Applying Markowitz Portfolio Weights with Whole Shares
Summary
The document explains how to translate a Markowitz allocation into stock purchases. Multiply the available capital by each asset’s portfolio weight to obtain target dollar allocations, then buy the closest feasible number of shares. If a broker does not support fractional shares, the resulting holdings will only approximate the target weights.
It also cautions that implementing an optimized allocation does not guarantee strong performance. The answer cites an out-of-sample comparison in which a Markowitz strategy had a lower Sharpe ratio than an equal-weight allocation over the study’s sample. This is a warning about the reported evidence, not a universal conclusion: the document gives no details about the study’s assumptions, assets, or period, and does not discuss transaction costs, rebalancing, or how integer share constraints affect the portfolio.
Key ideas
- Portfolio weights can be converted into target dollar amounts by multiplying them by investable capital.
- Whole-share purchases approximate target weights when fractional shares are unavailable.
- The cited study reported weaker out-of-sample Sharpe performance for Markowitz than equal weighting.
- One study’s reported result does not establish that optimized portfolios will always underperform.
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Full text
# Understanding portfolio weights and purchasing stock in modern portfolio theory # Understanding portfolio weights and purchasing stock in modern portfolio theory 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 by phdstudent (score 2, accepted) https://quant.stackexchange.com/a/19062 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.
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