Diversity Based Portfolios

Overview



Choueifaty and Coignard



Choueifaty and Coignard propose constructing a portfolio with the highest diversification ratio, where the diversification ratio is defined to be
{% DR = \frac{\vec{w}^T \vec{\sigma}}{\sqrt{\vec{w}^T \Sigma \vec{w}}} %}
(see braga pg 91) where {% w %} is a column vector of portfolio weights,

Fernholz





Portfolio Generating Functions Fernholz defines a function S to be the generating function of a portfolio {% \pi %} if
{% log(Z_{\pi}(t)/ Z_{\mu}(t)) = log S(\mu(t)) + \Theta(t) %}
where {% Z_{\mu} %} is the value of the market portfolio and {% Z_{\pi} %} is the value of the generated portfolio and {% \Theta %} is a measurable process of bounded variation.

Frenholz uses this definition to show how to create diversity generated portfolios using various definitions of diversity, such as the following:

  • Entropy
  • Gini Coefficient
  • Renyi Entropy
(see Fernholz)

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