Hi, I have read today a mention about the Treynor ratio. Apparently it is a similar thing to the Sharpe ratio but defined as follows:T_A=(r_A-r_f)/beta_A, in other words excess return divided by beta.The problem is this: Since according to CAPM r_A=r_f+beta_A*(r_M-r_f) we can rewrite the Treynor ratio asT_A=beta_A*(r_M-r_f)/beta_A=r_M-r_f. This is completely independent on A.So, can anyone explain me the deeper sense of this Treynor ratio? Thanks a lot.

No, that is not what the CAPM says.The CAPM says the EXPECTED return is E(r_A) =r_f+beta_A*(r_M-r_f) The actual return r_A is this plus a random term which has expectation zero.So: if the CAPM holds exactly, the expected value of the Treynor ratio is the same for all stocks, which makes sense since in a world of CAPM no stock is preferable to any other. In the real world (whether CAPM holds or not) the empirically measured Treynor ratios of different stocks will be different (the difference could be due to non_CAPMness or it could be due to sampling error in a CAPM world).

Agreed. Indeed, if CAPM holds then the average Treynor ratio is the same. Good point.But I still miss the usefulness of the ratio. It seems to tell us something about historical over/underperformance with respect to systematic risk? As it would be r_M-r_f + \frac{\epsilon}{\beta}. Any enlightenment here?