Auto balancing portfolio through new purchases

Question

I'd like to have an investment portfolio with a few items, each having a target proportion (e.g. I want to have stocks of Apple and the total value of those should be 25% of my portfolio's total worth). I also would like to invest periodically. My question is: Is there some sort of formula/simulation that I can use to figure out each period how much to invest in each stock so that everything stays balanced? I tried working it out with an LP, but couldn't wrap my head around it.

A little example to make things a bit more clear:

Say that there's currently 3 stocks in my portfolio with the following info:

Stock name                | S1     S2     S3
Current # of stock owned  | 10     8      9
Current price             | 10     9      11
Current size of portfolio | 36.9%  26.6%  36.5%
Target size in portfolio  | 40%    20%    40%

If I now want to invest 50 euros (or dollars, or what have you), is there a way to repeatedly determine how much of which stock to buy so that after buying those things, the portfolio is more balanced than before? I'd like to find a way to solve this problem in general, not just solve this particular example.

Answer 1

Finding the "optimal" solution (and even defining what optimal is) would probably take a lot of searching of all the possible combinations of stocks you could buy, given that you can't buy fractional shares.

But I'd guess that a simple "greedy" algorithm should get you close enough. For any given portfolio state, look at which stock is furthest below the target size - e.g. in your example, S3 is 3.5% away whereas S1 is only 3.1% away and S2 is over-sized. Then decided to buy one stock of S3, recalculate the current proportions, and repeat until you can't buy more stocks because you've invested all the money.

If you have to pay a transaction fee for each kind of stock you purchase you might want to calculate this in larger lot sizes or just avoid making really small purchases.