Solving for the least difference when buying shares and desired portfolio weights

Question

Quick and hopefully easy question. Let's say I'm looking to invest some money in stocks and I've got 5 I want in total, of these 5 I want to invest equally in all of them, 20% of the total capital weight each.

the problem being of course that each stock price has a different cost, so it's going to be unlikely that I can buy a combination of shares which will give me exactly 20% in each stock.

So the question is, is there a fast way to, or function, for solving non-linear problems like this, so that I can input the stock price and then the desired weight and get the solution for the least total difference?

Cheers fo the help!

Answer 1

May be use a MIP (Mixed Integer Programming) solver. The problem can be formulated as:

 min sum(i, abs(target(i) - price(i)*purchase(i)))
 subject to 
     sum(i, price(i)*purchase(i)) <= budget 
 purchase(i): integer variable

The absolute value can be linearized:

 min sum(i,z(i))
 -z(i) <= target(i) - price(i)*purchase(i) <= z(i)

Alternatively you can use a quadratic objective:

 min sum(i, (target(i) - price(i)*purchase(i))^2 )
 subject to 
     sum(i, price(i)*purchase(i)) <= budget 
 purchase(i): integer variable

This would require a MIQP solver.