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source: Reilly, Chapter One. Appendix
     source:
Variance and standard deviation … they measure how actual values differ from
the expected values for a given series.
Variance or s2 = S (Probability) (Possible return - Expected Return)2
= S (P) [(R - E(R)]2
for more
           financial market training, read www.fundmetrics.com

                  Example:

                 Probabilities          Possible returns
                        0.15                     0.20
                        0.15                    (0.20)
                        0.70                     0.10

                                                             =Expected Return, E(R)



      Possible                                                  x the
      Returns            E(R)     R - E(R)     Squared      Probability
        0.20            0.07        0.13       0.0169            0.15
       (0.20)           0.07       (0.27)      0.0729            0.15
        0.10            0.07        0.03       0.0009            0.70
                                                                  S=
Thus, the standard deviation, which is the square root of the variance, is:




                                          In percentage terms:
                 Expected return has already been calculated as:
Thus, the returns are in the range of 7% +/- 11.87 %




                                                       Most of them …




                              -11.87      7%      +11.87




 It answers the question of, quot;What's likely result?quot;. Does
        that completely answer the question? …


  So, far, that's how it works for expected returns and probabilities.
NOW:     Let's say you are given a data series:

    Returns      Average    Difference             Difference Squared
        0.07        0.04         0.03                        0.0009
        0.11        0.04         0.07                        0.0049
       -0.04        0.04        -0.08                        0.0064
        0.12        0.04         0.08                        0.0064
       -0.06        0.04        -0.10                        0.0100
       0.04                                       S=           0.03
    Average                                            Divide by 5 …
                               Equals the Variance =           0.01

  Extract the square root, and you get Standard Deviation:
=          0.08


Thus, the returns are in the range of 4% +/- 7.56%




                                                     Most of them …




                            - 7.56      4% +7.56

Your returns are sometimes negative, usually positive.
What is the range of possibilities, for most observations? ………………..



                                      Are you smarter than a fifth grader ?



What is the coefficient of variation?
               Return         Risk = Standard Deviation / Mean
     High          10            9         0.08 / 0.04
      Low            4           7
                                              =            1.89
              Range:           Low       -0.036
                              High        0.116 Mt . Makiling

Compare that with the volatility of your competitor: 10% +/- 9%
                                      = Standard Deviation / Mean

 comment                                         =           0.90
                Range:          Low          0.010           1.20
High   0.190 Mt. Banahaw

          Mt . Makiling




       -3.6           1%     4%          10% 11.6            19%


              Reaches
              negative                       Mostly positive
              territory                      returns, 1% to 19%




                           Coefficient of variation


Which is better:

           12% +/- 20%

                     (8)



                      or
                                                      Risk
                                                       20
           12% +/- 10%                                 10
+2


      -10           0           10           20                 30     40   %




      How about this:


                                                  Only positive returns
                                                  Using futures and options.
                                 Voila!




Covariance: is relevant to the investor, who seeks to maximize returns, while
minimizing the fluctuations in portfolio returns that cause uncertainty. The
secret is to add assets to the portfolio that have HIGH RETURNS but LOW or
NEGATIVE COVARIANCE (Low or negative CORRELATION) with the other assets
in that portfolio.

Covariance is important in the formula for portfolio variance.
Portfolio variance is reduced if asset covariances are zero or negative.

Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the
covariance by expressing the relationship as a number between +1 and -1.


 Correlation coefficient = r i j = Covariance i j /   s s
                                                       i    j


The   si
           = is expressed as = Square Root of S (I-I)]2/ N

Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the
covariance by expressing the relationship as a number between +1 and -1. It
allows you to compare the quot;covariance of this pair of assetsquot; with the
quot;covariance of another pair of assets.quot;
ij              i j       i       j


The   si
           = is expressed as = Square Root of S (I-I)]2/ N

Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the
covariance by expressing the relationship as a number between +1 and -1. It
allows you to compare the quot;covariance of this pair of assetsquot; with the
quot;covariance of another pair of assets.quot;


If the two data series moved together, the covariance would equal        ss
                                                                         i    j
because all the numbers would be exactly the same. In such a case,


 Correlation coefficient = rij = Covarianceij /    ss
                                                   i       j
                                                               =   1.0
>>> Flip to quot;examplequot; worksheet
ow actual values differ from

 xpected Return)2




                 PxR
                0.03
               (0.03)
                0.07
                0.07
=Expected Return, E(R)




                    =
                    0
                 0.01
                    0
            0.01410
            =Variance
oot of the variance, is:
                0.12


               11.87     st.dev
                0.07         ER
               or 7% what does this mean
of them …




and probabilities.
of them …




ions? ………………..



han a fifth grader ?




 10% +/- 9%
Mt. Banahaw




         19%




positive
, 1% to 19%




           Return
              12     High
              12     Low
30      40    %




positive returns
 futures and options.




o maximize returns, while
t cause uncertainty. The
GH RETURNS but LOW or
ATION) with the other assets


variance.
e zero or negative.

ardizequot; or quot;normalizequot; the
 ber between +1 and -1.


s   j


I)]2/ N

ardizequot; or quot;normalizequot; the
 ber between +1 and -1. It
of assetsquot; with the
j


I)]2/ N

ardizequot; or quot;normalizequot; the
 ber between +1 and -1. It
of assetsquot; with the



nce would equal  ss  i   j
e. In such a case,


 =     1.0
Observ.         i      j         i-Ī         j-ĵ Product    (i-Ī)2   (j-ĵ)2
     1        3      8 subtract        subtract
     2        6     10      average    average Column
     3        8     14 from i.bar from j.bar         dxe
     4        5     12
     5        9     13                        Exercise:
     6       11     15                            Do this by hand
           AVG? AVG?
                                                sum? sum? sum?
          Variance of I : sum of square deviations from mean divided by #observ
          Variance of j
          Covarianceij = quot;Product columnquot; divided by #observations

          Correlation quot;rquot;                    ??



          The solution is hidden in a secret place on this worksheet
divided by #observ
Observ.      i     j           i-Ī        j-ĵ Product          (i-Ī)2
     1      3      8           (4)        (4)    16              16
     2      6     10           (1)        (2)      2              1
     3      8     14            1           2      2              1
     4      5     12           (2)          0      0              4
     5      9     13            2           1      2              4
     6     11     15            4           3    12              16
  Avg 7           12                     S=      34             42

          Covarianceij =             34 / 6             Variance =
                                                             42
                           =          5.67                    6
                                                            equals
                                                               7.00
          Correlationij =        Covij / si sj          of i
                                                        St.Dev = sqrt of variance
                       = 5.67 / (2.65) (2.38)              2.65


                           =         0.90


          Conclusion: Data series are highly-correlated.
                Excatly how much, we calculated.
(j-ĵ)2
               16
                4
                4
                0
                1
                9
              34         add up. Sum of squared deviations from the mean.


riance =
              34
               6
           equals
            5.67
            of j
Dev = sqrt of variance
          2.38
rom the mean.

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how to teach covariance via excel, for CFA review course, Portfolio Management

  • 1. source: Reilly, Chapter One. Appendix source: Variance and standard deviation … they measure how actual values differ from the expected values for a given series. Variance or s2 = S (Probability) (Possible return - Expected Return)2 = S (P) [(R - E(R)]2 for more financial market training, read www.fundmetrics.com Example: Probabilities Possible returns 0.15 0.20 0.15 (0.20) 0.70 0.10 =Expected Return, E(R) Possible x the Returns E(R) R - E(R) Squared Probability 0.20 0.07 0.13 0.0169 0.15 (0.20) 0.07 (0.27) 0.0729 0.15 0.10 0.07 0.03 0.0009 0.70 S= Thus, the standard deviation, which is the square root of the variance, is: In percentage terms: Expected return has already been calculated as:
  • 2. Thus, the returns are in the range of 7% +/- 11.87 % Most of them … -11.87 7% +11.87 It answers the question of, quot;What's likely result?quot;. Does that completely answer the question? … So, far, that's how it works for expected returns and probabilities. NOW: Let's say you are given a data series: Returns Average Difference Difference Squared 0.07 0.04 0.03 0.0009 0.11 0.04 0.07 0.0049 -0.04 0.04 -0.08 0.0064 0.12 0.04 0.08 0.0064 -0.06 0.04 -0.10 0.0100 0.04 S= 0.03 Average Divide by 5 … Equals the Variance = 0.01 Extract the square root, and you get Standard Deviation:
  • 3. = 0.08 Thus, the returns are in the range of 4% +/- 7.56% Most of them … - 7.56 4% +7.56 Your returns are sometimes negative, usually positive. What is the range of possibilities, for most observations? ……………….. Are you smarter than a fifth grader ? What is the coefficient of variation? Return Risk = Standard Deviation / Mean High 10 9 0.08 / 0.04 Low 4 7 = 1.89 Range: Low -0.036 High 0.116 Mt . Makiling Compare that with the volatility of your competitor: 10% +/- 9% = Standard Deviation / Mean comment = 0.90 Range: Low 0.010 1.20
  • 4. High 0.190 Mt. Banahaw Mt . Makiling -3.6 1% 4% 10% 11.6 19% Reaches negative Mostly positive territory returns, 1% to 19% Coefficient of variation Which is better: 12% +/- 20% (8) or Risk 20 12% +/- 10% 10
  • 5. +2 -10 0 10 20 30 40 % How about this: Only positive returns Using futures and options. Voila! Covariance: is relevant to the investor, who seeks to maximize returns, while minimizing the fluctuations in portfolio returns that cause uncertainty. The secret is to add assets to the portfolio that have HIGH RETURNS but LOW or NEGATIVE COVARIANCE (Low or negative CORRELATION) with the other assets in that portfolio. Covariance is important in the formula for portfolio variance. Portfolio variance is reduced if asset covariances are zero or negative. Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the covariance by expressing the relationship as a number between +1 and -1. Correlation coefficient = r i j = Covariance i j / s s i j The si = is expressed as = Square Root of S (I-I)]2/ N Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the covariance by expressing the relationship as a number between +1 and -1. It allows you to compare the quot;covariance of this pair of assetsquot; with the quot;covariance of another pair of assets.quot;
  • 6. ij i j i j The si = is expressed as = Square Root of S (I-I)]2/ N Meanwhile the correlation statistic helps to quot;standardizequot; or quot;normalizequot; the covariance by expressing the relationship as a number between +1 and -1. It allows you to compare the quot;covariance of this pair of assetsquot; with the quot;covariance of another pair of assets.quot; If the two data series moved together, the covariance would equal ss i j because all the numbers would be exactly the same. In such a case, Correlation coefficient = rij = Covarianceij / ss i j = 1.0 >>> Flip to quot;examplequot; worksheet
  • 7. ow actual values differ from xpected Return)2 PxR 0.03 (0.03) 0.07 0.07 =Expected Return, E(R) = 0 0.01 0 0.01410 =Variance oot of the variance, is: 0.12 11.87 st.dev 0.07 ER or 7% what does this mean
  • 8. of them … and probabilities.
  • 9. of them … ions? ……………….. han a fifth grader ? 10% +/- 9%
  • 10. Mt. Banahaw 19% positive , 1% to 19% Return 12 High 12 Low
  • 11. 30 40 % positive returns futures and options. o maximize returns, while t cause uncertainty. The GH RETURNS but LOW or ATION) with the other assets variance. e zero or negative. ardizequot; or quot;normalizequot; the ber between +1 and -1. s j I)]2/ N ardizequot; or quot;normalizequot; the ber between +1 and -1. It of assetsquot; with the
  • 12. j I)]2/ N ardizequot; or quot;normalizequot; the ber between +1 and -1. It of assetsquot; with the nce would equal ss i j e. In such a case, = 1.0
  • 13. Observ. i j i-Ī j-ĵ Product (i-Ī)2 (j-ĵ)2 1 3 8 subtract subtract 2 6 10 average average Column 3 8 14 from i.bar from j.bar dxe 4 5 12 5 9 13 Exercise: 6 11 15 Do this by hand AVG? AVG? sum? sum? sum? Variance of I : sum of square deviations from mean divided by #observ Variance of j Covarianceij = quot;Product columnquot; divided by #observations Correlation quot;rquot; ?? The solution is hidden in a secret place on this worksheet
  • 15. Observ. i j i-Ī j-ĵ Product (i-Ī)2 1 3 8 (4) (4) 16 16 2 6 10 (1) (2) 2 1 3 8 14 1 2 2 1 4 5 12 (2) 0 0 4 5 9 13 2 1 2 4 6 11 15 4 3 12 16 Avg 7 12 S= 34 42 Covarianceij = 34 / 6 Variance = 42 = 5.67 6 equals 7.00 Correlationij = Covij / si sj of i St.Dev = sqrt of variance = 5.67 / (2.65) (2.38) 2.65 = 0.90 Conclusion: Data series are highly-correlated. Excatly how much, we calculated.
  • 16. (j-ĵ)2 16 4 4 0 1 9 34 add up. Sum of squared deviations from the mean. riance = 34 6 equals 5.67 of j Dev = sqrt of variance 2.38