The CAPM considers the relation between a mean-variance
efficient portfolio and an asset uncorrelated with this portfolio.
Let us denote this specific asset return by
.
The riskless asset with constant return
may be such an asset.
Recall from (17.4) the condition for a mean-variance efficient
portfolio:
In order to eliminate
, we can multiply
(17.4) by
to get:
Plugging this into (17.4), we obtain:
For the asset that is uncorrelated with the portfolio, equation (17.12)
can be written as:
since
is the mean return of this asset and is otherwise uncorrelated
with the risky assets. This yields:
and if (17.13) is plugged into (17.12):
with
The relation (17.14) holds if there exists any asset that is
uncorrelated with the mean-variance efficient portfolio
.
The existence of a riskless
asset is not a necessary condition for deriving (17.14).
However, for this special
case we arrive at the well-known expression
 |
(17.15) |
which is known as the Capital Asset Pricing Model (CAPM),
see Franke et al. (2001).
The beta factor
measures the relative performance with respect
to riskless assets or an index. It reflects the sensitivity of an asset with respect to
the whole market. The beta factor is close to 1 for most assets. A factor of
1.16, for example, means that the asset reacts in relation to movements of the whole
market (expressed through an index like DAX or DOW JONES) 16 percents stronger
than the index. This is of course true for both positive and negative fluctuations
of the whole market.
Summary

-
The weights of the mean-variance efficient portfolio satisfy

-
In the CAPM the mean of
depends on the riskless asset
and the pre-specified mean
as follows

-
The beta factor
measures the relative performance with respect
to riskless assets or an index and reflects the sensitivity of an asset with
respect to the whole market.