17.1 Portfolio Choice

Suppose that one has a portfolio of $p$ assets. The price of asset $j$ at time $i$ is denoted as $p_{ij}$. The return from asset $j$ in a single time period (day, month, year etc.) is:

\begin{displaymath}x_{ij}=\frac{p_{ij}-p_{i-1,j} }{p_{ij} }\cdotp\end{displaymath}

We observe the vectors $x_i=(x_{i1},\ldots,x_{ip})^{\top}$ (i.e., the returns of the assets which are contained in the portfolio) over several time periods. We stack these observations into a data matrix $\data{X}=(x_{ij})$ consisting of observations of a random variable

\begin{displaymath}
X\sim (\mu ,\Sigma ).
\end{displaymath}

The return of the portfolio is the weighted sum of the returns of the $p$ assets:
\begin{displaymath}
Q = c^{\top}X,
\end{displaymath} (17.1)

where $c = (c_1,\ldots,c_p)^{\top}$ (with $\sum ^p_{j=1}c_j=1$) denotes the proportions of the assets in the portfolio. The mean return of the portfolio is given by the expected value of $Q$, which is $c^{\top}\mu $. The risk or volatility of the portfolio is given by the variance of $Q$ (Theorem 4.6), which is equal to two times
\begin{displaymath}
\frac{1}{2}\;c^{\top}\Sigma c.
\end{displaymath} (17.2)

The reason for taking half of the variance of $Q$ is merely technical. The optimization of (17.2) with respect to $c$ is of course equivalent to minimizing $c^{\top} \Sigma c$. Our aim is to maximize the portfolio returns (17.1) given a bound on the volatility (17.2) or vice versa to minimize risk given a (desired) mean return of the portfolio.

Summary
$\ast$
Given a matrix of returns $\data{X}$ from $p$ assets in $n$ time periods, and that the underlying distribution is stationary, i.e., $X\sim(\mu,\Sigma)$, then the (theoretical) return of the portfolio is a weighted sum of the returns of the $p$ assets, namely $Q =
c^{\top}X$.
$\ast$
The expected value of $Q$ is $c^{\top}\mu $. For technical reasons one considers optimizing $\frac{1}{2}\;c^{\top}\Sigma c$. The risk or volatility is $c^{\top}\Sigma c=\mathop{\mathit{Var}}(c^{\top}X)$.
$\ast$
The portfolio choice, i.e., the selection of $c$, is such that the return is maximized for a given risk bound.