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I was asked this question the other day:

By having two uncorrelated portfolios, one with sharpe ratio 2 and the other with sharpe ratio 5, what is the max sharpe ratio we can achieve.

I tried computing the sharpe ratio of the combined portfolio, where portfolio 1 was assigned a weight and portfolio 2 -> 1- weight and then compute the derivative with respect to w to get the optimal weight. Unfortunately, I cannot seem to be able to get a numeric result. If anyone knows how to solve this, I'd really appreciate it!

Thank you in advance!

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Here is the numeric proof that a combination of the two assets will always get the same maximum Sharpe ratio (combined). I don't have the mathematical proof though:

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On a side note, here is the mathematical formula: Optimise the Sharpe ratio of a portfolio of uncorrelated assets

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