Overview
Diverity based portfolios are based on maximizing a measure of diversity of assets or risks. The basic assumption being that risk and return are hard to measure and/or forecast. Under this assumption, the best that a manager can do is just to build a diverse portfolio of risk, in particular, risk premia. Various measures of diveristy have been proposed.
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