Hi @RobKing I think the disconnect is that your first test should be "but - adjusted RAROC is RAROC - beta(rm-rf) = 17%-1.5(12%-4%) = 5% > risk-free rate." So you would accept. As discussed above and where this first caused confusion is that GARP (Crouhy) proposed two ARAROC tests, neither of which is terribly intuitive to me both correctly itemized by @Maged above. Using the 2016 practice exam question P2.30 assumptions of RAROC = 17.0%, Rf = 4.0%, β=1.50, and R(m) = 12%:
Previous: ARAROC = ( RAROC - Rf ) / Beta => to be compared with Rm-Rf. In this case, (17.0% - 4.0%)/1.50 = 8.67% which is greater than 8.0% = R(m) - Rf = 12.0% - 4.0%
Current: ARAROC = RAROC - Beta (Rm - Rf ) => to be compared with Rf. In this case, 17.0% - 1.5*(12.0% - 4.0%) = 5.00% which is greater than the riskfree rate of 4.0%.
It remains easiest, for me, to simply calculate the (Jensen's) alpha of the RAROC. In this case, Jensen's alpha = 17.0% - 4.0% - 1.50* (12.0%-4.0%) = +1.0%; i.e., above the SML, so to speak, and therefore "accept." I hope that clarifies!
Just to clarify, I did not mean to suggest the ARAROC values are identical. As my linked example above shows, the specific ARAROC values will differ based on the approach. But they serve the exact same test and the result of the test (mathematically) must be the same because, if you'll note my last sentence, this ARAROC test is essentially similar to asking "is the Jensen's alpha positive?" What I mean is, take Shweser's version (which is Crouhy's, whereas ours is Grinolds, but Crouhy's source on this is Grinold!):
Adjusted RAROC=RAROC-Beta*(E(R_m)-R_f) ... and we accept if Adj_RAROC > Rf rate; so subtracting Rf from both sides, this is a test of:
(Adjusted RAROC - Rf) = (RAROC-Beta*(E(R_m)-R_f)) - Rf > 0
But per CAPM, expected RAROC = Beta*(E(R_m)-R_f)) + Rf, and
Jensen's alpha α = E[RAROC] - (Beta*(E(R_m)-R_f)) + Rf) = E[RAROC] - (Beta*(E(R_m)-R_f)) - Rf.
So this will be accepted if alpha > 0, same as under Grinold's (our) version.
Visually both are testing whether the actual RAROC "plots above" the SML; i.e., has positive alpha.
The Crouhy version accepts if the vertical distance is positive
The Grinold version accepts is the slope of the actual line (implied by the distance from the Rf to the actual RAROC) is steeper than the slope of the SML, given by the equity risk premium, in which case there must be positive alpha