About ranking on risk-adjusted measures

So if A has a higher Treynor but lower Sharpe than B does,

does it mean A has less systemmatic risk and more individual-specific risk than B?

And which would you invest in if you held a balanced market fund?

A is giving you a better return given the systematic risk it takes on versus B, but could still have more systematic risk overall. It depends on what the fund is that I’m analyzing. If I want to put all my portfolio into a concentrated sector fund I’m going to use the sharpe ratio and analyze total risk and invest in portfolio B I don’t need to worry about the impact to my overall shapre given correlation since there’s no portfolio to combine with it). If I’m putting it into a market fund I’m going to use the Treynor ratio for analysis and likely go with portfolio A.

On a risk-adjusted returns basis, A has less systematic risk and more individual-specfic risk than B.

If we had a balanced market fund, and I assume you mean diversified so there was very little individual-specific risk then you would invest in the fund with the highest treynor.

If the fund wasn’t well diversified and you wanted to capture all risk in the risk adjusted returns metric then you would use the sharpe.