Haas School of Business
University of California at Berkeley
UGBA 103
Avinash Verma
C
APITAL
A
SSET
P
RICING
M
ODEL
Recall that in the last note, we discussed Capital Market Line (CML). CML gave us a relationship between risk
(as measured by the standard deviation of returns) and reward (as measured by the expected return). However,
CML holds
only
for
efficient portfolios
. In this note, we seek a relationship between risk and return that is true
more generally rather than only for efficient portfolios. Capital Asset Pricing Model provides such a relationship
between (a different measure of) risk and reward (as measured by expected return).
1.
We start by asking the question: When we expand a portfolio by adding a new
security to it, how much risk does the new security add to the portfolio risk? In other
words, what part of the total risk of the expanded portfolio can be attributed to the
security that has just been added? We know that the portfolio variance is given by:
∑∑
=
=
=
=
n
i
n
j
ij
j
i
p
p
x
x
R
V
1
1
2
)
~
(
σ
σ
We also know that this double sum is an operation in which covariance between
returns on any two securities in the unshaded matrix below is weighted with the
fraction invested in each security in the pair (the fractions or the portfolio weights are
in the yellow cells), and then a sum is taken over all possible pairs.
S
ECURITY
1
S
ECURITY
2
S
ECURITY
n
x
1
x
2
…
…
x
n
S
ECURITY
1
x
1
2
1
11
σ
σ
=
21
12
σ
σ
=
…
…
1
1
n
n
σ
σ
=
S
ECURITY
2
x
2
12
21
σ
σ
=
2
2
22
σ
σ
=
…
…
2
2
n
n
σ
σ
=
…
…
…
…
…
…
…
…
…
…
…
…
S
ECURITY
n
x
n
n
n
1
1
σ
σ
=
n
n
2
2
σ
σ
=
…
…
2
n
nn
σ
σ
=
2.
Entries either in the first row or the first column in the matrix above can be attributed
to our decision to add Security
1
to the portfolio. Therefore, the contribution that
Security
1
made to the portfolio risk can be thought of as a weighted sum of all the
entries in the first row [or, equivalently, the first column] where each entry is
weighted with the fraction invested in every other security that Security
1
is paired
with. Thus, we multiply (i) the portfolio weights,
x
1
,
x
2
, …
x
n
,
in the
white on black
row with entries in the row for Security
1
,
σ
11
,
σ
12
, …
σ
1
n
, and conclude that what
Security
1
contributed to
2
p
σ
, the portfolio risk, is given by (
x
1
times):
n
n
x
x
x
1
12
2
11
1
...
σ
σ
σ
+
+
+
.
3.
Now, it is one of the properties of covariance that the covariance between a variable,
say,
Z
~
, and a
weighted sum
, such as
return on a portfolio
, is the same weighted
sum of covariances between
Z
~
and the constituents of the weighted sum. Thus:
(
29
Zn
n
Z
Z
p
Zp
x
x
x
R
Z
COV
σ
σ
σ
σ
......
)
~
,
~
2
2
1
1
+
+
=
≡
.
If the other variable relative to which the covariance of portfolio returns is being
measured is defined to be the return on Security
1
, that is if
1
~
~
R
Z
≡
, then:
(
29
n
n
p
p
x
x
x
R
R
COV
1
12
2
11
1
1
1
......