nuveen Tax Advantaged Toatal Return
Strategy Fund -‐0.0008807 -‐0.0042821 Turner All Cap Growth Fd Inv Cl 0.001374 -‐0.0006277 Royce Focus Trust Incorporated -‐0.0009561 -‐0.0032877 Value Line Fd 0.00068 -‐0.0073301 Royce Micro Capital Trust Incorporated -‐0.0020002 -‐0.0009161 Waddell & Reed Advisors Vanguard Fd
Cl C 0.0007984 -‐0.0025183 Royce Value Trust Incorporated -‐0.0014709 -‐0.0036793 WFA Omega Growth Fund 0.0001339 0.0020274 Source Capital Incorporated -‐0.0019719 -‐0.0005154 William Blair Funds Growth Fund 0.0010072 -‐0.0001439 Special Opportunities Fund Inc -‐0.0008939 0.0010707 Wireless Fund 0.0024353 -‐0.0012021 Tri-‐Continental Corporation -‐0.0007514 -‐0.0076875 Zacks All-‐Cap Core Fund Class A 0.0001283 -‐0.0034738 Zweig Fund Incorporated New -‐0.0016247 -‐0.0038564 note: values with a yellow and pink
background indicate significance at the 5% and 10% test level,
Appendix 2: regression output for alpha calculation
This appendix reports regression output for grouped alpha calculation. I regress the grouped fund excess returns on the excess market returns and the two control variables. The constant I find in this regression is the value for alpha. The first regression is for open-end funds before the crisis, the second is for closed-end funds before the crisis. The third regression is for open-end funds during the crisis. Last, I regress for closed-end funds during the crisis.
regress RpRfopen0508 RmRf0508 SML HML, robust
Number of obs 177 F( 1, 30) 422.32 Prob > F 0.0000
R-squared 0.8977
Root MSE 0.00609
Open 0508 Coeff. Robust Std. Err. t. P>|t| [95% Conf. Interval] RmRf0508 SML HML _cons 0.8422047 -0.0202614 0.0170292 0.0006082 0.0342484 0.0632864 0.0131986 0.000464 24.59 -0.32 1.29 1.31 0.000 0.749 0.199 0.192 0.7746063 0.9098032 -0.1451743 0.1046515 -0.0090218 0.0430801 -0.0003076 0.001524
regress RpRfclosed0508 RmRf0508 SML HML, robust
Open 0508 Coeff. Robust Std. Err. t. P>|t| [95% Conf. Interval] RmRf0508 SML HML _cons 0.6872311 -0.1604441 0.0380407 -0.0010575 0.0503219 0.0907821 0.0232282 0.0006947 13.66 -1.77 1.64 -1.52 0.000 0.079 0.103 0.130 0.5879072 0.786555 -0.3396272 0.018739 -0.0078064 0.0838878 -0.0024286 0.0003137
regress RpRfopen0809 RmRf0809 SML HML, robust
Number of obs 177 F( 1, 30) 83.75 Prob > F 0.0000 R-squared 0.7035 Root MSE 0.00922 Number of obs 56 F( 1, 30) 331.40 Prob > F 0.0000 R-squared 0.9583 Root MSE 0.01113
Open0809 Coeff. Robust Std. Err. t P>|t| [95% Conf. Interval] RmRf0809 SML HML _cons 0.9943067 0.074233 -0.1582201 -0.0018415 0.0348869 0.0694191 0.0387041 0.0015669 28.50 1.07 -4.09 -1.18 0.000 0.290 0.000 0.245 0.924301 1.064312 -0.0650666 0.2135325 -0.2358855 -0.0805547 -0.0049858 0.0013027
regress RpRfclosed0809 RmRf0809 SML HML, robust
Appendix 3: Example of regression output when considering different benchmarks
This appendix reports Stata output for regressions with a different benchmark for one fund; the Liberty All Star Equity Fund. I regress fund excess returns on excess market returns and on the two control variables. I focus on the value of R-squared, which does not change significantly.
regress LibertyAllStarEquityFund SP500Index SML HML
Source | SS df MS Number of obs = 177 --- F( 3, 173) = 74.45 Model | .045044293 3 .015014764 Prob > F = 0.0000 Residual | .034889917 173 .000201676 R-squared = 0.5635 --- Adj R-squared = 0.5559 Total | .079934209 176 .000454172 Root MSE = .0142
Number of obs 56 F( 1, 30) 45.78 Prob > F 0.0000 R-squared 0.8051 Root MSE 0.03624 Closed 0508 Coeff. Robust Std. Err. t. P>|t| [95% Conf. Interval] RmRf0809 SML HML _cons 1.352013 0.3415981 -0.283295 -0.0023046 0.1276667 0.3444248 0.1590381 0.004541 10.59 0.99 -1.78 -0.51 0.000 0.326 0.081 0.614 1.095831 1.608195 -0.3495408 1.032737 -0.6024282 0.0358383 -0.0114249 0.0068157
regress LibertyAllStarEquityFund DowJonesIndustrialAverage SML HML
Source | SS df MS Number of obs = 177 ---+--- F( 3, 173) = 71.42 Model | .044225511 3 .014741837 Prob > F = 0.0000 Residual | .035708699 173 .000206409 R-squared = 0.5533 ---+--- Adj R-squared = 0.5455 Total | .079934209 176 .000454172 Root MSE = .01437
regress LibertyAllStarEquityFund Russell2000Index SML HML
Source | SS df MS Number of obs = 177 ---+--- F( 3, 173) = 69.88 Model | .043793889 3 .014597963 Prob > F = 0.0000 Residual | .036140321 173 .000208904 R-squared = 0.5479 ---+--- Adj R-squared = 0.5400 Total | .079934209 176 .000454172 Root MSE = .01445
regress LibertyAllStarEquityFund NasdaqComposite SML HML
Source | SS df MS Number of obs = 177 ---+--- F( 3, 173) = 73.31 Model | .044740354 3 .014913451 Prob > F = 0.0000 Residual | .035193855 173 .000203433 R-squared = 0.5597 ---+--- Adj R-squared = 0.5521 Total | .079934209 176 .000454172 Root MSE = .01426
