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Alternative Model to Evaluate Selectivity and Timing Performance of Mutual Fund Managers: Theory and Evidence Dr. Cheng Few Lee Distinguished Professor of Finance Rutgers, The State University of New Jersey Editor of Review of Quantitative Finance and Accounting Editor of Review of Pacific Basin Financial Markets and Policies Outline Part 1:Alternative Model to Evaluate Selectivity and Timing Performance of Mutual Fund Managers: Theory and Evidence 1. Introduction 2. Methodologies 3. Empirical Results 4. Conclusion 5. References Part 2: Potential problems of alpha and Performance Measurement A.Ferson’s Model (2008) B.Lee and Jen (1978) C.Chang, Hung and Lee (2003) Part 1 Alternative Model to Evaluate Selectivity and Timing Performance of Mutual Fund Managers: Theory and Evidence Introduction The investment of mutual funds has been extensively studied in finance. Early researchers (Treynor (1965), Sharpe (1966), and Jensen (1968)) employed a one parameter indicator to evaluate the portfolio - easily compare their performance by these estimated indicators. - assume the risk levels of the examined portfolios to be stationary through time. Fama (1972) and Jensen (1972) pointed out that the portfolio managers may adjust their risk composition according to their anticipation for the market. Fama (1972) suggested that the managers’ forecasting skills can be divided into two parts: the selectivity ability and the market timing ability. - selectivity ability (micro-forecasting): involving the identification of the stocks that are under- or over-valued relative to the general stocks. - timing ability (macro-forecasting): involving the forecast of future market return. Introduction (Cont.) Treynor and Mazuy (1966) used a quadratic term of the excess market return to test for market timing ability. - the extension of the Capital Asset Pricing model (CAPM). - If the fund manager can forecast market trend, he will change the proportion of the market portfolio in advance. Jensen (1972) developed the theoretical structure for the timing ability. Under the assumption of a joint normal distribution of the forecasted and realized returns, Jensen showed that the correlation between the managers’ forecast and the realized return can be used to measure the timing ability. - Bhattacharya and Pfleiderer (1983) extended Jensen’s (1972) work and used a simple regression method to obtain accurate measures of selectivity and market timing ability. - Lee and Rahman (1990) further corrected the inefficient estimated of parameters by a Generalized Least Squares (GLS) method. Introduction (Cont.) Henriksson and Merton (1981) used options theory, developed by Merton (1981), to explain the timing ability. In this paper, we empirically examined the mutual fund performance by using six models, proposed respectively by Treynor (1965), Sharpe (1966), Jensen (1968), Treynor and Mazuy (1966), Henriksson and Merton (1981), and Lee and Rahman (1990). In addition to examining the selectivity, timing, and overall performance, we also try to find some relationship between estimated parameters and the real investment. Methodologies It is well known that the return is not a sufficient indicator for valuing the performance, so it is necessary to consider its risk taken. - Markowitz (1952) was the first to quantify the link that exists between return and risk, and also built the foundation of modern portfolio theory. - Moreover, the Markowitz model contains the fundamental elements of the CAPM, which was also the basis for most of models adjusted in this paper. Treynor index Treynor (1965) uses the concept of the security market line drawn from the CAPM to get a coefficient β. Under the assumption of complete diversification of asset allocation, it means that we just have systematic risk measured by β. The Treynor index (TI) measuring the reward per unit of systematic risk for the portfolio can be showed as follows: r p rf TI , (1) p where r p is the average return of the pth mutual fund, and r f is defined as risk-free rate. The numerator of Treynor index can be viewed as excess return on the portfolio. This ratio is a risk-adjusted performance value. This indicator is suitable for valuing the performance of a well-diversified portfolio; this is because it just takes the systematic risk into account. Sharpe index Different from Treynor (1965), Sharpe (1966) argues the phenomenon that the fund managers will be in favor of fewer stocks. Therefore, it is impossible to diversify the individual risks completely. In other words, the excess return should be calculated based on the total risk (including systematic and nonsystematic risks). The Sharpe index (SI), applying the concept of the capital market line can be written as: r p rf SI , (2) p where p is the standard deviation of the portfolio, namely total risk. The Sharpe index is expressed as the reward per unit of total risk. The higher the two indices mentioned above, the better the fund’s performance. Because this measure is based on the total risk, it enables to measure the performance of the portfolio which is not very diversified. Jensen index Jensen (1968) proposes a regression-based view to measure the performance of the portfolio. The Jensen index (or called Jensen alpha) utilizes the CAPM to determine whether a fund manager outperformed the market. It’s formula is as follows: R p,t p p,t Rm,t u p,t , (3) where R p ,t and Rm,t are the excess returns ( Rt = rt r f ) at time t of the portfolio return and the market return, respectively. The term of u p ,t in the formula is the residual at time t. The coefficient p is used to measure the performance of mutual funds in the sense of the additional return due to the manager’s choice. It also represents the fund manager’s selectivity ability without considering timing ability. A significantly positive and high value of Jensen alpha indicates superior performance compared with the market index. Relationships between these three measures It should be noted that all three performance measures are interrelated. If pm pm / p m 1, then the Jensen index JI divided by p becomes equivalent to the Sharpe index SI. Since p pm / m and pm pm / p m , 2 the Jensen index must be multiplied by 1 / p in order to derive the equivalent Sharpe index: JI [rp r f ] [rm r f ] ( pm ) [ rp r f ] [ rm r f ] p p m m p p m . SI P SI m (commom constant) Relationships between these three measures (Cont.) If the Jensen index is divided by P , it is equivalent to the Treynor index TI plus some constant common to all portfolios: JI [rp rf ] [rm rf ] p TI p [rm rf ] TI p commom constant . p p p The types for the Treynor index and the Sharpe index are very similar. Based on a well-diversified portfolio, the p can be replace by p / m . Then the Treynor index can be written as TI (rp rf ) m / p SI / m . The measures for the Treynor index and Jensen alpha have the same criticism pointed by Roll (1977), the reference index. In addition, when considering a market timing strategy involving varying the beta according to anticipated movements in the market, the Jensen alpha often becomes negative and doesn’t reflect the true performance of the manager. Treynor-Mazuy model Treynor and Mazuy (1966), putting a quadratic term of the excess market return into equation (3), provide us with a better framework for the adjustments of the portfolio’s beta to test a fund manager’s timing ability. The fund manager with timing ability will be able to adjust the risk exposure from the market. To take a simple example, if a fund manager expects a coming up (down) market, he will hold a larger (smaller) proportion of the market portfolio. The equation can be given below: R p ,t p 1 Rm,t 2 Rm,t p ,t , 2 (4) where the coefficient 2 is used to measure the timing ability. When 2 is significantly larger than zero, it represents that, in a up (down) market, the increasing (decreasing) proportion in the risk premium of the mutual fund is larger than that in the market portfolio. This model was formulated empirically by Treynor and Mazuy (1966). It was then theoretically validated by Jensen (1972), and Bhattacharya and Pfleiderer (1983). Henriksson-Merton Model Henriksson and Merton (1981) used options theory to explain the timing ability. It consists of a modified version of the CAPM which takes the manager’s two objectives into account, and depends on whether he forecasts that the market return will or will not be better than the risk-free asset return. They view the coefficient as a binary variable. This means that a fund manager with market timing ability should have different values in the up and down markets. We can express the equation as: R p,t p 1 Rm,t 2 Max(0,Rm,t ) p,t . (5) If 2 0 , this shows that the manager has the ability to forecast a market to be down or up. For an up market (a down market), the equation (5) can be expressed as Rp,t p 1 Rm,t p,t ( Rp,t p (1 2 ) Rm,t p,t ). Jensen’s (1972) Viewpoint Jensen (1972) showed that the timing ability can be measured by the correlation between the managers’ forecast and the realized return. Bhattacharya and Pfleiderer (1983) modify Jensen’s (1972) model to propose a regression-based model to evaluate the market timing and selectivity abilities. Assume that t Rm,t E( Rm ) , where E ( Rm ) is the unconditional mean of Rm,t . Moreover, we assume t* is the conditional expected value of t , expressed as E ( t | t ) ) referring to the expected value of t under t , the manager’s information set prior to time t. Then the relationship between t* and t is t* ( t t ) . For minimizing the variance of the forecasting error, we get the optimal value of /( 2 ) , 2 2 representing the ratio of the forecasted change to the realized change in the market information. This ratio also reflects their relevance. min E[ t ( t t )]2 min E[ t2 2 t ( t t ) 2 ( t t ) 2 ] min E ( t2 2 t2 2 t t 2 t2 2 2 t t 2 t2 ) min E ( t2 2 t2 2 t2 2 t2 ) ( E ( t ) E ( t ) E ( t t ) 0 ) min E ((1 ) 2 t2 2 t2 ) min (1 ) 2 2 2 2 In the market equilibrium, the systematic risk coefficient βp,t is the same as the target risk coefficient βp,T. However, if the fund manager owns other specific information, this will make t and t* different. Then, the systematic risk βp,t of the portfolio will be effected by both t* and the fund manager’s sensitivity θ with specific information t . The relationship can be expressed as p ,t p ,T t* . Suppose that the target systematic risk of mutual funds is decided by the reaction for the net market return, i.e. βp,T = θE(Rm). We can rewrite the Jensen’s (1972) model (equation (3)) as: R p,t p {E( Rm ) [ Rm,t E( Rm ) t ]}(Rm,t ) u p,t . (6) Rearranging the equation (6), we get: R p ,t p E ( Rm )(1 ) Rm,t ( Rm,t ) 2 t Rm,t u p ,t . (7) It also can be shown as: R p ,t 0 1 Rm,t 2 ( Rm,t ) 2 t . (8) In a large sample setting, the estimated coefficients are: plim 0 p , plim 1 E ( Rm )(1 ) , (9) plim 2 . Bhattacharya and Pfleiderer (1983) also use the term p to evaluate the selectivity ability, 2 but use the correlation ( 2 2 ) between forecasted and real values of the 2 excess market return to measure the market timing ability. In short, is just decided by 2 and . The first term, 2 , can be calculated by t , i.e. t t Rm,t u p,t . We can 2 regress t2 on Rm ,t , 2 t2 2 2 2 ( Rm,t ) 2 t , (10) where t 2 2 ( Rm,t ) 2 ( t2 2 ) (u p ,t ) 2 2 ( Rm,t ) t u p ,t . Therefore, we can get 2 2 2 2 2 2 2 2 2 . 22 In addition, in the assumption of the stationary Wiener process of t , Merton (1980) proposes a simple estimator for without estimating expected return in advance, i.e. 2 1 n [ln(1 Rm,t )]2 . When (always positive) is significantly different from zero, it 2 n t 1 means that the fund manager can forecast the market trend. Lee-Rahman Model Lee and Rahman (1990) find that the residual terms of the above two equations ((8) and (10)) exists heteroscedasticity. This results the coefficients, 0 , 2 2 2 , and 2 , estimated from OLS are not efficient. The way to solve this problem is to calculate the variance of the residuals, t and t . They can be shown as: 2 2 2 ( Rm,t ) 2 u2 , 2 (11) 2 2 4 4 4 2 u4 4 2 2 2 ( Rm,t ) 2 u2 , (12) where u2 is the variance of the residual u as defined in equation (3). Using 2 and calculated earlier, we can get the variances and 2 . In order to get efficient 2 2 estimates, we utilize a GLS method with correction for heteroscedasticity to adjust the weights of the variables in equation (8) and (10) by and 2 . 2 Data We utilize different methods to examine the selectivity, market timing, and overall performance for the open-end equity mutual funds. The samples used in this study were the monthly returns of the 628 mutual funds ranging from January, 1990 to September, 2005, 189 monthly observations. The fund data were obtained from the CRSP Survivor-Bias-Free US Mutual Fund Database. Then, we use ICDI’s fund objective codes to sort the objectives of the mutual funds. In total, there are 23 types of mutual fund objectives. To simplify the empirical process, we just divide them as two groups, growth funds and non-growth funds. Finally, our empirical study consists of 439 growth funds and 189 non-growth funds. In addition to the CRSP fund data, the S&P 500 stock index obtained from Datastream is used for the return of the market portfolio. Moreover, we use the Treasure bill rate with a 3-months holding period as the risk-free return. The Treasury bill rate is available from the website of the Federal Reserve Board. Figure 1: The scatter between fund and market excess returns. This Figure shows the relationship between the growth and non-growth fund excess returns (net of risk-free rate), and the market excess return (S&P 500 index). Panel A: Growth Fund Excess Return V.S. Market Excess Return Excess Equal-weighted Return for the Growth Funds .20 .15 .10 .05 .00 -.05 -.10 -.15 -.20 -.20 -.10 .00 .10 .20 Excess Market Return Panel B: Non-growth Fund Return V.S. Market Excess Return Excess Equal-weighted Return for the Non-growth Funds .20 .15 .10 .05 .00 -.05 -.10 -.15 -.20 -.20 -.10 .00 .10 .20 Excess Market Return The fund return process is built by the equal weighted average of the funds in the same group. However, the difference of the scatter plots for the growth funds (Panel A) and the non-growth (Panel B) is small. Both of them can not display a convex relationship between the fund excess return and market excess return. From the previous explanation for it, a convex relationship represents that the fund has the market timing ability. Obviously, we need to take better look at the performance of the individual funds in the growth and non- growth funds. For the overall statistics, on average the growth funds have a high mean (0.0082) of the monthly returns than that of the non-growth funds (0.0072). Moreover, the means of the two groups are higher than the market (0.0066). Figure 2: S&P 500 stock index and treasure bill rate This figure shows the graphs for the market index (S&P 500) and risk-free rate (T-bill rate) in this study. S&P 500 Stock Index Treasury Bill Rate 1,600 8 1,400 7 1,200 6 5 1,000 4 800 3 600 2 400 1 200 0 1990 1992 1994 1996 1998 2000 2002 2004 1990 1992 1994 1996 1998 2000 2002 2004 It is interesting to compare the fund performance for the different market situations. We cut the entire sample into two subperiods by the end of 1997. As shown in Panel A of Figure 2, the former subperiod represents an obvious upward tendency, but the latter one does not have a clear trend. The fund performance between these distinct samples will give us a more informative and meaningful inference for our empirical study. Table 1: Mutual fund performance measured by Treynor index Objective Mean Std. Dev. Max Min Over the Market 1990:01 – 2005:09 Market Treynor index=0.0032 Growth 0.0052 0.0025 0.0193 -0.0024 360 (82.0%) Non-growth 0.0045 0.0070 0.0221 -0.0609 131 (69.3%) 1990:01 – 1997:10 Market Treynor index=0.0061 Growth 0.0082 0.0023 0.0230 -0.0039 388 (88.4%) Non-growth 0.0027 0.0196 0.1144 -0.1210 110 (58.2%) 1997:11 – 2005:09 Market Treynor index=0.0004 Growth 0.0024 0.0034 0.0154 -0.0082 302 (68.8%) Non-growth 0.0053 0.0059 0.0291 -0.0108 167 (88.4%) This seems to point out the growth funds are more valuable to be invested than the non-growth funds. For the subperiod 1990-1997, the apparent bull market strengthens the growth funds’ performance, but weakens the non-growth funds’. Compared with those in the subperiod 1997-2005, the market is not clear. The non-growth funds even have more outstanding performance than the growth funds. Clearly, the results indicate that the non-bull market has a negative effect on the growth funds. Table 2: Mutual fund performance measured by Sharpe index Objective Mean Std. Dev. Max Min Over the Market 1990:01 – 2005:09 Market Sharpe index=0.0774 Growth 0.1037 0.0439 0.2695 -0.0455 326 (74.3%) Non-growth 0.0824 0.0606 0.2860 -0.1296 115 (60.8%) 1990:01 – 1997:10 Market Sharpe index=0.1712 Growth 0.1976 0.0505 0.3631 -0.0327 321 (73.1%) Non-growth 0.1148 0.1116 0.3933 -0.2355 67 (35.4%)) 1997:11 – 2005:09 Market Sharpe index=0.0088 Growth 0.0392 0.0546 0.1958 -0.1429 300 (68.3%) Non-growth 0.0621 0.0544 0.2644 -0.0993 164 (86.8%) Table 3: Mutual fund performance measured by Jensen index Std. Objective Mean Max Min p 0 p 0 Dev. 1990:01 – 2005:09 Growth 0.0017 0.0020 0.0080 -0.0069 360 (108) 79 (1) Non-growth 0.0013 0.0029 0.0087 -0.0118 131 (22) 58 (1) 1990:01 – 1997:10 Growth 0.0020 0.0020 0.0101 -0.0104 388 (92) 51 (1) Non-growth -0.0002 0.0043 0.0117 0.0220 109 (16) 80 (2) 1997:11 – 2005:09 Growth 0.0015 0.0029 0.0084 -0.0125 302 (48) 137 (7) Non-growth 0.0030 0.0031 0.0104 -0.0055 167 (10) 22 (0) Most of the funds have positive Jensen alphas, especially for the growth funds of the subperiod 1990-1997 (88%) and the non-growth funds of the subperiod 1997-2005 (88%). As for the significance of the coefficient alpha, it depends on the investment length and the market trend. For the best of them, 30% of the growth funds for the entire period are significantly larger than zero with 95% confidence. In addition, for the growth funds of the subperiod 1997-2005, it has lower proportion than the non-growth funds and even 7 funds have significantly negative alpha values. Table 4: Measured by the Treynor-Mazuy’s model Panel A: Selectivity ability Objective Mean Std. Dev. Max Min p 0 p 0 1990:01 – 2005:09 Growth 0.0035 0.0029 0.0116 -0.0057 402 (169) 37 (0) Non-growth 0.0042 0.0029 0.0114 -0.0084 182 (49) 7 (0) 1990:01 – 1997:10 Growth 0.0027 0.0030 0.0130 -0.0163 375 (111) 64 (1) Non-growth 0.0023 0.0042 0.0131 -0.0182 152 (28) 37 (0) 1997:11 – 2005:09 Growth 0.0042 0.0042 0.0157 -0.0129 376 (96) 63 (4) Non-growth 0.0066 0.0054 0.0218 -0.0039 184 (44) 5 (0) Panel B: Timing ability Std. Objective Mean Max Min 2 0 2 0 Dev. 1990:01 – 2005:09 Growth -1.0379 1.1441 1.2844 -5.4791 68 (1) 371 (151) Non-growth -1.6293 1.4283 1.0747 -6.3699 23 (0) 166 (71) 1990:01 – 1997:10 Growth -0.6109 1.5554 4.6919 -6.1783 160 (21) 279 (56) Non-growth -1.9597 1.9150 3.7024 -7.9143 23 (1) 166 (50) 1997:11 – 2005:09 Growth -1.2081 1.2509 1.5759 -6.5961 60 (1) 379 (90) Non-growth -1.6374 1.7949 1.4023 -7.7479 28 (0) 161 (40) The growth funds in the subperiod 1990-1997 have little evidence of the timing ability, only sixteen of them have significantly positive estimates with 95% confidence. Except for these, the other results show no timing ability for nearly all funds. In fact, over 80% of them have negative values of timing ability and a considerable ratio among them have significantly negative estimates. Moreover, no one exhibits significantly positive estimates in both subperiods. For the selectivity ability, a very high ratio of the funds has positive estimates and many of them are significantly positive. None of the funds have significantly positive or negative estimates of selectivity or timing ability in both periods. For the correlations between the estimates of timing and selectivity ability, they are -0.62 for the entire period, -0.44 for the subperiod 1990-1997, and -0.77 for the subperiod 1997- 2005. Compared with Table 3, with considering the timing ability, the estimates for the selectivity ability have higher values. This is also consistent with Grant (1977) and Lee and Rahman (1990). Table 5: Measured by the Henriksson-Merton’s model Panel A: Selectivity ability Objective Mean Std. Dev. Max Min p 0 p 0 1990:01 – 2005:09 Growth 0.0049 0.0042 0.0173 -0.0108 404 (152) 35 (0) Non-growth 0.0069 0.0043 0.0228 -0.0062 186 (61) 3 (0) 1990:01 – 1997:10 Growth 0.0035 0.0045 0.0152 -0.0280 356 (102) 83 (1) Non-growth 0.0052 0.0051 0.0196 -0.0141 170 (42) 19 (0) 1997:11 – 2005:09 Growth 0.0060 0.0058 0.0212 -0.0091 381 (82) 58 (0) Non-growth 0.0094 0.0085 0.0382 -0.0054 175 (44) 14 (0) Panel B: Timing ability Objective Mean Std. Dev. Max Min 2 0 2 0 1990:01 – 2005:09 Growth -0.1956 0.2283 0.3341 -1.1735 75 (2) 364 (110) Non-growth -0.3435 0.3063 0.2302 -1.4967 23 (0) 166 (67) 1990:01 – 1997:10 Growth -0.1131 0.2563 1.2541 -0.9976 156 (13) 283 (45) Non-growth -0.3807 0.3782 0.7033 -1.6574 22 (1) 167 (55) 1997:11 – 2005:09 Growth -0.2442 0.2729 0.4193 -1.3411 74 (0) 365 (49) Non-growth -0.3494 0.4187 0.4233 -1.9448 36 (0) 153 (39) In general, the results are similar with those in Table 4. The estimates of the selectivity ability even display larger values. Furthermore, the correlations between the estimates of timing and selectivity ability are -0.85 for the entire period, -0.76 for the subperiod 1990-1997, and -0.90 for the subperiod 1997- 2005. They show a more considerable relation than the previous one. Table 6: Measured by the Lee-Rahman’s model Panel A: Selectivity ability Objective Mean Std. Dev. Max Min p 0 p 0 1990:01 – 2005:09 Growth 0.0033 0.0028 0.0109 -0.0062 399 (150) 40 (0) Non-growth 0.0041 0.0026 0.0101 -0.0080 183 (48) 6 (0) 1990:01 – 1997:10 Growth 0.0028 0.0030 0.0133 -0.0165 377 (123) 62 (1) Non-growth 0.0029 0.0042 0.0133 -0.0180 156 (34) 33 (0) 1997:11 – 2005:09 Growth 0.0038 0.0039 0.0156 -0.0086 361 (77) 78 (0) Non-growth 0.0061 0.0050 0.0201 -0.0048 178 (37) 11 (0) Panel B: Timing ability Objective Mean Std. Dev. Max Min 1990:01 – 2005:09 Growth 0.0816 0.0510 0.2274 0.0000 439 (36) Non-growth 0.0982 0.0621 0.2409 0.0069 189 (41) 1990:01 – 1997:10 Growth 0.1086 0.0732 0.4081 0.0007 439 (24) Non-growth 0.1442 0.0745 0.2896 0.0034 189 (19) 1997:11 – 2005:09 Growth 0.0997 0.0612 0.2599 0.0002 439 (7) Non-growth 0.1118 0.0672 0.2975 0.0004 189 (12) The result for the selectivity ability is very similar with that in Table 4. Because the Lee-Rahman model assumes no negative timing ability, it is worth it to discuss the result of it. The non-growth funds have better timing ability than the growth funds for all periods. Different from the previous two models, the correlations between the estimates of timing and selectivity ability are 0.43 for the entire period, 0.24 for the subperiod 1990-1997, and 0.45 for the subperiod 1997-2005. All of them are positive. Moreover, forty of the funds in the entire period have significantly positive estimates in both selectivity and timing ability. Nevertheless, none of the funds have significantly positive estimates in both selectivity and timing ability in both subperiods. Table 7: Return on the initial investment of $ 1.00 1990:01 – 2005:09 1990:01 – 1997:10 1997:11 – 2005:09 Growth Non-growth Growth Non-growth Growth Non-growth Mean 4.9902 4.5626 3.0965 2.5022 1.6080 1.8503 Std. Dev. 1.7289 2.5958 0.6060 1.3274 0.4373 0.5418 Max 13.1251 14.4838 5.4613 8.4894 2.9991 3.5658 Min 1.0856 0.2307 1.0115 0.1758 0.4177 0.7859 Market portfolio 3.4771 2.5881 1.3435 Risk-free securities 1.8921 1.4646 1.2919 Fund return greater 362 (82.5%) 119 (63.0%) 363 (82.7%) 71 (37.6%) 304 (69.2%) 163 (86.2%) than market portfolio Fund return greater than risk-free 436 (99.3%) 174 (92.1%) 437 (99.5%) 155 (82.0%) 331 (75.4%) 168 (88.9%) securities Figure 3: Growth and non-growth mutual fund returns with initial investment $1.0 for the entire period and two subperiods. Panel A: Mutual fund returns with initial investment $1.0, 1990:01 – 2005:09 Return of growth funds Return of non-growth funds 70 24 60 20 Number of funds Number of funds 50 16 40 12 30 8 20 4 10 0 0 0 2 4 6 8 10 12 14 0 2 4 6 8 10 12 14 16 Panel B: Mutual fund returns with initial investment $1.0, 1990:01 – 1997:10 Return of growth funds Return of non-growth funds 120 24 100 20 Number of funds Number of funds 80 16 60 12 40 8 20 4 0 0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 0 1 2 3 4 5 6 7 8 9 Panel C: Mutual fund returns with initial investment $1.0, 1997:11 – 2005:09 Rerurn of growth funds Return of non-growth funds 50 24 40 20 Number of funds Number of funds 16 30 12 20 8 10 4 0 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Table 7 shows the growth funds and the non-growth funds have very different performance with the market. In the subperiod 1990-1997, 83% of the growth funds perform better than the market, but only 38% of the non-growth funds do that. In the subperiod 1997-2005, 69% of the growth funds still perform better than the market. However, for the non-growth funds, 86% of them are better than the market. On average, for the initial investment $ 1.0 at the beginning of 1990, the growth funds and the non-growth funds will get $ 5.0 and $ 4.6 in the end, respectively. Both of them are more than the market ($ 3.5) and the risk- free asset ($ 1.9). How about the forty funds with significantly positive estimates in the selectivity and timing ability? The mean of the final amount for them is only $ 5.1, very close to the overall mean. From our empirical study, we seem to be able to conclude that the selectivity and timing abilities are not the key factors to decide the fund’s performance. Conclusions The findings support that the growth funds perform better than the non-growth funds in the long run. However, their performances are easily affected by the market condition. The performance for the real investment also supports this inference. As for the selectivity and timing abilities, about one-third of the funds have the selectivity ability, but very few have the timing ability. Moreover, a fund with both significantly positive selectivity and timing abilities does not guarantee to get a superior performance. References A. References for this paper Bhattacharya, S., and P. Pfleiderer, “A Note on Performance Evaluation.” Technical Report 714, Stanford, Calif.: Stanford University, Graduate School of Business (1983). Brinson, Gary P., B. D. Singer, and G. L. Beebower, “Determinants of Portfolio Performance II: An Update.” Financial Analysts Journal 47, 40-48 (1991). Fama, E. F., “Components of Investment Performance.” Journal of Finance 27, 551-567 (1972). Grant, D., “Portfolio Performance and the Cost of Timing Decisions.” Journal of Finance 32, 837-846 (1977). Henriksson, R. D. and R. C. Merton, “On Market Timing and Investment Performance. П. 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Roll, R., “A Critique of the Asset Pricing Theory's Tests, Part I: On Past and Potential Testability of the Theory.” Journal of Financial Economics 4, 126- 176 (1977). Sharpe, W. F., “Mutual Fund Performance.” Journal of Business 39, 119- 138 (1966). References A. References for this paper (Cont.) Stambaugh, R. F., “On the Exclusion of Assets from Tests of the Two- Parameter Model: A Sensitivity Analysis.” Journal of Financial Economics 10, 237-268 (1982). Treynor, J. L., “How to Rate Management of Investment Funds.” Harvard Business Review 13, 63-75 (1965). Treynor, J. L., and K. K. Mazuy, “Can Mutual Funds Outguess the Market?” Harvard Business Review 44, 131-136 (1966). References B. Additional References 1. Ferson, Wayne E. “The Problem of Alpha and Performance Measurement,” Paper presented at the 16th Annual Pacific Basin Economics, Accounting and Management Conference as keynote speech. 2. Lee, C. F . "Functional Form, Skewness Effect and the Risk-Return Relationship," Journal of Financial and Quantitative Analysis, March, 1977. 3. Lee, C. F . "Investment Horizon and the Functional Form of the Capital Asset Pricing Model," The Review of Economics and Statistics, August, 1976. 4. Lee, C. F . "On the Relationship between the Systematic Risk and the Investment Horizon," Journal of Financial and Quantitative Analysis, December, 1976. 5. Lee, C. F., and Frank C. Jen. "Effects of Measurement Errors on Systematic Risk and Performance Measure of a Portfolio," Journal of Financial and Quantitative Analysis, June, 1978. 6. Lee, C. F., and John K.C. Wei. "The Generalized Stein/Rubinstein Covariance Formula and Its Application to Estimate Real Systematic Risk," Management Science, October 1988. References B. Additional References (Cont.) 7. Lee, C. F., and S. Rahman. "Market Timing, Selectivity, and Mutual Fund Performance: An Empirical Investigation," Journal of Business, Vol. 63, April 1990. 8. Lee, C. F., and S. Rahman. "New Evidence on Timing and Security Selection Skill of Mutual Fund Managers," Journal of Portfolio Management, Winter 1991. 9. Lee, C. F., and Son N. Chen. "On the Measurement Errors and Ranking of Composite Performance Measures, Quarterly Review of Economics and Business, Autumn, 1984. 10. Lee, C. F., and Son N. Chen. "The Effects of the Sample Size, the Investment Horizon and Market Conditions on the Validity of Composite Performance Measures: A Generalization," Management Science, November, 1986. 11. Lee, C. F., and Son N. Chen. "The Sampling Relationship Between Sharpe's Performance Measure and Its Risk Proxy: Sample Size, Investment Horizon and Market Conditions," Management Science, June, 1981 12. Lee, C. F., C.C. Wu and K.C. John Wei. "Heterogeneous Investment Horizon and Capital Asset Pricing Model: Theory and Implications," Journal of Financial and Quantitative Analysis, Vol. 25, September 1990. References B. Additional References (Cont.) 13. Lee, C. F., F. Fabozzi and S. Rahman. "Errors-in-Variables, Functional Form and Mutual Fund Returns," Quarterly Review of Economics and Business, Winter, 1991. 14. Lee, C. F., Frank J. Fabozzi and Jack C. Francis. "Generalized Functional Form for Mutual Fund Returns," Journal of Financial and Quantitative Analysis, December, 1980. 15. Lee, C. F., Jow-Ran Chang and Mao-Wei Hung. “An Intertemporal CAPM Approach to Evaluate Mutual Fund Performance” Review of Quantitative Finance and Accounting, Vol. 20, No. 4, 415-433, 2003. 16. Lee, C. F., K.C. John Wei and Alice C. Lee. “Linear Conditional Expectation, Return Distributions, and Capital Asset Pricing Theories,” The Journal of Financial Research, Volume XXII, Number 4, Winter 1999.” Part 2 Potential problems of alpha and Performance Measurement Ferson’s Model (2008) Lee and Jen (1978) Chang, Hung and Lee (2003) Ferson’s Model (2008) -1 Market Timing Models When you think about performance measures in terms of their OE benchmarks some new insights emerge. Two examples are the most popular classical models of market timing ability. I think that the interpretation of performance in these models is not well understood, and that by using the OE portfolio concept, new understanding is possible. Formal models of market timing ability were first developed in the 1980s, following the intuitive regression model of Treynor and Mazuy (1966). In the simplest example, a market timer has the ability to change the market exposure of the portfolio in anticipation of moves in the stock market. When the market is going up, the timer takes on more market exposure and generates exaggerated returns. When the market is going down, the timer moves into safe assets and minimizes losses. Merton and Henriksson (1981) model this behavior as like put option on the market. A successful market timer can be seen as producing "cheap" put options. Ferson’s Model (2008) -2 The Merton-Henriksson market timing regression is: rpt+1 = ap + bp rmt+1 + Λp Max(rmt+1,0) + ut+1. (10) The coefficient Λp measures the market timing ability. If Λp = 0, the regression reduces to the market model regression used to measure Jensen's alpha, and the intercept measures performance as in the CAPM. However, if Λp is not zero the interpretation is different. Ferson’s Model (2008) -3 The intercept of (10) has been naively interpreted in may studies as a measure of "timing-adjusted" selectivity performance. This only makes sense if the manager has with "perfect" market timing, defined as the ability to obtain the option-like payoff at zero cost. But in reality no one has perfect timing ability, and the interpretation of ap as timing adjusted selectivity breaks down. For example, a manager with some timing ability who picks bad stocks may be hard to distinguish from a manager with no ability who buys options at the market price. Indeed, without an estimate of the market price of a put option on the market index, the intercept ap has no clean interpretation. In the model of Merton and Henriksson, the OE portfolio is a combination of the market index, the risk-free asset and options on the market index. The OE portfolio has a weight equal to bp in the market index returning Rm, a weight of Λp P0 in an option with beginning-of-period price P0 and return {Max(Rm- Rf,0)/P0 -1} at the end of the period, and a weight of (1-bp-ΛpP0) in the safe asset returning Rf. The option is a one-period European call written on the relative value of the market index, Vm/V0 = 1+Rm, with strike price equal to the end of period value of the safe asset, 1+Rf. Ferson’s Model (2008) -4 Given a measure of the option price P0 it is possible to estimate returns in excess of the OE portfolio. In practice, the price of the option must be estimated from an option pricing model. For equity options the Black Scholes (1972) option pricing model is a simple choice. Let r0 be the return on the option measured in excess of the safe asset. The difference between the excess return of the fund and that of the OE portfolio may be computed as: αp = E(rp) - bp E(rm) - Λp P0 E(r0). (11) The measure αp captures "total" performance in the following sense. If an investor holds the OE portfolio he obtains the same market beta and nonlinear payoff with respect to the market as the fund. The difference between the fund's expected return and that of the OE portfolio reflects the manager's ability to deliver the same beta and nonlinearity at a below-market cost, and thus with a higher return. The essence of successful market timing is the ability to produce the convexity at below-market cost. Ferson’s Model (2008) -5 Note that the measure αp is not the same as the intercept in regression (10), so the intercept does not measure the fund's return in excess of the OE portfolio. The problem is that the term Max(rmt+1,0) in the regression is not an excess return, so the intercept is not an alpha. Taking the expected value of (10) and comparing it with the expression for alpha in (11), the intercept in (10) is related to the "right" alpha in this model as: αp = ap + Λp P0 Rf. (12) Only if the fund has perfect timing ability does the intercept in the regression (10) measure timing-adjusted selectivity. If a manager had perfect timing ability she would deliver the same payoff as the OE portfolio, while "saving" the cost of the option, ΛpP0. Increasing the position in the safe asset by this amount leaves the beta unchanged and produces the additional return, ΛpP0Rf. The additional return is the difference between the intercept, ap, and the alpha in (11). If a manager had perfect timing ability and could generate a higher return in excess of the OE portfolio than ΛpP0Rf, the extra return could then be presumed to be attributed to selectivity. Ferson’s Model (2008) -6 Under this interpretation, when αp > ΛpP0Rf, then ap > 0 measures the selectivity-related excess return, on the assumption of perfect market timing ability. Since in general the return to timing activity will be less than ΛpP0Rf, then for a given total performance the intercept in (10) is less than the selectivity performance. The literature typically finds that funds with positive timing coefficients have negative intercepts, consistent with understated selectivity performance. Ferson’s Model (2008) -7 Treynor-Mazuy model The Treynor-Mazuy (1966) market-timing model is a quadratic regression: rpt+1 = ap + bp rmt+1 + Λp rmt+12 + vt+1. (13) Treynor and Mazuy (1966) argue that Λp>0 indicates market-timing ability. Like the intercept of Equation (10), the intercept in the Treynor-Mazuy model has been naively interpreted as a "timing-adjusted" selectivity measure. However, as in the Merton-Henriksson model, the intercept does not capture the return in excess of an OE portfolio because rm2, in this case, is not a portfolio return.8 However, the model can be modified to capture the difference between the return of the fund and that of an OE portfolio. 8 Note that the intercept in (13) can be interpreted as the difference between the fund's average return and that of a trading strategy that holds the market index and the safe asset, with a time-varying weight or beta in the market index equal to bp + Λprmt+1. However, this weight is not feasible at time t without foreknowledge of the future market return, so this strategy is not a feasible OE portfolio. Ferson’s Model (2008) -8 Let rh be the excess return of the maximum correlation portfolio to the random variable rm2 and let Λh be the portfolio's regression coefficient on rm2. The OE portfolio that replicates the beta and convexity of rp has a weight of Λp/Λh in rh and bp in rm, with 1 - bp - Λp/Λh in the safe asset, assuming Λh≠0.9 The fact that the OE portfolio has the same beta and convexity coefficient as rp can be seen by substituting the regression for rh on rm2 into the combination of rm and rh that defines the OE excess return. The means of rp and the OE portfolio excess returns differ by αp = ap - ah Λp/Λh, where ap is the intercept of (13). Thus, αp measures the total return performance, in the presence of timing ability, on the assumption that timing ability may be captured by a quadratic function. A modified version of the model is the system: 9 As Λh approaches zero the weight of the OE portfolio in rh becomes infinite. If Λh=0, no portfolio can be formed with a nonzero correlation with rm2. In this unlikely event the model of Equation (14) is undefined. Ferson’s Model (2008) -9 rp = [αp + ah Λp/Λh] + bp rm + Λp rm2 + εp, (14) rh = ah + Λh rm2 + εh. This system is easily estimated using the Generalized Method of Moments (Hansen, 1982). In general, it is not possible to separate the effects of timing and selectivity on return performance without making strong assumptions about one of the components. But it is possible to measure the combined effects of timing and selectivity on total performance when there is some timing ability. The measures described above capture the excess return of the fund over the OE portfolio, and thus the total abnormal performance, when the nonlinearity implied by market timing fits the particular model. No study has yet examined these performance measures empirically, so it would be interesting in future research to implement the measures in (11) and (14) using data on managed portfolios where market timing is likely to be present. Lee and Jen (1978) -1 Effects of Measurement Errors on αj and βj. In this paper we will analyze equation (2), a form of CAPM used empirically by Jensen [9], Friend and Blume [6], and Miller and Scholes [11]. Using hat to denote sample estimate, the estimated regression line can be written as (3) ˆ Rjt RTt j j ( Rmt RTt ), ˆ where RTt is used as a proxy for Rft. We will now derive the properties of j and j when Rmt and Rft are ˆ ˆ either with or without errors. Lee and Jen (1978) -2 Measurement Errors on Rft Let us examine first the possible sources of measurement error on Rft. As has been argued by Roll [12] and Jen [8], the treasury bill rate is only a proxy for the risk-free rate. We therefore postulate (4) R ft RTt U t where RTt is treasury bill rates used as a proxy for R ft ,U t N (0, u2 ) and is i.i.d. Lee and Jen (1978) -3 In addition, it is well known that one of the unrealistic assumptions used to derive CAPM is that an investor can borrow freely at the riskless rate. Violation of this assumption has been ˆ hypothesized by Friend and Blume to have caused j to be negatively correlated with j . ˆ Allowing for the fact that the borrowing rate is higher than the riskless rate, Brennan [3] showed that the relationship between return and systematic risk of a capital asset is still linear. He further showed that the only difference between the traditional CAPM and this version is to replace Rf by Rb, the latter represents a weighted average of market's lending and borrowing rate. After considering the traditional element of the borrowing rate, Brennan derived this new form of CAPM: Lee and Jen (1978) -4 (5) E( Rj ) Rb [ E( Rm ) Rb ] j Following (2) and (5), we can obtain a new regression model as (6) Rjt Rbt ( Rmt Rbt ) wjt j j where and j are true parameters of the model and wjt j N (0, w ). 2 We now postulate that: (7) Rbt RTt B Vbt , where B is positive constant[3] Vbt is distributed with zero mean and finite variance and is i.i.d.. Substituting (7) and (4) into (6), defining ebt = Ubt －Vbt, we have this new theoretical model of CAPM model [3] Recall Rb > Rf in the market. Lee and Jen (1978) -5 Lee and Jen (1978) -6 Lee and Jen (1978) -7 Lee and Jen (1978) -8 Lee and Jen (1978) -9 Lee and Jen (1978) - 10 Chang, Hung and Lee (2003) -1 Chang, Hung and Lee (2003) -2 Chang, Hung and Lee (2003) -3 Chang, Hung and Lee (2003) -4 Chang, Hung and Lee (2003) -5 Chang, Hung and Lee (2003) -6 Chang, Hung and Lee (2003) -7