formula differences

From a quick back of the envelope calculation it does not look like ${\displaystyle M^{2}\equiv {\overline {R_{C}}}-{\overline {R_{M}}}=({\overline {R_{P}}}-{\overline {R_{f}}}){\frac {\sigma _{M}}{\sigma _{P}}}-({\overline {R_{M}}}-{\overline {R_{f}}})}$ and ${\displaystyle M^{2}\equiv S\times \sigma _{B}-{\overline {R_{M}}}}$ are equivalent. For the two equations to both give ${\displaystyle M^{2}}$ then ${\displaystyle \sigma _{B}={\frac {{\overline {R_{f}}}\sigma _{P}}{{\overline {R_{P}}}-{\overline {R_{f}}}}}+\sigma _{M}}$. Does anyone have citations for both of these to be the same or both being used as the Modigliani risk-adjusted performance? If they aren't the same, which is more widely considered the right choice? Zfeinst (talk) 14:56, 3 August 2012 (UTC)