Portfolio Optimization for 10 Securities Using Lagrange Multipliers, No Short-Selling, Weights Sum to 1

Problem:
Construct the Optimal Portfolio that:
delivers the target return (mu_Target)
with minimum risk
Minimize the risk of the portfolio (in this case, measured as half the variance)
While maintaining an expected return target of (mu_Target)
By adjusting the investment weights on each asset
Subject to the budget constraint that the weights sum to 1
Method:
Since constraints are equalities => We can use Method Lagrange
Supports up to 10 securities.
Able to do more if requested. Please contact us.

Constraints:
No short-selling (ie. No negative weights)

Solution 00:
Basic MPT with only budget constraint that weights sum to 1
Solution 01:
Tweaked solution where no negative weights are allowed,
but budget contraint fails, as sum of weights exceed 1.
Solution 02:
Maintain that no negative weights are allowed,
but normalize weights such that they sum to 1.
This yields a practical solution, but usually unable to meet target return.

Portfolio Optimization for 5 Securities Using Lagrange Multipliers, No Short-Selling, Weights Sum to 1

Problem:
Construct the Optimal Portfolio that:
delivers the target return (mu_Target)
with minimum risk
Minimize the risk of the portfolio (in this case, measured as half the variance)
While maintaining an expected return target of (mu_Target)
By adjusting the investment weights on each asset
Subject to the budget constraint that the weights sum to 1
Method:
Since constraints are equalities => We can use Method Lagrange
Supports up to 5 securities.
Able to do more if requested. Please contact us.

Constraints:
No short-selling (ie. No negative weights)

Solution 00:
Basic MPT with only budget constraint that weights sum to 1
Solution 01:
Tweaked solution where no negative weights are allowed,
but budget contraint fails, as sum of weights exceed 1.
Solution 02:
Maintain that no negative weights are allowed,
but normalize weights such that they sum to 1.
This yields a practical solution, but usually unable to meet target return.

Construct the Optimal Portfolio that:
delivers the target return (mu_Target)
with minimum risk
Minimize the risk of the portfolio (in this case, measured as half the variance)
By adjusting the investment weights on each asset
Subject to
Expected return target = mu_Target (specified by user)
Weights sum to 1 (ie. fully invested)
Weights can be negative

This Excel spreadsheet calculates Value-at-Risk (VaR) for a portfolio of up to 12 tickers
Parametric, Historical, Monte-Carlo simulated Value-at-Risk (VaR)
Mean, Standard Deviation, Variance, Correlation, Covariance
Incremental VaR for each ticker in the portfolio (without rebalancing)
Marginal VaR for each ticker in the portfolio (with rebalancing)
Cholesky Decomposition of the covariance matrix using built-in VBA function