Multifactor Mutual Fund Performance Evaluation Based on the Panel Data Estimation

Transcription

1 6. Multifactor Mutual Fund Performance Evaluation 6 Multifactor Mutual Fund Performance Evaluation Based on the Panel Data Estimation Joanna Olbryś * 6.1. Introduction Fama (197) suggests that the portfolio manager s forecasting ability could be split into two separate activities: microforecasting (which is referred to in literature as selectivity) and macroforecasting (which is referred to in literature as market-timing). The traditional performance measurement literature has attempted to distinguish security selection, or stock-picking ability, from markettiming, or the ability to predict overall market returns. However, the literature finds that it is not easy to separate ability into such dichotomous categories. Fama and French (1993) find that two variables, market value (MV) and the ratio of book value to market value (BV/MV) capture much of the cross-section of average stock returns. They form portfolios meant to mimic the underlying risk factors in returns related to size and book-to-market equity. These mimicking portfolios (SMB and HML) have been introduced as explanatory variables into regressions of Polish equity mutual funds portfolios excess returns in Olbryś (010b). The market-timing and selectivity abilities of funds managers have been evaluated for the period January 003 December 009, using modified three-factor market-timing models with the Fama and French s spread variables SMB and HML. The Newey-West (1987) procedures HAC (heteroscedasticity and autocorrelation consistent covariance method) have been * Ph.D., Assistant Professor, Faculty of Computer Science, Bialystok University of Technology. Financial support in from the Polish Ministry of Science and Higher Education within the grant no. N N is gratefully acknowledged. 107

2 Joanna Olbryś used. In Olbryś (009) we have examined the usefulness of the conditional market-timing models for the investment managers performance evaluation. Ferson and Schadt (1996) use a collection of public information variables: (1) the lagged level of the one-month Treasury bill yield, () the lagged dividend yield of the CRSP value-weighted NYSE and AMEX stock index, (3) a lagged measure of the slope of the term structure, and (4) a lagged quality spread in the corporate bond market (Freson and Schadt 1996, p. 437). In Poland, the suitable variables are: (1) the lagged monthly dividend yield of the WSE stock index, () the lagged monthly level of the 1M WIBOR, (3) the lagged monthly measure of the slope of the term structure (Olbryś 009, p. 5). The evidence on Polish market shows that the quality of these models is rather low. The main goal of this chapter is to present a comparative analysis of funds performance evaluation based on the panel data estimation technique: the seemingly unrelated regression model (SUR). The purpose is an analysis of market-timing and selectivity skills of Polish equity open-end mutual funds managers using three-factor modified T-M-FF and H-M-FF models with additional Fama and French s variables. We compare the regression results of the models and investigate their statistical properties. According to the literature, the method most widely applied in market-timing models estimation is the one proposed by Newey-West in 1987 (e.g. Ferson and Schadt 1996; Prather and Middleton 006; Romacho and Cortez 006). Some previous publications also describe applications of the GLS procedure with correction for heteroscedasticity (e.g. Henriksson 1984; Henriksson and Merton 1981) or the Fama-MacBeth cross-sectional regression approach from 1973 (Carhart 1997). A few authors apply the Hansen s (198) Generalized Method of Moments (GMM). The literature search has not, however, revealed any example of applying the SUR method or another panel data estimation method. 6.. A Brief Literature Review As already mentioned, Fama (197) was the first to propose a formalized theoretical methodology for the decomposition of total returns into the components of timing and selectivity. Treynor and Mazuy (1966) develop a procedure for detecting timing ability that is based on a regression analysis of the managed portfolio s realized returns, which includes a quadratic term. Merton (1981) and Henriksson and Merton (1981) propose a theoretical structure that allows for the formal distinction of managers forecasting skills into timing and selectivity. By assuming that the market timer s forecasts take two possible predictions: either stocks will outperform bonds or bonds will outperform stocks, Merton (1981) derives an equilibrium theory that shows that 108

3 6. Multifactor Mutual Fund Performance Evaluation the returns pattern resulting from a market-timing strategy is similar to the returns pattern of an option strategy (of the put-protective type). Based on this model, Henriksson and Merton (1981) develop statistical procedures (both parametric and nonparametric tests) to investigate market-timing abilities of portfolios managers. Some others researchers develop models that allow the decomposition of managers performance into market-timing and selectivity skills. The majority of empirical studies seem to suggest that significant positive timing ability is rare Classical Market-Timing Models The T-M Model Treynor and Mazuy (1966) (T-M model) develop a procedure for detecting timing ability that is based on a regression analysis of the managed portfolio s realised returns, which includes a quadratic term, as follows (Olbryś 010b, p. 96): r P, t α P + β P rm, t + γ P ( rm, t ) + ε P, t =, (6.1) r P, t RP, t RF, t r M, t RM, t RF, where: = is the simple excess return on portfolio P in the period t, = t is the simple excess return on market portfolio M in the period t, R, is the one-period return on portfolio P, R, is the one-period P t return on market portfolio M, Jensen s portfolio P, P F t M t R, is the one-period return on riskless securities, α (Jensen 1968) measures the selectivity skills of the manager of β P is the systematic risk measure of portfolio P, market-timing skills of the manager of portfolio P, γ P measures the ε is a residual term, with the following standard CAPM conditions: E( ε, ) = 0, E( ε ε 1 ) 0. P t P, t P, t P, t = If the portfolio manager has the ability to forecast security prices, the intercept α P in equation (6.1) will be positive. On the one hand, a passive strategy can be expected to yield a zero intercept. On the other hand, if the manager is doing worse than a random selection buy-and-hold policy, α P will be negative. If a mutual fund manager increases (decreases) the market exposure of the portfolio prior to a market increase (decrease) then the portfolio return will be a convex function of the market return and γ P will be positive. The size of the estimate γˆ P informs about the manager s market skills. 109

4 Joanna Olbryś The H-M Model Henriksson and Merton (1981) (H-M model) assume that in each period the managers forecast whether stocks will outperform riskless bonds or vice-versa. They can choose between two target levels of systematic risk (Henriksson and Merton 1981, pp ): η 1 when they predict that riskless securities outperform the market, R M, t RF, t, η when they predict that the market outperforms riskless securities, R M, t > RF, t. Since the managers forecasts are not observable, the risk of the portfolio at time t, β t, should be a random variable for a market timer, assuming a value η 1 or η depending on whether the manager forecasts a down market or an up market. If the forecaster is rational, then η > η1. Under the assumption that beta is not observable, the return of portfolio P in period t is given by: R P, t RF, t + ( b + θ t ) rm, t + λ + ε P, t =, (6.) where: R P, t, R M, t, R F, t, r M, t, ε P, t are as in equation (6.1), b is the unconditional (on the forecast) expected value of β t, θ t = βt b is dependent on the forecast expected value of β t, λ is the expected excess return from selectivity. In this form, η 1 = b + θ t (6.3) is the target level of systematic risk when the forecaster predicts R M t RF, t, and η = b + θ t (6.4) is the target level of systematic risk corresponding to a forecast of R M, t > RF, t (Henriksson 1984, p.77). Simply put, the market-timer will want to: (1) hold a high-beta portfolio, when R M, t > RF, t, and () hold a low-beta portfolio when R M, t < RF, t (Sharpe et al. 1999, p. 850). 110

5 6. Multifactor Mutual Fund Performance Evaluation Using the return process described in (6.), LSR analysis can be used to estimate the separate contributions from security analysis and market-timing, as follows: r P, t α P + β P rm, t + γ P ym, t + ε P, t =, (6.5) where: r P, t, r M, t, α P, β P, γ P, ε P, t are as in equation (6.1), y = max{0, R R } = max{0, r }. M, t F, t M, t M, t Equation (6.5) is motivated by Merton s analysis of the value of markettiming. Merton (1981) shows that the returns obtained by a timing strategy would be similar to the returns obtained from an option investment strategy of the put protective type. In this way, αˆ P measures the contribution of security selection to portfolio performance, which corresponds to testing the null hypothesis: H : α 0 (6.6) 0 P = i.e., the manager does not have microforecasting ability (Romacho and Cortez 006, p. 354). The estimate βˆ P represents the proportion invested in the market portfolio when following the option investment strategy. The estimate γˆ P represents the number of free put options on the market due to the manager s market-timing skills. In this context, the evaluation of market-timing skills is carried out by testing the null hypothesis: H : γ 0 (6.7) 0 P = i.e., the manager does not possess any timing ability or does not act on his forecast ( η 1 = η ) (Henriksson 1984, pp ). A negative value for the regression estimate γˆ P would imply a negative value for market-timing (Olbryś 010b, p. 99) Three-Factor Market-Timing Models In 1994 Grinblatt and Titman showed that tests of mutual fund performance are quite sensitive to the chosen benchmark. For this reason, we run three-factor analogs of equations (6.1) and (6.5), in which the two new additional 111

6 Joanna Olbryś explanatory variables are Fama and French s SMB and HML factors. Performance evaluation in terms of a modified three-factor versions of the T-M or H-M models might allow a better assessment of manager s selectivity and timing skills (Olbryś 010b, p. 100) Fama and French s Spread Variables SMB and HML on the Polish Market The three fundamentals used by Fama and French are: (1) the overall market return, () the performance of small stocks relative to large stocks (SMB), and (3) the performance of value stocks relative to growth stocks (HML) (Tsay 010, p. 483). The size (SMB) and book-to-market (HML) mimicking portfolios on the Polish market have been constructed by Olbryś (010b), using the Fama and French s (1993) procedure. The 61 Warsaw Stock Exchange companies were entered into the database. The stock closing prices were obtained from Annual reports were obtained from (based on Notoria Service). We sort all firms according to the market capitalization at the end of June each year, beginning on June 8, 00. We take the market capitalization MV to be the number of shares as of the end of June (per WSE) multiplied by the end of June WSE share price. We also sort these same firms according to their end of calendar year book-to-market ratio BV/MV. In June of each year t from 00 to 009, all stocks were ranked according to the size of MV. The median size is then used to divide these stocks into two groups, S Small and B Big, where the big group includes all the firms with market capitalization greater than or equal to the median. Next we divide the stocks into three book-to-market equity groups based on the breakpoints for the bottom 30% (Low), middle 40% (Medium) and top 30% (High) of the ranked values of BV/MV for stocks. Proceeding in this way, we construct six portfolios for each year: BH, BM, BL, SH, SM, SL from the intersections of the two MV and the three BV/MV groups. The daily value-weighted returns on the six portfolios are calculated from July of year t to June of (t+1), and the portfolios were reformed in June of (t+1). In the next step we form the size portfolio SMB by taking the difference between an equally weighted combination of three small market capitalization indices and three big market capitalization indices (Olbryś 010b, p. 94). The rate of return on this portfolio is equal to (6.8): 1 RSMB = ( RSH + RSM + RSL RBH RBM RBL ). (6.8) 3 In the last step we form the book-to-market portfolio HML by taking the difference between an equally weighted combination of the two high book-to- 11

7 6. Multifactor Mutual Fund Performance Evaluation market indices and two low book-to-market indices. The rate of return on HML portfolio is equal to (6.9): 1 RHML = ( RBH + RSH RBL RSL ). (6.9) Three-Factor T-M Model With F&F Spread Variables (T-M-FF Model) In (Olbryś 010b, p. 100) the modified three-factor T-M-FF model has been expressed as: r P, t α P + β P rm, t + δ1p rsmb, t + δ P rhml, t + γ P ( rm, t ) + ε P, t =, (6.10) where: r P, t, r M, t, α P, β P, γ P, ε P, t are as in equation (6.1), = R R is the simple excess return on the portfolio SMB in the r SMB, t SMB, t F, t period t, r HML, t RHML, t RF, t in the period t, = is the simple excess return on the portfolio HML δ is the sensitivity measure of the returns on portfolio P to 1P changes in the SMB factor returns, δ P is the sensitivity measure of the returns on portfolio P to changes in the HML factor returns Three-Factor H-M Model With F&F Spread Variables (H-M-FF Model) In a way analogous to (10), Olbryś (010b) expressed the modified three-factor H-M-FF model as: r P, t α P + β P rm, t + δ1p rsmb, t + δ P rhml, t + γ P ym, t + ε P, t =, (6.11) r P, t, r M, t, r SMB, t, r HML, t, y M, t, equations (6.5) or (6.10), as appropriate. where: α P, β P, γ P, 1P δ, δ P, ε P, t are as in 6.5. Estimation Method Seemingly Unrelated Regression Model (SUR) A panel of data consists of a group of cross-sectional units that are observed over time. The number of cross-sectional units may be denoted by N and the number of time periods by T. There is one method of panel data estimation 113

8 Joanna Olbryś employed in this analysis: SUR technique. The acronym SUR stands for seemingly unrelated regression equations which was described by Zellner (196). SUR is a way of estimating panel data models that are long (large T) but not wide (small N). When estimating an SUR model, the data need to be arranged as a time series (not a panel) with different funds variables listed separately (Adkins 009, pp ). In the basic SUR model, the errors are assumed to be homoscedastic and linearly independent within each equation. The error of each equation may have its own variance. Each equation is correlated with the others in the same time period. The latter assumption is called contemporaneous correlation, and it is this property that sets SUR apart from other models (Adkins 009). Given that it is very likely that equity funds portfolios from the same market are contemporaneously correlated, the SUR model seems to be appropriate for this case. If contemporaneous correlation does not exist, the LSR method applied separately to each equation (fund s portfolio) is quite efficient (Olbryś 010a, p. 13) Data The creation of investment funds in Poland was made possible by the legislative act of March, The first balanced open-end mutual fund Pioneer was created in 199. It was the only open-end investment fund until 1995, when it was joined by the stable growth open-end mutual fund Korona. The first equity open-end mutual fund Pioneer was created in A proliferation of funds in Poland was made possible by the legislative act of August 8, For this reason, we have examined the performance of 15 selected equity open-end Polish mutual funds which were created up to the end of 00. We use daily data following Bollen and Busse s (001) evidence that daily data provide better opportunity for inference regarding timing ability than monthly data. This evidence has been examined in the case of Polish open-end equity mutual funds by Olbryś (008). We study daily ordinary excess returns from January 003 to December 009. Daily returns on the main index of the Warsaw Stock Exchange companies are used as the returns on the market portfolio. The daily average of returns on 5-week Treasury bills are used as the riskless asset returns. Daily rates of return on spread variables SMB (6.8) and HML (6.9) are used as the values of the additional exogenous variables in the T- M-FF (6.10) and H-M-FF (6.11) models. We have detected (based on the ADF test) that the analyzed series r M, t, r SMB, t and r HML, t are stationary. In the data panel the number of funds is equal to N=15 and the number of time periods is T=

9 6. Multifactor Mutual Fund Performance Evaluation 6.7. Empirical Results As mentioned above, in (Olbryś 010b) the market-timing and selectivity abilities of funds managers have been evaluated for the period January 003 December 009, using classic T-M (6.1) and H-M (6.5) market-timing models, as well as modified three-factor market-timing models T-M-FF (6.10) and H-M-FF (6.11), with the Fama and French s spread variables SMB and HML. Following literature, the robust HAC estimators have been used. Tables provide details on the estimated T-M-FF and H-M-FF market-timing models, respectively. The SUR method has been used for considering the contemporaneous correlation effects. In the case of all models and all funds, we have received the same estimator values as when using HAC method (Olbryś 010b, pp ), but the standard estimator errors have been smaller. Table 6.1. Three-factor T-M-FF model (6.10) (the period from January, 003 to December 31, 009) Equity funds ˆP α ˆP β ˆ δ 1P ˆ δ P ˆP γ R 1 Arka BZ WBK FIO Subfundusz Arka Akcji *** 0.730*** 0.070*** 0.046*** -.03*** 0.6 Aviva Investors FIO Subfundusz Aviva Polskich Akcji *** 0.751*** * -1.9*** BPH FIO Parasolowy Subfundusz BPH Akcji *** *** -0.96*** DWS Polska FIO Top 5 Małych Spółek *** 0.16*** 0.070*** -1.54*** DWS Polska FIO Akcji Dużych Spółek *** ** DWS Polska FIO Akcji Plus *** 0.09*** ** ING Parasol FIO Subfundusz Akcji *** ** -0.9** Legg Mason Akcji FIO *** *** -1.05*** Millennium FIO Subfundusz Akcji *** 0.033*** 0.050*** -1.08*** Novo FIO Subfundusz Novo Akcji *** 0.085*** *** Pioneer FIO Subfundusz Pioneer Akcji Polskich 0.814*** *** -1.5*** PKO Akcji - FIO *** 0.09* 0.08* -.19*** PZU FIO Akcji KRAKOWIAK *** *** -1.39*** Skarbiec FIO Subfundusz Akcji Skarbiec - Akcja *** 0.068*** UniFundusze FIO Subfundusz UniKorona Akcje *** 0.091*** * 0.31 Mean *Significant at 0.1; **significant at 0.05; ***significant at Source: Author s calculations (using Gretl 1.8.5). 115

10 Joanna Olbryś 116 Table 6.. Three-factor H-M-FF model (6.11) (the period from January, 003 to December 31, 009) Equity funds ˆP α ˆP β ˆ δ 1P ˆ δ P ˆP γ R 1 Arka BZ WBK FIO Subfundusz Arka Akcji 0.001*** 0.650*** 0.071*** 0.046*** -0.17*** 0.6 Aviva Investors FIO Subfundusz Aviva 0.001*** 0.67*** Polskich Akcji 0.01* -0.17*** BPH FIO Parasolowy Subfundusz BPH Akcji * 0.673*** *** -0.09*** DWS Polska FIO Top 5 Małych Spółek *** 0.17*** 0.071*** -0.1** DWS Polska FIO Akcji Dużych Spółek *** * DWS Polska FIO Akcji Plus *** 0.09*** ** ING Parasol FIO Subfundusz Akcji *** ** -0.09*** Legg Mason Akcji FIO ** 0.650*** *** -0.10*** Millennium FIO Subfundusz Akcji * 0.631*** 0.03*** 0.049*** -0.11*** Novo FIO Subfundusz Novo Akcji * 0.46*** 0.085*** ** Pioneer FIO Subfundusz Pioneer Akcji Polskich * 0.746*** *** -0.14*** PKO Akcji - FIO ** 0.467*** 0.09* 0.08* -0.19*** PZU FIO Akcji KRAKOWIAK ** 0.640*** *** -0.13*** Skarbiec FIO Subfundusz Akcji Skarbiec - Akcja *** 0.067*** UniFundusze FIO Subfundusz UniKorona Akcje * 0.475*** 0.091*** * 0.31 Mean *Significant at 0.1; **significant at 0.05; ***significant at Source: Author s calculations (using Gretl 1.8.5). Results of the T-M-FF tests (Table 6.1) show that the estimates of Jensen s measure of performance ( αˆ P ) are positive, but not significant for almost all of the funds, i.e., the null hypothesis (6.6) is not rejected. We can observe that in the case of H-M-FF models (Table 6.) ten out of fifteen funds reject the null hypothesis (6.6), i.e. they present a significant positive estimate of selectivity. According to Jensen s interpretation of αˆ P value, this measure could be positive for two reasons: (1) the extra returns actually earned on the portfolio due to the manager s ability, and () the positive bias in the estimate of αˆ resulting from the negative bias in estimate of βˆ P (Jensen 1968, p. 396). The levels of the systematic risk ( βˆ P ) are significantly positive and high. As for the influence of the size (SMB) and book-to-market (HML) factors, the results in Tables 6.1 and 6. are similar. As for the sensitive measure of the fund s portfolio P returns due to the changes in the SMB factor returns, only eight out of fifteen funds exhibit positive and statistically significant coefficients ˆ δ. The P 1P

11 6. Multifactor Mutual Fund Performance Evaluation spread variable HML is positive and statistically significant in the case of ten out of fifteen funds (coefficients ˆ δ P in Tables ). Unfortunately, the empirical results show no statistical evidence that Polish equity funds managers have outguessed the market. Almost all of the funds (except Skarbiec FIO Subfundusz Akcji Skarbiec Akcja in Tables ) reject the null hypothesis (6.7), but they present significantly negative estimates of market-timing skills ( ˆ γ P < 0 ). We find pronounced evidence of negative market-timing. Significant negative estimates of market-timing indicate that, contrary to what would be expected of rational investors, the managers increase the exposition of their portfolios to the market in down markets and act inversely in up markets (Romacho and Cortez 006). There is a statistically significant negative relationship between selectivity ( αˆ P ) and timing ( γˆ P ). With respect to the adjusted determination coefficient, the funds present different values of the R coefficient. The mean values of R are equal to 0.5 (in Tables ). To summarize, basing on the empirical analysis it can be concluded that in the case of all funds, we have received better models as when using robust HAC estimators (Olbryś 010b). The Breusch-Pagan (1980) Lagrange multiplier statistic was used to test for the existence of contemporaneous correlation among the cross-sectional units (Marshall and Young 003). Our aim is to test if the covariance matrix of disturbances in an SUR system is diagonal. The LM statistic is a natural candidate for this. The null hypothesis of diagonality involves OLS applied to each equation. The LM statistic for H 0 can be formed as (Breusch and Pagan 1980, p. 47): LM = T N i= 1 i 1 r ij j= 1 (6.1) where ij r = σ σ ij i σ j is the squared correlation computed based on the residuals from the least squares estimator, not SUR (Adkins 009, p. 18). The LM statistic will be distributed as χ ( ( 1)) N N if the null hypothesis of diagonality is correct. The results are equal to: LM = (the T-M-FF 1 model) and LM = 80.1 (the H-M-FF model), χ (105) = 19.9, α = Strong statistical evidence to reject the null hypothesis of no contemporaneous correlation was found, lending support to the use of the SUR model. 117

12 Joanna Olbryś 6.8. Conclusions The purpose of this chapterwas to evaluate the market-timing models of Polish equity mutual funds portfolios using the SUR method. We have received better models as when using robust HAC estimators (Olbryś 010b). The empirical results obtained using three-factor models do not support the hypothesis that mutual fund managers are able to follow an investment strategy that successfully times the return on the market portfolio. This evidence is consistent with most of the literature on mutual fund performance, e.g.: Treynor and Mazuy (1966), Henriksson (1984), Bollen and Busse (001), Prather and Middleton (006), Romacho and Cortez (006). A possible direction for further investigation would be the performance evaluation in terms of a hybrid multifactor market-timing model. References Adkins L. C. (009), Using Gretl for Principles of Econometrics, Version 1.31, July 0. Bollen N. P. B., Busse J. A. (001), On the Timing Ability of Mutual Fund Managers, The Journal of Finance, 56 (3), Breusch T. S., Pagan A. R. (1980), The Lagrange Multiplier Test and Its Applications to Model Specification in Econometrics, Review of Economic Studies, 47, Carhart M. M. (1997), On Persistence in Mutual Fund Performance, The Journal of Finance, 5 (1), Fama E. F. (197), Components of Investment Performance, The Journal of Finance, 7, Fama E. F., French K. R. (1993), Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, 33, Ferson W. E., Schadt R. W. (1996), Measuring Fund Strategy and Performance in Changing Economic Conditions, The Journal of Finance, 51, Grinblatt M., Titman S. (1994), A Study of Monthly Mutual Fund Returns and Performance Evaluation Techniques, Journal of Financial and Quantitative Analysis, 9, Henriksson R. (1984), Market Timing and Mutual Fund Performance: An Empirical Investigation, Journal of Business, 57, Henriksson R., Merton R. (1981), On Market Timing and Investment Performance. II. Statistical Procedures for Evaluating Forecasting Skills, Journal of Business, 54, Jensen M. (1968), The Performance of Mutual Funds in the Period , Journal of Finance, 3, Marshall B. R., Young M. (003), Liquidity and Stock Returns in Pure Order-Driven Markets: Evidence from the Australian Stock Market, International Review of Financial Analysis, 1, Merton R. (1981), On Market Timing and Investment Performance. I. An Equilibrium Theory of Value for Market Forecasts, Journal of Business, 54, Newey W. K., West K. D. (1987) A Simple, Positive Semi-Definite, Heteroskedasticity and Autocorrelation Consistent Covariance Matrix, Econometrica, 55, Olbryś J. (008), Data Frequency Effects Inference Regarding Market-Timing Ability of Mutual Fund Managers (in Polish), Studia i Prace Wydziału Nauk Ekonomicznych i Zarządzania Uniwersytetu Szczecińskiego, 10,

13 6. Multifactor Mutual Fund Performance Evaluation Olbryś J. (009), Conditional Market-Timing Models for Mutual Fund Performance Evaluation, Prace i Materiały Wydziału Zarządzania Uniwersytetu Gdańskiego, 4 (), Olbryś J., (010a), Orthogonalized Factors in Market-Timing Models of Polish Equity Funds, Quantitative Methods in Economics, 11 (1), Olbryś J. (010b), Three-Factor Market-Timing Models with Fama and French s Spread Variables, Operations Research and Decisions,, Prather L. J., Middleton K. L. (006), Timing and Selectivity of Mutual Fund Managers: An Empirical Test of the Behavioral Decision-Making Theory, Journal of Empirical Finance, 13, Romacho J. C., Cortez M. C. (006), Timing and Selectivity in Portuguese Mutual Fund Performance, Research in International Business and Finance, 0, Sharpe W. F., Alexander G. J., Bailey J. V. (1999), Investments, Upper Saddle Rive: Prentice Hall. Treynor J., Mazuy K. (1966), Can Mutual Funds Outguess the Market?, Harvard Business Review, 44, Tsay R. S. (010), Analysis of Financial Time Series, New York: John Wiley & Sons. Zellner A. (196), An Efficient Method of Estimating Seemingly Unrelated Regressions and Tests for Aggregation Bias, Journal of American Statistical Association, 57,

14 Joanna Olbryś 10