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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. Statistical Procedure for Evaluating Forecasting Skills.”
    Journal of Business 54, 513-534 (1981).
   Jensen, M.C., “The Performance of Mutual Funds in the Period 1945-1964.”
    Journal of Finance 23, 389-416 (1968).
References
A. References for this paper (Cont.)
   Jensen, M. C., “Optimal Utilization of Market Forecasts and the Evaluation
    of Investment Performance.” In G. P. Szego and Karl Shell(eds.),
    Mathematical Methods in Investment and Finance Amsterdam : Elsevier
    (1972).
   Lee C. F., and S. Rahman, “Market Timing, Selectivity, and Mutual Fund
    Performance: An Empirical Investigation.” Journal of Business 63, 261-278
    (1990).
   Markowitz, H., “Portfolio Selection.” Journal of Finance 7, 77-91 (1952).
   Merton, R. C., “On Market Timing and Investment performance. I. An
    Equilibrium Theory of Value for Market Forecasts.” Journal of Business 54,
    363-406 (1981).
   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

				
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