This Solver model uses the QUADPRODUCT function at cell I14 to compute the portfolio variance.
It can be solved for the minimum variance using either the GRG nonlinear solver or the Quadratic Solver.
  Stock 1 Stock 2 Stock 3 Stock 4 Stock 5   Total
Portfolio % 20.00% 20.00% 20.00% 20.00% 20.00% 100.00%
Expected Return 7.00% 8.00% 9.50% 6.50% 14.00%  
Linear QP Terms 0 0 0 0 0    
Variance/Covariance Matrix        
  Stock 1 Stock 2 Stock 3 Stock 4 Stock 5
Stock 1 2.50% 0.10% 1.00% -0.50% 1.60%
Stock 2 0.10% 1.10% -0.10% 1.20% -0.85%
Stock 3 1.00% -0.10% 1.20% 0.65% 0.75% Variance 0.68%
Stock 4 -0.50% 1.20% 0.65% 0.40% 1.00% Std. Dev. 8.22%
Stock 5 1.60% -0.85% 0.75% 1.00% 2.00% Return 9.00%
An investor wants to put together a portfolio consisting of up to 5 stocks. Using the Markowitz method, what is the
best combination of stocks to minimize risk for a given return? The variances are known for each stock, as are the
covariances between all stocks. The returns for all stocks are also known.        
1) The variables are the percentage allocations of our funds to invest in each stock. In this worksheet, the variables
are given the name Allocations. The sum of the allocations (which must be 100%) is computed in the cell named
2) The constraints are very simple. First there are the logical constraints:        
  Allocations >= 0 via the Assume Non-Negative option      
  Total_Portfolio = 1            
Then there is a constraint that the portfolio return should be at least a certain target value (9% in this example). This
return is calculated in the cell named Portfolio_return:          
  Portfolio_Return >= 0.09            
3) The objective is to minimize portfolio variance, which is calculated according to the Markowitz method in the cell
named Portfolio_Variance.              
In this worksheet, we use the QUADPRODUCT function to compute the portfolio variance. If you see #NAME? on
this worksheet, you need to open the add-in (DOTPRD32.XLL or DOTPROD.XLL) that provides QUADPRODUCT.
In the Full Markowitz worksheet, we calculate the portfolio variance 'manually' without using QUADPRODUCT.
The Markowitz method can only be used if all the variances of individual stocks, and the covariances between each
pair of stocks are known. In this model we assumed that all the terms are given. In the Full Markowitz worksheet we
actually calculate the variances and covariances from a history of stock prices.