Portfolio Variance with Many Risky Assets
1
Case 1: Unsystematic risk only.
Recall that when the correlation
ρ
between two securities equals zero, the portfolio vari
ance is given by:
σ
2
p
=
X
2
1
σ
2
1
+
X
2
2
σ
2
2
A simple generalization of this formula holds for many securities
provided that
ρ
= 0
between all pairs of securities
:
σ
2
p
=
X
2
1
σ
2
1
+
X
2
2
σ
2
2
+
···
+
X
2
N
σ
2
N
.
(1)
We will prove the following result. As
N
→ ∞
, the portfolio standard deviation
σ
p
→
0.
To make the notation simpler, assume that
σ
1
=
σ
2
=
···
=
σ
N
=
σ
. This means that
each asset is equally risky. Under those circumstances, we try the simple diversiﬁcation
strategy of dividing our wealth equally among each asset such that
X
i
=
1
N
.
These assumptions allow us to rewrite expression (1) as
σ
2
p
=
±
1
N
¶
2
σ
2
+
±
1
N
¶
2
σ
2
+
···
+
±
1
N
¶
2
σ
2
(2)
There are
N
identical terms in expression (2), which means:
σ
2
p
=
N
±
1
N
¶
2
σ
2
σ
2
p
=
1
N
σ
2
The expression in (3) shows that as
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Fall '09
 Finance, Standard Deviation, Variance, Probability theory, Systemic risk, Ri

Click to edit the document details