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					     Chapter 19

Performance Evaluation




                         1
And with that they clapped him into irons and hauled
      him off to the barracks. There he was taught
     “right turn,” “left turn,” and “quick march,”
  “slope arms,” and “order arms,” how to aim and
     how to fire, and was given thirty strokes of the
  “cat.” Next day his performance on parade was a
  little better, and he was given only twenty strokes.
  The following day he received a mere ten and was
           thought a prodigy by his comrades.

                      - From Candide by Voltaire
                                                     2
                 Outline
 Introduction
 Importance  of measuring portfolio risk
 Traditional performance measures
 Performance evaluation with cash deposits
  and withdrawals
 Performance evaluation when options are
  used
                                              3
             Introduction
 Performance  evaluation is a critical aspect
 of portfolio management

 Proper performance evaluation should
 involve a recognition of both the return and
 the riskiness of the investment


                                                 4
         Importance of
     Measuring Portfolio Risk
 Introduction
A  lesson from history: the 1968 Bank
  Administration Institute report
 A lesson from a few mutual funds
 Why the arithmetic mean is often
  misleading: a review
 Why dollars are more important than
  percentages
                                         5
               Introduction
 When  two investments’ returns are
  compared, their relative risk must also be
  considered
 People maximize expected utility:
  • A positive function of expected return
  • A negative function of the return variance

          E (U )  f  E ( R),  2 
                                   
                                                 6
        A Lesson from History
    The 1968 Bank Administration Institute’s
     Measuring the Investment Performance of
     Pension Funds concluded:
    1) Performance of a fund should be measured by
       computing the actual rates of return on a
       fund’s assets
    2) These rates of return should be based on the
       market value of the fund’s assets
                                                  7
A Lesson from History (cont’d)
 3) Complete evaluation of the manager’s
    performance must include examining a
    measure of the degree of risk taken in the
    fund
 4) Circumstances under which fund managers
    must operate vary so great that indiscriminate
    comparisons among funds might reflect
    differences in these circumstances rather than
    in the ability of managers
                                                     8
           A Lesson from
        A Few Mutual Funds
 Thetwo key points with performance
 evaluation:
  • The arithmetic mean is not a useful statistic in
    evaluating growth
  • Dollars are more important than percentages

 Considerthe historical returns of two
 mutual funds on the following slide
                                                       9
       A Lesson from
 A Few Mutual Funds (cont’d)
       44 Wall   Mutual          44 Wall   Mutual
Year    Street   Shares   Year    Street   Shares
1975   184.1%    24.6%    1982     6.9      12.0
1976    46.5      63.1    1983     9.2      37.8
1977    16.5      13.2    1984    -58.7     14.3
1978    32.9      16.1    1985    -20.1     26.3
1979    71.4      39.3    1986    -16.3     16.9
1980    36.1      19.0    1987    -34.6     6.5
1981    -23.6     8.7     1988    19.3      30.7
                          Mean   19.3%     23.5%
                                                    10
                   A Lesson from
             A Few Mutual Funds (cont’d)
                             Mutual Fund Performance
                   $200,000.00
                   $180,000.00
                   $160,000.00
Ending Value ($)




                   $140,000.00                                             44 Wall
                   $120,000.00                                             Street
                   $100,000.00                                             Mutual
                    $80,000.00                                             Shares
                    $60,000.00
                    $40,000.00
                    $20,000.00
                          $-
                                      77        80          83        86
                                 19        19          19        19
                                                     Year


                                                                                 11
       A Lesson from
 A Few Mutual Funds (cont’d)
   Wall Street and Mutual Shares both had
 44
 good returns over the 1975 to 1988 period

 Mutual  Shares clearly outperforms 44 Wall
 Street in terms of dollar returns at the end of
 1988


                                               12
   Why the Arithmetic Mean
     Is Often Misleading
 The arithmetic mean may give misleading
 information
  • E.g., a 50% decline in one period followed by a
    50% increase in the next period does not return
    0%, on average




                                                  13
   Why the Arithmetic Mean
  Is Often Misleading (cont’d)
 The proper measure of average investment
 return over time is the geometric mean:
                           1/ n
                     n
                          
           GM   Ri   1
                    i 1 
        where Ri  the return relative in period i



                                                     14
   Why the Arithmetic Mean
  Is Often Misleading (cont’d)
 Thegeometric means in the preceding
 example are:
  • 44 Wall Street: 7.9%
  • Mutual Shares: 22.7%

 The geometric mean correctly identifies
 Mutual Shares as the better investment over
 the 1975 to 1988 period
                                           15
    Why the Arithmetic Mean
   Is Often Misleading (cont’d)
                       Example

A stock returns –40% in the first period, +50% in the
second period, and 0% in the third period.

What is the geometric mean over the three periods?




                                                        16
    Why the Arithmetic Mean
   Is Often Misleading (cont’d)
                       Example

Solution: The geometric mean is computed as follows:
                             1/ n
                      n
                          
              GM   Ri   1
                    i 1 
                  (0.60)(1.50)(1.00)  1
                    0.10  10%

                                                       17
    Why Dollars Are More
  Important than Percentages
 Assume   two funds:
  • Fund A has $40 million in investments and
    earned 12% last period

  • Fund B has $250,000 in investments and earned
    44% last period



                                                18
     Why Dollars Are More
   Important than Percentages
 The  correct way to determine the return of
  both funds combined is to weigh the funds’
  returns by the dollar amounts:

 $40, 000, 000        $250, 000             
 $40, 250, 000 12%    $40, 250, 000  44%   12.10%
                                            


                                                      19
          Traditional
     Performance Measures
 Sharpe and Treynor measures
 Jensen measure
 Performance measurement in practice




                                        20
Sharpe and Treynor Measures
 The   Sharpe and Treynor measures:
                            R  Rf
        Sharpe measure 
                              
                            R  Rf
        Treynor measure 
                              
               where R  average return
                    R f  risk-free rate
                       standard deviation of returns
                       beta
                                                         21
        Sharpe and
  Treynor Measures (cont’d)
 The Treynor measure evaluates the return
 relative to beta, a measure of systematic risk
  • It ignores any unsystematic risk


 The Sharpe measure evaluates return
 relative to total risk
  • Appropriate for a well-diversified portfolio, but
    not for individual securities
                                                    22
          Sharpe and
    Treynor Measures (cont’d)
                       Example

Over the last four months, XYZ Stock had excess returns
of 1.86%, -5.09%, -1.99%, and 1.72%. The standard
deviation of XYZ stock returns is 3.07%. XYZ Stock has
a beta of 1.20.

What are the Sharpe and Treynor measures for XYZ
Stock?

                                                      23
          Sharpe and
    Treynor Measures (cont’d)
                  Example (cont’d)

Solution: First compute the average excess return for
Stock XYZ:

         1.86%  5.09%  1.99%  1.72%
      R
                       4
        0.88%

                                                        24
          Sharpe and
    Treynor Measures (cont’d)
                  Example (cont’d)

Solution (cont’d): Next, compute the Sharpe and Treynor
measures:

                       R  Rf     0.88%
    Sharpe measure                      0.29
                                 3.07%
                       R  Rf     0.88%
  Treynor measure                       0.73
                                  1.20
                                                      25
             Jensen Measure
    Jensen measure stems directly from the
 The
 CAPM:

        Rit  R ft     i  Rmt  R ft 
                                           




                                                26
    Jensen Measure (cont’d)
 The   constant term should be zero
  • Securities with a beta of zero should have an
    excess return of zero according to finance
    theory


 According to the Jensen measure, if a
 portfolio manager is better-than-average,
 the alpha of the portfolio will be positive
                                                    27
    Jensen Measure (cont’d)
 TheJensen measure is generally out of
 favor because of statistical and theoretical
 problems




                                                28
   Performance Measurement
          in Practice
 Academic   issues
 Industry issues




                             29
           Academic Issues
    use of traditional performance
 The
 measures relies on the CAPM

 Evidence continues to accumulate that may
 ultimately displace the CAPM
  • APT, multi-factor CAPMs, inflation-adjusted
    CAPM

                                                  30
              Industry Issues
 “Portfolio managers are hired and fired
  largely on the basis of realized investment
  returns with little regard to risk taken in
  achieving the returns”

 Practicalperformance measures typically
  involve a comparison of the fund’s
  performance with that of a benchmark
                                                31
     Industry Issues (cont’d)
 Fama’s decomposition can be used to assess
 why an investment performed better or
 worse than expected:
  • The return the investor chose to take
  • The added return the manager chose to seek
  • The return from the manager’s good selection
    of securities

                                                   32
33
 Performance Evaluation With
 Cash Deposits & Withdrawals
 Introduction
 Dailyvaluation method
 Modified Bank Administration Institute
  (BAI) Method
 An example
 An approximate method


                                           34
                  Introduction
   The owner of a fund often taken periodic
    distributions from the portfolio and may
    occasionally add to it

   The established way to calculate portfolio
    performance in this situation is via a time-
    weighted rate of return:
    • Daily valuation method
    • Modified BAI method

                                                   35
     Daily Valuation Method
 The   daily valuation method:
  • Calculates the exact time-weighted rate of
    return
  • Is cumbersome because it requires determining
    a value for the portfolio each time any cash
    flow occurs
     – Might be interest, dividends, or additions and
       withdrawals

                                                        36
            Daily Valuation
            Method (cont’d)
 The   daily valuation method solves for R:
                         n
                Rdaily   Si  1
                        i 1

                      MVEi
            where S 
                      MVBi

                                               37
              Daily Valuation
              Method (cont’d)
   MVEi = market value of the portfolio at the end of
    period i before any cash flows in period i but
    including accrued income for the period

 MVBi    = market value of the portfolio at the
    beginning of period i including any cash flows at
    the end of the previous subperiod and including
    accrued income

                                                        38
        Modified BAI Method
 The   modified BAI method:
  • Approximates the internal rate of return for the
    investment over the period in question

  • Can be complicated with a large portfolio that
    might conceivably have a cash flow every day



                                                     39
Modified BAI Method (cont’d)
 It   solves for R:
                        n
             MVE   Fi (1  R ) wi
                       i 1

          where F  the sum of the cash flows during the period
            MVE  market value at the end of the period,
                    including accrued income
               F0  market value at the start of the period
                    CD  Di
                wi 
                     CD
              CD  total number of days in the period
               Di  number of days since the beginning of the period
                       in which the cash flow occurred
                                                                       40
            An Example
 Aninvestor has an account with a mutual
 fund and “dollar cost averages” by putting
 $100 per month into the fund

 Thefollowing slide shows the activity and
 results over a seven-month period


                                              41
42
        An Example (cont’d)
 Thedaily valuation method returns a time-
 weighted return of 40.6% over the seven-
 months period
  • See next slide




                                              43
44
         An Example (cont’d)
 The   BAI method requires use of a computer

 The BAI method returns a time-weighted
 return of 42.1% over the seven-months
 period (see next slide)



                                            45
46
     An Approximate Method
 Proposed by the American Association of
  Individual Investors:

    P  0.5(Net cash flow)
 R 1                      1
   P0  0.5(Net cash flow)


 where net cash flow is the sum of inflows and outflows

                                                          47
            An Approximate
            Method (cont’d)
 Using   the approximate method in Table 19-
 6:
             P  0.5(Net cash flow)
          R 1                      1
            P0  0.5(Net cash flow)
              5,500.97  0.5( 4, 200)
                                      1
              7,550.08  0.5(-4, 200)
             0.395  39.5%

                                            48
     Performance Evaluation
     When Options Are Used
 Introduction
 Incremental    risk-adjusted return from
  options
 Residual option spread
 Final comments on performance evaluation
  with options


                                             49
              Introduction
 Inclusion  of options in a portfolio usually
  results in a non-normal return distribution

 Beta and standard deviation lose their
  theoretical value of the return distribution is
  nonsymmetrical


                                                 50
       Introduction (cont’d)
 Consider two alternative methods when
 options are included in a portfolio:
  • Incremental risk-adjusted return (IRAR)

  • Residual option spread (ROS)




                                              51
   Incremental Risk-Adjusted
      Return from Options
 Definition
 An IRAR example
 IRAR caveats




                               52
                Definition
 The incremental risk-adjusted return
 (IRAR) is a single performance measure
 indicating the contribution of an options
 program to overall portfolio performance
  • A positive IRAR indicates above-average
    performance
  • A negative IRAR indicates the portfolio would
    have performed better without options
                                                53
          Definition (cont’d)
 Usethe unoptioned portfolio as a
 benchmark:
  • Draw a line from the risk-free rate to its
    realized risk/return combination

  • Points above this benchmark line result from
    superior performance
     – The higher than expected return is the IRAR

                                                     54
Definition (cont’d)




                      55
             Definition (cont’d)
  The   IRAR calculation:

     IRAR  ( SH o  SH u ) o


where SH o  Sharpe measure of the optioned portfolio
      SH u  Sharpe measure of the unoptioned portfolio
        o  standard deviation of the optioned portfolio

                                                        56
         An IRAR Example
A  portfolio manager routinely writes index
  call options to take advantage of anticipated
  market movements
 Assume:
  • The portfolio has an initial value of $200,000
  • The stock portfolio has a beta of 1.0
  • The premiums received from option writing are
    invested into more shares of stock
                                                 57
58
  An IRAR Example (cont’d)
 The    IRAR calculation (next slide) shows
 that:
  • The optioned portfolio appreciated more than
    the unoptioned portfolio

  • The options program was successful at adding
    about 12% per year to the overall performance
    of the fund
                                                    59
60
             IRAR Caveats
 IRAR  can be used inappropriately if there is
 a floor on the return of the optioned
 portfolio
  • E.g., a portfolio manager might use puts to
    protect against a large fall in stock price
 The standard deviation of the optioned
 portfolio is probably a poor measure of risk
 in these cases
                                                  61
      Residual Option Spread
 The residual option spread (ROS) is an
  alternative performance measure for portfolios
  containing options
 A positive ROS indicates the use of options
  resulted in more terminal wealth than only holding
  stock
 A positive ROS does not necessarily mean that the
  incremental return is appropriate given the risk

                                                  62
            Residual Option
            Spread (cont’d)
 Theresidual option spread (ROS)
 calculation:
                 n         n
        ROS   Got   Gut
                t 1      t 1



  where Gt  Vt / Vt 1
          Vt  value of portfolio in Period t
                                                63
          Residual Option
          Spread (cont’d)
 Theworksheet to calculate the ROS for the
 previous example is shown on the next slide

 TheROS translates into a dollar differential
 of $1,452



                                             64
65
                The M2
          Performance Measure
 Developed   by Franco and Leah Modigliani
  in 1997
 Seeks to express relative performance in
  risk-adjusted basis points
  • Ensures that the portfolio being evaluated and
    the benchmark have the same standard
    deviation

                                                     66
               The M2 Performance
                Measure (cont’d)
 Calculate the risk-adjusted portfolio return
  as follows:
                                benchmark
  Rrisk-adjusted portfolio                Ractual portfolio
                                portfolio
                                   benchmark       
                                1                  Rf
                                      portfolio    
                                                    
                                                               67
           Final Comments
 IRAR and ROS both focus on whether an
 optioned portfolio outperforms an
 unoptioned portfolio
  • Can overlook subjective considerations such as
    portfolio insurance




                                                 68

				
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