# Hansen-Jagannathan bounds derivation: last step is not clear

Pennachi's "Asset Pricing" chapter 4 derives:

$$\frac{E[R_{i}-R_{f}]}{\sigma_{R_{i}}}=-\rho_{m_{01},R_{i}}\frac{\sigma_{m_{01}}}{E[m_{01}]}$$

Then, he states that the fact that $-1\leq \rho_{m_{01},R_{i}} \leq 1$ implies that: $$\left | \frac{E[R_{i}-R_{f}]}{\sigma_{R_{i}}} \right | \leq \frac{\sigma_{m_{01}}}{E[m_{01}]}$$

This last step is not clear to me, could you please explain how it follows? Wikipedia says that it follows from Cauchy–Schwarz inequality, but I cannot figure out how.

P.S. There is a question about H-J bounds already, but there is an intuitive explanation, and I couldn't find an answer over there.

• I might be a bit dense now. I don't think that sentence on Wikipedia relates to that last step. Basically in the first equation $a = r b$ and we have $|r| \leq 1$ so certainly $b > a$ as we have in the second equation. Taking absolutes just drops the signs. Commented May 13, 2014 at 11:36
• Thank you! Your explanation indeed clearly answers my question! Commented May 13, 2014 at 14:46
• @BobJansen maybe you just put it as an answer? We have so many unanswered questions in the forum ... thanks! Commented May 13, 2014 at 14:52
• @BobJansen, vote up for putting your comment as an answer. This way I will be able to mark your answer as "the answer". Commented May 13, 2014 at 14:59

Basically in the first equation $a=rb$ and we have $|r| \leq 1$ so certainly $b>a$ as we have in the second equation. Taking absolutes just drops the signs.