Understanding portfolio weights and purchasing stock in modern portfolio theory

Question

Recently I've been learning about the markowitz algorithm. It's pretty interesting, but I'm curious how we apply this in practice. Lets say I have some optimal portfolio:

$R_p = x_aR_a + x_bR_b$

Which for simplicity's sake, we will say is just a simple two asset portfolio. This question will apply to a portfolio with a riskless asset and a tangency portfolio as well, and the n-asset case.

Let's suppose the markowitz algorithm says

$$ x_a = 0.34938 $$ $$ x_b = 0.65062 $$

This is fine in theory, but now how do we buy the stock? If I have $10,000 to invest then my dollar value given to asset A is

$10,000 * 0.34938$

and my dollar value in asset B is

$10,000 * 0.65062$

So if I go to my broker and say "buy me $3493.8 dollars worth of asset A", I will most likely be buying some fraction of a share of a company to get this exact value. I'm not aware of a case where you can buy fractional shares of a company via a broker.

Is there a rule to go by here to use these weights, or am I not understanding how to apply them correctly?

Thank you!

Answer 1

That's the way you apply. Usually you get the closest number of shares possible. However, if you use that strategy you are very likely to underperform the market. Check table 3 on this paper for the Out of sample performance of the Markowitz strategy. Over their sample the Sharpe Ratio is 0.07 whereas a simple naive strategy 1/N yielded 0.18.