" Jurnal Reksadana dalam Bhs Inggris "

MUTUAL FUNDS PERFORMANCE PERSISTENCE: USING MARKOV TRANSITION PROBABILITY MATRIX


Wen-Tao Lee
Department of Finance, National Sun Yat-sen University
No.70, Lien-Hai RD. Kaohsiung City, 80424, Taiwan ROC
capm2000@ms63.hinet.net


ABSTRACT

We explore performance persistence in Taiwan’s mutual funds using absolute and relative performance. We use the Markov transition probability instant of traditional Spearman order correlation or winner-loser method. Our sample, largely free of survivorship bias, that the performance of mutual funds persists; however, persistence most due to relative performance. A probit analysis indicates that poor performance increases the probability of disappearance.

Keywords: Mutual Funds, Performance, Persistence, Transition Probability Matrix

1. INTRODUCTION

Since the beginning in 1986, there are now a total of close to 700 mutual funds raised in Taiwan by the end of 2006. At the end of 2006, 522 mutual funds still survived. Owing to great increase of number of mutual funds in Taiwan, a lot of investors are unable to select appropriate funds. Most of the investors take past performance of funds as the major reference. Starting from 1996, Securities Investment and Consulting Association of the R.O.C. published periodical ranks of funds issued by domestic investment trust companies . A lot of wealth management companies also offer return performance of mutual funds in the past. Yet, is the historical performance persistent? If the answer is yes, how strongly the persistence is ? Can investors take the past performance as the reference in investment decision making?

In earlier research on mutual funds performance persistence, both foreign and domestic empirical studies adopted dichotomy of winners and losers or grades of historical performance ranks in evaluation. The two methods do provide evidence of whether mutual funds performance is persistent. Nevertheless, dichotomy of winners and losers fail to provide investors with sufficient information. Coefficients of historical ranks show the relation between current and past performance of mutual funds. Investors still are unable to make investment decision with such information only.

To increase the reference value in decision making, the researcher expanded the dichotomy model of winners and losers by adding the number of divisions of mutual funds performance ranks. With estimation of Markov Chain Transition Probability matrix, the research provides richer transition probability matrix in funds performance ranks so that investors can have an overview of changes of mutual funds performance ranks in the past. Through Markov Chain joint test and likelihood ratio test, the research examines whether transition probability of mutual funds performance in the past obeys random and has clarified whether domestic mutual funds performance is persistent.

Earlier evaluation on funds performance persistence in Taiwan mostly ignored survivorship bias. For the first time, the research takes survivorship bias into consideration, making measurement of mutual funds performance persistence more accurate. Also, the research makes statistics of the performance of funds before liquidation or merge.

This research aims at 315 mutual funds raised in Taiwan and invested in domestic stocks between 1990 and 2006. Estimation of performance indexes each year is made through daily funds net value and dividends in the past years to verify the several common fund performance indexes are persistent. Such indexes include absolute return and three relevant indexes--Sharpe index, Jensen index and Treynor index. The empirical results show the three relevant indexes are significantly persistent while absolute return persistence is not very significant. Mutual funds with poorer performance are more likely to be liquidated or merged than those the historical average.
The major contribution of the research is the use of the most comprehensive and rich domestic mutual funds data to evaluate performance persistence of mutual funds raised in Taiwan and invested in domestic stocks. With transition probability matrix for estimation funds performance ranks in contingency tables, the paper provides more information than earlier studies using dichotomy, regression analysis or grades to serve as reference for investment decision making.
In addition to the foreword, the paper includes literature discussions and study methods in the second part, sample data and funds performance evaluation in the third part, empirical results of funds performance ranks stability in the fourth part, sub-sample test and performance of liquidated or merged funds in the fifth part and conclusions and suggestions in the sixth part.

2. LITERATURE REVIEW AND STUDY METHODS

2.1 Measurement of funds performance

Treynor (1965) adopted the idea of Security Market Line (SML) to propose the mutual funds model. The performance model is the index value of funds return after deduction risk free interest rate in excess return to be divided by unit system risk:
Treynor Indices (1)
Is the value from the market model?
Treynor index represents how many units of excess return a single unit system risk can have. Greater value means better performance of mutual funds allocation. Treynor (1965) made ranks of mutual funds through Treynor index and examined the performance was persistent with Spearman index. Carlson (1970) studied 57 mutual funds in the U.S. with not only Sharpe index but also Treynor index to evaluate funds performance. The conclusion also showed lack of performance persistency in funds market.

Sharpe (1966) developed the performance evaluation model based on Capital Market Line (CML). Different from Treynor, Sharpe believed mere consideration of system risk failed to include all risk. Impacts of total risks on performance must be taken into consideration. Estimation of Sharpe index is:

(2)is the average return; is the return variable; is average risk free return.
Sharpe index evaluation refers to the return of each unit total risk. Greater value means better performance. Sharpe (1966) studied 34 open mutual funds between 1944 and 1963 in the U.S. and made ranks with Sharpe index. Test was made through Spearman order correlation and the results showed funds in the two cycles were not in significant positive correlation. Carlson (1970) studied 57 funds in the U.S. by using Sharpe index and Treynor index. The conclusion showed lack of performance persistency in funds market.

Jensen (1968) proposed Market Model of absolute performance evaluation index to evaluate whether funds had abnormal return. The abnormal return means performance higher orlower the performance of benchmark allocation. Also, Jensen (1968), with this model, evaluated ability of fund managers to predict individual stock prices fluctuation and general pricing level of securities in the future. The measurement model is:

(3)is the return of funds in ; is the risk free return in ; is the return of benchmark allocation in ; is intercept of regression estimation, representing the excess return of the fund; is coefficient of regression estimation, representing the system risk of the fund; regression residual variance. As , fund performance exceeds performance of market benchmark allocation; when , fund performance in inferior to performance of market benchmark allocation. Therefore greater value means better fund performance.

2.2 Funds performance persistence examination

In study of funds performance persistence, some found persistency of mutual funds performance and some found evidence of persistent depended on the period. Grinblatt and Titman (1992) examined mutual funds performance from 1974 to 1987. The results showed funds performance was persistent in the long run.
Goetzmann and Ibbotson (1994) used Jensen index and regression to eamine whether funds performance was persistent. Empirical results showed, in most periods, performance was statisticcally remarkable. In absolute return and Jensen index and winner/loser performance persistent, the results showed winners in this period are 60% likely to be winners again in the next period. For funds performance persistence, funds with good performance in the past were persistent. Those with poor performance may continue to perform poorly. The results only could suggest investors avoid funds with poor performance. It still does not greatly help in selecting funds.

Brown and Goetzmann (1995) and Malkiel (1995) all pointed out funds performance persistence was related to sample periods. Most of the empirical results showed more apparent evidence of persistent performance of funds in 1970’s. Performance of funds in 1980’s was not persistent with evidence of reversed performance. Chiu Xianbi and Lin Qinpei (1999) believed better classification would show abilities of fund managers. Reverse of performance might be due to inaccruate classification.
Kahn and Rudd (1995) used regeression to review relation of performance of fixed income and stock funds in earlier and later periods. The results showed bond market had significant persistent. Different classification affects study of funds performance persistence.

Other than dichotomy of winners and losers or regression estimation, test can be made with absolute return or performance indexes in Spearman order correlation. For example, Williamson (1972) studied 180. funds between 1961 and 1970 in the U.S. After grade test, the conclusion supported lack of persistency in fund performance.
In Taiwan, Wu Jinting (1998) used multiple performance indexes and Spearman grade correlation. It was found not all domestic funds performance was persistent. Wong Shihuei (2002) found funds performance persistence was related to performance period. In annual performance, funds performance was not persistent. Xu Qinjun and Jiang Zhijiang (1993) used data from 1998 to 2002 to estimate persistency of different funds through Spearman rank rank-order correlation. The empirical results showed not each period had persistency. More classified funds had more significant performance persistency.

Chen Anlin, Hong Jialing and Lee Wenzhi (2001) also used Spearman order correlation to estimate persistency of performance ranks of 64 funds prior to 1999. The results show funds performance in persistency. Lin Xiuwei and Wang Jiazhen (2003) studied 248 domestic funds from 1994 to 2001 through traditional Spearman order correlation to estimate funds performance persistence under different indexes. The empirical results showed relevant performance indexes were not stabily persistent. Indexed related to CPAM beta were mmore persistent in teh long run.
Neither Sharpe (1966), Jensen (1968), Carlson (1970) and Williamson (1972) aborad nor domestic studies with Spearman order correlation failed to provide sufficient information. The dichotomy of winners and losers offer basic persistence probability to investors . However, such diviiosn of winners and losers fail to meet needs of investors. Although both Spearman order correlation and dichoromy can be used to examine whether funds performance is persistent, they provide too little information of funds performance persistence. The research seeks a bettter methods than the two in simplicity and direct perception to provide investors with more information on persistency of funds performance.

In Contingency Table by Carhart (1997), ranks of funds performance was divided into ten divisions. The author estimated year-by-year rank division transition probability matrix. Fig. II is transition probability matrix by Carhart. The method included more information than dichotomy and enabled iinvestors to have a better view of funds performance persistence. Carhart only estimatd funds with best and worst performance in investment allocation to see if they had excess return. Statistics test was not made on transition probability to see if rank division was random in past years.

Earlier studies on domestic funds performance often had insuffient samples. Survivorship bias was often ignored in funds performance persistence studies. This study is the first one to take liquidation or merge of domestic mutual funds into consideration. Therefore, there is no survivorship bias issue in measurement of funds performance persistence.

2.3 Markov Chain Transition Probability statistics test

The greatest benefit of dichotomy to evaluate funds performance is simplicity. It is relevantly simple in null hypothesis test. Use of more than two divisions in transition probability to evaluate funds performance persistence requires joint test of persistency of funds performance. Markov Chain by Anderson and Goodman (1957) can have statistics test of transition probability matrix of funds performance ranks to examine whether transference of ranks obeys random or annual ranks are non-random but persistent. Simple introduction to Markov Chain transition probability matrix is as follows. See Appendix I for inference of transition probability.
Presuming relevant performance of same type funds is random and obeys interrupted uniform distribution, in N funds, we divide into m grades of performance from large to small funds. For example, we divided funds performance ranks into 10 grades. The 1st grade includes funds in the first 10%; the 2nd grade covers funds ranked between the first 11% and 20%. The 10th grade contains funds ranked between 91% and 100%.

Based on annual funds performance, we gave grades. Annual funds performance grade transference matrix had states. If time has funds, any random and year maximum number of funds is . Presuming is fund in state; in , it belongs to state, under , random variable obeys asymptotically normal distribution and expected value is 0; variable is . Under funds performance state and specific , to provide null hypothesis , one must have:

(4)to obey incremental Chi-square distribution.
Freedom is . To test all ( ), one can sum up all . The results will obey freedom Chi-square distribution .

2.4 Probability test of likelihood ratio

Other than Markov process transition probability test, Kupiec (1995) presumed events in binominal distribution. Provided null hypothesis is event occurrence ratio = , through Likelihood Ratio (LR), one can examine whether occurrence ratio of events equals to null hypothesis. Measurement of LR is as follows:

(5)and statistics obeys liberty 1 Chi-square distribution. The advantage of likelihood ratio test is direct test on occurrence ration. Provided ranks of funds performance is random, likelihood ratio tests whether transition probability occurrence ration is the same as random null hypothesis to verify whether absolute ranks of funds performance is random.

3. SAMPLE DATA AND FUNDS PERFORMANCE EVALUATION

3.1 Sample source

The source in this research is mutual funds database of Taiwan Economic Journal (TEJ) covering 1990 to 2006. The researcher selected two types of funds--“raised in Taiwan and invested in domestic stocks” and “raised abroad and invested in domestic stocks” investing four targets small stocks, tech stocks, stocks and stocks/bonds. Table 1 lists overview of funds raised in Taiwan and investment target in past years. This research aims at 315 funds from the sample data.
Funds performance index is evaluated with net values on trading days of the year. If new funds were raised in January, they are included in our funds performance ranks. Those raised after January are included in the next year performance. Funds liquidated or merged in December are included in the performance ranks of the year. Those prior to December are not included in ranks and evaluation of the year. There are 315 funds between 1990 and 2006 meeting the sorting principle.
The risk free return is based on agent variable of risk free interest rate of one-year time deposit interest rate on the website of Central Bank of Republic of China (Taiwan). The Benchmark Index is weighted stock price index by Taiwan Stock Exchange.

3.2 Funds performance estimation

From simulation study of funds performance evaluation, Chiu Xianbi (1994) found Sharpe, Treynor, Jensen indexes and fund return before risk adjustment did not have significant difference in accuracy of distinguishing profitability of funds. The most widely used Sharpe, Treynor, and Jensen relevant performance indexes are also used in this study. Daily net values and dividends from TEJ are used to estimate daily return of funds in continuous time. The daily return is:

(6)is net value in t period.

is cash dividend issued in t period.
Treynor index, Sharpe index and Jensen index for measurement of relevant performance of funds are in formulas (1), (2) and (3):

3.3 Measurement of survivorship bias

Few domestic empirical studies of funds performance measured survivorship bias. To accurately measure whether funds performance is persistent, probability of fund liquidation or merge is considered. In the 315 funds, 74 were liquidated or merged during the sample period. To measure survivorship bias or probability of liquidation or merge, there a total 2,017 pieces of effective samples in the 315 funds between 1990 and 2006. Probability of funds to be liquidated or merged each year is 3.67%.

3.4 Statistics test and null hypothesis

For funds performance transition matrix, other than annual funds performance rank transition probability matrix, the research also includes new funds and liquidated or merged funds each year into transition probability matrix estimation. That is, funds raised this year are listed in the previous year as new funds. Those existing in the previous year but liquidated or merged this year are listed in liquidation division this year. Null hypothesis is set as funds performance obedience random. Fund ranks are not necessarily affected by ranks in previous year. In consideration of survivorship bias and presumption of funds performance rank obeying random, likelihood ratio test by Kupiec (1995) is null hypothesis as:

means probability of funds being liquidated or merged each year; is number of divisions of funds performance ranks. In ranks under two divisions, provided probability of liquadatoin or merge in the past , the likelihood ratio test null hypothesis is Pij=48.18%. In Markov Chain transition probability joint test, performance of funds in division in the previous year have the same probability of falling into each division in the next year. Probability of liquidation or merge ratio is . That is, null hypothesis is:

4. FUNDS PERFORMANCE RAND STABILITY EMPIRICAL RESULTS

4.1Performance persistent of two divisions

Goetzman and Ibbotson (1994), Brown and Boetzman (1995) and Malkiel (1995) used winner and loser dichotomy, which is also used here. See estimated winner and loser transition probability matrix in Table 2 and likelihood ratio test. Table 3 lists transition probability joint test of whether ranks of funds performance in the previous year obey random in the next year.
S1 refers to funds ranked 0～50% in the previous year; S2 shows funds ranked in the 2nd 50% in the previous year. From Table 2, one can see both absolute return and Sharpe, Jensen and Treynor indexes refuse transition probability as random null hypothesis . In the 4th column, funds ranked in the first 50% are less likely to be liquidated or merged than the historical average. The probability of their refusal to be liquidated or merged is the same at 3.46% of historical average in liquidation or merge. Those in the 2nd 50% are more likely to be liquidated or merged than historical average. The refusal to be liquidated or merged is also 3.46%.
Table 3 offers joint test of Markov Chain transition probability. Through test of transition probability matrix, no matter what the state of funds performance was in the previous year, all states refused transition probability equal null hypothesis, meaning transition of funds performance in the previous and next years is not random. Test of transition probability matrix where all probabilities are equal also refuses null hypothesis.

4.2 Performance persistence of ten divisions

Probability estimation and test in two divisions tell whether funds performance is persistent. The information is too simple. Carhart (1997) used funds performance Contingency Table. He divided absolute performance ranks of funds into ten divisions in the unit of one year to estimate funds abosulte performance rank transition probability. Through transition possibility table, one can better grab funds performance transition probability. With transition probability in contingency table of Carhart (1997), the research estimates funds performance index rank transition probability. With Markov Chain by Anderson and Goodman (1957), the paper tests transition probability, including the process of funds performance being random anmd not persistent and funds performancetransition probability in stationary Markov Chain.

With rank division method by Carhart (1997), funds performance is ranked from top to bottom in 10 divisions. Funds with performance in the first 10% are in the 1st division; those in the first 10%~20% are in the 2nd division. Those in the last 10% are in the 10th division. To begin, if funds performance is in random effect, it is not persistent. Each transition probability in transition probability matrix is equal. Providing is transition probability of division in the previous year and this year, after giving previsous year performance , Markov Chain joint test null hypothesis is 且 and Kupiec (1995) likelihood ratio test is .
Table 4 lists test results of transition probability matrix and likelihood ratio of the ten performance rank division in absolute return and Sharpe, Jensen and Treynor relevant performance indexes. The first colmn S1 to S10 refer to funds performance with ten quantile in the first to tenth divisions in the previous year. R1 to R10 in teh first row mean funds performance in the next year in ten quantile in the first to tenth divisions. Table 5 is joint test results of Markov Chaintransition probability matrix.

In Table 4 a. Absolute Performance, Funds S1 ~ S3 in teh first 30% in the previous year have transition probability to be still in the first 30% the next year. Likelihood ratio in S3 to R1 test refuse random null hypothesis. No matter in any area between S1 and R3 of the performance in the previous year, probability of funds performance the next year in the last division R10 refuses null hypothesis. This means, in absolute performance index, funds ranked in the first 30% in the previous year have lower probability to be inthe last division than random probability. In Table 4 a. Absolute Performance, funds in S10 are very likely to be in the last division R10 the next year, significantly refusing performance as random null hypothesis. In absolute return performance, although funds with good performance do not have significant persistency, funds with poor performance are very likely to keep the poor performance the next year.

In Table 4 b. Sharpe Index, funds with either good or poor performance have a lot of evidence of persistency. Funds with good performance have greater probability than random hypothesis to keep good performance, especially those with Sharpe index ranked in the first 10% in S1 division. They have as high as 16.8% probability to be in R1 the next year and 15.2% of S2 into R2. Funds with good performance in the previous year have lower probability than random hypothesis to be in the second half divisions. Example: funds in S1~S3 in the previous year have around 5.5% probability to be in the last 10% of R10 the next year. This is lower than random null hypothesis Pij = 9.645%. Table 4 c. Jensen Index and d. Treynor Index have similar performance as Sharpe index does. Funds with good performance in the previous year still have good performance the next year. Probability from S1 to R1 is over 15%. That from S1 to S2 to R10 is below or around 6%. Funds with poor performance as in S10 have probability as high as over 17% to be in the last 10% of R10 division the next year.
Brown and Goetzmann (1995) pointed out funds with poor performance were more likely to be terminated. Similar results are found in this resonance. In either absolute performance or relevant performance index, most of funds in S1～S3 refuse liquidation or merge equaling to historical average; those in S9～S10 are over 8% likely to be liquidated or merged the next year. Probability of refusal to be liquidated or merge equals to historical average.
Table 5 is the joint test of the 10 divisions. In the joint test of performance obedience random the next year, funds ranked in the first 10% only have Sharpe, Jensen and Treynor indexes significantly refusing random null hypothesis the next year. Absolute return performance fails to refuse transition probability non-random. Absolute return performance persistency is not distinctive. Funds with poor performance such as in the last 20% of S9～S10 refuse funds performance as random null hypothesis the next year. This means they are more likely to keep bad performance.

4.3 Classification sub-sample performance persistency

To understand whether nature of funds lead to different results, funds are divided into four sub-samples based on the investment target to reevaluate persistency of funds performance. The sub-samples are tech stocks, small stocks, stocks, and stocks + bonds. Tech stocks have 41 funds including 264 samples; small stocks and stocks have 260 stocks including 1,592 samples; stocks + bonds have 87 funds including 339 samples.
Tech stocks and small stocks do not cover many samples. With too many rank divisions in the sub-sample ttransition probability matrix, transition probability will have poor estimation representation and test performance of statistics is not high. As a result, only 3 divisions are used . S1~S3 means fund ranked in the first to third group in the previous year; R1~R3 refer to funds ranked in the first to third divisions the next year. The empirical results of tech stocks and small stocks in Table 5 show absolute performance transition probability matrix do not have distinctive transition probability different from random. On the contrary, new funds in small stocks had good performance in the first year. In absolute return, new funds are 51.9% likely to be in the first 1/3 ranks. Sharpe, Jensen and Treynor indexes show similar results. In either absolute performance or relevant performance indexes, the probability of funds in the first 1/3 of S1 division in the previous year to be in the first 1/3 of R1 division is not higher than random probability.

Empirical results of stocks sub-samples in Table 6 show absolute return performance is similar to Sharpe, Jensen and Treynor indexes. The four performance indexes have quite high persistency evidence. In absolute performance, funds in S1 in the previous year have 39.7% probability to be in the first 1/3 of R1 the next year and 21.8% probability to be in the last 1/3 rank. The two reach distinctive refusal probability of null hypothesis in statistics. This proves stocks funds with good performance have high persistency. In absolute performance, funds ranked in the last 1/3 in the previous year have only 22.1% probability to be in the first 1/3 rank the next year. They have 41.9% probability to be in the last 1/3 rank the next year. This shows stocks funds with poor performance in the previous year still tend to do poorly the next year.
In Table 6 of stock + bonds funds do not show evidence of persistence in absolute return. The threerelevant performance indexes do have high persistence. What is worth attention is funds of the four indexes ranked the last 1/3 of S3 in the previous year have probability of as high as 10% to be liquidated or merged than the historical average 3.46%.

Table 7 lists joint test of sub-samples empirical transition probability. Except for small stocks funds, the rest three sub-samples performance ranked in S1 or S3 mostly refuse randomnull hypothesis the next year. Funds with better performance in the previous year have grezter probability to perfrom well under good ranks. Those with poor performance under bad ranks have greater probability of liquidation or merge than thehistorical average. The probability of being in the poor ranks is higher than random hypothesis.

5. PERFORMANCE OF LIQUIDATED OR MERGED FUNDS

In the 229 fund samples in the study, 85 were liquidated or merged between 1990 and 2006. No estimation was available in earlier studies on performance of liquidated or merged funds. Scale and history of mutual funds in Taiwan already suffice statistics of liquidated or merged funds. The research made statistics of absolute return performance rank percentage of these 85 funds of the two years prior to liquidation or merge, or then rank of these funds in tow years, the previous year and the year of liquidation or merge with other funds. Table 8 shows average performance of these funds are in 65% of total fund market, which means 65% of the total funds over-performed the liquidated funds. Both liquidated or merged funds were defeated by over 70% of funds prior to liquidation or merge. Liquidated funds were defeated by 77.1% of funds in average; merged funds were beaten by 74.4% of funds in average.
In Table 8, merging funds had performance rank of 53% and 61.4% in the two years prior to and previous of the merge year in overall market. In the year of merging other funds, their average performance ranked at 57.9% of overall market. After merging other funds, these funds had performance rank at 54.3% and 52.6% in the 1st and 2nd year after merge. From such data, one can learn these merging funds had around the same performance of the overall market average. After merging other funds, the performance significantly improved or declined.

6. CONCLUSIONS AND SUGGESTIONS

This paper aims at funds raised in Taiwan and invested in domestic stocks between 1990 and 2006. Through Markov Transition Probability matrix, statistics is made to see if funds performance were persistent in the 17 years. To better estimate funds performance persistency, the research for the first time takes survivorship bias into domestic funds performance persistence consideration to reduce survivorship bias and more faithfully estimate funds performance ranks persistency.
Empirical results show that, in absolute return performance, persistence is lower. Sharpe, Jensen and Treynor relevant performance indexes have more distinctive persistence. Funds with good performance in the previous year had lower probability than historical average of liquidation or merge; on the contrary, those with poor performance will have greater probability than historical average of liquidation or merge.
In empirical results of tech stocks, small stocks, stocks and stock + bond funds, small stocks funds have less performance persistency. Stocks and stock + bond funds have more distinctive performance persistency. For stock + bond funds, if they performed poorly in the previous year, they had very high probability of liquidation or merge.
Statistics were made on performance of liquidated or merged companies prior to liquidation or merge. The performance was mostly below the average in the previous year or two years ahead. The performance in the year of liquidation or merge is below 70% of market rank. This means liquidated or merged funds were mostly those with poor performance. Before and after merging other funds, merging funds did not have great changes in performance.
In light of past performance of funds, performance did have certain persistency evidence, although it still fails to serve as the final single criterion for fund investment. What is clear is investors shall avoid funds with bottom performance in the same type, as they may continue perform poorly and have high probability of being liquidated or merged.

APPENDIX I: PERFORMANCE RANK MARKOV PROCESS

To give grades of each fund performance, each funds performance grade transition matrix has state. If in time there are funds, any and maximum number of funds is . Provided is fund at in state to in state, under of number of funds in state. When Individual funds is at time 0,1,…, is in states , ,…, . Given initial state , its performance rank state joint probability is:
(Annex 1)
Presuming transition probability is stationary, when ( ) means number of observation of in state and in state, will have a total of observation value; when is fund transition state after , ,…, number,
(Annex 2)
It is the sum of state and state. In dimension space, to describe all funds in initial state, funds transition state joint probability is:
(Annex3)
Statistics quantity of is in the preceding formula. According to Anderson and Goodman (1957), providing actual distribution is (Annex1) to be multiplied by a factorial function. Under the conditions of , , conditional probability distribution is:
(Annex4)
In which

This distribution will be the same. Providing number of observation is probability multi normal distribution and number of observation is and in conditional probability distribution:
(Annex5)
If Markov Chain transition probability is stationary transition probability, formula (Annex 1) can rewrite sufficient statistics. Therefore, if

Annex 3 can be rewritten as
(Annex6)
When transition probability is not required to be stationary, is the minimum assembly of sufficient statistics.
Stationary transition probability can be estimated with formula (9) maximum likelihood. Under probability axiom limit, all and
(Annex7)
Supposing transition probability distinctively has the same form and does inter-depend on state ( ), in sample, and, under probability, there is consistent multinomial trials distribution. Under such a sample, maximum likelihood estimate leads to:
(Annex8)
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Table 1: Mutual Funds Raised in Taiwan between 1990 and 2006 and Investment Target Classification
Investment target Raised and invested in Taiwan Raised in Taiwan and invested in domestic and foreign target Raised outside Taiwan and invested in Taiwan Total
Small stocks 29 1 30
Real property securitization 8 8
Real estate 8 1 9
Stocks 164 67 4 235
Stocks + bond 85 25 110
Principle guarantee 11 2 13
Index stocks 6 1 1 8
Tech stocks 41 10 51
Energy stocks 2 2
Funds 8 74 82
Currency 2 2 4
Bones 93 28 121
Asset securitization 5 5
Total 447 226 5 678
Remark: Data of fund raising and investment target are from TEJ database.
Table 2: Funds Performance Transition Probability Matrix in Two Divisions
1990~2006 ( R1 ) ( R2 ) ( C )
a. absolute return
( S1 ) 52.7*** 44.6** 2.67
( S2 ) 45.3* 48.8 5.98***
(New) 44.6 55.4** 0***
b. Sharpe index
( S1 ) 56.8*** 40.9*** 2.32**
( S2 ) 42.4*** 51.2* 6.33***
(New) 40.9** 59.1*** 0***
Jensen index
( S1 ) 53*** 44.5** 2.56*
( S2 ) 45* 48.9 6.1***
(New) 44.6 55.4** 0***
Treynor index
( S1 ) 56.4*** 41.2*** 2.32**
( S2 ) 42.6*** 51.1* 6.33***
(New) 41.6** 58.4*** 0***
Remark: Unit is %. Symbols *, ** and *** refer to refusal of the transition probability as (1-Pc)/2 null hypothesis at 10%, 5% and 1% significance levels in likelihood ratio test. Pc is past average of liquidation or merge. New means funds not raised in the previous year; C refers to next funds liquidated or merged. Statistics sample period is from 1990 to 2006.
Table 3: Joint Test of Whether Dichotomy Funds Performance Transition Probabilities Equal
Null hypothesis Absolute return Sharpe index Jensen index Treynor index
P(S1,R1)=P(S1,R2) 8.3*** 27*** 9.5*** 25.1***
P(S2,R1)=P(S2,R2) 14*** 23.9*** 15.5*** 23.6***
P(New,R1)=P(New,R2) 15.3*** 21.9*** 15.3*** 20.5***
All probabilities equal.等 37.6*** 72.9*** 40.3*** 69.1***
Remark:1. P (S1, R1) is the probability of being winners this and previous year; P (S1, R2) refers to probability of being the winners in the previous year and losers this year;; P (S2, R2) shows probability of being losers in the two years in a row; P (S2, R1) is the probability of being losers in the previous year and winners this year. 2 The last column null hypothesis refers to all transition matrix transition probabilities being equal. 。


Table 4: Funds Performance Transition Probability Matrix in Ten Divisions
1990~2006 ( R1 ) ( R2 ) ( R3 ) ( R4 ) ( R5 ) ( R6 ) ( R7 ) ( R8 ) ( R9 ) ( R10 ) ( C )
a. absolute return
( S1 ) 12.8 8.94 13.4 7.26 13.4 10.1 8.38 8.38 10.6 5.59** 1.12**
( S2 ) 8.19 13.5 11.1 7.6 8.77 12.3 11.7 7.02 12.9 5.26** 1.75
( S3 ) 14.5** 9.83 9.25 9.25 8.67 7.51 15** 12.1 7.51 5.2** 1.16**
( S4 ) 10.6 13.5 10.6 8.82 14.1* 6.47 11.8 7.06 5.88* 8.24 2.94
( S5 ) 8.93 11.9 10.7 10.7 7.14 7.14 8.93 11.9 8.33 7.74 6.55*
( S6 ) 9.25 12.7 9.83 9.83 9.25 6.36 13.3 8.67 11.6 5.78* 3.47
( S7 ) 9.83 7.51 12.1 7.51 9.25 11.6 8.09 15** 7.51 6.94 4.62
( S8 ) 11.8 4.71** 6.47 11.8 9.41 14.7** 8.24 9.41 11.2 6.47 5.88
( S9 ) 8.62 8.05 9.77 13.8* 8.05 9.2 12.1 8.05 4.02*** 10.9 7.47**
( S10 ) 5.52* 6.13 8.59 8.59 7.36 12.9 4.91** 7.36 11.7 18.4*** 8.59***
(New) 9.57 9.24 6.93* 9.9 8.91 9.24 6.6* 9.9 13.5** 16.2*** 0***
b. Sharpe index
( S1 ) 16.8*** 11.2 10.6 10.6 9.5 7.82 10.6 8.94 8.38 5.03** 0.56***
( S2 ) 9.94 15.2** 11.1 9.36 10.5 8.77 10.5 11.7 6.43 5.26** 1.17**
( S3 ) 10.4 10.4 10.4 8.09 13.3 12.7 11.6 9.83 4.05*** 6.94 2.31
( S4 ) 16.5*** 10.6 11.2 11.8 10 8.82 8.24 5.88* 5.88* 5.88* 5.29
( S5 ) 7.74 10.1 15.5** 14.9** 8.33 7.14 5.95* 11.9 9.52 6.55 2.38
( S6 ) 9.83 9.25 7.51 12.1 10.4 8.09 8.09 9.25 15** 6.94 3.47
( S7 ) 7.51 5.78* 8.09 8.67 9.83 12.7 13.9* 11 13.3 4.62** 4.62
( S8 ) 5.88* 10 10.6 11.8 10.6 12.4 8.82 10.6 7.06 6.47 5.88
( S9 ) 8.05 8.62 12.1 5.75* 8.62 9.77 12.6 8.05 6.32 11.5 8.62***
( S10 ) 6.75 4.91** 7.36 5.52* 6.13 9.2 8.59 7.36 16.6*** 18.4*** 9.2***
(New) 9.9 9.57 5.61** 7.92 7.92 9.57 8.58 10.2 12.9* 17.8*** 0***


Table 4 (continued)
1990~2006 ( R1 ) ( R2 ) ( R3 ) ( R4 ) ( R5 ) ( R6 ) ( R7 ) ( R8 ) ( R9 ) ( R10 ) ( C )
c. Jensen index
( S1 ) 15.1** 12.3 14* 7.26 9.5 12.3 7.26 8.94 6.7 6.15* 0.56***
( S2 ) 9.94 13.5 11.1 9.94 8.19 9.36 6.43 14.6** 10.5 4.68** 1.75
( S3 ) 12.1 8.67 11 7.51 10.4 7.51 13.9* 12.1 9.25 6.36 1.16**
( S4 ) 12.9 9.41 10.6 11.2 9.41 8.24 7.06 8.24 8.82 10.6 3.53
( S5 ) 10.1 7.74 11.9 13.7* 7.14 8.33 7.74 8.33 10.1 8.93 5.95
( S6 ) 8.67 12.1 9.25 11 10.4 8.67 7.51 9.25 10.4 9.83 2.89
( S7 ) 9.83 9.25 7.51 9.83 10.4 10.4 10.4 12.1 10.4 4.05*** 5.78
( S8 ) 9.41 8.82 8.82 10.6 9.41 14.7** 12.4 8.82 7.65 6.47 2.94
( S9 ) 6.32 10.9 7.47 8.62 9.77 11.5 10.9 8.05 4.6** 12.6 9.2***
( S10 ) 5.52* 5.52* 8.59 8.59 7.98 9.2 10.4 5.52* 11 17.8*** 9.82***
(New) 9.57 8.25 7.92 8.25 10.6 7.92 11.6 9.24 14.5*** 12.2 0***
d. Treynor index
( S1 ) 16.2*** 12.8 12.8 8.94 10.6 7.82 10.6 7.82 6.7 4.47*** 1.12**
( S2 ) 9.36 12.3 11.1 12.3 8.19 9.94 9.36 13.5 7.02 5.26** 1.75
( S3 ) 13.3 10.4 8.67 8.67 13.3 9.25 11 11 5.78* 6.94 1.73
( S4 ) 12.9 10 14.7** 10 8.24 12.9 8.24 6.47 5.29** 6.47 4.71
( S5 ) 9.52 11.9 11.9 13.7* 10.1 10.1 7.14 9.52 7.14 6.55 2.38
( S6 ) 8.09 8.67 12.7 9.25 12.1 8.09 5.2** 11.6 16.2*** 5.2** 2.89
( S7 ) 8.09 5.78* 7.51 9.25 10.4 13.3 13.3 11.6 11 5.78* 4.05
( S8 ) 4.71** 8.82 8.82 13.5 8.24 13.5 11.8 8.24 7.06 8.24 7.06**
( S9 ) 6.32 9.77 6.9 8.05 9.77 9.77 12.1 8.62 8.05 11.5 9.2***
( S10 ) 9.2 4.91** 9.2 7.36 4.91** 6.13 8.59 5.52* 17.8*** 17.8*** 8.59***
(New) 10.9 9.9 5.61** 6.6* 8.58 7.59 9.57 10.6 13.2** 17.5*** 0***
Remark: S1 through S10 refer to funds performance ranked to the 1st and 10th divisions in ten decimal; S1 has the best performance while S10 has the poorest performance. R1 through R10 refer to the first ten divisions of fund performance this year in. R1 has the best performance; R10 has the worst performance. New means funds not raised in the previous year; C means funds liquidated or merged the next year. Statistics sample period is from 1990 to 2006. Unit is %. Symbols *, ** and *** refer to refusal of the transition probability as (1-Pc)/10 null hypothesis at 10%, 5% and 1% significance levels in likelihood ratio test. Pc is past average of liquidation or merge.

Table 5 Joint Test of Whether Dichotomy Funds Performance Transition Probabilities Matrixes
Equal in the Ten Divisions
States absolute return Sharpe index Jensen index Treynor index
( S1 ) 15.4 20.1** 22.4** 23.3**
( S2 ) 14.4 15.2 15.8 12.2
( S3 ) 18.9* 13.3 12.6 11.7
( S4 ) 13.5 18.8* 4.7 16.5
( S5 ) 9.3 18.5* 9.2 9.3
( S6 ) 9.6 9.5 3.2 19.6**
( S7 ) 11.6 16.7 10.1 12.6
( S8 ) 17* 10.4 9.1 18.2*
( S9 ) 18.6* 20.9** 25.4*** 20.6**
( S10 ) 37.1*** 46.8*** 37.9*** 46.9***
( New ) 34.8*** 42.9*** 25.3*** 45***
(All) 200.2*** 233.2*** 175.7*** 235.8***
Remark: S1 through S10 refer to funds performance ranked to the 1st and 10th divisions in ten decimal in the previous year. S1 has the best performance; S10 has the worst performance. The last column (All) shows whether all transition probabilities are equal. Symbols *, ** and *** refer to refusal of the joint test transition probability of null hypothesis of Pij, j=1,2,…10 all equaling under previous states (S1~S10) at 10%, 5% and 1% significance levels in likelihood ratio test. Pc is past average of liquidation or merge. (All) means joint test in which probabilities in transition probability matrix equaling to (1-Pc)/10.


Table 6 Performance Transition Probability Matrix of Sub-samples
1990~2006 ( R1 ) ( R2 ) ( R3 ) ( C ) ( R1 ) ( R2 ) ( R3 ) ( C )
a. tech stocks b. small stocks
1.absolute return
( S1 ) 39.7 38.5 21.8* 0*** 36.7 34.7 26.5 2.04
( S2 ) 36 34.7 28 1.33 29.5 36.4 27.3 6.82
( S3 ) 25.4 26.8 42.3* 5.63 26.8 29.3 39 4.88
(New) 35 30 35 0** 51.9** 22.2 25.9 0
2.Sharpe index
( S1 ) 41* 39.7 19.2** 0*** 40.8 26.5 30.6 2.04
( S2 ) 34.7 38.7 26.7 0*** 27.3 36.4 29.5 6.82
( S3 ) 23.9 22.5* 46.5*** 7.04 26.8 36.6 31.7 4.88
(New) 37.5 27.5 35 0** 48.1* 25.9 25.9 0
3.Jensen index
( S1 ) 41* 37.2 20.5** 1.28* 36.7 30.6 30.6 2.04
( S2 ) 32 40 28 0*** 31.8 38.6 22.7 6.82
( S3 ) 25.4 23.9 45.1** 5.63 26.8 29.3 39 4.88
(New) 40 27.5 32.5 0** 48.1* 25.9 25.9 0
4.Treynor index
( S1 ) 44.9** 34.6 20.5** 0*** 36.7 32.7 28.6 2.04
( S2 ) 28 41.3* 30.7 0*** 31.8 34.1 27.3 6.82
( S3 ) 25.4 26.8 40.8 7.04 26.8 31.7 36.6 4.88
(New) 40 25 35 0** 48.1* 25.9 25.9 0
c. stocks d. stocks + bonds
1.absolute return
( S1 ) 39.7*** 32.5 25.8*** 1.97*** 39.8 30.1 25.8 4.3
( S2 ) 32.7 31 31.6 4.65 28.6 36.9 31 3.57
( S3 ) 29.6 30.7 32.3 7.4** 40 22.5* 25 12.5***
(New) 22.9*** 35.2 41.9*** 0*** 26.8 34.1 39 0***
2.Sharpe index
( S1 ) 39.5*** 36.5** 22.3*** 1.75*** 46.2*** 32.3 18.3*** 3.23
( S2 ) 33.4 29.6 32.7 4.2 33.3 32.1 29.8 4.76
( S3 ) 28.7 30.5 32.7 8.07*** 27.5 32.5 27.5 12.5***
(New) 23.7*** 30.5 45.8*** 0*** 26.8 26.8 46.3*** 0***
3.Jensen index
( S1 ) 39.7*** 32.3 26.4** 1.53*** 45.2*** 26.9 23.7* 4.3
( S2 ) 32.7 29.4 33 4.87 28.6 35.7 32.1 3.57
( S3 ) 28.5 32.1 31.8 7.62*** 33.8 31.2 22.5* 12.5***
(New) 25** 36 39** 0*** 26.8 30.5 42.7** 0***
4.Treynor index
( S1 ) 40*** 33.6 24.5*** 1.97*** 48.4*** 26.9 21.5** 3.23
( S2 ) 32.5 32.3 31 4.2 31 38.1 23.8 7.14
( S3 ) 28.3 31.6 32.3 7.85*** 26.2 32.5 31.2 10*
(New) 25.4** 28.8 45.8*** 0*** 28 26.8 45.1** 0***
Remark:1. S1 through S3 refer to divisions 1 to 3 in the 3 decimal of performance rank in the previous year. S1 has the best performance; S3 has the worst performance. R1 through R3 refer to divisions 1 to 3 in the 3 decimal of performance rank this year. R1 has the best performance; R3 has the worst performance. New funds not raised in the previous year; C means funds liquidated or merged the next year. Units are %. Symbols *, ** and *** refer to refusal of the transition probability as (1-Pc)/3 null hypothesis at 10%, 5% and 1% significance levels in likelihood ratio test. Pc is past average of liquidation or merge.
2. Tech stocks include 41 funds. In the sample period, there were 264 samples; small stocks have 29 funds and there were 161 samples in the sample period. Stocks contain 260 funds with 1,592 samples in the sample period. Stocks + bonds cover 87 funds of 339 samples in the sample period.

Table 7 Joint Test of Sub-samples
States absolute return Sharpe index Jensen index Treynor index
a. Tech stocks
( S1 ) 8.8** 11.2*** 7.9** 11.2***
( S2 ) 2.8 5.5 5.5 6.1*
( S3 ) 4.1 8.9** 6.4* 4.1
(New) 2.2 2.7 3 3.5
(All) 17.8** 28.3*** 22.7*** 24.9***
b. Small stocks
( S1 ) 1.7 2.5 1.2 1.3
( S2 ) 1 1 2.2 0.8
( S3 ) 1.1 0.6 1.1 0.6
(New) 5.8 4.1 4.1 4.1
(All) 9.6 8.3 8.6 6.8
c. Stocks
( S1 ) 21.9*** 33.5*** 23.3*** 25.3***
( S2 ) 0.2 1.4 1.1 0.5
( S3 ) 7.6** 12.2*** 9.4** 10.9***
(New) 25.6*** 30.6*** 19.8*** 29.4***
(All) 55.3*** 77.8*** 53.6*** 66.1***
c. Stocks + bonds
( S1 ) 3 11.9*** 7.9** 12.3***
( S2 ) 1.2 0.2 0.9 3.8
( S3 ) 15.2*** 11.1*** 12.5*** 5.5
(New) 6* 10.6*** 7.6** 9.5**
(All) 25.5*** 33.8*** 28.9*** 31.1***
Remark: S1 through S3 refer to divisions 1 to 3 in the 3 decimal of performance rank in the previous year. S1 has the best performance; S3 has the worst performance. (All) examines whether all transition probabilities are equal. Symbols *, ** and *** refer to refusal of the joint test transition probability of null hypothesis of Pij, j=1, 2, 3 all equaling under previous states (S1~S3) at 10%, 5% and 1% significance levels in likelihood ratio test. Pc is past average of liquidation or merge. (All) means joint test in which probabilities in transition probability matrix equaling to (1-Pc)/3.

Table 8 Average Ranks of Liquidated or Merged Funds
Classification Two years prior to liquidation One year before liquidation Liquidation/merge Number Average survival years
Liquidation 65.1% 65.5% 77.1% 33 4.09
Merged funds 61.5% 67.9% 74.4% 52 4.14

Merging funds performance Two years prior to liquidation One year before liquidation Liquidation/merge One year after liquidation Two years after liquidation
53% 61.4% 57.9% 54.3% 52.6%
Remark: As number of funds each year differed, rank is made in by percentage. Lower percentage means higher performance rank.