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I am very new to this field, and have very recently started doing some self study on this topic. After reading some papers and reproducing some of the results in them, I am not very clear about what objective exactly is fulfilled by doing portfolio optimization. I assume that having a minimum variance portfolio is of interest or relevance for investors because it tells them objectively which stocks to buy and which ones to sell, is it not? If that is not the case, then could you please explain why would an optimized portfolio be of interest to investors?

Also, if we say that investors should invest in stocks according to the optimized portfolio weights, does it mean that investors should invest in all the stocks anyway – and this method just gives the proportion of each to be invested in? If yes, it is a bit confusing, because in real life, although I myself have not traded, I assume that an investor would typically buy or sell only a small subset of stocks at any given time. Is this not true? So, the more meaningful problem to solve would be, in my mind, to find out which stocks to buy or/and sell at any given time, based on the studied stock returns, is it not?

EDIT: For example, we can use www.portfoliovisualizer.com to check which assets to invest in (according to whatever theory they are using):

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The algorithm is not giving as output a portfolio of all the investments that the investor initially had. It is giving a subset of them (I assume the ones most likely to make profit). But portfolio management theory doesn't seem to talk about subsets of assets. Which one is correct, or better?

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  • $\begingroup$ The idea that at any time there is only a small number of stocks worth buying is very much against the findings of Portfolio Theory (although it was widely believed before PT was developed). A portfolio consisting of only a few stocks will in general have insufficient diversification. So yes, people who use Portfolio Theory invest in large numbers of assets. Of course there are many obstacles to the direct practical application of the theory. But the general idea of diversification is very very important and useful in practice. $\endgroup$ – noob2 Jun 8 '16 at 15:38
  • $\begingroup$ But still, it is still a small subset of all the available stocks to invest in, is it not? I mean, according to one lecture (youtube.com/watch?v=tL7Lcl90Sc0), there are 8000+ total stocks available today. So, any investor would not invest in more than, say, 1000 stocks, right? $\endgroup$ – Kristada673 Jun 8 '16 at 15:47
  • $\begingroup$ Anyway, that's beside the point. My question would remain the same even if investors invest in all 8000+ stocks - what specific information is it giving to the investor that the investor can exploit in making more returns? Does it say which ones to buy and which ones to sell? If not, how does it help? $\endgroup$ – Kristada673 Jun 8 '16 at 15:49
  • $\begingroup$ It is telling you IN THEORY (if the assumptions of the theory are true and the numbers you plug into the formulas are correct) exactly what ypu said: how much to invest (what to buy and what to sell) of the given stocks to achieve a calculated return and level of risk. $\endgroup$ – noob2 Jun 8 '16 at 15:57
  • $\begingroup$ Ok. So if you go to www.portfoliovisualizer.com, and select, say, 10 assets, it outputs a subset of the assets as the recommended portfolio: i.stack.imgur.com/AHRRE.png So, does it tell us that I should invest in these 4 stocks? Or should I sell the other 6? $\endgroup$ – Kristada673 Jun 8 '16 at 16:06
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It means you should buy the 4 stocks. The model you are using seems to restrict short selling, i.e. by removing this restriction you could get negative weights on certain assets with all assets adding up to 100%.

Re your question on limiting your asset selection to stocks that are expected to outperform the population: this has to do with your assumptions behind expected returns. Remember, that the important inputs for your mean variance optimization are expected return, variance and correlations between the assets. Generally historical data is used for this purpose, but this doesn't necessarily reflect the expected future performance of the assets you are about to invest in. You might just as well substitute the historical average returns with your personal expectation of the returns, and if you e.g. expect certain assets to yield negative returns with positive correlation to your total portfolio, then obviously it doesn't really matter whether you include that stock in the selection or not, if short selling restriction is imposed then 0% weight will be allocated to that stock.

Bottom line: PT maximizes your Sharpe, that is the expected excess return of your total portfolio divided by the expected total variance. Assuming that you are risk averse, the returns are normally distributed (i.e. heavy fat tails don't limit the extent to which you can lever your portfolio, and hence Sharpe is reasonable measure of the risk-return trade off) and the inputs are reasonable, then theoretically PT should yield the optimal composition of your portfolio. Relax these assumptions, and MPT runs in to some of the problems that you are wondering about.

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