17.4 The Capital Asset Pricing Model (CAPM)

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 $y_0$. The riskless asset with constant return $y_0 \equiv r$ may be such an asset. Recall from (17.4) the condition for a mean-variance efficient portfolio:

\begin{displaymath}
2 \Sigma c - \lambda_1 {\bf\mu} - \lambda_2 {1_p} = 0.
\end{displaymath}

In order to eliminate $\lambda_2$, we can multiply (17.4) by $c^{\top}$ to get:

\begin{displaymath}
2c^{\top} \Sigma c - \lambda_1 \bar{\mu} = \lambda_2.
\end{displaymath}

Plugging this into (17.4), we obtain:
$\displaystyle 2 \Sigma c - \lambda_1 {\bf\mu}$ $\textstyle =$ $\displaystyle 2c^{\top} \Sigma c
{1_p} - \lambda_1\bar{\mu}{1_p}$  
$\displaystyle {\bf\mu}$ $\textstyle =$ $\displaystyle \bar{\mu} {1_p} +
\frac{2}{\lambda_1}(\Sigma c -
c^{\top} \Sigma c {1_p}).$ (17.12)

For the asset that is uncorrelated with the portfolio, equation (17.12) can be written as:
$\displaystyle y_0$ $\textstyle =$ $\displaystyle \bar{\mu} - \frac{2}{\lambda_1} c^{\top} \Sigma c$  

since $y_0 = r$ is the mean return of this asset and is otherwise uncorrelated with the risky assets. This yields:
$\displaystyle \lambda_1$ $\textstyle =$ $\displaystyle 2 \frac{c^{\top} \Sigma
c}{\bar{\mu}-y_0}$ (17.13)

and if (17.13) is plugged into (17.12):
$\displaystyle {\bf\mu}$ $\textstyle =$ $\displaystyle \bar{\mu} {1_p} + \frac{\bar{\mu}-y_0}{c^{\top} \Sigma
c} (\Sigma c - c^{\top} \Sigma c {1_p})$  
$\displaystyle {\bf\mu}$ $\textstyle =$ $\displaystyle y_0 {1_p} + \frac{\Sigma {c}}
{c^{\top} \Sigma c} (\bar{\mu} - y_0)$  
$\displaystyle {\bf\mu}$ $\textstyle =$ $\displaystyle y_0 {1_p} + {\bf\beta}
(\bar{\mu} - y_0)$ (17.14)

with

\begin{displaymath}
{\bf\beta} \equiv
\frac{\Sigma c} {c^{\top} \Sigma c}.
\end{displaymath}

The relation (17.14) holds if there exists any asset that is uncorrelated with the mean-variance efficient portfolio ${c}$. 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

\begin{displaymath}
{\bf\mu} = r {1_p} + {\bf\beta} (\bar{\mu} - r),
\end{displaymath} (17.15)

which is known as the Capital Asset Pricing Model (CAPM), see Franke et al. (2001). The beta factor $\beta$ 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
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The weights of the mean-variance efficient portfolio satisfy $2 \Sigma c - \lambda_1 \mu - \lambda_2 1_p = 0.$
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In the CAPM the mean of $X$ depends on the riskless asset and the pre-specified mean $\overline \mu$ as follows $\mu=r1_p+\beta (\overline \mu -r).$
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The beta factor $\beta$ 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.