Optimal Growth Portfolio Derivations
Overview
The solution to the optimal growth portfolio is a straightforward application of
Lagrange multipliers.
The following derivations are states using
Linear Algebra
to simplify the calculations.
Optimal Portfolio with Target Volatility
We solve the problem of finding the portfolio with the highest growth rate, given a
chosen portfolio standard deviation.
Maximize
{% \vec{w}^T \vec{\mu} - \vec{w}^T \Sigma \vec{w} %}
subject to
{% \vec{w}^T \vec{1} = 1 %}
{% \vec{w}^T \Sigma \vec{w} = \sigma^2 %}
Solution
Introduce
Lagrange multipliers
{% \lambda %} and {% \gamma %}
the Lagrangian is given by
{% L = \vec{w}^T\vec{\mu} - \vec{w}^T \Sigma \vec{w} - \lambda(\vec{w}^T \vec{1} - 1) - \gamma(\vec{w}^T \Sigma \vec{w} - \sigma^2) %}
then we have first order conditions
{% \vec{\mu} - \Sigma \vec{w} - \lambda \vec{1} - 2 \gamma \Sigma \vec{w} = 0 %}
{% \vec{w} = \frac{1}{1+2\gamma} \Sigma ^{-1} (\vec{\mu} - \lambda \vec{1}) %}
The two Lagrange coefficients are set so that the original constraints are satisfied.
(see
Luenberger pg 431)
Optimal Portfolio
If {% \gamma %} is set to zero, then the second constraint is ignored and the resulting
portfolio is the growth optimal portfolio.
{% \vec{w} = \Sigma ^{-1} (\vec{\mu} - \lambda \vec{1}) %}
To solve, solve for {% \lambda %} using the constraint on weights, and then plug back in
{% [S^{-1}\vec{\mu} - \lambda S^{-1} \vec{1}]^T \vec{1} = 1 %}
{% [\vec{\mu}^T S^{-T} - \lambda \vec{1}^T S^{-T}] \vec{1} = 1 %}
{% \lambda \vec{1}^T S^{-1} \vec{1} = \vec{\mu}^T S^{-T} \vec{1} - 1 %}
{% \lambda = \frac{1}{\vec{1}^T S^{-T}\vec{1}} (\vec{\mu}^T S^{-T}\vec{1}-1) %}
API
The growth optimal portfolio is implement in the
portfolio module
