Assume we have a family of \(n\) assets whose daily return is vector \(\mu\) and the covariance matrix is \(\Sigma\). If we make a portfolio whose asset weight is \(w_k\) for asset \(k\) and the vector of all weights is \(w\), then the portfolio’s volatility is \(\sigma = (w^\top\Sigma w)^{1/2}\).

Now, for matrix \(\Sigma\), we can have spectral decomposition

\[\Sigma V = V \Lambda\]where \(V\) is matrix of columns of eigenvectors and \(\Lambda\) is a diagonal matrix of eigenvalues (in descending order), both of \(\Sigma\). Hence we can write \(\Sigma = V \Lambda V^\top\). Substitute this into the formula for the square of volatility, we have

\[\begin{aligned} \sigma^2 &= w^\top \Sigma w \\ &= w^\top (V\Lambda V^\top) w \\ &= (w^\top V) \Lambda (V^\top w) \end{aligned}\]if we substitute \(u = V^\top w\), we have

\[\begin{aligned} \sigma^2 &= (w^\top V) \Lambda (V^\top w) \\ &= u^\top \Lambda u \end{aligned}\]Now since \(\Lambda\) is diagonal,

\[\Lambda = \text{diag}\begin{pmatrix}\lambda_1 & \lambda_2 & \dots & \lambda_n\end{pmatrix}\]with \(\lambda_1 \ge \lambda_2 \ge \dots \ge \lambda_n\), therefore

\[\sigma^2 = \sum_{k=1}^n u_k^2\lambda_k\]This is significant because normally the variance \(\sigma^2\) is not additive but we can make it so by looking at the eigenvector dimensions. Also, if \(\Sigma\) is positive definite (usually the case for covariance matrix) then \(\lambda_k > 0\) for all \(k\). This is a useful information on the problem of finding the minimal \(\sigma^2\) subject to some constraint on \(u\). Assume we have the condition that \(\sum_k u_k=C\) for some constant \(C>0\), it is trivial to see that the way to minimize \(\sigma^2\) is to make \(u_k=0\) for \(k=1,\dots,n-1\) and \(u_n=C\). In this case, the portfolio volatility is \(\sigma = u_n\sqrt{\lambda_n} = C\sqrt{\lambda_n}\). In this case, \(C\) is just a scaling factor for the size of the portfolio. If we have \(u\), it is easy to reverse it to get

\[w = V u.\]Since \(V\) and \(\Lambda\) are the eigenvector and eigenvalues of \(\Sigma\), we can consider the mutually orthogonal eigenvectors are different “risk dimensions” and a portfolio is constructed so as to distribute the total risk to these different dimensions. In this case, we can write \(R=\begin{pmatrix}r_1 & r_2 & \dots & r_n\end{pmatrix}\) as a vector of risk proportions to each dimension and the risk distribution satisfies

\[\begin{aligned} r_k &= \frac{u_k^2 \lambda_k}{\sigma^2} \\ u_k &= \sqrt{\frac{r_k\sigma^2}{\lambda_k}} = \sigma\sqrt{\frac{r_k}{\lambda_k}} \end{aligned}\]Hence we can construct a portfolio with arbitrary allocation of risk to each dimensions.