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
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