That's counter-intuitive, but I could see it making sense if you are simply averaging many draws of individual stock picks as opposed to holding an index. I tend to look at it that the individual draws, would have much wider (expected volatility and) confidence intervals over the future, than a diversified index. The old adage that more potential reward comes with more potential risk. @OP I don't know that there's a simple analytic answer, but if you look back at some of Sharpe's older books and papers, there's often a chart that shows that after about 7 or 10 stocks, the diversification benefits quickly taper off (i.e. the biggest boost in diversification comes from the first several components). So you potentially have the benefits of concentration, without losing too much of the diversification benefits in return. I think a lot also depends on which stocks you select as the 10 or 5 (volatility, correlation, characteristics, etc); a useful book that might give you some ideas is Fred Piard's recipe book. It at least allows you to see some simple portfolio based systems (growing in assets from very small to 10+) that are compared to index benchmarks in terms or risk, reward, etc... It can at least give you a better emperical sense of what might be achievable. His components are more ETF based, than individual based, however.
Depends on what you are worried about. If you are trying to minimise tracking error then you'd choose the largest market cap stocks in each sector. If you are trying to maximise return you'd probably use some factors, eg value, momentum ... GAT
Making a lot of assumptions: 27% individual stock standard deviation, 5% arithmetic return on all stocks (i.e. no skill in stock picking), 0.85 correlation. The latter figure is most important; a higher correlation will reduce the benefit of diversification. These are then plugged into the standard formula for return and risk. It's 1.2% less in geometric return; see the blog post for an explanation of what that does or doesn't mean. GAT PS this is the subject of my new book so if you wait a few months you can get a few hundred pages of explanation
You do get a natural statarb style benefit from re-balancing - you, in general, will be picking up more of cheaper stocks and disposing of rich ones.
Geometric mean is more independent from the values and context than arithmetic mean. Arithmetic mean will more easily skew to extreme outliers, while geometric mean will auto-normalize (given reasonably comparable sets of factors > 0). Also, geometric mean may make more sense for percentage returns. However, in actual practice it's hard to scale trading/investing so might be an unreasonable expectation to exactly reproduce X years of returns. Arithmetic mean being more dependent on context assumes a perfect copy, while geometric mean assumes further growth/decay independent of absolute values. Using median for one observer is a fair point, though median will completely remove outliers (you really want to see/grok the real distribution). It doesn't make much sense to compare the different means in and of itself. They are after all incomplete descriptions of a more complex distribution. You can use percentiles or weighted averages and myriads of other more interesting stuff as well, so depends on desired function and perspective. For me, diversification is useful because the future is unknowable and my trust for finance is healthy = low.