The expected value of the sum of two random variables is the sum of their individual expected values. The variance of the sum is the sum of the individual variances plus twice the covariance.

So, if you have two different assets with the same expected value and same variance, but with zero correlation, then all mixtures of the two will have the same expected returns, but a 50/50 basket of the two assets will have 0.707 (1/sqrt(2)) times the variance of either of the two assets alone.

More generally, for a basket of non-correlated assets, a basket that maximizes expected returns divided by standard deviation of returns will never be 100% one asset. Returns are a linear function of the individual weights, but std. deviation of returns are a non-linear function. (This is true, regardless of statistical distribution of the random variables, as long as std. deviation is well-defined... no assumptions about normal distribution are involved.)

Nice writeup. The catch is it's very hard to measure correlation reliably for badly behaved random variables (like the ones we have in finances).

I've worked in the industry, and yeah is a good explanation. The problem is that there is a reason why some strategies are uncorrelated to the market ; because they are not trading liquid vanilla products. Institutional investors cannot invest in crazy stuff that easily. Renaissance and other unicorns are not always open to new institutional investors, and certainly not one with tons of compliance to deal with.

1) Portfolio size, 2) low market correlation, 3) above market performance.

Choose two. You cannot have all three.