Overview - 2023.2 English

Vitis Libraries

Release Date
2023.2 English

A Portfolio consists of a holding in a number of risky assets (shares or funds) and optionally a risk free asset (bonds).

Portfolio Optimisation seeks to select the best asset distribution according to an objective. The objective is assumed to be maximizing the return for a given amount of risk (or minimizing the risk for a given required return). The asset risk is quantified by the variance of the asset. This is basically Modern Portfolio Theory or the Markowitz Model.

The starting point for Portfolio Optimisation is a list of daily prices of the available assets.

The daily returns are calculated as: \(\frac{price_n-price_{n-1}}{price_{n-1}}\)

The excess returns are calculated as: \(excess_n = r_n - average\, return\, for\, the\, asset\)

The covariance matrix \(\boldsymbol{\Sigma}\) of the excess returns matrix is calculated as:

\(\boldsymbol{\Sigma} = [\sigma_{ij}] = \frac{\boldsymbol{A^TA}}{M-1}\)

Where \(\boldsymbol{A} =\) excess return matrix \(= \begin{bmatrix} r_{11}-\overline{r_1} & ... & r_{N1}-\overline{r_N} \\ r_{12}-\overline{r_1} & ... & r_{N2}-\overline{r_N} \\ \wr & ... \wr \\ r_{1M}-\overline{r_1} & ... & r_{NM}-\overline{r_N}\end{bmatrix}\)

\(\boldsymbol{A^T}\) is the transpose of \(\boldsymbol{A}\)

\(\overline{r_i}\) is the mean return for asset \(i\)

\(r_{ij}\) is the \(j^{th}\) excess return for asset \(i\)

\(N\) is the number of assets

\(M\) is the number of returns