New Evidence of Persistence in Open-Ended Mutual Funds over
Different Time Horizons
Yufeng Zhu，S2260395
Supervisor: Prof. Dr. Roberto Wessels
Master of Finance, Faculty of Economics and Business, University of Groningen
June 2013
Abstract
This thesis investigates performance persistence in equity funds in the US over five- and two-year intervals in a sample free of survivorship bias. Using standardized performance measure-Sharpe ratio, no support is found for previous conclusions of long-term persistence in mutual funds. Moreover, persistence over short-term intervals is weaker than determined by previous research, with two out of four sub-periods exhibiting reversal. Further analysis of the predictability of being a winner or loser in the first period for future performance indicates that both positive past performance and negative past performance has predictability for future performance when persistence is present.
JEL classification: G10; G11; G23; C14
Keywords: Persistence; Survivorship bias; Contingency table; Mutual funds;
1. Introduction
Mutual funds have been one of the most popular investment choices in recent years. By the end of 2012, there were $13 trillion of assets under management in the US, double the amount in 2002 (Investment Company Institute, hereafter ICI, 2013). Most investors characteristically take the usefulness of past performance for granted in the belief that past performance of a mutual fund is a good predictor of future performance. But the existence of performance persistence, defined as a positive relation between performance in an initial ranking period and a subsequent period, is still an unresolved question.
Studies on performance persistence differ in how it is measured. Grinblatt and Titman (1992) use abnormal returns as their performance measure and report positive persistence in mutual fund performance of US market for an observation interval of five years.1 Goetzmann
and Ibbotson (1994) demonstrate that performance persistence is robust to both raw and risk-adjusted returns, which they measure using Jensen’s alpha with intervals of one month to three years. Carhart (1997), using a four-factor model, also reports performance persistence at one year in risk-adjusted returns. Moreover, Hendricks, Patel, and Zeckhauser (1993) indicate that performance persistence exists for various risk-adjusted performance measures, including Jensen’s alpha with different benchmarks and the Sharpe ratio. Most of these empirical studies tend to favour conventional investment wisdom, where past performance contains useful information for predicting future performance, especially in the short-term period. However, hardly any studies test whether general persistence is the result of the predictability of positive or negative past performance or some combination thereof. This thesis contributes to the literature by attempting to fill this gap.
This paper also differs from previous studies in its selection of performance measure. It uses the Sharpe ratio to adjust risk and measure the performance of mutual funds. Compared with Jensen’s alpha or the three- or even four-factor model, the Sharpe ratio suffers from less bias in adjusting risk. This is because the appropriate indexes used by the former models when attempting to explain risk are unknown, since fund data are limited, which can lead to unknown bias in performance estimation. Using the Sharpe ratio to adjust excess returns based on total risk lowers the diversification of funds and thus renders them more comparable among themselves.
1 The performance measure used by Grinblatt and Titman (1992) is the abnormal return relative to the P8 benchmark, which
The remainder of the paper is structured as follows. Section 2 reviews the literature. Section 3 introduces the Sharpe ratio as a performance measure and the methodology of the contingency table in studying persistence. Section 4 describes the data. The empirical results for general persistence are presented in Section 5. Section 6 employs the del-statistic to analyse the predictability of being a winner (loser) in the first period for performance in the second period. Section 7 concludes the study.
2. Literature review
Persistence has been tested empirically by studying the relation between the past performance of a portfolio of funds and its performance in the next period (see Grinblatt and Titman (1992) for a long-term study and Goetzmann and Ibbotson (1994) for a short-term analysis). The existence of persistence implies that this relation should be positive. Various methods have been employed to test this relation in the literature, which is briefly reviewed below.
Grinblatt and Titman’s (1992) study of the persistence of performance in US mutual funds (from December 31, 1974 to December 31, 1984) with abnormal returns first splits their sample into two five-year sub-periods and calculates the abnormal returns2 of funds in each
sub-period. They then regress the abnormal returns in the second period on the abnormal returns in the first period. The authors conclude that the past performance of mutual funds can report valuable information to mutual fund investors.
Goetzmann and Ibbotson (1994) also test persistence but use a different, nonparametric method. Using 728 mutual funds from 1976 to 1988, the authors focus on the short-term persistence of mutual fund performance, especially at two-year intervals. Sorting funds into winners and losers according to the median performance of the full sample, the authors find a repeating performance pattern in two consecutive periods. Based on past winning performance, the probability of selecting a winner is more than 50%. Moreover, this repeating-winners pattern occurs in four out of five two-year periods and the authors conclude that past returns and relative rankings can predict future performance.
However, besides persistence, Brown and Goetzmann (1995) find reversals in performance during certain years for a sample of equity funds from 1976 to 1988. Defining a fund as a winner that year if its return is above or equal to the median return of all reported funds in the sample, the authors employ a nonparametric methodology based on contingency tables to
examine the repeating pattern of mutual fund performance. By analyzing the cross-product ratio (CPR), that is, the number of repeating performers over the number of non-repeating performers, the authors find that seven out of 12 periods exhibit positive persistence. However, two years of the sample exhibit significant reversals, meaning that a winning fund in the first period ended up being a loser in the second period. The authors conclude that persistence depends on the study’s period.
This thesis follows a similar methodology as that of Brown and Goetzmann (1995), except that it examines differences in performance measures and the classification of winners and losers. Unlike previous studies, however, it measures the performance of mutual funds using the Sharpe ratio to adjust for return risk, which is not common in previous studies. The main models applied by previous researchers are the capital asset pricing model (CAPM) with various market indexes (Malkiel (1995)), the three-index model (Brown and Goetzmann (1995)), and the four-factor model (Carhart (1997)).
One problem related to applying these models is that the appropriate adjustment for measuring abnormal performance is unknown. Therefore, the adjustment easily leads to performance estimation bias if the explanation variables are not appropriately selected. For example, Elton, Gruber, and Hlavka (1993) find that excluding the index of firm size as a risk index leads to overstatement of the performance of mutual funds holding small stocks. As a result, this approach incorrectly estimates average performance. To improve the explanatory power of the three-index model, Elton, Gruber, and Blake (1996 a) add a new index to explain the performance of growth versus value stocks. Thus, ignoring this index can misleadingly indicate the temporary strong performance of certain types of funds if the sample consists of numerous funds with growth or value as their objective. However, the appropriate adjustments are unknown most of the time; consequently, the biases they produce are also unknown. Similarly, in the CAPM, different benchmarks lead to different results in terms of performance measurements. Malkiel (1995) indicates that the average alpha is negative for the Standard & Poor’s (S&P) 500 but positive for the Wilshire 5000 Stock Index. This is because during the author’s sample period, in the 1980s, small stocks had a tendency to underperform the S&P 500. Mutual funds performance tends to be better for the Wilshire 5000 because the index includes smaller companies. It would be worse if a positive alpha were used as a benchmark in contingency tables to define winners and losers. The results of repeating performance may be biased.
future performance. Malkiel and Saha (2003) investigate repeated winning performance in hedge fund returns with a nonparametric method and find that only 50% of winners in the first period repeat their performance in the next period. The authors measure repeat winners as the percentage of counts of winner winner observations over the total number of winner -winner and -winner-loser observations.
This study differs from prior research in several ways. First, unlike most studies, it uses a standardized performance measure instead of raw returns. Risk is adjusted through the Sharpe ratio, which measures the reward per unit of risk and avoids the introduction of unknown biases from inappropriate risk adjustments. Second, employing contingency tables, this study defines funds as winners (losers) if they rank in the upper (bottom) quartile, instead of above (below) the median. Compared with division by half, division by quartiles results in more convincing persistence in mutual fund performance rankings. Last but not least, this study tests the hypothesis that being a winner (loser) in the first period predicts being a winner in the second period.
3. Methodology
The use of contingency tables is one of the most common methods to study performance persistence in mutual funds as it provides a clear and direct view of persistence for different time horizons. This thesis employs the 2 × 2 contingency table that classifies winners and losers based on quartiles instead of the median. This method delivers more convincing persistence test results. Before testing performance persistence, it is necessary to set up a standard for measuring mutual fund performance. This thesis uses the Sharpe ratio, which acts as a standardized measurement. The remainder of this section illustrates the methodology.
3.1 Performance measures
such as those based on the CAPM or APT lead to persistence bias because they favour small-cap and high dividend yield stock holdings, respectively. As a result, small-small-cap and income-oriented funds tend to demonstrate greater persistence over other types of funds.
Given the shortcomings of these risk adjustment techniques, this study uses the Sharpe ratio to measure mutual fund performance. Unlike the CAPM, the Sharpe ratio does not refer to the market index and is therefore unaffected by different market portfolio choices. Since the Sharpe ratio is based on total portfolio risk, which includes both market risk and unsystematic risk, it allows the evaluation and comparison of the performance of different mutual funds. Following Morningstar’s calculation process, the Sharpe ratio is expressed as S =∑(𝑅𝑖,𝑡−𝑅𝑓,𝑡) 𝑁⁄
σ , (1)
where Ri,t is the monthly return of mutual fund i at time t, Rf,t is the return of a one-month US
Treasury bill at the corresponding time t, and σ is the standard deviation of the monthly excess returns of the sample period. This ratio measures a fund’s return in excess of the risk-free rate compared to the fund’s total risk, which is measured by its standard deviation. By definition, the Sharpe ratio determines the reward per unit of risk. The higher the Sharpe ratio, the better the mutual fund’s historical risk-adjusted performance. A negative Sharpe ratio implies an inability to produce more return per unit of risk than the risk-free rate. The next stage uses the Sharpe ratio as a criterion to rank and compare the performance of mutual funds.
3.2 Persistence tests
the persistence of ranking in mutual fund performance, determining whether past performance contains information predictive of future performance.
Similar to a traditional winner–loser 2 × 2 contingency table, an improved winner–loser 2 × 2 contingency table that counts the top (bottom) quartile of all sorted funds as winners (losers) is constructed. Funds in the second and third quartiles are classified as other. For the long-term persistence analysis, the monthly returns of all funds from January 2003 to December 2012 are first divided into two sub-period samples, one from January 2003 to December 2007 and the other from January 2008 to December 2012. The first period is referred to as the ranking period and the second period is the evaluation period,3 during which
fund performance (ranking) is evaluated. The Sharpe ratio is calculated with equation (1) in each period and acts as a fund’s risk-adjusted performance measure. In both the ranking and evaluation periods, all funds are ranked and sorted according to their Sharpe ratios. The quartiles are formed based on fund rank. The funds in the top quartile are defined as winners, those in the bottom quartile are defined as losers, and funds ranked in the second and third quartiles are in the category referred to as other. According to their performance (ranking) in two consecutive periods, mutual funds are classified into one of nine categories, each denoting their status in the two consecutive periods: winner–winner (WW), winner–loser (WL), loser–winner (LW), loser–loser (LL), winner–other (WO), loser–other (LO), other– other (OO), other–winner (OW), and other–loser (OL). However, this study is only interested in four categories: WW, WL, LW, and LL.
The CPR is used to test the persistence of performance (rankings) in the two consecutive periods. The CPR measures the number of repeating performers to the number of non-repeating performers, expressed as (WW*LL)/(WL*LW). If there were no persistence in the performance ranks in the two consecutive periods, the CPR would equal one. The significance of the CPR is tested with a Z-statistic, which equals the natural logarithm of the CPR divided by its standard deviation.4
In addition, the Spearman rank correlation is introduced to test the persistence of mutual fund performance. It measures the correlation between performance ranks in the ranking and
3 For the two-year interval, the performance two years prior is used to predict performance in the next two years; that is,
performance in 2003–2004 is used to predict performance in 2005–2006. Subsequent periods are also defined similarly, with four pairs: 2003–2004/2005–2006, 2005–2006/2007–2008, 2007–2008/2009–2010, and 2009–2010/2011–2012. For each pair, the ranking period refers to the first two years and the evaluation period to the next two years.
evaluation periods. Since the Spearman rank correlation5 treats each quartile equally, it
presents only a general view of the performance persistence of mutual funds over consecutive periods.
4. Data
Open-ended equity mutual funds alive at the end of 2002 and trading on the US market during 2003–2012 with the three major investment objectives of growth, income, and blend were obtained from the Survivor-Bias-Free US Mutual Fund Database of the Center for Research in Security Prices (CRSP). This database contains data on mutual fund monthly returns and general information, such as fund name, investment style, fee structure, holdings, and asset composition. In this database, monthly returns are calculated as change in net asset value (NAV), which assumes dividends are included and invested from one period to the next.6 To calculate the returns, net asset value is deducted from all management expenses and
12b fees. Front and rear load fees are also deducted. This study selects equity domestic income funds, equity domestic growth funds, and equity domestic growth and income mutual funds.
Survivorship bias has always been a hot issue in the study of mutual fund performance persistence. There are two main views in the literature. One suggests that survivorship bias upwardly biases persistence performance. The view represented in this study is that of Brown, Goetzmann, Ibbotson and Ross (1992), who demonstrate that survivorship truncation, even where no corresponding event exists, leads to performance persistence. If funds with poor performance are omitted from the sample; funds with better performance tend to have higher risk. Given their survival, higher-risk funds would have higher expected performance. Consequently, a winner in the first period would win in the second period in the survivor sample. Selecting the sample of mutual funds from the Survivor-Bias-Free US Mutual Fund Database helps avoid this bias. On the other hand, Grinblatt and Titman (1992) and Hendricks et al. (1993) find a reversal effect for the survivorship bias if fund survival depends on average performance over multiple periods.
5 The Spearman rank correlation is calculated as 𝜌 = 𝑆𝑎𝑏
𝑆𝑎∗𝑆𝑏, where Sab is the covariance of two ranks in consecutive periods
and Sa and Sb are the standard deviations of the ranks in each period, respectively. The significance of the Spearman rank
correlation can be tested by using a z-test when the number of samples is larger than 30, where z = 𝜌 ∗ √𝑁 − 1.
6 The Survivor-Bias-Free US Mutual Fund Database guide calculates the returns as Rt = 𝑁𝐴𝑉 𝑡∗𝑐𝑢𝑚𝑓𝑎𝑐𝑡
𝑁𝐴𝑉 𝑡−1 - 1, where the variable
cumfact is an adjustment factor that starts out as one in a given day and is then modified depending on the types of dividends
A number of studies measure survivorship bias empirically by comparing the results from a survivorship-biased sample and a sample free from such bias. For instance, Grinblatt and Titman (1989) and Wermers (1997) show that the impact of survivorship bias is modest. Grinblatt and Titman (1989) indicate that survivorship bias accounts for 10 basis points to 40 basis points, on average, per year. Moreover, Wermers (1997) shows that there is only a 23-basis point difference between surviving funds and all funds, including both live and dead funds from 1974 to 1994. However, survivorship bias is not trivial from the perspective of Malkiel (1995) and Carhart (1997). The mean return of surviving funds is significantly larger than that of non-surviving funds. Carhart (1997) compares the results of a full sample and a survivor-biased sample. In the full sample, all funds remain in the portfolio until they disappear and the sample is then rebalanced. In the survivor-biased sample, only funds that survive until the end of the sample period are included. By comparing the results, the author finds strong persistence in the full sample but only weak persistence in the survivor-biased sample. Consequently, if inactive funds are excluded from the sample, the results are potentially misleading and deliver an incomplete picture of performance persistence.
The sample in this thesis includes 324 equity funds from January 2003 to December 2012, both active and inactive, which helps avoid any strong effects of survivorship bias on the persistence measure. For funds that stopped reporting, missing return observations were entered manually. This process was conducted as follows: a) If a fund stopped reporting in the ranking period, the return of the month in which the fund stopped reporting is assumed to be zero and the average monthly Sharpe ratio is calculated with the available and assigned returns for the first period. For example, the Hartford Value Opportunities Fund stopped reporting its return in February 2007, which means that it stopped reporting in the ranking period for the five-year evaluation. It is assumed that in February 2007 its return was zero, as for the remaining 10 months as well. In the next period, this fund is excluded from the sample pool when using the contingency table. b) If a fund stopped reporting in the evaluation period, similar to the first situation, the return of the month when the fund stopped reporting is assumed to be zero and the average monthly Sharpe ratio is calculated with the available and assigned returns. For instance, the AIM Large Cap Basic Value Fund stopped reporting its returns in May 2011, so zeros are assigned to its missing returns in and after May 2011. The assumption of zero returns for missing observations is overestimated for funds that stopped reporting, since it assumes investors lose only part of their investment.7 A summary
7 If -1 were assigned as the return of a fund that stopped reporting, performance would be underestimated. Take the same
of manual entries for missing observations is shown in Table I for both the two- and five-year studies.
Table I
Statistical Summary of Manual Entries for Missing Observations
This table summarizes the number of manual entries for missing return observations. The total number of funds indicates the observable number of funds in a specific study period. Total unit refers to the number of observations of returns, which equals the number of funds times the length of time. The percentage of manual entry observations reports the percentage of manual entries in the total unit.
Total Number
of Funds Total Unit
Number of Manual Entries for Returns Percentage of Manual Entry Observations Five years 2003–2007 324 19440 1110 5.71% 2008–2012 275 16500 1241 7.52% Two years 2003–2004 324 7776 55 0.71% 2005–2006 313 7512 259 3.45% 2007–2008 289 6936 301 4.34% 2009–2010 269 6456 212 3.28% 2011–2012 254 6096 298 4.89% 5. Empirical results
Funds are ranked at the beginning of the ranking period and at the beginning of the evaluation period. If a fund stopped reporting before those times, it is not available for ranking. With the use of the contingency table, the CPR and Spearman rank correlation for the sorted funds in two consecutive sub-periods are calculated to illustrate the degree of performance (ranking) persistence in mutual funds.
Tables II and III show the numbers of repeating performers and reversal performers in two consecutive periods, according to their Sharpe ratio rankings. The intersection of winners and winners indicates the frequency of funds that rank in the top quartile in the first period as well as in the top quartile in the next period, based on their Sharpe ratio. The results of contingency table tests show whether information about past ranking predicts future rankings for mutual funds.
Table II
Test of Fund Long-Term Performance Persistence
This table reports the two-way contingency table of the repeating performance of mutual funds. The winners in each period are funds that rank in the top quartile based on their Sharpe ratios and losers are funds ranking in the bottom quartile. The intersection of winners in the initial years and winners in subsequent years comprises funds that were winners in 2003–2007 and remained winners in 2008–2012. The same principle is applied to the other categories. The CPR is calculated as (WW*LL)/(WL*LW). The Z-statistic is measured as the natural logarithm of the CPR divided by its standard error and is asymptotically normally distributed under the assumption of the independence of the observations. The Spearman rank correlation is the correlation between the ranks of all funds in two sub-periods. The Z-statistic is calculated as the product of the Spearman rank correlation and the square root of the sample size minus one. The Z-statistic tests whether the CPR is statistically different from one and whether the Spearman rank correlation is statistically different from zero. Z-Statistics of 1.645, 1.960, and 2.575 correspond to significance at the 10%, 5%, and 1% levels, respectively. The superscripts ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
2008–2012
Winners Losers
2003–2007 Winners Losers 22 15 12 14
CPR 1.711
Z-Statistic 1.040
Spearman Rank Correlation 0.115
Z-Statistic 2.060**
Table III
Test of Fund Short-Term Performance Persistence
This table reports the two-way contingency table of the repeating performance of mutual funds. Winner–winner indicates the total number of funds that rank in the top quartile according to their Sharpe ratio in the first two-year period who are also in the top quartile in the next two-year period. Loser–loser represents the number of funds that rank in the bottom quartile according to their Sharpe ratio in the first two-year period who are also in the bottom quartile in the next two-year period. The same principle is applied to other categories. The CPR is calculated as (WW*LL)/(WL*LW). The Z-statistic is measured as the natural logarithm of the CPR divided by its standard error and is asymptotically normally distributed under the assumption of the independence of the observations. The Spearman rank correlation is the correlation between the ranks of all funds in the two sub-periods. The Z-statistic is the calculated as the product of the Spearman rank correlation and the square root of the sample size minus one. The Z-statistic tests whether the CPR is statistically different from one and whether the Spearman rank correlation is statistically different from zero. Z-Statistics of 1.645, 1.960, and 2.575 correspond to significance at the 10%, 5%, and 1% levels, respectively. The superscripts ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
Year Total WW LW WL LL CPR Z-Statistic
Table III reports the results of the same analysis when persistence is measured at two-year observation intervals. The outcomes are quite different from those of the long-term study. As presented in Table III, short-term persistence was not a consistent phenomenon in 2003–2012. Only two out of four sub-periods exhibited persistence in performance ranking, while the other two sub-periods displayed strong reversal effects. The CPR for 2005–2006/2007–2008 is 0.051, significant at the 1% confidence level. Around one-third of top-ranking funds in 2005–2006 ended up in the bottom in 2007–2008. The CPR is also smaller than one in 2009–2010/2011– 2012, only 0.316, and significant at the 1% level as well. This finding counters the hypothesis that persistence exists continuously in the performance of equity funds. Using the Spearman rank correlation also leads to a similar conclusion. Consequently, it is difficult to conclude that past performance indicates future performance in mutual funds over the short term, since a reversal in performance persistence is found from January 2005 to December 2008 and from January 2009 to December 2012.
This result shows that short-term persistence prevails when the economic environment is stable, similar to the results of Malkiel (1995), who discovered that persistence was not significant in the late 1980s and even exhibited a reversal in 1987. Through 1987 to early 1990, the US stock market experienced a sharp collapse and the crisis then expanded to the whole economy. There was what appeared to be a small recovery in 1989 but it turned out to be illusory. Affected by the economic environment, the performance of equity mutual funds shows no sign of persistence in 1987, 1988, or 1990 in Malkiel’s research, using a one-year interval.
The global economy experienced a severe recession from 2007 to 2012, considered the worst crisis since the Great Depression of the 1930, with side-effects magnified on a global scale. The US stock market fell sharply in 2007 and fluctuated in 2011–2012 with reference to the Dow Jones Index. Consequently, an alternative reason for the absence of persistence is that the market environment affects the measure of persistence. Different observation periods lead to different performance persistence results. Thus, the phenomenon of persistence in the performance of mutual funds is not stable.
6. Test of winner/loser predictability
24 = 16 distinct prediction forms in the 2 × 2 table. Thus, the CPR does not indicate whether
only winners or losers or both have predictability for future performance. Consequently, a new indicator, the ∇-statistic (or del-statistic) is introduced to test the predictability of winners and losers. Developed by Hildebrand, Laing, and Rosenthal (1977), the ∇-statistic compares the rate of prediction errors obtained by applying a specific prediction theory versus prediction without the theory. In other words, it measures the power of the prediction theory by comparing its error rate with that of the unconditional prediction. Compared with the CPR, this technique helps discriminate between alternative prediction forms.
The hypothesis of the predictability of being a winner (loser) can be translated into the statement that a fund tends to be a winner (loser) in the second period if it is a winner (loser) in the first period. As in the contingency table method, paired observations of a fund’s ranking (winner or loser) in the first period and its ranking in the second period are required to test predictability. From the sample, 324 paired observations are obtained initially. Winners (losers) are defined according to the same principle as in the contingency table. In each period, funds are classified as winners (losers) and non-winners (non-losers) and another 2 × 2 table is built. Table IV shows the results of a case where being a winner in the past two years, in 2003–2004, is a predictor of being a winner in the subsequent two-year period.
The results presented in Table IV are convincing evidence of the predictability of being a winner in the first period for still being a winner in the second period. Among 81 funds that were winners in the first period, 33 funds remained winners in the second period. The other 48 funds, conversely, can be considered prediction errors, since they are cases in which the prediction is being falsified. In other words, the prediction of being a winner in the second period conditional on being a winner in the first period is wrong 59.3% of the time. When a prediction is made in terms of the error rate on the unconditional probability of being a winner in the first period, the probability is 75.9%, which is higher than the error rate achieved under the predictability of being a winner in the first period.
This can be expressed by the ∇-statistic, which measures the predictability of being a winner in the first period by taking the ratio of the error rate when a fund is a winner in the first period to the error rate when a fund’s previous performance is not considered and subtracting the result from one. In other words,
∇ = 1 − 𝑃21
where the ratio 𝑃21⁄ is the estimated error rate when prediction is based on the condition that 𝑃.1
a fund is a winner in the first period and P2. is the estimated error rate when prediction is based
on a fund’s unconditional position in the first period. The value of the ∇-statistic is the difference in errors between knowing the fund is a winner in the first period and not knowing the fund’s previous position. If being a winner in the past yields zero prediction errors, the ∇-statistic is equal to one, indicating a 100% increase in error reduction. In the specific case of Table IV, the ∇-statistic is 0.22, which means a 22% reduction in errors with the knowledge that a fund is a winner in the first period over not knowing this information. The estimated ∇-statistic can be normalized with the standard deviation, using the formula of Hildebrand et al. (1977: 200). This process is similar to the calculation of t-statistics.
Table IV
Cross-Classification of Being a Winner in the First and Second Periods at Five-Year Observation Intervals
This table reports the cross-classification of being a winner in the first and second periods at five-year observation intervals. The proportions of cells and columns and rows are displayed in parentheses.
2003–2007
Winners Non-Winners Total
2008–2012 Winners 33 45 78 (0.102) (0.139) (0.241) Non-Winners 48 198 246 (0.148) (0.611) (0.759) Total 81 243 324 (0.250) (0.750) (1.000)
winner in the first period predict being a winner in the second period. The same is the case for using information on being a loser in the past. However, in two other cases, past information significantly worsens prediction over no information. Thus, conclusion that being a winner or a loser in the past both can predict its future performance if persistence is present can be drawn from this test.
Table V
Summary of the Predictability Cross-Classification of Past Positive Performance for Future Performance
This table reports the predictability of being a winner in the first period for performance in the second period. The frequency counts simulate the process described in Table IV. The ∇-statistic can be considered a normally distributed variate, approximately. If the variate exceeds 1.96, the hypothesis of no effect is rejected at the 95% level of confidence.
Analysis Period Total
Number Frequency Counts per Cell Statistic
Table VI
Summary of the Predictability Cross-Classification of Past Negative Performance for Future Performance
This table reports the predictability of being a loser in the first period for performance in the second period. The frequency counts simulate the process described in Table IV. The ∇-statistic can be considered a normally distributed variate, approximately. If the variate exceeds 1.96, the hypothesis of no effect is rejected at the 95% of level of confidence.
Analysis Period Total
Number Frequency Counts per Cell Statistic
N11 N12 N21 N22 CRP ∇ 2003-2007/2008-2012 324 14 55 67 188 0.714 -0.051 (-1.015) (-0.972) 2003-2004/2005-2006 324 37 41 44 202 4.143 0.285 (5.054) (4.741) 2005-2006/2007-2008 313 4 68 74 167 0.133 -0.232 (-3.788) (-6.208) 2007-2008/2009-2010 289 32 36 40 181 4.022 0.274 (4.651) (4.284) 2009-2010/2011-2012 269 12 50 55 152 0.663 -0.067 (-1.147) (-1.167) 7. Conclusion
The primary focus of this thesis is the performance persistence of mutual funds. Conducting tests based on a contingency table to analysis persistence in mutual funds performance ranking over five- and two-year intervals, this study finds that pass five-year performance provides less valuable information for performance in the following five years. However, the probability of observing repeating performance pattern is 50–50 for an observation interval of two years. This phenomenon indicates that short-run persistence in mutual fund performance depends greatly on the period studied and therefore the economic environment.
performance in the first period. It is therefore difficult to conclude that persistence exists among winner or loser funds.
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Appendix I: Test of the Long-Term Performance Persistence of Funds with the Assumption of Negative Returns
This table reports the two-way contingency table of the repeating performance of mutual funds. Winners in each period are funds that rank in the top quartile according to their Sharpe ratio. Losers are funds ranked in the bottom quartile based on their Sharpe ratio. The intersection of winners in the initial years and winners in subsequent years consists of funds that were winners in 2003–2007 and remained winners in 2008–2012. The same principle is applied to the other categories. The CPR is calculated as (WW*LL)/(WL*LW). The Z-statistic is measured as the natural logarithm of the CPR divided by its standard error and is asymptotically normally distributed under the assumption of the independence of the observations. The Spearman rank correlation is the correlation between the ranks of all funds in the two sub-periods. The Z-statistic is calculated as the product of the Spearman rank correlation and the square root of the sample size minus one. The Z-statistic tests whether the CPR is statistically different from one and whether the Spearman rank correlation is statistically different from zero. Z-Statistics of 1.645, 1.960, and 2.575 correspond to significance at the 10%, 5%, and 1% levels, respectively. The superscripts ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
2008–2012 Winners Losers 2003–2007 Winners 23 10 Losers 14 7 CPR 1.150 Z-Statistic 0.234
Spearman Rank Correlation 0.122
Appendix II: Test of the Short-Term Performance Persistence of Funds with the Assumption of Negative Returns
This table reports the two-way contingency table of the repeating performance of mutual funds. Here winner–winner indicates the total number of funds that rank in the top quartile according to their Sharpe ratio in the first two-year period who are also in the top quartile in the next two-year period. Loser– loser represents the number of funds that rank in the bottom quartile according to their Sharpe ratio in the first two-year period who are also in the bottom quartile in the next two-year period. The same principle is applied to the other categories. The CPR is calculated as (WW*LL)/(WL*LW). The Z-statistic is measured as the natural logarithm of the CPR divided by its standard error and is asymptotically normally distributed under the assumption of the independence of the observations. The Spearman rank correlation is the correlation between the ranks of all funds in the two sub-periods. The Z-statistic is calculated as the product of the Spearman rank correlation and the square root of the sample size minus one. The Z-statistic tests whether the CPR is statistically different from one and whether the Spearman rank correlation is statistically different from zero. Z-Statistics of 1.645, 1.960, and 2.575 correspond to significance at the 10%, 5%, and 1% levels, respectively. The superscripts ***, **, and * indicate significance at the 1%, 5%, and 10% levels, respectively.
Year Total WW LW WL LL CPR Z-Statistic
