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Alternative Model to Evaluate Selectivity and Timing Performance of Mutual Fund Managers: Theory and

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Title: Alternative Model to Evaluate Selectivity and Timing Performance of Mutual Fund Managers: Theory and


1
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
    JerseyEditor of Review of Quantitative Finance
    and Accounting
  • Editor of Review of Pacific Basin Financial
    Markets and Policies

2
Outline
  • Part 1Alternative 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
  • Fersons Model (2008)
  • Lee and Jen (1978)
  • Chang, Hung and Lee (2003)

3
Part 1
  • Alternative Model to Evaluate Selectivity and
    Timing Performance of Mutual Fund Managers
    Theory and Evidence

4
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.

5
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 Jensens (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.

6
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.

7
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.

8
Treynor index
9
Sharpe index
10
Jensen index
11
Relationships between these three measures
12
Relationships between these three measures (Cont.)
13
  • 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 doesnt reflect
    the true performance of the manager.

14
Treynor-Mazuy model
15
Henriksson-Merton Model
16
Jensens (1972) Viewpoint
17
(No Transcript)
18
(No Transcript)
19
Lee-Rahman Model
20
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 ICDIs 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 SP 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.

21
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 (SP 500 index).
Panel A Growth Fund Excess Return V.S. Market
Excess Return
22
Panel B Non-growth Fund Return V.S. Market
Excess Return
23
  • 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).

24
Figure 2 SP 500 stock index and treasure bill
rateThis figure shows the graphs for the market
index (SP 500) and risk-free rate (T-bill rate)
in this study.
25
  • 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.

26
Table 1 Mutual fund performance measured by
Treynor index
27
  • 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.

28
Table 2 Mutual fund performance measured by
Sharpe index
29
Table 3 Mutual fund performance measured by
Jensen index
30
  • 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.

31
Table 4 Measured by the Treynor-Mazuys model
32
  • 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).

33
Table 5 Measured by the Henriksson-Mertons
model
34
  • 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.

35
Table 6 Measured by the Lee-Rahmans model
36
  • 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.

37
Table 7 Return on the initial investment of
1.00
38
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, 199001 200509

39
Panel B Mutual fund returns with initial
investment 1.0, 199001 199710

40
Panel C Mutual fund returns with initial
investment 1.0, 199711 200509
41
  • 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
    funds performance.

42
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.

43
ReferencesA. 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. ?. 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).

44
ReferencesA. 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).

45
ReferencesA. 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).

46
ReferencesB. Additional References
  • 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.
  • Lee, C. F . "Functional Form, Skewness Effect and
    the Risk-Return Relationship," Journal of
    Financial and Quantitative Analysis, March, 1977.
  • Lee, C. F . "Investment Horizon and the
    Functional Form of the Capital Asset Pricing
    Model," The Review of Economics and Statistics,
    August, 1976.
  • Lee, C. F . "On the Relationship between the
    Systematic Risk and the Investment Horizon,"
    Journal of Financial and Quantitative Analysis,
    December, 1976.
  • 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.
  • 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.

47
ReferencesB. Additional References (Cont.)
  • Lee, C. F., and S. Rahman. "Market Timing,
    Selectivity, and Mutual Fund Performance An
    Empirical Investigation," Journal of Business,
    Vol. 63, April 1990.
  • Lee, C. F., and S. Rahman. "New Evidence on
    Timing and Security Selection Skill of Mutual
    Fund Managers," Journal of Portfolio Management,
    Winter 1991.
  • 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.
  • 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.
  • 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
  • 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.

48
ReferencesB. Additional References (Cont.)
  • 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.
  • 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.
  • 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.
  • 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.

49
Part 2
  • Potential problems of alpha and Performance
    Measurement
  • Fersons Model (2008)
  • Lee and Jen (1978)
  • Chang, Hung and Lee (2003)

50
Fersons 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.

51
Fersons Model (2008) - 2
  • The Merton-Henriksson market timing regression
    is
  • rpt1 ap bp rmt1 ?p Max(rmt1,0)
    ut1. (10)
  • The coefficient ?p measures the market timing
    ability. If ?p 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 ?p is not zero the
    interpretation is different.

52
Fersons 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 ?p 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-?pP0) 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 1Rm, with
    strike price equal to the end of period value of
    the safe asset, 1Rf.

53
Fersons 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
  • ap E(rp) - bp E(rm) - ?p P0 E(r0).
    (11)
  • The measure ap 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.

54
Fersons Model (2008) - 5
  • Note that the measure ap 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(rmt1,0) 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
  • ap ap ?p 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, ?pP0. Increasing the position
    in the safe asset by this amount leaves the beta
    unchanged and produces the additional return,
    ?pP0Rf. 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 ?pP0Rf, the extra return could
    then be presumed to be attributed to selectivity.

55
Fersons Model (2008) - 6
  • Under this interpretation, when ap gt ?pP0Rf,
    then ap gt 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
    ?pP0Rf, 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.

56
Fersons Model (2008) - 7
  • Treynor-Mazuy model
  • The Treynor-Mazuy (1966) market-timing model is a
    quadratic regression
  • rpt1 ap bp rmt1 ?p rmt12 vt1. (13)
  • Treynor and Mazuy (1966) argue that ?pgt0
    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 ?prmt1. 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.

57
Fersons Model (2008) - 8
  • Let rh be the excess return of the maximum
    correlation portfolio to the random variable rm2
    and let ?h be the portfolio's regression
    coefficient on rm2. The OE portfolio that
    replicates the beta and convexity of rp has a
    weight of ?p/?h in rh and bp in rm, with 1 - bp -
    ?p/?h in the safe asset, assuming ?h?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 ap ap - ah
    ?p/?h, where ap is the intercept of (13). Thus,
    ap 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 ?h approaches zero the weight of the OE
    portfolio in rh becomes infinite. If ?h0, no
    portfolio can be formed with a nonzero
    correlation with rm2. In this unlikely event the
    model of Equation (14) is undefined.

58
Fersons Model (2008) - 9
  • rp ap ah ?p/?h bp rm ?p rm2
    ep, (14)
  • rh ah ?h rm2 eh.
  • 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.

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Lee and Jen (1978) - 1
  • Effects of Measurement Errors on aj and ßj.
  • 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)
  • where RTt is used as a proxy for Rft.
  • We will now derive the properties of and
    when Rmt and Rft are either with or without
    errors.

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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)
  • where RTt is treasury bill rates used as a proxy
    for
  • and is i.i.d.

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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 hypothesized by Friend and Blume to have
    caused to be negatively correlated with
    .
  • 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

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Lee and Jen (1978) - 4
  • (5)
  • Following (2) and (5), we can obtain a new
    regression model as
  • (6)
  • where and are true parameters of the
    model and .
  • We now postulate that
  • (7)
  • where B is positive constant3 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 gt Rf in the market.

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