# Difference between Treynor ratio and market premium

The definition of Treynor ratio is given by $$T = \frac{r_i-r_f}{\beta_i},$$ where $$r_i$$ is the portfolio $$i$$'s return, $$r_f$$ is the risk-free rate and $$\beta_i$$ is the portfolio $$i$$'s beta. I am stunned after reading this definition. Isn't it exactly the market premium?

The CAPM model says that $$E[r_i] - r_f = \beta_i (E[r_m] - r_f).$$ Compare the above two equations we then conclude that $$T$$ is universal for all $$i$$ as $$T$$ is nothing but the market premium $$r_m - r_f$$. Could you point out what I missed? thank you guys

So, the Treynor Ratio $$T_p$$ is intended to be used for the appraisal of portfolios (not only individual securities). If all securities in the portfolio satisfy the CAPM (with $$\alpha = 0$$) then indeed the Treynor Ratio achieved will be equal the the Risk Premium on the Market. But Treynor thought if someone really smart comes along (George Soros, Warren Buffet, chici, etc.) they will achieve a TR higher than this. So "superior performance" is $$\alpha > 0$$ if you are using the CAPM or $$T_p > R_{PM}$$ if you are using Treynor's model.
• Thanks for including chichi in the list of smart investors! The answer makes sense. By the way, what is $R_{PM}$? Jul 31, 2022 at 4:27
• In at least one textbook it is the abbreviation for risk premium on the market, or maybe it was $RP_M$ and I mis-spelled it. In other words the Market Risk Premium. Jul 31, 2022 at 5:56