Asset Covariance
Overview
When calculating a portfolio variance, one often looks to use
the portfolio holdings and an asset covariance matrix to produce
a result. But this can present a challenge for portfolios of
a reasonable size.
The covariance matrix of n assets contains roughly {% \frac{1}{2} n^2 %}
covariances. Suppose you decide that you will need at least k data points
per covariance to get a good statistical estimate, this will require
roughly {% \frac{1}{2} n^2 k %} data points to estimate.
For small portfolios, this may be achievable, but the problem
becomes untractable at a certain point. The following
models seek to impose additional assumptions that mitigates this
problem.
Calculations
The portfolio variance can be calculated as:
{% \sigma _{portfolio} ^2 = \sum_{i,j} ^n cov(r_{i port}, r_{j port}) = \sum_i \sum_j w_i w_j \sigma_i \sigma_j \rho_{ij} %}
where {% w_i %} is the weight of the portfolio invested in the given asset.
{% \sigma_i %} is the standard deviation of the ith asset
{% \rho_{ij} %} is the correlation between asset i and j.
Stated in Matrix terms
{% Portfolio \; Variance = \textbf{w}^T \Sigma \textbf{w} %}
where
- {% \Sigma %} is the asset covariance matrix
- {% \textbf{w} %} is the vector of portfolio weights.
