Diversification Method

Diversification Method

Why Diversification Method exists?

Back to the Markowitz model, we can gain an efficient frontier. But what will happen if we have some little variations to the estimations of returns or covariance matrix? In this way, we have a ‘well-diversified model’, which can be also called Corvalan method.

The existence of ‘well-diversified’ portfolios ^[1] – in the sense that the distribution of wealth among a large number of assets – in an infinitesimal neighborhood of the efficient frontier. In other words, the return and the risk of the final portfolio shouldn’t be considered exact mathematical quantities, but rather as measurable variables within some range. The key feature of this behavior is that many of the possible portfolios are crowding towards the efficient frontier.

Note that the meaning of the diversified method. Just because of little variation, the outcome differences for consumers is so negligible. But it largely expands the combination situations for investors. In this way, it works.

Math Forms

Suppose the optimal solution of the original Markowitz problem is \(\omega\), the corresponding expected return is \(\mu =\omega^T r\) and variance is \(\sigma = \omega^T \Sigma \omega\). Then Corvalan method solves the model below:

\(\underset{\upsilon}{min}f(\upsilon)\)

\(s.t. \upsilon^T r \geq \mu(1- \Delta r), \upsilon^T \Sigma \upsilon \leq \sigma(1-\Delta \sigma) \quad  and \quad other \quad constraints, \)

where \(f(\upsilon)\) is a function that will be very large if \(\upsilon\)  fails to be well-diversified and \(\Delta r, \Delta \sigma\) respectively are the variance of  \(r\) and  \(\sigma\).

Reference

[1] Corvalán, A. (2005). Well diversified efficient portfolios. Documentos de Trabajo (Banco Central de Chile), (336), 1.