Minimum-Variance Portfolio Optimization with a Budget Constraint

Summary

The document formulates the minimum-variance portfolio problem for a universe of stocks. It assumes the portfolio is fully invested, represents asset risk through a covariance matrix, and seeks weights that minimize portfolio variance while requiring the weights to sum to one. The objective is written as a quadratic function of the portfolio weights, with a single budget constraint.

The author asks how to solve this optimization using Lagrange multipliers, but the document stops before deriving the solution. It gives no expected-return target, short-sale restrictions, estimation method for the covariance matrix, or empirical portfolio results. As stated, the setup describes a risk-only allocation problem: it does not balance expected return against risk, and the practical outcome would depend on the covariance inputs and any additional constraints chosen by an implementer.

Key ideas

• Portfolio variance is expressed as a quadratic form in the asset weights and covariance matrix.
• The fully invested constraint requires the weights to sum to one.
• The stated objective minimizes variance without including expected returns.
• The document poses a Lagrange multiplier solution but does not derive it.

Tags

# Minimum Variance Portfolio problem


# Minimum Variance Portfolio problem












Minimum Variance Portfolio Suppose there are N stocks in the investmentable universe and we have a fully invested portfolio investing 100% of the capital. The Covariance matrix is denoted as ∑. We are interested in finding the portfolio with minimum variance. An investor choosing this portfolio is only concerned about the risk of the portfolio. Denoting a vector of ones by i=(1,….,1)^' , we have the following optimization problem: $$Minimize \frac{1}{2}w'∑w$$

$$Subject to : w’ .i = w_1 + w_2 +……+ w_N =1$$

I would like to solve above problem using Lagrangian multipliers.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.