Chapter 3A
Why Diversification Is a Good Idea
Portfolio Construction, Management, & Protection, 5e, Robert A. Strong
Copyright ©2009 by South-Western, a division of Thomson Business & Economics. All rights reserved.
1
Variance of a Linear Combination:
The Practical Meaning
 One
measure of risk is the variance of
return
 The variance of an n-security portfolio is:
n
n
   xi x j ij i j
2
p
i 1 j 1
where xi  proportion of total investment in Security i
ij  correlation coefficient between
Security i and Security j
2
General Results
Portfolio Variance
Number of Securities
Source: Adapted by Edwin J. Elton and Martin J. Gruber, “Risk Production and Portfolio Size: An Analytical Solution,” Journal of Business,
October 1977, 415–437.
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Strength in Numbers

Portfolio variance (total risk) declines as the
number of securities included in the portfolio
increases
• On average, a randomly selected ten-security portfolio
will have less risk than a randomly selected threesecurity portfolio
• Risk-averse investors should always diversify to
eliminate as much unsystematic risk as possible
4
Covariance and Correlation
 Covariance
is a measure of how much two
random variables change together:
 Correlation is just the translation of
covariance into a UNITLESS measure that
we can understand (-1.0 to 1.0)
 Covariance of itself (A with A) is variance.
5
Calculations
6
Measure of Market Risk β
 Beta
of a Portfolio:
n
 p   xi  i
i 1
 Calculating
beta of a stock:
i 
COV ( Ri , Rm )
 m2
where Rm  return on the market index
 m2  variance of the market returns
Ri  return on Security i
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