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The Geometric Average

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					The Geometric Average


     Measuring Investment
     Returns
Average Returns

    When measuring returns for an asset
     class over time, it is useful to be able
     to calculate the ‘average’
    There are two choices…
    1.   Arithmetic average, and
    2.   Geometric average
The Bias Inherent in the
Arithmetic Average

   Arithmetic averages can yield incorrect
    results because of the problems of
    bias inherent in its calculation.
    Example
      Consider an investment that was purchased for $10, rose
       to $20 and then fell back to $10.
      Let us calculate the HPR in both periods:
                              $20  $10 $10
                      HPR1                  100%
                                $10      $10
                              $10  $20  $10
                      HPR 2                   50%
                                 $20      $20
The Bias Inherent in the
Arithmetic Average

Example Continued ...
      Consider an investment that was purchased for $10, rose
       to $20 and then fell back to $10.
      Let us calculate the HPR in both periods:
                              $20  $10 $10
                      HPR1                  100%
                                $10      $10
                              $10  $20  $10
                      HPR 2                   50%
                                 $20      $20


      The arithmetic average return earned on this investment
       was:
                                   100%  50% 50%
                         Average                     25%
                                        2        2
The Bias Inherent in the
Arithmetic Average

Example Continued ...
      The answer is clearly incorrect since the investor started
       with $10 and ended with $10.
      The correct answer may be obtained through the use of
       the geometric average:                n
                         GeometricAverage  n           (1  r )  1
                                                       i 1
                                                               i


                                           1/ n
                             n        
                             
                            (1  ri )
                             i 1     
                                                  1

                           [(1  100%)(1  (50%))]1/ 2  1
                           [(2)(.5)]1/ 2  1
                           (1)1/ 2  1  1  1  0
The Geometric Average

Example Continued ...
      The correct answer may be obtained through the use of
       the geometric average:
                                               n
                 GeometricAverage  n          (1  r )  1
                                              i 1
                                                      i


                                  1/ n
                    n        
                    
                   (1  ri )
                    i 1     
                                         1

                  [(1  100%)(1  (50%))]1/ 2  1
                  [(2)(.5)]1/ 2  1
                  (1)1/ 2  1  1  1  0
Measuring Returns

   When you are trying to find average
    returns, especially when those returns
    rise and fall, always remember to use
    the geometric average.
When Should You Use the
Geometric Average?

   When the returns each year are
    consistent…the geometric and
    arithmetic averages will produce the
    same results…as follows
Example – stable returns

      Year        Returns
      1998         7.0%
      1999         7.0%     When returns
                             are stable,
      2000         7.0%         each
      2001         7.0%      approach
      2002         7.0%       gives the
      2003         7.0%     same result.
      2004         7.0%

   Arthimetic =   7.00%
   Geometric =    7.00%
When Should You Use the
Geometric Average?

   When the returns each year are very
    different…the arithmetic average will
    overstate the actual average
    return….as seen in the following:
Example – volatile returns

      Year        Returns
      1998          7.0%
      1999        -15.0%       The greater
                              the volatility
      2000         29.0%       the greater
      2001         33.0%     the difference.
      2002        -12.0%
      2003         50.0%
      2004         -3.0%

   Arthimetic =   12.71%
   Geometric =    10.39%
Summary

   Use the geometric average when
    measuring average returns when
    those returns are volatile.
   The greater the volatility…the greater
    the difference between arithmetic
    average and geometric average.

				
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posted:4/26/2011
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