Portfolio Optimization with quadratic programming

Portfolio optimization with quadratic optimization in R

Background:

A model to find optimal asset allocations (%) by minimizing portfolio risk while maximizing expected returns, subject to investment constraints. Here we are using quadratic programming to balance the trade-off between expected returns and portfolio volatility, based on previously calculated expected returns and the covariance matrix of asset returns.

We use the following model:

$$ min \quad -w^T *E(r) + \lambda \cdot w \cdot \Omega \cdot w^T $$ $$ s.t.\text{ (subject to)} lb \leq w \leq up $$

$$ \sum w = 100 % $$ $$ 0.5 \leq w^T \cdot B \leq 1.1 $$

Here:

• w = Vector of the weight of every asset in portfolio
• E(r) = Vector of expected returns for each asset
• λ = Risk aversion parameter (higher values yields more risk-averse)
• Ω = Covariance matrix of asset returns (measures risk/volatility)
• -w' E(r) = Negative expected return (what we want to minimize to maximize returns)
• w' Ω w = Portfolio variance (risk measure)
• lb and up is the lower and upper bounds of portfolio weighting

Using packages:

• quadprog

Sources:

• https://www.youtube.com/watch?v=aYi-mCsNC08
• https://www.abiranalytic.com/mv-optimization
• https://stackoverflow.com/questions/62576639/using-quadprog-for-portfolio-optimization

For further work/research:

• https://www.youtube.com/watch?v=5lWltAyaEXA