Why techniques for portfolio optimization do not take into account the non-fractionability of stock prices?

Question

In a market with 3 stocks:

1. Stock A with price 25.00 USD;
2. Stock B with price 32.50 USD;
3. Stock C with price 50.75 USD;

Any portfolio optimization technique results in a vector of asset weights $\textbf{w}$ such that $0 \leq w_i \leq 1$ and $\sum_{i=1}^N w_i =1$. If I consider an equally weighted portfolio then: $$ w_i = \frac{1}{N}=\frac{1}{3} \approx 0.33 $$ If I am willing to invest 100.00 USD then I should buy:

1. $q_1 = \frac{33.00\\\$}{25.00\\\$}=1.32$ quantity of stock A;
2. $q_2 = \frac{33.00\\\$}{32.50\\\$} \approx 1.02$ quantity of stock B;
3. $q_3 = \frac{33.00\\\$}{50.75\\\$} \approx 0.65$ quantity of stock C;

If stock prices are not fractionable (are they?) then I should buy, for example:

1. ${\lfloor}q_1{\rfloor} = 2$ quantity of stock A;
2. ${\lceil}q_2{\rceil} = 1$ quantity of stock B;
3. ${\lceil}q_3{\rceil} = 0$ quantity of stock C;

With 17.50 USD of non-investable liquidity. Obviously the more one is capable to invest the more accurate will be the asset weights in relation to the absolute value of the stock price. How can retail investors deal with such a problem? Are stock prices fractionable? I can't find literature about that.

Answer 1

There are a few related reasons:

1. The optimization becomes a lot harder when only discrete values are considered. Mean variance has a closed form solution for the continuous case but the case with discrete holdings is quite hard;
2. For small retail investors fixed trading costs will swamp the rounding in your example but also for larger amounts (at least before Robinhood and others);
3. The literature is written with institutional investors in mind and they can round with little impact. This makes sense as they have a lot more to invest than retail;
4. ETF’s or mutual funds allow easy fractional investing for retail investors I. Just the way I want for no other costs. I personally just buy those and never bought single stocks in small amounts.

As noob2 points out in the comments, in the US nowadays it is possible to own fractional shares. In Europe I don’t know of any brokers that allow this.

Answer 2

As correctly mentioned in an earlier answer, portfolio optimization is something used for books in hedge funds and other institutions. As a smooth utility function changes slowly near its maximum, perturbation by rounding is inconsequential. If you would like to adopt a utility approach to personal investing, the following algorithm could work: (1) solve for optimal positions in floats; (2) while there are non-integer positions: for each asset and rounding up and down: find the rounding with the smallest utility deterioration and do that rounding. You should probably pay more attention to transaction costs than rounding, though.

Answer 3

In practice, creating a portfolio with (equally) weighted shares is doable as you will invest on "a lot" of shares. Suppose you're an advisor in a private bank and you're creating a portfolio with your client considering your shares example :

You will not invest USD 100.00 but USD 100,000.00, thus you would buy :

1333 shares of stock A => USD 33,325.00 // 1025 shares of stock B => USD 33,312.50 // 656 shares of stock 6 => USD 33,292.00

You would spend USD 99,929.50 in total and stay with USD 70.50 of cash. Globally, your portfolio is equally weighted (at some USD close). I guess that it doesn't change much the theoretical results you can have by supposing fractionable shares as your portfolio will probably react the same way regarding market movements.

Hope it helps