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Portfolio Management - Chapter 19 by RushenChahal


									     Chapter 19

Performance Evaluation

              Prof. Rushen Chahal

       Prof. Rushen Chahal          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

                        - From Candide by Voltaire

                     Prof. Rushen Chahal                  2
• Introduction
• Importance of measuring portfolio risk
• Traditional performance measures
• Performance evaluation with cash deposits
  and withdrawals
• Performance evaluation when options are

                   Prof. Rushen Chahal        3
• 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

                     Prof. Rushen Chahal           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

                   Prof. Rushen Chahal           5
• When two investments’ returns are
  compared, their relative risk must also be
• People maximize expected utility:
  – A positive function of expected return
  – A negative function of the return variance

           E(U )  f  E ( R),  2 
                                   

                       Prof. Rushen Chahal       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
    2) These rates of return should be based on the
       market value of the fund’s assets

                        Prof. Rushen Chahal               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

                    Prof. Rushen Chahal            8
             A Lesson from
          A Few Mutual Funds
• The two key points with performance
  – The arithmetic mean is not a useful statistic in
    evaluating growth
  – Dollars are more important than percentages

• Consider the historical returns of two mutual
  funds on the following slide

                       Prof. Rushen Chahal             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%

                        Prof. Rushen Chahal                      10
                         A Lesson from
                   A Few Mutual Funds (cont’d)
                             Mutual Fund Performance
Ending Value ($)

                   $140,000.00                                              44 Wall
                   $120,000.00                                              Street
                   $100,000.00                                              Mutual
                                      77        80          83         86
                                 19        19          19         19

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

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

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

                      Prof. Rushen Chahal             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

                         Prof. Rushen Chahal        14
     Why the Arithmetic Mean
    Is Often Misleading (cont’d)
• The geometric 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

                    Prof. Rushen Chahal         15
       Why the Arithmetic Mean
      Is Often Misleading (cont’d)

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?

                           Prof. Rushen Chahal                 16
       Why the Arithmetic Mean
      Is Often Misleading (cont’d)

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%

                         Prof. Rushen Chahal           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

                     Prof. Rushen Chahal            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%
                                           

                       Prof. Rushen Chahal             19
       Performance Measures
• Sharpe and Treynor measures
• Jensen measure
• Performance measurement in practice

                  Prof. Rushen Chahal   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

                          Prof. Rushen Chahal          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

                       Prof. Rushen Chahal              22
              Sharpe and
       Treynor Measures (cont’d)

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?

                          Prof. Rushen Chahal                 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%
          0.88%

                           Prof. Rushen Chahal                24
              Sharpe and
       Treynor Measures (cont’d)
                     Example (cont’d)

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

                         R  Rf            0.88%
      Sharpe measure                             0.29
                                          3.07%
                         R  Rf            0.88%
    Treynor measure                              0.73
                                           1.20
                          Prof. Rushen Chahal               25
             Jensen Measure
• The Jensen measure stems directly from the

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

                       Prof. Rushen Chahal       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

                      Prof. Rushen Chahal               27
       Jensen Measure (cont’d)
• The Jensen measure is generally out of favor
  because of statistical and theoretical problems

                    Prof. Rushen Chahal         28
    Performance Measurement
           in Practice
• Academic issues
• Industry issues

                    Prof. Rushen Chahal   29
             Academic Issues
• The use of traditional performance measures
  relies on the CAPM

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

                     Prof. Rushen Chahal           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

• Practical performance measures typically
  involve a comparison of the fund’s
  performance with that of a benchmark

                     Prof. Rushen Chahal          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

                     Prof. Rushen Chahal              32
Prof. Rushen Chahal   33
Performance Evaluation With Cash
     Deposits & Withdrawals
• Introduction
• Daily valuation method
• Modified Bank Administration Institute (BAI)
• An example
• An approximate method

                    Prof. Rushen Chahal          34
• 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

                         Prof. Rushen Chahal             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
     • Might be interest, dividends, or additions and

                         Prof. Rushen Chahal            36
            Daily Valuation
            Method (cont’d)
• The daily valuation method solves for R:

               Rdaily   Si  1
                             i 1

           where S 

                    Prof. Rushen Chahal      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

                       Prof. Rushen Chahal             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

                      Prof. Rushen Chahal              39
   Modified BAI Method (cont’d)
• It solves for R:
            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  total number of days in the period
              Di  number of days since the beginning of the period
                      in which the cash flow occurred
                               Prof. Rushen Chahal                    40
                An Example
• An investor has an account with a mutual fund
  and “dollar cost averages” by putting $100 per
  month into the fund

• The following slide shows the activity and
  results over a seven-month period

                    Prof. Rushen Chahal        41
Prof. Rushen Chahal   42
          An Example (cont’d)
• The daily valuation method returns a time-
  weighted return of 40.6% over the seven-
  months period
  – See next slide

                     Prof. Rushen Chahal       43
Prof. Rushen Chahal   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)

                   Prof. Rushen Chahal        45
Prof. Rushen Chahal   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

                        Prof. Rushen Chahal                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%

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

                      Prof. Rushen Chahal       49
• 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

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

  – Residual option spread (ROS)

                      Prof. Rushen Chahal     51
Incremental Risk-Adjusted Return
         from Options
• Definition
• An IRAR example
• IRAR caveats

                    Prof. Rushen Chahal   52
• 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
  – A negative IRAR indicates the portfolio would have
    performed better without options

                      Prof. Rushen Chahal            53
            Definition (cont’d)
• Use the 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

                        Prof. Rushen Chahal               54
Definition (cont’d)

      Prof. Rushen Chahal   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

                         Prof. Rushen Chahal            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

                     Prof. Rushen Chahal             57
Prof. Rushen Chahal   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

                     Prof. Rushen Chahal               59
Prof. Rushen Chahal   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

                      Prof. Rushen Chahal             61
         Residual Option Spread
• The residual option spread (ROS) is an alternative
  performance measure for portfolios containing
• 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

                       Prof. Rushen Chahal               62
               Residual Option
               Spread (cont’d)
• The residual 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

                            Prof. Rushen Chahal   63
             Residual Option
             Spread (cont’d)
• The worksheet to calculate the ROS for the
  previous example is shown on the next slide

• The ROS translates into a dollar differential of

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

                      Prof. Rushen Chahal            66
                  The M2 Performance
                   Measure (cont’d)
• Calculate the risk-adjusted portfolio return as

                                  benchmark
    Rrisk-adjusted portfolio                Ractual portfolio
                                  portfolio
                                      benchmark      
                                   1                 Rf
                                         portfolio   
                                                      
                                Prof. Rushen Chahal              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

                      Prof. Rushen Chahal            68

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