Risk Parity
Overview
Riks parity is an approach to portfolio construction (particularly in the asset allocation space) that looks
to allocate risk to the portfolio assets, rather than dollars. For example, rather than choosing to invest
a certain dollar amount in an asset, the risk parity approach would look to allocate a certain amount of
volatility, for instance.
The typical risk parity based portfolio allocates an equal amount of risk to each asset.
Naive Implementation
The naive implementation of risk parity parity ignores the dependence (correlations) between assets, and just
uses a simple measure, such as return variance, to proportion out portfolio capital.
for example , assuming that asset i has a return standard deviation of {% \sigma_i %}. Then each asset is
given a portfolio weight of
{% w_i = \frac{\sigma_i}{\sum_i \sigma_i} %}
The naive strategy is easy to compute and implement.
Optimal Implementation
When including dependence between assets, such as correlations, the solutin to the risk parity problem cannot be solved
exactly, except for simple situations. That means that
numeric optimization
must be employed to approximate the solution.
The optimization needed can be stated as
{% Minimize \, \sum \sum (w_i \frac{\partial{\sigma}}{\partial{w_i}} - w_j \frac{\partial{\sigma}}{\partial{w_j}})^2 %}
such That
{% \sum w_i = 1 %}
{% 0 \leq w_i \leq 1 %}
(see
braga pg 55)
