Gains from Diversification:
A TwoSecurity Illustration
W. L. Silber
The nice thing about diversification is that it almost always produces gains to a
portfolio in the form of increased return that exceeds the cost in terms of increased
standard deviation.
The only exception is the case of correlation equal to one.
This is
nicely illustrated with a simple numerical example.
Recall that the expected return,
R,
on a twoasset portfolio is:
(1)
2
2
1
1
R
X
R
X
R
+
=
,
where
R
1
and
R
2
are the expected returns on security 1 and 2 and
X
1
and
X
2
are the
weights invested in each.
The standard deviation
( 29
s
on the portfolio is:
(2)
2
/
1
2
1
2
1
2
2
2
2
2
1
2
1
]
2
[
r
s
s
s
s
s
X
X
X
X
+
+
=
,
where
1
s
and
2
s
are the standard deviations of asset 1 and 2 and
r
is the correlation
of returns.
Suppose we have the following information:
24
.
13
.
2
1
=
=
R
R
136
.
064
.
2
1
=
=
s
s
If
X
1
=1 and
X
2
=0, then the return on the portfolio is .13 and the standard
deviation is .064, all the same as asset 1.
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 Spring '08
 Miyakawa
 Standard Deviation, Mean, W. L. Silber

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